Emergent memory and kinetic hysteresis in strongly driven networks

Stochastic network-dynamics are typically assumed to be memory-less. Involving prolonged dwells interrupted by instantaneous transitions between nodes such Markov networks stand as a coarse-graining paradigm for chemical reactions, gene expression, molecular machines, spreading of diseases, protein dynamics, diffusion in energy landscapes, epigenetics and many others. However, as soon as transitions cease to be negligibly short, as often observed in experiments, the dynamics develops a memory. That is, state-changes depend not only on the present state but also on the past. Here, we establish the first thermodynamically consistent -- dissipation-preserving -- mapping of continuous dynamics onto a network, which reveals ingrained dynamical symmetries and an unforeseen kinetic hysteresis. These symmetries impose three independent sources of fluctuations in state-to state kinetics that determine the `flavor of memory'. The hysteresis between the forward/backward in time coarse-graining of continuous trajectories implies a new paradigm for the thermodynamics of active molecular processes in the presence of memory, that is, beyond the assumption of local detailed balance. Our results provide a new understanding of fluctuations in the operation of molecular machines as well as catch-bonds involved in cellular adhesion.

In the presence of a time-scale separation the coarsegraining of continuous-space dynamics to transitions on a network yields memory-less, Markovian kinetics. Long dwell periods in potential minima allow the system to reach a local equilibrium, thereby erasing all memory of the past prior to a transition to the next minumum (see Fig. 1a). These Markov-jump models are ubiquitous 1-12 but assume transitions to occur instantaneously 13,14 . Prolonged transition times [15][16][17][18][19][20][21][22] resulting, e.g. from spatial transport of molecules in chemical reactions under imperfect mixing 23,24 , the presence of a rugged energy landscape 25 as shown in Fig. 1b, or external forces that destabilize local minima as in Fig. 1c, are therefore bound to cause "mild" violations of Markovianity. Moreover, coexistence of slow and fast parallel transitions 21,26 depicted in Fig. 1d may cause "strong" violations of Markovianity, manifested, e.g. as so-called catch bonds in cellular adhesion 27,28 .
To account for transitions with a finite duration in complex networks as shown in Fig. 1e, we here develop a theory embodying an exact projection of continuous dynamics on a graph onto a network with discrete states. Diffusion on a graph arises quite generally from the averaging of fast degrees of freedom in Hamiltonian dynamics weakly coupled to a heat bath 29 , and includes both, a position dependent force and a position dependent diffusion coefficient 30 . The coarse-grained dynamics evolve as jumps between the nodes. A state-change occurs once the trajectory enters a new node for the first time.

RESULTS
Coarse-graining. We first consider a sub-graph with 5 states highlighted in Fig. 1e. A continuous trajectory on the graph is depicted in Fig. 2a, where the time runs from bright to dark. Consider a gedanken experiment in which we record a 'blinking' whenever the continuous trajectory enters a node, which changes color upon each state-change ( Fig. 2a and b). The time-series of state-changes arising from such a forward-in-time coarsegraining is shown in Fig. 2b alongside recurrences, i.e. re-visitations of nodes (see solid line and crosses, respectively). We measure the (local) joint probability density to exit state i after a time t and enter state j, ℘ loc j|i (t). Its marginal over time -the so-called splitting probability -is defined as φ loc j|i = ∞ 0 ℘ loc j|i (t)dt and is normalized according to j φ loc j|i = 1. Whenever ℘ loc j|i (t) deviates appreciably from an exponential function as in Fig. 2c, the continuous trajectory does not locally equilibrate in i before changing state to j, giving rise to memory.
If we instead coarse-grain the same trajectory backward in time we discover, somewhat surprisingly, a kinetic hysteresis. That is, the time-reversed coarsened trajectory (see dotted gray line Fig. 2b) differs from the forward one. This hysteresis allows for a unique decomposition of each residence time t in any given node into a dwell time τ -the interval in which the forward and timereversed coarsened trajectory coincide -and a transition time δt -the interval in which they differ (see Fig. 2b). This decomposition is the key step towards understanding the emergence and manifestations of memory in network dynamics. As our first main result we prove the statistical independence of local dwell and transition times (proof shown in the Supporting Information, SI), i.e.
where ℘ tr j|i (t − τ ) and ℘ dwell i (τ ) are the probability densities of transition and dwell time, respectively. Eq. (1) embodies the following symmetries: (i) the dwell time is a state variable -it does not depend on the final state j -and (ii) the transition time is reflection-symmetric, ℘ tr i|j (t) = ℘ tr j|i (t). We prove both symmetries in Supporting note 4 and illustrate symmetry (i) in Fig. S14 while symmetry (ii) is exemplary tested in Fig. S17 and Table S7. The splitting probability in turn obeys a reflec-arXiv:2011.04628v1 [cond-mat.stat-mech] 9 Nov 2020  Each exit-event from state 2 to state 1 is highlighted in blue. (right) Histogram (shaded region) of the exit time from state 2 that is well approximated by a memory-less single exponential decay (dashed line). Orange bars below the trajectory highlight dwell periods in reduced state 2, and black bars the duration of transitions from state 2 to state 1. (b) (left) Diffusion in a potential with a diffusive barrier. (right) Histogram (shaded region) of the exit time from state 2 and a single exponential decay with the same mean exit time (dashed line) that only poorly approximates the statistics of exit. (c) Double-well potential from panel (a) "tilted" by an additional pulling-force that destabilizes the (folded) state 2. (d) Schematic of rupture-pathways of a "catch-bond" under force. The bond can rupture along two possible pathways: a fast pathway 1 (double arrow), and a slow pathway 2 that involves an intermediate conformational change. As before the orange and black bars denote dwell and transition periods, respectively. The probability density of the life-time of the bond is shown below, whereby the probability densities depicted by the histogram (shaded region) and dashed line have the same mean. The stark disagreement between the two reflects that the rupture is not memory-less. (e) Schematics of a general network with a sub-network with 5 states highlighted in black. Transitions between states 1 and 3 (dashed line) are assumed to be slow. tion identity -the generalization of local detailed balance (see Methods and SI): where k B T is the thermal energy and the quantity ∆g ij (defined in Eq. (14) in Methods) is strictly conservative. The local force F j|i (x) integrated between the nodes i and j separated by a distance l j|i may yield a globally non-conservative contribution, and alone encodes any violation of microscopic reversibility. Explicit results for the moments of transition-and dwell-time are given in Eqs. (11)- (13) in Methods. Memory in state-changes emerges non-locally as a result of long recurrence and transition times. Long recurrence times arise whenever the continuous trajectory becomes trapped in the legs of the subgraph without changing state (see vertical arrows in Fig. 2b). Long transition times are due to slow dynamics between two nodes. Imagine that only one leg in Fig. 1e, say 3 → 1, displays slow dynamics, e.g. because of slow diffusion and/or the absence of an energy barrier. Then, not only ℘ loc 1|3 is clearly non-exponential (see blue line in Fig. 2c) but strikingly also ℘ loc 2|3 and all others become non-exponential (see green line in Fig. 2c). Note that the dynamics in our example obeys detailed balance. Therefore, the discrepancy between curves in Fig. 2c is not a non-equilibrium effect as erroneously concluded in Ref. 31.
In order to understand these anisotropic local 'waiting times' we dissect fluctuations in the probability density of time required to exit state i, ℘ exit i (t) ≡ j ℘ loc j|i (t). The independence of dwell and transition times in Eq. (1) implies three independent contributions to fluctuations σ 2 exit = σ 2 dwell + σ 2 tr,int + σ 2 tr,ext , where σ 2 ≡ t 2 − t 2 denotes the variance, and we further decomposed fluctuations of transition time into intrinsic fluctuations along the respective legs of the subgraph, σ 2 tr,int = j φ loc j|i σ 2 tr,j|i , and the extrinsic scatter of mean transition times among distinct legs, σ 2 tr,ext ≡ j φ loc j|i ( δt tr j|i ) 2 − ( j φ loc j|i δt tr j|i ) 2 . The three contributions in Eq. (3) are explained in Fig. 3a and given explicitly in Methods.
When σ 2 tr,ext vanishes, i.e. δt tr j|i = δt tr k|i for all j, k (see Fig. 3a left), the fluctuations of exit time are sub-Markovian since σ exit ≤ t exit j . In turn, super-Markovian fluctuations, that is σ exit ≥ t exit , necessarily imply the existence of multiple exit-pathways with distinct tran- sition times (see Fig. 3a, right). This proves that one can infer in general the existence of parallel transition pathways without actually resolving individual pathways, which is our second main result.
Super-Markovian exit dynamics. In a first demonstration of the practical implications of our results we address the counter-intuitive catch-bond phenomenon 34 depicted in Fig. 1d (see also Refs. 27 and 28). A ligand bound to a receptor is pulled by a constant force F until the bond ruptures. The time of rupture corresponds to the exit time from the bound state. A characteristic of catch-bonds is that they rupture along two possible pathways. One pathway involves a conformational change of the receptor that prolongs the transition time. In turn this gives rise to a non-monotonic force-dependence of the rupture-time (see Fig. 3c). That is, within a certain interval of F -the so-called catchbond phase 27,28,35 -the bond counter-intuitively survives longer if we pull stronger. The mean life-time t exit and its standard deviation σ exit reconstructed according to Ref. 34 are depicted in Fig. 3c, where the lines denote exact results (see Methods) and symbols were deduced from 500 simulated rupture events. A larger pulling-force increases the likelihood of choosing the slow path (see black line in Fig. 3b) and in turn amplifies extrinsic noise (see shaded areas reflecting relative noise contributions in Fig. 3b as well as red line in Fig. 3a). The observed fluctuations are evidently super-Markovian, i.e. σ exit ≥ t exit , and therefore imply the existence of at least two rupture pathways that are not equally fast, σ tr,ext = 0.
Sub-Markovian exit dynamics. We now consider the scenario where extrinsic transition noise vanishes, implying σ exit ≤ t exit . Particularly important examples are the steady-state operation of driven molecular machines and the more abstract "stopping times" of the thermodynamic entropy production 36 . We consider an ATPase operating under the influence of a nonequilibrium torque M , were M = 0 refers to the torque at which the ATPase stalls 37 . The rotation of the ATPase evolves as diffusion in a periodic potential with period 2π/3, reflecting the 120 • rotational symmetry of the motor, and a barrier height of 5 k B T separating the welldefined minima (see Fig. 1d). The torque is accounted for by tilting the potential. The continuous rotation is coarse-grained into a uni-cyclic network with three rotational states, whereby the statistics of rotational statechanges remain exact. The statistics of exit time from either minimum are depicted Fig. 3e.
The probability densities to make a step in the forward (+) and backward (−) direction after time t are given by ℘ loc ± (t) = φ ± ℘ exit (t), yielding a mean squared angular deviation 38 , δθ 2 t ≡ θ 2 t − θ t 2 (see proof in SI) where denotes asymptotic equality in the limit t → ∞. Memory-less, Markovian kinetics would predict σ M exit ≡ t exit ≥ σ exit and thus systematically overestimate fluc-  The partitioning of noise-contributions mapped onto a triangle; the center of the triangle represents the equi-partitioning of noise-sources, σ 2 dwell = σ 2 tr,int = σ 2 tr,ext = σ 2 exit /3. The left corner corresponds to Markov kinetics, σ 2 tr,ext = σ 2 tr,int = 0, and the left edge (blue line) to "sub-Markov" kinetics, σ 2 tr,ext = 0. The top corner corresponds to a vanishing dwell-and extrinsic transition-noise, σ 2 dwell = σ 2 tr,ext = 0, whereas the right corner depicts the limit of vanishing dwell-and intrinsic transition-noise, σ 2 dwell = σ 2 tr,int = 0. The boxed histograms are shown for illustrative purposes. The kinetic hysteresis increases along the gray arrow. The red circles depicts results of the catch-bond example for different pulling forces F shown in panel (b) and (c) (results depicted by larger symbols are additionally illustrated in (c)), and orange symbols show the results for the driven ATPase shown in panel (f). (b, c) Reconstructed catch-bond experiment; (b) shows φ 2|0 -the probability of taking the slow pathway 2 (black line); the shaded areas depict the fraction of dwell (σ 2 dwell /σ 2 exit ), extrinsic (σ 2 tr,ext /σ 2 exit ), and intrinsic (σ 2 tr,int /σ 2 exit ) noise, respectively. (c) depicts the mean, t exit , and standard deviation, σexit, of the bond's life-time. Lines correspond to exact results and symbols are deduced from 500 rupture experiments. Red circles depict pulling forces considered in panel (a). The density of the life-time of the bond at F = 20 pN is shown in Fig. 1d. (d-f ) Driven molecular motor displaying a vanishing extrinsic transition-noise, σtr,ext = 0. (d) Free energy landscape as function of the angle θt with a barrier-height of 5 kBT (blue line; see Methods for details) that becomes tilted due the action of the torque M (red arrow); dotted lines denote network-states, i.e. free energy minima. (e) Scaled probability density of exit-time from a state (i.e. first passage-time to an angular displacement of ±2π/3 = ±120 • ) as a function of the dissipation per step (i.e. torque M multiplied by the rotation-step 120 • ) of magnitude 0, 2, 5, 10 and 20 BT . (f) Steady-state mean squared angular deviation δθ 2 t compared with the a Markov-jump approximation δθ 2 t M as a function of dissipation; The full line depicts the lower bound (18) derived in Methods using the Thermodynamic Uncertainty Relation (TUR) 32,33 ; the individual noise-contributions for selected points (open symbols) are shown in panel (a). Fig. 3f). Memory is particularly pronounced in the regime of strong driving, i.e. φ + φ − or φ − φ + . Notably, we find the so-called thermodynamic uncertainty relation 32,33 to bound fluctuations from below (Fig. 3f, solid line). Our results thereby yield a "sandwich" bound on actual fluctuations in general unicyclic driven networks.

tuations (dotted line in
Networks with general topology. The exact coarse-graining of continuous paths to a network with N states Ω = {1, . . . , N } is completely described by ℘ A a|i0 (t), the joint probability density that the continuous trajectory starting from node i 0 ∈ Ω\A = A c arrives at time t for the first time in state a ∈ A without having visited any other state within A. The probability density ℘ A a|i0 (t) is normalized according to a∈A ∞ 0 ℘ A a|i0 (t)dt = 1 and is in turn completely specified with ℘ loc j|i (t) for all i and j (see proof in SI). Passing to Laplace space,g(s) ≡ ∞ 0 e −st g(t)dt, and writing all Laplace images℘ loc j|i (s) in the hollow matrixP(s) ji ≡℘ loc j|i (s) = φ loc j|i℘ tr j|i (s)℘ dwell i (s) that has non-zero elements for any pair i, j = i of connected neighbors we prove (see SI), upon extensive algebra, that ℘ A a|i0 (t) is the inverse Laplace transform of where 1 is the identity matrix, 1 A c is the indicator function on the subset A c , and k| = |k is a column vector with elements k| i = δ ik , whereby δ ik is Kronecker's delta. From Eq. (5) follows the splitting probability, i.e. the probability to reach a from i 0 before reaching any other state within A which reads where we have defined Φ ij ≡ φ loc i|j . Introducing the matrix T ij ≡ φ loc i|j t loc i|j the first mean passage time from i 0 to a, conditioned not to visit any state j ∈ A\{a}, in turn reads Eqs. (9)- (13) in Methods render Eqs. (6) and (7) fully explicit. Unique fingerprints of memory in state-to-state kinetics emerge already under minimal assumptions. Consider the kinetics from state 3 to the pair of target states A = {1, 2} in the network depicted in Fig. 4a (see also subnetwork in Fig. 1e). All direct transitions from each state are equi-probable, i.e. φ loc 1|3 = φ loc 2|3 = φ loc 4|3 = φ loc 5|3 = 0.25. Note the link 1−3 contains a diffusive barrier, that is, the diffusion landscape D(x) along the link 1 − 3 has a pronounced minimum, mimicking the effect of an entropic bottle-neck. In order to infer the waiting time distribution ℘ loc for the exit from states 3, 4 and 5, respectively, we simulated 4 × 10 5 exits from each state (see Fig. 4bd). The normalized waiting time distribution for the exit from state 3 is genuinely heterogeneous 39 , i.e. it shows strong variations between the respective legs (compare orange, blue and black symbols in Fig. 4b and see Fig.  S18 for a more detailed analysis of the complete network).
We now inspect the the probability to reach the target state 1 (3) within A before reaching state 3 (1). Note that a trajectory may reach state 1 through the link 1−3 or via state 4. Such conditioned transition kinetics quantify non-local effects and are particularly important for marginal observations, i.e. when we do not monitor all states but instead only a subset (in this case states 1, 2 and 3) while the remaining states are left as part of the "heat bath". This scenario is very relevant from an experimental point of view, since we can typically monitor only a limited number of states.
In our example in Fig. 4a we find, respectively, φ A 1|3 = 0.4 and φ A 2|3 = 0.6 while t A 1|3 ≈ 8.26, t A 2|3 ≈ 6.80. That is, the conditioned dynamics along 3 → 1 is slower and concurrently also more likely than the dynamics along 3 → 2. Note that both, Markov-state kinetics 1-12 as well as isotropic (decoupled) renewal processes 38,40-42 would invariably infer transition 3 → 1 to be erroneously faster than 3 → 2. This is because the direct transition 3 → 1 takes longer than the paths 3 → 2 that involve the detour through intermediate state 4. Markov models, for example, do not allow for this to happen because they assume transitions to occur instantaneously. Similarly, symmetric waiting time distributions in renewal processes render the duration of all transitions out of any state equal, such that any path involving a detour is bound to take longer on average.
A correct interpretation of marginal observations in the presence of memory therefore requires the thermodynamically consistent coarse-graining derived in our work. More severe situations are found in driven networks (see catch-bond analysis in Fig. 1d or Table S5).

DISCUSSION
Emerging from the mapping of continuous dynamics are three elementary, independent sources of fluctuations in state-transitions on a network: dwell-time fluctuations and the intrinsic and extrinsic noise arising from random transition times. The balance of these noise channels, depicted in the noise triangle in Fig. 3a, yields Markovian, sub-or super-Markovian fluctuations and thus sets the 'flavor of memory'. A vanishing extrinsic transition-noise causes sub-Markovian dynamics as in the driven AT-Pase (Fig. 3a, orange symbols). Markovian dynamics is dominated by dwell-noise (left corner). Super-Markovian fluctuations (observed e.g. in catch-bond dynamics) are dominated by extrinsic transition-noise (right corner). The noise triangle allows for a conclusive inference of underlying dominant, hidden continuous paths in general networks solely from the observed fluctuations in statetransitions. The kinetic hysteresis between forward-and time-reversed state-trajectories that arises in the pres-ence of transition-noise (grey arrow) provides a new understanding of the breaking of time-reversal symmetry in the presence of memory 31,41,43 . The widely adopted principle of local detailed balance is found to be a peculiarity of the Markovian limit, not a general feature time-reversal symmetry. Our results pave the way towards a deeper understanding of network-dynamics far from equilibrium including current-fluctuations in active molecular systems 32,33 .

Diffusion on a graph
The full dynamics is assumed to evolve as piece-wise continuous space-time Markovian diffusion on a graph with potential (weak) discontinuities at the set of all nodes i ∈ Ω. We denote all neighbor-nodes of i by N i ⊂ Ω. At any time t between the last passage by i in the direction of j at time t ini and the next visit of a node j ∈ N i or the return to i at time t fin , i.e. t ini < t < t fin , the system is assumed to evolve according to the anti-Itô Langevin equatioṅ where x t denotes the instantaneous distance from node i in the leg i − j with 0 < x t < l j|i , D j|i (x) and F j|i (x) are the diffusion landscape and force field along the leg directed from i to j, respectively, β ≡ 1/(k B T ), ξ t is standard Gaussian white noise with zero mean, i.e. ξ t = 0 and ξ t ξ t = δ(t − t ). The symbol " " denotes the anti-Itô product (see Supporting Information, SI). For any i ∈ Ω and j ∈ N i the force translates into the local potential, . If a global potential exists, that is U j|i (x) = U (x, j, i) − U (0, j, i), ∀i ∈ Ω, j ∈ N i the dynamics is said to obey detailed balance. Conversely, if no such potential exist microscopic reversibility is said to be broken (see SI, Supporting note 7).
Once node i is reached from within a leg the consecutive leg is chosen, without loss of generality, randomly from the set of all neighbors j ∈ N i with equal probability. Thereupon the dynamics again evolves according to Eq. (8) until the next visit of a node, which fully specifies the full system's dynamics. In Supporting note 8.B we explain in detail how Eq. (8) can account for discontinuities in the diffusion landscape and force-field.
Coarse-graining to state-changes on a network According to the gedanken experiment outlined in Fig. 2 the continuous trajectory is coarse-grained into a time-series of recurrences and state-changes on a network. Consecutive visits of the continuous trajectory of the same node correspond to recurrences (see colored crosses in Fig. 2a and b), whereas transitions between distinct nodes yield state-changes (see black crosses in Fig. 2a and line in Fig. 2b). In-between two consecutive state-changes the reduced network-state remains in the initial state (see Fig. 2b). This exactly specifies a coarse-grained trajectory on the network.
The dwell time τ corresponds to the sum of all recurrence times t r since the last state-change. The transition time δt corresponds to the time between the last recurrence and the instance of the state-change. The local waiting time t for a transition i → j is formally the sum of the dwell time and transition time, t = δt + τ , and corresponds to the time-interval between two consecutive first entrances of nodes. Since the complete dynamics is stochastic these quantities correspond to random variables. The joint probability density of a waiting time at i and consecutive transition to j corresponds to ℘ loc j|i (t), and the dwell and transition time are distributed according to ℘ dwell i (τ ) and ℘ tr j|i (δt), respectively. Precise formal definitions of the waiting, dwell and transition time functionals and the proof of Eq. (1) are given in the SI.

Statistics of waiting, dwell and transition time
A straightforward translation of Eq. (8) into a Fokker-Planck equation with appropriate boundary and internal continuity conditions allows us to obtain explicit results for the splitting probability and the statistics of dwell and transition time, which follow from some tedious algebra (see SI for the derivations). For convenience we introduce the following essential auxiliary integrals where g (k) j|i are depicted in Tab. I. In the following we require only the first five integrals I (k) j|i (k = 1, . . . , 5). Using the auxiliary integrals in Eq. (9) the splitting probabilities read and the first two moments of the transition time become where the second moment is generally sub-Markovian δt 2 tr j|i ≤ 2( δt tr j|i ) 2 . See also Ref. 26 for an alternative proof, where δt 2 tr j|i ≤ 2( δt tr j|i ) 2 corresponds to a coefficient of variation being smaller than one. Some further Table I. Integrands entering Eq. (9) at a glance.
extended algebra yields first two moments of the average local dwell time wherefrom follows the variance of the dwell time The independence of dwell and transition times in Eq. (1) immediately yields the binomial sum for the n-th moment of the local first passage time where the forward/backward symmetry implies δt l loc j|i = δt l loc i|j . The n-th moment of the exit time is then simply given by t n exit i = k φ loc k|i t n loc k|i , yielding the variance σ 2 exit,i = t 2 exit i − ( t exit i ) 2 . According to Eq. (13) the latter can be decomposed into three noise contributions σ 2 exit,i = σ 2 dwell,i + σ 2 tr,int,i + σ 2 tr,ext,i , where σ 2 dwell,i = τ 2 dwell i − ( τ dwell i ) 2 , the intrinsic noise due to transition time is given σ 2 tr,int,i = k φ loc k|i [ δt 2 tr k|i − ( δt tr k|i ) 2 ] and the extrinsic noise among different transition paths is given by Thermodynamic consistency Using Eq. (9) and the splitting probability in Eq. (10) one arrives at Eq. (2) of the main text, where (see SI for explicit derivation) Equation (2) generalizes the local detailed balance relation 6 and yields a thermodynamically consistent j|i . Local equilibration occurs if the local free energy barriers are high, meaning that B j|i kBT and B i|j kBT .
coarse-graining (see SI, Supporting note 7 for more details). It is important to note that Eq. (2) relies on the kinetic hysteresis of the coarse-graining, that is, the projection onto the reduced state-space and the time-reversal do not commute (see Fig. 2b).
In the case of high local (free) energy barriers, corresponding to B j|i → ∞ and B i|j → ∞ in Fig. 5, the full trajectory locally equilibrates in each node before any transition. In this situation the transition-rate to jump from node i to node j becomes w i→j w M i→j ≡ φ loc j|i / t exit i and Eq. (2) then implies (see proof in SI) where U i denotes the potential energy of node i, the first term on the right denotes the externally provided free energy input along the transition (i.e. "work"), and the free energy of state i is defined by The symbol " " denotes asymptotic equality "=", here taken in the limit of high (free) energy barriers (B j|i → ∞ and B i|j → ∞). Eq. (15) reflects the so-called local detailed balance 6 . When there is no work performed along the transition local detailed balance also implies global detailed balance. Local detailed balance is violated as soon as a single barrier B j|i ceases to be high. Interestingly, local detailed balance (Eq. (15)) can be violated even in systems obeying detailed balance globally. In this case the waiting time distribution becomes non-exponential (see Fig. 2c).
where the pulling-force is assumed to increase linearly between x = 0 and x = = 0.0414 nm, i.e., U load (F, The solid lines correspond to the free energy landscape at zero force F = 0, i.e. j|0 (x). The precise form of U 1|0 and U 2|0 is given in Supporting note 8.D.

Catch-bond analysis
We employed a so-called switch catch-bond model 34 with parameters chosen to reproduce experimental results on bacterial adhesion bonds 27 (see also Ref. 28 for related experiments). The potential along the jth path- is the (free) energy profile at zero pulling-force and U load (F, x) accounts for a nonzero pulling-force F . The potential along pathway 1, U 1|0 (x), and along pathway 2, U 2|0 (x), is depicted in Fig. 6, where solid lines represent potential values at zero pulling-force F = 0, dashed lines show the truncated potential under a moderate force F = 30 pN, and the dotted line corresponds to a pulling force F = 60 pN. Blue lines depict the potential along the fast pathway 1, and orange lines the potential along the slow pathway 2. The theoretical lines in Fig. 3b,c were obtained by numerical integration of Eq. (9) along both pathways (1 and 2) and consecutive use of Eqs. (10)- (13). The symbols were obtained from 500 (numerical) rupture-experiments, carried out by a Brownian Dynamics simulations using a stochastic Runge-Kutta scheme of strong order 1.5 with time-increment ∆t = 10 −6 s (see SI for more details). The error bars denote the standard deviation of 500 rupture events. As expected, an increase of the number of simulated rupture events reduces the discrepancy between theory and experiment (see SI, Supporting note 8.D). Notably, a simple change of units (see SI) renders the results to agree with other, presumably incompatible experiments 28 . For example, the maximum average lifetime of cadherin-catenin binding to actin was reported to be ≈1.2 s (instead of ≈ 30 s in Fig. 3c) at a pulling force of F ≈ 10 pN (instead of F ≈ 30 pN in Fig. 3c). Using the same model albeit with a re-scaled potential allows to quantitatively discuss quite different experiments 27,28 by means of Fig. 3c (see SI, Supporting note 8.D).

ATPase and unicyclic machines
Let x = θ/120 • such that x = l ± = 1 corresponds to a 120 • turn (one third of a revolution). The potential at driving M is assumed to be given by U ± (x) = B[1 − cos(2πx/l ± )]/2 ∓ M x, where B denotes the height of the rotational free energy barrier separating minima under equilibrium conditions M = 0 (see blue line in Fig. 3d). The non-equilibrium driving M , which denotes the difference between torque M and chemical energy input 37 (for more details see SI), effectively tilts the potential. In Fig. 3e,d the barrier is set to B = 5 k B T . The scenario B = 0 corresponding to simple biased diffusion can be solved analytically (see SI) and generally yields a lower bound on the angular diffusion. Using Eq. (4) we obtain the ratio of the true angular mean squared deviation, δθ 2 t (see Eq. (4)), and the one deduced from a Markov-jump model that corresponds to where equality holds as t → ∞, and the superscript "M" denotes the Markov-jump limit (see also Ref. 43).
Vanishing extrinsic noise renders the kinetics sub-Markovian, σ 2 exit ≤ ( t exit ) 2 and immediately yields δθ 2 t / δθ 2 t M ≤ 1. This implies the dotted line in Fig. 3f to be a general upper bound on angular diffusivity. Conversely, a lower bound to the diffusivity can be deduced from the so-called thermodynamic uncertainty relation (TUR) 32,33 , which for unicyclic networks in the limit t → ∞ states that where θ t /120 Inserting Eqs. (4) and (17) into Eq. (16) yields where in the last step we defined f ≡ M × 120 • /(k B T ) and used e f ≡ φ + /φ − which follows from Eq. (2). The right hand side of the inequality (18) is depicted in Fig. 3f by the solid gray line and coincides with the result of plain biased diffusion (i.e. with the barrier set to zero, B = 0; see SI for more details). Note that in all unicyclic networks the mean squared angular deviation (angular diffusivity) must lie between the dotted and solid gray lines in Fig. 3f.
Authorship contributions. DH and AG designed research, performed research, analyzed the data, and wrote the paper.
Competing interest. The authors declare no competing interests.
Data availability. The data that support the findings of this study are available from the corresponding author upon reasonable request. In this Supporting Information (SI) we provide detailed proofs of our results. We provide details about the methods and describe numerical schemes that were used to simulate individual trajectories of the full continuous space dynamics. On page 12 we provide preliminary comments outlining the main conceptual ideas the proofs rely on. The remainder of the SI is organized as follows.
In Supporting note 1 we derive a decomposition of paths on a graph including conditioned and unconditioned first passage paths 1 . We generalize the classical renewal theorem 2 to a matrix form. In the spririt of l'Hospital's rule we "invert" the singular renewal theorem to obtain splitting probabilities, moments of conditional first passage time for multi-target search problems in networks with both, general topology and star-like topology, in terms of moments of simpler unconditioned first past passage times. The generalized renewal theorem offers a convenient way to derive higher moments of exit time from any state.
In Supporting note 2 we show that any network can be decomposed into its star-like sub-graph components, which allows for a systematic analysis of large-scale network dynamics. We show how star-like graphs can be used as building blocks for constructing any non-star-like graph according to the theory of renewal networks 3 (see also . In Supporting note 3 we provide details on the equations of motion listed in the Method section and give some background on stochastic calculus. In the first part we display the stochastic equation of motion corresponding to a Langevin equation including the Itô and anti-Itô representation, and give details on their numerical implementation involving a stochastic Milstein and a higher order stochastic Runge-Kutta scheme. In the second part we translate the stochastic Langevin equation into a partial differential equation for diffusive dynamics on a graph -the Fokker-Planck equation. We provide boundary conditions for the Fokker-Planck equation for all quantities entering the generalized renewal theorem in Supporting note 2, which includes boundary conditions for both conditioned and unconditioned first passage problems. Equation 1 in the main text, which represents Eq. (S2) below, is proven in Supporting note 4. Using Green's function theory we prove both the forward/backward-symmetry of the transition time statistics and state-symmetry of the dwell time statistics.
In Supporting note 5 we calculate the first three unconditioned moments of first passage times. The first two moments of transition times are listed at the end of this section in Supporting note 5.D.
In Supporting note 6 we prove the main practical result, which represents the first two moments of dwell time, conditional first passage time, and exit time. The explicit results allow us to conveniently prove the second main result, namely that extrinsic transition noise dictates the amplitude of fluctuations (see Supporting note 6.E).
In Supporting note 7 we prove the thermodynamic consistency of the coarse-graining. The kinetic hysteresis is essential to guarantee the entropy production rate along both, the coarse-grained trajectory and the detailed continuous trajectory, to be identical. We relate forward and backward splitting probabilities exactly to the force field integrated along the path connecting two states, which generalizes the well known local detailed balance relation 7 and proves equation 2 in the main text.
Additional examples and data supporting the examples presented in the main text can be found in Supporting note 8. Theoretical results are thereby corroborated by numerical simulations. We also address therein the problem of both, discontinuous local potentials (diverging "force kicks") and discontinuous diffusion landscapes. All symmetries of dwell-and transition-time statistics are tested in this section.

Preliminary comments 12
Supporting note 1. Generalized renewal theorem (path decomposition) 13 1.A. Renewal theorem and conditional renewal theorem on a graph 13 1.B. Deducing moments of first passage times from the renewal theorem (optional background) 15  Before we start with the rigorous derivation of our results let us briefly highlight the most important conceptual ideas. We first establish that the state-to-state kinetics on a network is characterized by the local statistics of residence time (or waiting time) to remain in a state before transiting to another state. Any state-change is fully specified by the answer to the following two questions: 13 Q1: How long does it take until the state changes?
("residence time") Q2: Which is the next state? ("new state") At a first glance one might be tempted to believe that the order of answering these two question is not crucial. However, surprisingly it turns out that we simplify the problem if we swap the order. That is, we first answer Q2 and only then Q1, which in terms of conditional probabilities means Prob("residence time", "new state") = Prob("residence time"|"new state")Prob("new state"). (S1) The swapping of the order of these questions represents the first crucial conceptual step towards our main results. It offers the following two advantages as compared to the order in which the questions are typically addressed in the classic literature 3 . The first advantage of this alternative order is that we separate the state-changes from the complex statistics of "residence time". As we show below the statistics of the sequence of state changes characterized by "new state" in this order can be treated fully analytically. The second advantage is that we are now able to map a continuous dynamics onto state-to-state dynamics, which allows dissecting the duration of any state change -the residence time -further into a "dwell" and a "transition" period (see Fig. S1 and next paragraph). That is "residence time" = "dwell time" + "transition time" (S2) is a sum of independent random numbers for a given "new state", where "residence time" represents the duration of a state-change, "dwell time" the duration of dwelling in the vicinity of a state, and "transition time" represents the duration of a state-to-state transition. Here the swapping of the order of the questions is crucial since the dwell time and transition time become independent given that the "new state" is known, that is, Q2 is already answered. Eq. (S2) (or equivalently Eq. 1 in the main text) will be proven and discussed in Supporting note 4. In the following Supporting note 1 we present a decomposition of paths representing the third most important conceptual breakthrough that with combination of the finding sketched Eq. (S2) allows to conveniently gain access in the nature of non-Markovian yet renewal fluctuations.
It is important to stress that the functionals representing each term in Eq. (S2) assign the emergent memory in the coarse-grained dynamics to the underlying microscopic dynamics. To illustrate their exact meaning let us briefly consider the example depicted in Fig. S1. Here the detailed coordinate evolves along a triple-well free energy potential as function of time as shown as the "wriggling" thin line in Fig. S1a. Each colored cross therein represents a visit of a state, which is here chosen to be a local minimum of the potential. We exactly map the continuous dynamics into a coarse-grained digitized one as depicted Fig. S1b (see solid dark-gray line), which represents last visited potential minimum (i.e., state). The coarse-grained trajectory is fully characterized by the sequence of visited states (i.e. the sequence of colors) and the corresponding residence times t of each state visit (the first residence time interval in state 2 is highlighted in Fig. S1b). Comparing the coarse-grained dynamics with the detailed one in Fig. S1b one can decompose each residence time interval t = τ + δt into a dwell time τ -the duration between the first visit of a state and the last visit of the same state before visiting another one -and the transition time δt -the time-span between last visit of a state and the first visit of another one. At finite transition times the coarse-graining and the time reversal do not commute leading to what we call kinetic hysteresis (see also Fig. 2b in the main text). The kinetic hysteresis in the coarse-graining is crucial to obtain a thermodynamic consistent coarse-graining that preserves the entropy production rate of driven networks (e.g., molecular machines) in the presence of memory (see Supporting note 7 for the proof). This result is crucial for a correct description of state-to-state dynamics in networks driven far from equilibrium.

1.A. Renewal theorem and conditional renewal theorem on a graph
The classical renewal theorem 2 connects the first passage time density to the propagator of a system's dynamics. It can be understood as a decomposition of paths: any system that starts at position i 0 and arrives at position i T = a at time T must have reached said position a for the first time at a time t ≤ T (i t = a) for the first time and then either stayed there or returned again after time T − t, which mathematically means the following. Denoting the probability to find the system at time T in state i T = a given that it started initially in state i 0 by P (a, T |i 0 ), the renewal theorem states 2 where p a|i0 (t)dt denotes the probability that the process starting from i 0 reaches the position i t = a for the first time within the interval t ≤ t ≤ t + dt. We call p a|i0 (t) the unconditioned first passage time density to the target state a if the system initially started from i 0 . We call a first passage problem "unconditioned" if there is just one target state a as in Eq. (S3). The renewal theorem (S3) has been routinely used to study first passage phenomena 1,8 and connects the propagation of a system, characterized by P , to unconditioned first passage functionals that are embodied in the probability density p a|i0 (t).
To study conditional first passage problems 1 , which contain more then a single target state a, we need to generalize the renewal theorem (S3) in the following way. Let us consider a set of target states A corresponding to a subset of all network states Ω = {1, . . . , N }, that is, A ⊂ Ω. A conditional first passage problem asks for the first time until the system reaches the target state a ∈ A given that it has not yet visited any of the other target states from A\{a}. The problem is characterized by the joint density, ℘ A a|i0 (t), starting from i 0 to enter the set of target A for the first time at time t and hitting the specific target a ∈ A, with normalization a∈A ∞ 0 ℘ A a|i0 (t)dt = 1. In the spirit of the classical renewal theorem (S3) we find that the conditional first passage density ℘ A density to any subset A is related to simpler unconditioned first passage time densities according to which is a generalization of the renewal theorem to conditioned first passage problems; in the last step we introduced " * " as the one-sided convolution operation. An illustration of the generalized renewal theorem for a network with five states Ω = {1, 2, 3, 4, 5} and two target states A = {1, 2} with initial condition i 0 = 3 is shown in Fig. S2. In the simplest case, when the subset A contains just one element A = {a}, we trivially obtain p a|i0 (t) = ℘ all functions are of exponential order), we obtaiñ We note that the convolution in the last term of Eq. (S4) becomes a product after the Laplace transform in Eq. (S5).
In the following subsection we show how the generalized renewal theorem Eq. (S5) can be used to deduce explicit conditional moments of first passage time, which correspond to a multi-target search problem, in terms of simpler unconditioned "single-target" quantities. The following Supporting note 1.B is illustrates the renewal theorem in its most general form as presented in Eq. (S5). As we will explain in Supporting note 2 one can in fact construct any network problem by solving for networks with a specific and simpler star-like topology (see Supporting note 1.C). This sequential strategy allows for a systematic study of general networks network.

1.B. Deducing moments of first passage times from the renewal theorem (optional background)
The Taylor expansions of the functions entering the renewal theorem (S5) are connected to the moments of first passage times viã is the kth (unconditional) moment of first passage time to reach a single target j, and t k A j|i is the kth moment of the conditional first passage time towards state j given that none of the other states A\{j} have been reached before. A naïve Taylor expansion of Eq. (S5) using Eq. (S6) yields . . .
p a n−1 |a 1 (s)p a n−1 |a 2 (s) . . . 1p a n−1 |an (s) Notably, Eq. (S7) leads to an underdetermined system of equations, that is, the conditioned moments t A a|i0 and splitting probabilities φ A a|i0 cannot directly be obtained from unconditioned first passage moments highlighted by the superscript "single". Below we show how one can nevertheless deduce the splitting probabilities and conditioned moments of first passage time. Once the splitting probabilities are known, the k-th moment of the exit time, t k exit i0 = a φ a|i0 t k A a|i0 , can, somewhat surprisingly, be determined from Eq. (S7) as which holds for any a ∈ A. It is remarkable that we are able to obtain the k-th moment of the exit time from "just" the first k unconditioned moments of first passage time and the first k − 1 conditional moments -as if we get an additional moment of exit time apparently "for free".
To access the splitting probability as well as conditioned moments by means of the generalized renewal theorem (S5) we need an alternative strategy since Eq. (S7) is an underdetermined system of equations. In other words, we need to invert a singular equation. First, we define the subset of target states A = {a 1 , a 2 . . . , a n } ⊂ {1, 2, . . . , N } with n elements (n < N ). Second, we rewrite the renewal theorem (S5) in form of a matrix product (i, j = 1, . . . , n) and , where x(s) encodes the conditioned moments of first passage times according to Eq. (S6); Eq. (S9) is explicitly shown in Fig. S3.
According to Cramer's rule the γth component of x(s) is given by that is, we replace the γth column of M by y(s) to obtain M γ (s). Eq. (S10) solves Eq. (S9). The matrix M γ (s) is illustrated in Fig. S4. Since the matrices M(s) and M γ (s) both have all entries equal to 1 for s = 0, the limit s → 0 in Eq. (S10) seems to be undetermined (i.e., yields "zero divided by zero"). To avoid a division by zero we first identify the dyadic product M γ (0) = M(0) = νν with ν ≡ (1, . . . , 1) and then use the matrix determinant lemma to remove the singularity p a n−1 |a 1 (s)p a n−1 |a 2 (s)p a n−1 |a γ−1 (s)p a n−1 |i 0 (s)p a n−1 |a γ+1 (s) . . . 1p a n−1 |an (s) where in the first step we used Cramer's rule in Eq. (S10), the second step involves a division by s n of both numerator and denominator as well as an addition of zero; in the third step we employed the matrix determinant lemma. According to Eqs. (S6), (S9) and (S10) we have and where i, j = 1, . . . , n and i 0 / ∈ A = {a 1 , . . . , a n }. Notably, the zeroth moment of the conditioned first passage timethe splitting probability -requires the first unconditioned first passage times. Inspecting Eq. (S11) we generally find that the kth moment of the conditional first passage time can be expressed in terms of the first k + 1 unconditioned moments, which indicates that conditional first passage problems are notoriously more difficult to solve.

1.C. Renewal theorem on star-like graphs
Let us for now focus on graphs with a star-like topology, where all n = N − 1 "outer nodes" are target states, that is, A = {1, 2, . . . , n}, and the starting node is the "inner state"i 0 = N as depicted in Fig. S5. In the case of a star-like topology the renewal theorem Eq. (S5) simplifies, meaning that it can be inverted more easily. In a first crucial step we realize that each path on a star-like graph, which starts from one end of the star a to another end a = a, must pass through the center i 0 . That is, the unconditioned first passage time from a to a is the sum of first passage time from a to i 0 and the first passage time from i 0 to a, which effectively impliesp a|a (s) =p a|i0 (s)p i0|a (s) (see Fig. S5c). Usingp a|a (s) =p a|i0 (s)p i0|a (s) for a = a, the renewal theorem (S5) in matrix form becomes where u(s) and v(s) are vectors with elements u a (s) ≡p a|i0 (s) and v a (s) ≡p i0|a (s), respectively, and D(s) denotes a diagonal matrix with elements D ii (s) = 1 − u i (s)v i (s), which corrects forp a|a (s) = 1 =p a|i0 (s)p i0|a (s). Moreover, we added the superscript "loc" to℘ A in order to emphasize that this results holds only in the case of a star-like topology, which will later be used to represent the local kinetics in the vicinity of a node in general, non-star-like topology.
Using the Sherman-Morrison-Woodbury formula we are able to invert the matrix D + uv to get Figure S5. Star-like graph. (a) Star-like graph with N states from which n = N − 1 are "outer states" a, a = 1, . . . , n and one state is called the "inner state" N . (b) Conditional first passage paths belonging to℘ loc a|i 0 (s) (c) Illustration of p a|a (s) =p a|i 0 (s)p i 0 |a (s), which holds for a = a and effectively means that each path starting from a must pass through the center i0 to reach the other end a (a = a ). The matrix D(s) in Eq. (S14) accounts and corrects forp a|a (s) = 1 =p a|i 0 (s)p i 0 |a (s).
which is the central result of this section that allows us to obtain conditional many-target first passage time distributions from simpler unconditioned single-target first passage time densities. The local splitting probability which formally reads φ loc a|i0 =℘ loc a|i0 (0), can be obtained by taking the limit s → 0. Defining the forward and backward unconditioned single target moments, f (S16) Using the definitions the numerator of the right hand side of Eq. (S15) satisfies and consequently using the product rule of differentiation the denominator satisfies where Since the Laplace transform of the local first passage time density is given bỹ we directly obtain the splitting probability in the limit s → 0 yielding where in the last step we inserted the coefficients A (0) a and B (0) from equations (S18) and (S20), respectively. The first derivative of (S21) at s = 0 yields the conditional mean first passage times where we inserted A (0) a and B (0) from Eqs. (S18) and (S20) in the first step, and finally identified the splitting probability φ loc j|i0 from Eq. (S22) and used j φ loc j|i0 = 1. Using Eqs. (S22) and (S23) the mean exit time from node i becomes which is the first term in the result of Eq. (S23). The second moment of the conditional first passage time after differentiating Eq. (S21) twice at s = 0 yields and hence where in the second last step of the first line we used j φ loc j|i0 = 1 and Eqs. (S22) and (S23) and (S24); in the last step of the first line we used the third line of Eq. (S20) and the last step in the second line we inserted the first two lines of Eq. (S20) as well as Eq. (S24).

Supporting note 2. CONDITIONAL FIRST PASSAGE STATISTICS ON NETWORKS FROM STAR-LIKE SUBGRAPHS
The simplest network topology is a star-like topology, for which we are able to conveniently express moments of conditional first passage times in terms of simple unconditioned first passage moments as explained in Supporting note 1.C. In the following we show that according to Ref. 3 (see also Refs. 4-6) each network can be decomposed exactly into a full set of subnetworks with a star-like topology. Thereby, each star-like sub-graph characterizes the local kinetics on a graph in the vicinity of a network state. Hence we will use all star-like sub-graphs as building blocks to built up and describe a general network. We suppose that we have a large scale network with a set of N states, such that for each state i ∈ Ω = {1, . . . , N } there exist a non-empty set of neighboring states N i ⊂ Ω with i / ∈ N i . A fully connected network corresponds to The probability density that starting from state i a nearest neighboring state j ∈ N i will be reached for the first time at time t is distributed according to the probability density ℘ loc j|i (t); its Laplace transform reads℘ loc dt is the (splitting) probability that starting from i the neighboring state visited next will be state j. We now define the matrix P(s) asP where Φ ij = φ loc i|j and T ij = φ loc i|j t loc i|j for i ∈ N j and j = 1, . . . , N . We emphasize that the Laplace transform allows to conveniently add independent random variables due to the following well known reason: Suppose two independent random variables t 1 and t 2 are distributed according to the densities f 1 and f 2 with their Laplace transformf i (s) = Having established the local kinetics we can now determine the first passage time to a set of target states A starting from state i 0 / ∈ A (i 0 ∈ Ω\A) for a general network as follows. To select a target state and non-target states we first define the projection matrix onto target state A and non-target states A c ≡ Ω\A, which are given by respectively, where i| = |i is a unit column vector with all elements zero except the ith component and 1 is the identity matrix. The matrices 1 A and 1 A c are the indicator functions of A and A c , respectively. For example, for all target states α ∈ A we find 1 A |α = |α and 1 A c |α = 0, whereas for all non-target states β ∈ A c we find 1 A |β = 0 and 1 A c |β = |β . Starting from i 0 ∈ A c the Laplace transform of the probability density to hit the target state a ∈ A "after the first step" (without having visited a single non-target state) is given by a|Q (1) (s)|i 0 = a|P(s)|i 0 ; similarly, if we select all elements that perform exactly one jump into a state j ∈ A c and then enter a in the second jump we obtain a|P(s)|j j|P(s)|i 0 , which after summing over all intermediate non-target states with Eq. (S29), yields a|Q (2) (s)|i 0 = a|P(s)1 A cP (s)|i 0 . More generally, the Laplace transform of the probability density to hit target a for the first time exactly after k-th transitions while transiting k − 1 times between non-target states is given by Summing now over all possible numbers of intermediate transitions we obtain a geometric sum that yields 3 which proves Eq. 5 in the main text. Therefore, the conditional first passage time towards any set of targets can be represented by the local first passage time densities according to Eq. (S31). The probability density ℘ A a|i0 (t) corresponds to the inverse Laplace transform (s → t) of Eq. (S31). The trivial case in which A contains all neighbors of i 0 , that is A = N i0 , one immediately obtains 1 A cP (s)|i 0 = 0, which simplifies Eq. (S31) to℘ A a|i0 = a|P(s)|i 0 =℘ loc a|i0 (s) for all a ∈ A = N i0 .
Using Eq. (S31) one can conveniently deduce the splitting probability and moments of the conditional first passage time as follows. Using Eq. (S28) and (S31) the splitting probability becomes, which is Eq. 6 of the main text. Analogously, the conditional mean first passage time reads where we have used the product rule of differentiation "∂(f g) = (∂f )g +f ∂g" and the formula d the third line, which finally leads to Eq. 7 in the main text. Note that the conditional mean first passage time, , is obtained by dividing, Eq. (S33) by the splitting probability Eq. (S32). Higher moments can formally be obtained along the same line via Eq. (S31), such that the kth moment satisfies Hence, using Eq. (S31) any moment of the first passage time within the network can effectively be deduced from P(s), which is why the remainder of the Supporting information revolves solely aroundP(s). Before we deducẽ P(s), which is defined in Eq. (S27), let us briefly comment on the limiting case when transition are instantaneous and a time-scale separation allows for a local equilibration prior to any transition 7 . In this case the network dynamics becomes memoryless (i.e., Markovian). Once the network can be described by memory-less jump dynamics as used, for instance, in the celebrated Gillespie algorithm 9,10 , the transitions between network states can be characterized by constant transition rates w i→j from state i to state j. According to the Gillespie algorithm starting from state i the time until the state changes is exponentially distributed with the rate of leaving state i, r i = j∈Ni w i→j , yielding the same exit time distribution   Table S2. Detailed description of the update conditions of the microstate from time step t to time step t + dt. The process it is the coarse grained renewal process (last visited state). Each randomly generated βt is uniformly chosen among the network states adjacent to state j, which are from the set of neighboring states Nj with j = αt or j = it, respectively. Thereby, each element from Nj chosen at random with a uniform probability 1/|Nj|, where |Nj| denotes the number of elements in the set Nj. We note that (xt, αt, it) and (l α t |i t − xt, it, αt) represent the same microstate configuration, even though the last visited state it is replaced by the currently targeted state αt = it.
update conditions at the border currently targeted new network state generate randomly βt ∈ Nα t and set αt → βt with generate randomly βt ∈ Ni t and set αt → βt with

3.A. Stochastic differential equation on a graph
We parametrize the micro-state Γ t at time t in such a way that the reduced state i t represents the last visited mesostate in the network i t ∈ Ω = {1, . . . , N }. The micro-state is assumed to be fully characterized by Γ t = (x t , α t , i t ), where x t denotes the distance from the last visited meso-state i t along the currently targeted meso-state α t as shown in Fig. S7a. The variable Γ t = (x t , α t , i t ) fully determines the micro-state configuration on the graph. Denoting the distance between two nodes i and j by l i|j = l j|i , the distance function x t must lie within the interval 0 ≤ x t ≤ l αt|it . The currently targeted state α t and the last visited state i t remain constant, unless the distance x t reaches either the "inner boundary" x t = 0 or the "outer boundary" x t = l αt|it , after which the variables change according to the rule in Tab. S2. As an illustration we use the trajectory from Fig. S1 (see also Fig. 2 from the main text), where the detailed trajectory corresponds to Γ t = (x t , i t , α t ). In Fig. S1a each colored cross corresponds to a visit of a state, where each revisit of the state with the same color corresponds to all incidents x t = 0 in which the "inner boundary is hit". Conversely, the first visit of a state α t with a different color corresponds to hitting the outer boundary x t = l αt|it after which the latest visited state becomes i t+0 = α t and the newly targeted state is chosen, without loss of generality, equally likely among the neighbors α t+0 ∈ N αt . It should be noted that the micro-state description deliberately contains a redundancy since the micro-states Γ t = (x t , α t , i t ) andΓ t = (l αt|it − x t , i t , α t ) correspond to exactly the same micro-state configuration, even though the last component ofΓ t does not represent the last visited state (see Fig. S7).
The micro-state Γ t = (x t , α t , i t ) evolves such that both the last visited state i t and the currently targeted state α t remain constant during each period, in which the distance x t lies within the interval 0 < x t < l αt|it , which corresponds to x t = 0 and x t = l αt|it . During such periods when both α t and i t are constant, the distance x t between two connected nodes evolves according to the anti-Itô Langevin equatioṅ where D j|i (x) denotes the diffusivity at location x with D j|i (x) = D i|j (l i|j − x), ξ t is Gaussian white noise with zero mean and covariance ξ t ξ t = δ(t − t ), " " denotes the anti-Itô product, and F j|i (x) is a drift force pointing from state i to state j with F j|i (x) = −F i|j (l i|j − x). Furthermore, β is the inverse temperature, which according to the fluctuation dissipation theorem allows to express the inverse friction coefficient (mobility) as µ(x) = βD(x), which is already inserted in the first term of Eq. (S35). The anti-Itô differential equation can also be written as an Itô equatioṅ where D i|j (x) = ∂ x D i|j (x), which is equivalent to Eq. (S35). Eqs. (S35) and (S36) describe the time evolution of the first component of the micro-state Γ t = (x t , α t , i t ). Numerical schemes to evolve Eqs. (S36) and Eq. (S36) are presented below, where Supporting note 3.A(i) shows a naive simple Euler method, and Supporting note 3.A(ii) the celebrated Milstein scheme 11 , which for systems with additive noise can be conveniently improved further into the explicit stochastic Runge-Kutta scheme 12 , which is provided in Supporting note 3.A(iii). For systems with multiplicative noise (non-constant noise amplitude), we generate trajectories using the Milstein scheme from Supporting note 3.A(ii). For systems with additive noise (constant noise amplitude, that is D j|i (x) = const.), we use the scheme shown in Supporting note 3.A(iii) (adopted from Ref. 12). Functionals of trajectories such as the dwell and transition time periods are always evaluated irrespective of the chosen numerical intergration scheme of the Langevin equation as will be explained in Supporting note 3.A(iv). 3.A(i). Naïve anti-Itô Euler scheme (strong order 0.5) The simplest way to numerically integrate the anti-Itô Langevin equation Eq. (S35) in time from t to time t + ∆t is the following anti-Itô Euler schemẽ where σ i|j (x) ≡ 2D i|j (x)∆t andẐ t is a standard normally distributed random number, i.e.,Ẑ t ∼ N (0, 1). The first line of Eq. (S37) estimates the updated position with the valuex t after which the second line effectively "replaces" the last term of the first line by σ αt|it (x t )Ẑ t . Eq. (S37) becomes the well known Euler-Maruyama method if D i|j (x) is constant, since the second line then simplifies to x t+∆t =x t . Once the position exceeds the outer boundary x t+∆t ,x t > l αt|it or the inner boundary x t+∆t ,x t < 0, α t and i t are updated according to Tab. S2. We note the pathwise error of the Euler scheme (S37) (i.e. the strong error) 11 is expected to scale as ∝ ∆t 0.5 , i.e. the scheme is of the strong order 0.5.

3.A(ii). Milstein scheme (strong order 1.0)
Since to our knowledge higher order stochastic Runge-Kutta schemes can only be found in the literature for Itô integrals or its Stratonovich variants 11 , we will now use the Itô representation of the equation of motion (S36). In the case of multiplicative noise D i|j (x) = 0 Euler scheme from Supporting note 3.A(i) can be improved according to the Milstein scheme 11 , which is of strong order 1.0 (i.e., pathwise error scales as ∝ ∆t 1.0 ), which propagates the system from time t to time t + ∆t according to where Z t ∼ N (0, 1). The last term in the first line reduces the pathwise error from ∝ ∆t 0.5 to ∝ ∆t 1.0 . In the second line of Eq. (S38) we solely combined the terms containing the derivative of the diffusion coefficient. Once the position exceeds the outer boundary x t+∆t > l αt|it or the inner boundary x t+∆t < 0, α t and i t are updated according to Tab. S2. For systems with non-constant diffusion coefficient D i|j (x) (i.e., multiplicative noise) we use the Milstein scheme from Eq. (S38).

3.A(iii). Stochastic Runge-Kutta with additive noise (strong order 1.5)
For the simulation of stochastic trajectories with a constant diffusion coefficient D i|j (x) = const. = D, which renders the noise additive, we use an explicit stochastic Runge-Kutta scheme of strong order 1.5 from Ref. 11 (see Table S3. Update of dwell and transition time functionals. The initial position is assumed to be x0 = 0, time of the last visit is initially set to Tvisit = 0 along with the time of the last transition T last = 0. Each passage accross an outer boundary results in a transition time δt and dwell time τ that correspond to one transition event along a single transition γ. update conditions at the boundary step functional outer boundary (xt ≥ l α t |i t ) inner boundary (xt ≤ 0) 1 st set Tvisit = t 2 nd splitting transition store one transition γ = (it → αt) 3 rd dwell time τ store τ = Tvisit − T last in transition γ 4 th transition time δt store δt = t − Tvisit in transition γ 5 th set T last = t also explicitly Ref. 12), which involves the following steps assuming a time increment ∆t. In order to update the distance variable from x t to x t+∆t , we first generate two independent standard normally distributed random numbers, Z t ∼ N (0, 1) and ζ t ∼ N (0, 1), calculateR t =Ẑ t /2 + ζ t · √ 3/6 and then update the position according to 12 where σ = √ 2D∆t. We emphasize that this stochastic Runge-Kutta scheme is of strong order 1.5 and assumes the diffusion coefficient to be constant. Moreover, this scheme requires to generate two random numbers instead of one random number in each iteration step. A quite comprehensive collection of further higher order stochastic integration schemes can be found in Ref. 11, which in contrast to Eq. (S39) require to generate non-Gaussian random numbers.

3.A(iv). Evaluation of dwell and transition time functionals
The residence time interval in one reduced network state spans the time period between the first entering a network state on a graph until the first entrance to another state (see Fig. S1), i.e., the time between two state changes. The dwell and transition time dissect the residence time interval into two separate intervals, in which the last recurrence (revisit) to the same state before changing to another state terminates the dwell time τ and the triggers the beginning of the transition time period δt which in turn spans the remaining time until the state changes. To evaluate dwell and transition time functionals numerically we perform the following computational steps.
Whenever the position x t exceeds the "outer boundary" x t ≥ l αt|it (state change) or "inner boundary" x t ≤ 0 (recurrence), which represent a state visit, the variables α t , i t are updated according to Tab. S2. Any update of α t , i t according to Tab. S2 is accompanied with a change of dwell time τ and transition time δt according to Tab. S3. Thereby, T visit denotes the last visit of a network state and T last the time when the last transition to another network state was completed. Each transition event is stored in a list for each transition γ (see second step in Tab. S3).

3.B. Fokker-Planck equation on local star-like graph
The preceding subsection dealt with single trajectories. Let us now pick without loss of generality one state of interest, i, and focus us on a local star-like graph spanned by the i-th state. For a pair of neighboring states α, β ∈ N i at distances x and y within 0 ≤ x ≤ l α|i and 0 ≤ y ≤ l β|i the probability density to find the system in the state (x, α, i) after time t starting initially from (y, β, i) denoted by P i (x, α, t|y, β) satisfies the Fokker-Planck equation where U α|i (x) = − x 0 F α|i (x )dx and without any loss of generality we assume the diffusion constant to be continuous D α|i (x) = D α |i (0) for all α, α ∈ N i . Note that J i (x, α, t|y, β) ≡ D α|i (x)[βF α|i (x) − ∂ x ]P i (x, α, t|y, β) denotes the 25 probability flux away from i andL F α|i (x) denotes the (forward) Fokker-Planck operator. The initial probability density is set to P i (x, α, t|y, β) = δ(x − y)δ αi , where δ(x − y) is the delta-function and δ αi the Kronecker-delta. The inner boundary conditions read P i (0, α, t|y, β) = P i (0, α , t|y, β), ∀α ∈ N i , and α J i (0, α, t|y, β) = 0, which mean that trajectories are continuous and fluxes are conserved according to Kirchhoff's law. Note that the generalization to both, diverging force kicks and discontinuous diffusion landscapes, is explicitly discussed in Supporting note 8.B. Hence, we derive all of the results assuming without loss of generality assuming the validity of Eq. (S41) and, thereby, render the derivation of the main results less tedious. There are two distinct boundary conditions at the outer end of the αth leg adjacent to node i (i.e. α ∈ N i ), which correspond to P i (l α|i , α, t|y, β) = 0 if the α-th outer boundary is absorbing, J i (l α|i , α, t|y, β) = 0 if the α-th outer boundary is reflecting. (S42) For all ends absorbing P i (x, α, t|y, β)dx is the probability that a trajectory starting from distance y from state i in direction towards state β will be at time t within the interval x and x+dx having never reached any of the neighboring states = i. In this case the survival probability, S i (t|y, β) = α l α|i 0 P i (x, α, t|y, β), decays monotonically in time from S i (0|y, β) = 1 to S i (∞|y, β) = 0. More precisely, we obtain 1,13 where ℘ loc α|i is the local state-to-state kinetics with −∂ t S i (t|0, β) = α ℘ loc α|i (t) = ℘ exit i (t) for all β; the Laplace transform of ℘ loc α|i is given in Eq. (S15).

Supporting note 4. PROOF OF SYMMETRY AND INDEPENDENCE OF DWELL AND TRANSITION TIME USING GREEN'S FUNCTION THEORY
In the following we prove that diffusive dynamics on a graph (S40) renders dwell and transition times conditionally independent functionals. We first show that the aforementioned conditional independence follows directly from the definition of the coarse-graining (last visited state) based on the gedanken experiment from Fig. S1 (see also Fig. 2 in the main text). Using Green's function theory we then prove the following two symmetries: (i) the dwell-time statistics solely depend on the initial state, and (ii) the transition time obeys a forward/backward symmetry.

4.A. Proof of conditional independence between transition and dwell time
The independence of transition and dwell time follows immediately from the coarse-graining of the micro-state trajectory once we identify each state-visit by an "erasure of memory" as follows.
The micro-state Γ t = (x t , α t , i t ) is characterized by the last visited state i t , and the distance x t from the last visited state in direction to the currently targeted state α t . Each recurrence in Fig. S1 (see also Fig. 2 in the main text), highlighted by colored crosses, represents a state-visit x t = 0, which in turn fully determines the micro-state via Γ t = (0, α t , i t ) = (0, β, i t ) = (l αt|it , i t , α t ), where the symbol " =" refers to parameters corresponding to the same micro-state. Since the micro-state Γ t is assumed to have a Markov kinetic, we find that the future state-visit depends only on the last state-visit not on the state-visits before the last one, which triggers a renewal of the dynamics. Since a transition spans the time after the last revisit of a state (recurrence) and the dwell time spans the time before the last revisit of a state (see Fig. S1a), said revisit of a state causes their statistical independence. This completes the proof of independence between transition and dwell time. In the following we derive symmetries of transition and dwell times using the underlying Fokker-Planck equation on a graph.

4.C. Transition-time statistics from Green's function along a single leg with absorbing boundary conditions
The transition path from node i to node α starts with the last recurrence to node i and ends with the first visit of another node α conditioned that i has not been visited in between 14 . Suppose that x t denotes the distance from node i towards node α satisfying the Langevin equation (S35) with fixed α t = α and i t = i. Then the transition time is described by the random variable whose probability density function is denoted by ℘ tr α|i (δt), which we now determine. Note that an unsuccessful transition attempt terminates as soon as x t = 0, whereas a transition is successfully completed once x t = l α|i . Since the transition eventually only results from successful attempts we need to discard all unsuccessful attempts in the following manner by introducing the transition Green's function, G tr α|i , defined as follows. The probability density starting from x 0 = y to find the trajectory after time t at distance x from node i in direction to node α, while never having either returned to state i or reached state α, will be given by G tr α|i (x, t|y). The probability density satisfies initially G tr α|i (x, 0|y) = δ(x − y) and evolves as function of time t according to the Fokker-Planck equation ∂ t G tr α|i (x, t|y) =L F α|i (x)G tr α|i (x, t|y) = −∂ xĴ F α|i (x)G tr α|i (x, t|y) with absorbing boundary conditions G tr α|i (0, t|y) = G tr α|i (l α|i , t|y) = 0, which account for the termination by either a successful or an unsuccessful transition attempt. The absorbing boundaries effectively terminate the process once either of the nodes i or α is reached. The transition-time statistics of will be determined by taking the limit y → 0 (starting from node i) and x → l α|i (ending in node α) from successful trajectories in Eq. (S50), i.e., Before taking the limit we Laplace transform the Green's functionG tr α|i (x, s|y) = ∞ 0 e −st G tr α|i (x, t|y)dt, which can be written using the solutions Eq. (S46) in the form [15][16][17] where we defined the Wronskian satisfying 15,17 At the boundaries the Wronskian becomes w α|i (l α|i , s) = ψ in α|i (l α|i , s)J out α|i (l α|i , s) and w α|i (0, s) = −J in α|i (0, s)ψ out α|i (0, s) due to ψ out α|i (l α|i , s) = ψ in α|i (0, s) = 0. Using for x > y, the Laplace image of the probability density of the transition time becomes J out α|i (y, 0) w α|i (y, s) + J in α|i (y, s)ψ out α|i (y, s) J out α|i (y, s) w α|i (y, 0) + J in α|i (y, 0)ψ out α|i (y, 0) × where we performed the following algebraic steps. From the first to the second line of Eq. (S55) we rewrote the first fraction, which formally gives "0/0" in the limit y → 0, by first using Eq. (S53) with x = 0, inserting the resulting ψ in (y, s), and using lim y→0 w α|i (y, s) = w α|i (0, s). Since J out α|i (y, s) does not have a singularity in the limit lim y→0 J out α|i (y, s), the singularity "0/0" is solely encoded in the bracketed term "[· · · ] → 0", and cancels in the limit y → 0 in both numerator and denominator. Employing l'Hospitals rule (on the bracketed terms "[· · · ]") we now determine their first derivative with respect to y D α|i (0) ∂ ∂y w α|i (y, s) + J in α|i (y, s)ψ out α|i (y, s) where U α|i (y) = ∂ y U α|i (y), and we have deduced ∂ y w α|i (y, s) = −U α|i (y)w α|i (y, s) from the left side of Eq. (S53). Furthermore, we used ∂ y J in α|i (y, s) = −sψ in α|i (y, s) following from Eq. (S46), and finally employed J out α|i (y, s) = −D α|i (y)βU α|i (y)ψ out α|i (y, s) − D α|i (y)∂ y ψ out α|i (y, s). Inserting Eq. (S56) into Eq. (S55) and applying l'Hospital's rule finally yields the Laplace transform of the probability density of the transition timẽ Eq. (S57) fully characterizes the statistics of transition time.

4.D. Forward/backward symmetry of transition-time statistics
The statistics of the corresponding backward transition can be obtained in an analogous manner as Eq. (S57). Identifying ψ out i|α (y, s) = ψ in α|i (l α|i − y, s) and J out i|α (y, s) = −J in α|i (l α|i − y, s) the backward transition time statistics ℘ tr α|i (s) become℘ where the first step follows from Eq. (S57) and in the second step we used Eq. (S53); in the last step we identified Eq. (S57), which completes the proof of℘ tr i|α (s) =℘ tr α|i (s). In other words, we hereby have proven that the duration of both transitions i → α and α → i are identically distributed.
A similar derivation can be found in Ref. 18 for underdamped systems in which the momentum is assumed to be equilibrated.

4.E. Dwell-time statistics before a transition obeys a state-symmetry
Let t denote the time of exiting from state i towards state α and δt the corresponding transition time, which are distributed according to the probability densities ℘ loc α|i (t) and ℘ tr α|i (δt), respectively. The Laplace transform of ℘ loc α|i (t) and ℘ tr α|i (δt) will be denoted by℘ loc α|i (s) and℘ tr α|i (s). The transition-time statistics ℘ tr (or℘ tr ) does not depend on the time at which a transition path starts and, hence, is independent on the time interval before τ = t − τ is called the dwell-time period. Therefore, we can obtain the statistics of the dwell-time via de-convolution which in Laplace space becomes a simple divisioñ where in the last step of the first line we inserted℘ loc α|i (s) from Eq. (S48), φ loc α|i from Eq. (S49) and℘ tr α|i (s) from Eq. (S57). In the second line of Eq. (S59) we merely canceled out equal terms appearing in numerator and denominator, respectively. We strikingly found that the result does not depend on the final state α, which is why the dwell-time statistics obeys a state symmetry, meaning that only the initial state i enters the statistics. Therefore, we can writẽ ℘ dwell α|i (s) as℘ dwell i (s) in the last step of Eq. (S59). Since the product℘ loc α|i (s) = φ loc α|i℘ tr α|i (s)℘ dwell α|i (s) in Laplace space becomes a convolution in the time domain, we have hereby completed the proof of Eq. 1 in the main text.

4.F. Concluding remarks on the proofs
To summarize this section we have shown in Supporting note 4.A that each change of state i → α in a network, which is taken with splitting probability φ loc α|i , has a corresponding distribution of residence time t being a sum of conditionally independent dwell time τ and transition time δt = t − τ (for a given transition between the pair of states i → α). We have proven two symmetries. First, we proved in Supporting note 4.D that the statistics of transition time obeys a forward/backward symmetry ℘ tr α|i (δt) = ℘ tr i|α (δt). Second, the statistics of dwell time is proven in Eq. (S59) to depend solely on the initial state i, that is, the dwell-time statistics does not depend on the state α to which the trajectory is going to transit.

Supporting note 5. UNCONDITIONED MOMENTS FROM BACKWARD FOKKER-PLANCK EQUATION ON STAR-LIKE GRAPHS AND MOMENTS OF TRANSITION TIME ALONG A SINGLE LEG
In this supporting note we focus on star-like graphs spanned by the i-th node such that each state is taken from the set of neighboring states N i . We determine unconditioned moments of the first passage time that are used in Supporting note 1.C.

5.A. Backward Fokker-Planck equation
In this subsection we determine the unconditioned first passage time to node a ∈ N i starting from a point that lies between node i and a neighboring node j ∈ N i . Before determining the unconditioned moments of first passage time it proves convenient to translate the forward Fokker-Planck equation (S40) into its adjoint, backward form, which reads 13 and the boundary conditions at the i-th inner node from Eq. (S41) become Furthermore we consider first the unconditioned first passage problem to state a by setting P i (l a|i , a, t|y, j) = 0 (ath node absorbing) and the remaining links are made reflecting, meaning that ∂ y P (x, α, t|y, j)| y=l j|i = 0 holds for all α ∈ N i and j ∈ N i \{a}. Note that the forward and backward Fokker-Planck equations both satisfy ∂ t P i (x, α, t|y, j) = L B j|i (y)P i (x, α|y, j) =L F α|i (x)P i (x, α|y, j). The backward Fokker-Planck equation allows to conveniently determine the moments of the unconditioned first passage time using standard methods as follows (see also textbooks in Refs. 1 and 13 or Ref. 19 for a discussion on graphs). We denote the survival probability by which decays monotonically in time, −∂ t S a|i (t|y, j) ≥ 0, due to the single absorbing end at a and x = l a|i . From the backward Fokker-Planck equation (S60) follows the evolution equation for the survival probability with S a|i (t|l a|i , a) = 0 for all t ≥ 0 and ∂ y S a|i (t|y, j)| y=lj = 0 if j = a as well as S a|i (0|y, j) = 1 for all j and y < l j|i . Since the first passage time density is the negative derivative of the survival probability, −∂ t S a (t|y, j), we can write the unconditioned kth moment as where the last identity follows from partial differentiation. Operating from the left by the backward operatorL B j|i (y) in Eq. (S60) yields the hierarchical connection between the moments 13 with T for example, T j , which follows from the decomposition of trajectories on star-like graphs as explicitly depicted in Fig. S5c. Note that a from Fig. S5c here plays the role of j (j = a) and the inner node i 0 here becomes i.

5.B. Hierarchy of moments
In the following we translate Eq. (S65) into a hierarchical integration formula that allows to deduce T (k) a|i (y, j) = 1. First, for j = a we use Eq. (S65) and set therein y = y 2k−1 such that the integral y 0 dy 2k−1 (· · · ) over both sides yields which in turn allows us to obtain Similarly, for j = a we get where we used T The second set of moments with j = a follows from Eq. (S70) where we used the first condition of Eq. (S66), which reads T (k) a|i (0, j). We note that Eqs. (S72) and (S73) determine all unconditioned moments of the first passage time for diffusion graphs with potentials. In the following subsection we provide explicit results for first two moments, i.e., for k = 1 and k = 2.

5.C. Explicit unconditioned moments of the first passage time
To render the rather convoluted calculation more transparent we now perform a notational simplification by removing within this subsection the subscript i referring to current, tagged node i. More precisely, in this subsection we use the shorthand notation l j ≡ l j|i , U j ≡ U j|i , D j ≡ D j|i and j =a ≡ j∈Ni\{a} .
We calculate the first two unconditioned moments using Eq. (S72) and Eq. (S73). Starting with the first moment, we use Eq. (S72) to obtain the first moment of the unconditioned "forward" first passage time j . Note that each "backward" first passage moment is effectively a "simple" 1-dimensional first passage problem. One can, for example, use Ref. 13 for an alternative derivation of Eq. (S77).
Any of the aforementioned integrals can be expressed in terms of the following auxiliary and elementary integrals j . Analogously, the second moments from Eqs. (S76) and (S77) become For completeness we also list in Tab. S5 the third moments, which allow for an alternative derivation of the main result, which is not pursued here. We used them, however, to verify independently the correctness of results in Supporting note 6.C. Table S4. First six auxiliary functions at a glance. Since in this section we omitted for convenience the initial state i in the subscript, the general case is recovered as g α|i and h α|i along with Dα = D α|i and Uα = U α|i , which is used in the main text and the remainder of the Supporting information.  Table S5. Unconditioned third moments. After some quite extended tedious but straight forward calculations we obtain the third moments. Third moments are listed for the sake of completeness.

5.D. Transition time along one leg
Let us now focus on the segment between pair of nodes, an initial node i and a target node a. Suppose that we start at a distance x from node i and ask for the n-th moment of the first passage time, δT To make the calculations more efficient and to avoid redundant integrals we introduce the shorthand notation from Eq. (S78) along with the definitions such that I (k) a|i (l a|i ) and R (k) a|i (l a|i ). After some algebra we obtain the first moment by means of the following calculation δT (1) a|i (x) = 1 I (1) a|i l a|i x e βU a|i (y) D a|i (y) dy x 0 e βU a|i (y) D a|i (y) dy where in the first step we simply used (S81) with δT Similarly, the second moment of the transition time reads, using Eq. (S81) and some tedious but straightforward algebra, where in the last step of the first line we inserted Eq. (S83) and afterwards inserted the auxiliary integrals from Eqs. (S78) and (S82) and removed redundant terms that cancel in the limit x → 0. Note that all integrals of the type "R (k) a|i " cancel. From Eqs. (S84) and (S85) we establish immediately that the transition time is generally sub-Markovian, i.e. δt 2 loc a|i ≤ 2[ δt loc a|i ] 2 .

Supporting note 6. EXPLICIT CONDITIONAL MOMENTS OF THE FIRST PASSAGE TIME, DWELL TIME AND TRANSITION TIME
In this section each sum over j (or k) runs over all neighboring states of the fixed state i, i.e., we use the short-hand notation j ≡ j∈Ni and k ≡ k∈Ni , where N i denotes the set of states adjacent to the fixed initial state i.

6.A. Splitting probability
Inserting the first line of Eq. (S17) and Eq. (S79) into Eq. (S22) results in the local splitting probability where a ∈ N i . The first moment of the exit time is obtained by inserting the first line of Eq. (S17) and Eq. (S79) into Eq. (S24) yielding where we first used Eq. (S24) and then inserted Eqs. (S17) and (S79); in the last step we used Eq. (S86). The first moment of the dwell time τ i can now be deduced from the conditional independence of transition and dwell time (see Supporting note 4), implying t exit ]. Upon using j φ loc j|i = 1 we obtain The exact expression for the second moment of the exit time is obtained after some tedious algebra and reads where in the first step we adopted Eq. (S26), in the second line we inserted Eqs. (S79) and (S80); in the last line we canceled equal terms and finally inserted Eqs. (S86) and (S89). From Eq. (S90) follows that a vanishing extrinsic noise, which corresponds t loc a|i = t exit i , immediately renders fluctuations sub-Markov (see following subsection Supporting note 6.E for more details).
Using the conditional independence of dwell and transition times, t 2 loc j|i = δt 2 tr j|i + 2 δt tr j|i τ dwell where in the first step we used the definition t 2 exit i ≡ j φ loc j|i t 2 loc j|i and in the second step we inserted the transition time moments δt tr j|i and δt 2 tr j|i from Eqs. (S84) and (S85), respectively, the splitting probability φ loc j|i from Eq. (S86) as well as the first moment of the dwell time τ dwell i from Eq. (S88) into the second moment of the exit time Eq. (S90). In the final step of Eq. (S88) we combined the two sums. Eq. (S91) precisely proves the second line of Eq. 12 in the main text.

6.E. Second main result: Vanishing extrinsic noise renders fluctuations sub-Markov
Let us briefly prove that vanishing extrinsic noise implies sub-Markov fluctuations. Vanishing extrinsic noise means that all transition from state i to a neighboring state a take on average equally long, that is, δt tr a|i = δt tr j|i for all a, j ∈ N i . Let us from now on (in this subsection) assume that the extrinsic noise vanishes. According to Eq. (S87) and Eq. (S89) we obtain t loc j|i = t exit i , which inserted into Eq. (S90) gives where in the last step we again used Eq. (S87). Since the auxiliary integrals are defined to be positive I j|i ≥ 0, we generally find t 2 exit i ≤ 2[ t exit i ] 2 , which proves that fluctuations become sub-Markov. We note that t 2 exit This completes the proof of the main result of this paper, which states that vanishing extrinsic noise implies fluctuations to become sub-Markov.
We note that the following converse equivalent conclusion can be drawn from the proven statement. Whenever the fluctuations are pronounced akin super-Markov, that is t 2 exit i ≥ 2[ t exit i ] 2 , there must exist parallel transitions i → a and i → j that are unequally fast t tr a|i = t tr j|i .

Supporting note 7. THERMODYNAMIC CONSISTENCY OF COARSE-GRAINING
In this supporting note we prove the thermodynamic consistency of the coarse-graining. Thermodynamic consistency refers to the entropy production rate 7,21 being equal for both, the coarse grained process -the last visited state i t -and the detailed "microscopic" process, Γ t = (x t , α t , i t ). We show that the validity or the violation of detailed balance is entirely encoded in the splitting probability, which is explicitly evaluated in Eq. (S86). At the end of the subsection we discuss the special limit of local detailed balance. Figure S8. Directed cycle. Detailed balance is broken if and only if one can find at least one directed closed cycle (see thick red arrow lines) leading to a non-vanishing work integral C F · dx = 0. Conversely, detailed balance is satisfied if C F · dx = 0 holds for all cycles C in the network. The number of state in the cycle C here are |C| = 6, whereas the total number of network states are Ω = 9

7.A. Thermodynamic consistency is embodied by the splitting probability
It turns out that the functional from of ln(φ loc j|i /φ loc i|j ) determines the steady state entropy production rate. As we show explicitly in the following paragraph the ratio of between forward and backward splitting probability along a single path can be cast into the form where the last term can be identified by the entropy produced in the surrounding medium at a inverse temperature β = 1/(k B T ), which equals the integral of the friction force along the path from meso-state i to meso-state j, defined by i→j F (x) · dx ≡ l j|i 0 F j|i (x)dx. The first two terms "g(i) − g(j)" are a discrete gradient term, which only in the limit of local detailed balance (e.g., see Ref. 7) relates to the free energy difference between the states j and i.
In a long trajectory each state will on average be entered and left equally often such that "g(i) − g(j)" contributes to a zero entropy production rate in the long time limit. The non-conservative last term in Eq. (S93) is the only term that can account for a non-zero entropy production rate in the steady state. Since the last term in Eq. (S93) equals the entropy production in the surrounding medium according to the exact underlying dynamics on a graph we have, therefore, identified an exact link between the microscopic entropy production -i.e., the right hand side of equation (S93) -and the coarse-grained dynamics, which is represented by the left hand side of Eq. (S93).
To illustrate that we pick a directed cycle of states in the network as depicted in Fig. S8. The integral of the force along a closed cycle is called the affinity 21 (S94) The affinity accounts for non-conservative (rotational) force fields. Detailed balance is broken if one can find at least one cycle C with A[C] = 0.
Before we prove Eq. (S93) in the following subsection, let us briefly comment on its connection to the breaking of time-reversal symmetry. At a first glance Eq. (S93) seems to be in stark contrast to the finding of Ref. 22, which in addition to the affinity from Eqs. (S93) and Eq. (S94) identified also a so-called spurious "waiting time contribution" to the entropy production rate. The reason why the "waiting time contribution" generally vanishes, here, is the kinetic hysteresis in the coarse-graining (see Fig. 2b), that is, the time reversed coarse-grained process does not coincide with coarse-grained time reversed process. In the following we prove Eq. (S93), which is equivalent to the corresponding integral form Eq. (S94).
where in the second step we defined βg(α) ≡ − ln[ k∈Nα 1/I (1) k|α ] for α = i, j and in the second line we inserted Eq. (S78), that is, I is the force pointing away from node i in direction to node j. The reflected force is minus the forward one, where we also used D i|j (x) = D j|i (l j|i − x) in the first step, carried out the integrals in the second step and finally

7.C. The peculiar local equilibration
In the limit of high free energy barriers, which implies B j|i → ∞ in Fig. S9, the first two auxiliary integrals become where is assumed to be small compared to l j|i . The average time to exit from state i according to Eq. (S87) for high barriers becomes approximately equal to The rate of jumping from state i to state j in turn becomes Figure S9. Local potential assuring local equilibration. Local potential between nodes i and j. The two states are separated by a single maximum at x * j|i of the local potential characterized by j|i . Local equilibration occurs if the local (free) energy barriers are high enough, meaning that B j|i kBT and B i|j kBT .
where we defined the local free energy of state i by dy. Note that the auxiliary potential is defined by U k|i (0) = 0, meaning that it represents the local potential relative to the value at node i. For networks that are not externally driven the work along any transition in Eq. (S100) vanishes, which in addition to local equilibrium assumed in this subsection would also yield global equilibrium.
Since, high free energy barriers between any pair of state will eventually render all higher order integrals I j|i is negligibly short compared to the mean exit time from state i. More precisely, it has been found for a parabolic barrier that the transition time scales logarithmically 23,24 with B j|i , i.e. ∝ ln B j|i , while the exit time grows much faster 13,25,26 , i.e. ∝ e βB j|i . One can show that a rectangular shaped potential with a barrier height B j|i in fact yields a finite transition time δt tr j|i = I j|i /I (1) j|i in the limit B j|i → ∞ while at the same time the exit time diverges ∝ e βB j|i → ∞. The shape of the potential barrier may therefore decide whether or not the transition time grows with increasing barrier height.

8.A. Summary of the results
Let us briefly summarize this section. We first explain in Supporting note 8.B the extension of our theory to both discontinuous local potential landscapes and discontinuous diffusion landscapes.
As a first supporting example in Supporting note 8.C, which is not discussed in the main text, we analyzed a well studied bare diffusion on a graph with vanishing force field. We confirm that the transition time is generally sub-Markov, and show that vanishing extrinsic noise indeed renders the exit time sub-Markov.
In Supporting note 8.D we provide a detailed analysis of the catch-bond system presented in Figs. 1d and 3b,c in the main text. We further corroborate the exact theory by improving the statistics of simulations beyond the values typically accessible in experiments 27,28 . We verify the symmetry (i) in the main text, which states that the dwell time solely depends on the initial state and not on the pathway along which it ruptures.
In Supporting note 8.E we provided details of the energetics of the ATPase 29 and related them to the splitting probability. We provide a detailed analysis to the data that leads to Fig. 3e in the main text, and verify the main practical result derived in Supporting note 6. We verify the symmetry (ii) in the main text, which implies the transition-time indeed indeed obeys the reflection-symmetric, i.e., obeys a forward/backward symmetry.
In Supporting note 8.F we discuss the biased diffusion -the ATPase model from Supporting note 8.E in the limit of all barriers set to zero. We showed that even under detailed balance conditions the waiting time appreciably differs from a memory-less exponential distribution. The biased diffusion model is shown to saturate the lower bound plotted i j ∆U j|i Figure S10. Discontinuous potential. Local potential between state i three neighbor-states. Along the leg from state i to state j the potential has a discontinuity of strength ∆U j|i .
in Fig. 3f in the main text and is derived from the thermodynamic uncertainty relation 30,31 .
Finally, we provide in Supporting note 8.G an additional analysis on the synthetic network used in Fig. 2 in the main text to introduce the non-Markovian coarse-graining. We provided the local waiting time distributions between all pairs of states and determined the matrices entering Eqs. 6 and 7 in the main text. Interestingly, at long times the local waiting time becomes exponentially decaying with a possibly non-normalized pre-factor that paves the way of further studies of the long-time asymptotics. This section illustrates the straightforward and conclusive extension to possibly larger scale networks using the star-like sub-graphs as building blocks (see Supporting note 2).

8.B. Generalization to discontinuous local potentials and discontinuous diffusion landscapes
Before we apply our theory to examples let us briefly discuss certain generalizations to include both, discontinuous local potentials and discontinuous diffusion landscapes. We will first explain how one deals with discontinuous local potentials in general. Next, we account for possible discontinuous diffusion landscapes by removing them through a linear stretch of coordinates. Therefore, discontinuous diffusion landscapes can always be accounted for by mapping the coordinate system onto a continuous diffusion landscapes, but with possible discontinuities in the local potential. In the following first discuss discontinuous local potentials, and then explain the scaling of coordinates that allows to account for discontinuous diffusion landscapes.
Discontinuous local potential. Let us begin with a discontinuous "diverging force kick" at the node i towards state j which effectively means F j|i (x) = F cont j|i (x) + ∆U j|i δ(x), where F cont j|i (x) is some continuous force field, δ(x) denotes the Dirac delta-function, and ∆U j|i denotes the strength of the discontinuity. The "force kick" yields the potential U j|i (x) = −∆U j|i − x 0 F cont j|i (x )dx . The local potential has a discontinuity once U j|i (0) = −∆U j|i = 0. A single discontinuity between states i and j is schematically depicted in Fig. S10 (see blue line). The transition time is not affected by such "kicks" since the transition path spans the time interval after the last passage of state i until the first entrance into state j, which can be confirmed by the following argument. To formally avoid a discontinuity we replace the discontinuity ∆U j|i δ(x) by a smoothened force ∆U j|i / within the interval 0 ≤ x ≤ and afterwards take the limit → 0. The auxiliary integrals according to Eq. (S78) become lim →0 I j|i | ∆U j|i =0 (for k = 1, 2 . . .). Since all the odd-valued k auxiliary integrals are affected by the discontinuity in precisely the same manner "I (2k−1) j|i ∝ e −β∆U j|i ", we find that first two moments of transition time, Eqs. (S84) and (S85), are not affected by the discontinuity.
Importantly, a kick of strength ∆U j|i affects the splitting probability φ loc j|i of choosing a transition due to φ loc j|i ∝ 1/I (1) j|i ∝ e β∆U j|i [cf. Eqs. (S78) and (S86)]. Since the dwell time is affected by both splitting probability and transition time [cf. Eqs. (S88) and (S91)] a force-kick of strength ∆U j|i does affect the dwell-time statistics. As a interim summary we find that force-kicks arising from a discontinuous local potential (see Fig. S10) affect both the splitting probability and the dwell-time statistics, whereas the transition time is not affected.
Discontinuous diffusion landscape. Discontinuous diffusion landscapes, i.e. D j|i (x) satisfies D j|i (0) = D k|i for some k = i, are dealt with in the following way. First, we locally re-scale the coordinate system such that the discontinuity disappears. Specifically, we locally stretch the coordinates between nodes i and j, l j|i , homogeneously by a factor α j|i (l j|i → α j|i l j|i ) to obtain a re-scaled diffusion landscape, α 2 j|i D j|i (x/α j|i ), and a correspondingly re-scaled local potential, U j|i (x/α j|i ) − β −1 ln α j|i , where β −1 = k B T is the thermal energy. By choosing α j|i such that the diffusion landscape becomes continuous, we obtain a mapping from a discontinuous diffusion landscape onto a continuous one. Hence, discontinuous diffusion landscapes can be removed entirely via a linearly change of local coordinates. Such a re-scaling (mapping) give rise to a discontinuous potential which can be dealt with as explained above. In this sense all of the results presented here (main text and Supporting Information) apply to dynamics on a graph with both, discontinuous diffusion and discontinuous local potentials equally well. Notably, the results derived Supporting note 6, i.e. Eqs. 10-12 in the main text, can be used unaltered in the case of discontinuities in the potential and diffusion landscapes. The simplest kinetics on a graph corresponds to a vanishing force F j|i (x) = −∂ x U j|i (x) = 0 rendering the dynamics purely diffusive. If the dynamics is purely diffusive with D j|i (x) = D (for convenience we set it to D = 1) the integrals from Eq. (S78) yield I (k) where · · · is the floor function that yields the closest integer that is smaller than "· · · ". Using Eq. (S86) the splitting probabilities in a zero-force network are given by (S101) Using I Hence we hereby confirm with this example one of our main result, which is proven in Supporting note 6.E and states that vanishing transition noise renders the exit time fluctuations sub-Markov. Exact potential. In the analysis of the catch-bond dynamics we consider a local free energy given by U j|0 (x) = U (0) j|0 (x) denotes the free energy along path j without pulling, and U pull (x, F ) denotes change of potential under force F . Defining the scaled dimensionless distancex ≡ x/(4.14 nm), the potentials are given by where l 1|0 = 0.06 × 4.14 nm = 0.248 nm and l 2|0 = 0.397 × 4.14 nm = 1.64 nm. The dimensionless unit-length x = x/(4.14 nm) is used to connect thermal energy and force according to k B T /(1 pN) = 4.14 nm. The diffusion landscape is set to be constant D j|0 (x) = (4.14) 2 nm 2 s −1 = 17.1 nm 2 s −1 along both pathways j = 1 and j = 2.
In contrast to the experiment 27 we assumed here that all trajectories instead of 99.2 % start from x = 0. We note that the fit experimental data carried out in Ref. 27 found the initial binding to take place with 99.2 % in what was called state 1, which corresponds here to the distance x = 0. Correspondingly, about 0.8 % of pulls carried out in Ref. 27 were estimated to start in the first intermediate minima along the slow path 2 (potential is depicted in Fig. 6 in the main text).
Simulation with improved statistics. We simulated individual pulling experiments using the strong order 1.5 stochastic Runge-Kutta scheme from Supporting note 3.A(iii) with a time increment of ∆t = 10 −6 s. In Fig. 3c of the main text we show the results for 500 rupture events. Since at low pulling-force the probability φ 2|0 of the bond to rupture along the slow pathway 2 is so small, none or only few ruptures along the slow paths are recorded. Therefore, the average life-time is expected to be particularly poorly estimated at small pulling force. To illustrate the validity of our exact theory we reduced the experimental error further by using more rupture events (in our analysis up to 10 000 ruptures), and thereby decreased the statistical error. In Fig. S11 we show the results from Fig. 3c with subsequently increased number of rupture events. To better illustrate the effect of the error at small pulling force (and short life-times) we also show the results from Fig. S11 on a semi-logarithmic scale in Fig. S12. Note that Fig. S11b represents Fig. 3b,c in the main text. Comparing the average life time of the bond along pathway 1, t 1|0 , and pathway 2, t 2|0 we find a strong asymmetry (more than two orders of magnitude) of the mean life time of the bond (see highlighted entries in Table S6).        (deviations are merely arising from finite statistics). This example illustrates that the dwell-time statistics does not depend on the pathway of the rupture (states 1 and 2) but only on the initial state 0, i.e. the dwell-time statistics solely depends on initial state (not on the final one). This example corroborates Eq. 1 in the main text.
Alternative experiment from Ref. 28. Finally, we want to comment on the effect of changing the length-scale. Suppose the length x is stretched by a factor λ such that U j|i (x) → U j|i (x/λ), i.e. F j|i (x) → F j|i (x/λ)/λ, which implies that the load force F becomes equivalent to the load force F/λ after scaling. To address a related experiment 28 with quite different time-and length-scales we need to scale the length by a factor λ (λ ≈ 3) such that the maximum life-time is found at F ≈ 10 pN as reported in Ref. 28 instead of F ≈ 30 pN, which is shown here in Fig. S11 (see also Ref. 27). Moreover, scaling the diffusion constant D → αD corresponds to an accelerated time, which re-scales the bond life-time ∝ α −1 λ −2 . To shift the maximum life time from 30 s (see Fig. S11) to 1.2 s = 30 s/25 from the experiment in Ref. 28 we, in addition to the scaled location of the maximum, scale the diffusion constant by α = 25 × λ −2 ≈ 2.78. With this scaled units we obtain the same plots as shown in Fig. S11 (see also Fig. 6b,c in the main text) but with the x-axis scaled by a factor of 1/3 and the y-axis is scaled by a factor of 1/25 to quantitatively account for different experiment reported in Ref. 28.
Summarizing, in this subsection we further confirmed Eqs. 10-13 in the main text, by numerical experiments, which are shown in Figs. S11 and S12 using more statistics (up to 10 000 rupture events). We tested the decomposition of the bond life-time into its dwell-and transition-period according to Eq. 1 in the main text and we corroborated the theoretical result that the dwell time indeed solely depends on the initial state but not on the final one (see Fig. S14).

8.E. Additional information about ATPase with sine-wave potential
Energetics. We assume the dynamics of the ATPase to be described by the following model. The ATPase rotates stochastically about one axis and feels an angle dependent torque at rotation angle θ t at time t. The torque is assumed to have the following two contributions: (i) a rotational free energy potential (see blue shaded lines Fig. S15a) separates three favorable rotational states (minima) that are separated by 120 • and triggers a conservative torque proportional to the slope of the blue line. The potential is given by U rot (θ) = B 2 [1 − cos(θ/3)] with the implied conservative torque given by −∂ θ U rot (θ). (ii) an external non-equilibrium torque M embodies sum of a mechano-chemical force arising from the hydrolysis of an ATP molecule and a mechanical torque that is applied to the shaft. More precisely, we assume tight coupling with M = ∆µ/120 • − M mech , where ∆µ = µ ATP − µ ADP − µ Pi is the chemical free energy released along a hydrolysis reaction ATP → ADP + P i and M mech reflects a mechanical torque 29 .
Coarse-graining. Using the scaled coordinate x = θ/120 • the local potential which accounts for both torque (i) and mechano-chemical force (ii) is given by U ± (x) = B 2 [1 − cos(2πx)] ± (M × 120 • )x with l ± = 120 • /120 • = 1, where "+" accounts for the potential along the counterclockwise direction and "−"corresponds to the potential along the opposite direction. Detailed balance is established whenever the chemical free energy released per 120 • step is balanced by the mechanical torque (multiplied by 120 • ), i.e. M = 0 (see item (ii) above).
For convenience, we restrict our analysis to a periodic rotation which has a sine wave shape with barriers of height B that separate two minima. Counting the minima in the counterclockwise direction yields the set of states Ω = {1, 2, 3} such for each state i ∈ Ω the local potential formally reads U i±1|i ≡ U ± with the convention "i − 1 = 3 if i = 1" and "i + 1 = 1 if i = 3". The sets of neighboring states are then N 1 = {2, 3}, N 2 = {1, 3} and N 4 = {1, 2}, that is, the The periodicity in each 120 • step and the forward/backward symmetry of the transition time "℘ tr + (δt) = ℘ tr − (δt)", which is proven in Supporting note 4, implies that the local first passage time is given by ℘ loc ± (t) = φ ± ℘ exit (t), such that t k ± = t k exit . Therefore, the extrinsic transition-noise vanishes which according the proof of our second main result shown in Supporting note 6.E implies the fluctuations to become sub-Markov: σ 2 exit ≡ t 2 exit − ( t exit ) 2 ≤ ( t exit ) 2 , that is, t 2 exit ≤ 2( t exit ) 2 . The symmetry for the mean first passage time t + = t − , was, to the best of our knowledge, first discovered in Ref. 32 for lattice models of kinesin motors (see also Ref. 33). The extension to the entire distribution ℘ loc ± (t) = φ ± ℘ exit (t) was later found in studies for the stopping-time of the thermodynamic entropy production in active molecular processes 34 . The symmetry allows us to simplify the discussion by merely focusing on the splitting probability φ ± and the exit time distributions ℘ exit (t).
Splitting probability. Let us begin with the splitting probability which involves the auxiliary integrals where we used the first line of Eq. (S78) with k = 1 and inserted U ± (x) = B 2 [1 − cos(2πx)] ∓ β(M × 120 • )x and l ± = 1. Substituting x → 1 − x in one of the integrals one can easily show, after some short algebra, that the two integrals are related via I where we defined f = β(M × 120 • ). Exit-time statistics and implied number of transitions. The number of exits after time t, n t , where one exit corresponds to the event of leaving one minima and reaching any other minima for the first time is stochastic and solely influenced by the exit time ℘ exit (t). As explained in above the distribution of the waiting-time the same along both Table S7. Comparing theory to simulation with 5 kBT barriers. Non-equilibrium driving is quantified in terms of f ≡ M × 120 • /(kBT ). Each experimental value is deduced from 500 000 Brownian dynamic simulations using the stochastic Runge Kutta scheme with time increment ∆t = 10 −4 in dimensionless simulation units βD = D = 1. Even though the numerical value in the table would change for another value of D, we would still obtain for any D the same result presented in Fig. 3f, since σexit/ t exit remains unaffected. The theoretical values for the splitting probability follow from Eq. (S108). By evaluating the auxiliary integrals in the first line of Eq. (S78) and using Eqs. (S87), and (S90), we obtain the theoretical values for mean first exit time t exit and the second moment of the exit time t 2 exit , and therefrom the standard deviation σexit = t 2 exit − ( t exit ) 2 . Note that for f = 20 the system is driven so strongly that no backward transition is expected to be observed in 500 000 trajectories, which is why we experimentally determine φ− = 0. directions "+" and "−", i.e. ℘ loc ± (t)/φ loc ± = ℘ exit (t). At long times the central limit theorem for renewal processes 35 renders n t asymptotically normally distributed with mean n t t/ t exit and variance var(n t ) ≡ n 2 t − n t since there forward and backward transition are trivially symmetric.
In this subsection we showed that the splitting probability for the ATPase modeled by a tilted periodic potential is fully determined by the external driving f and is given by Eq. (S108), which notably holds for any 120 • periodic potential. We relate the number of state-to-state transitions to the exit time via the well-established central limit theorem for renewal processes 35 (see also Ref. 3). We illustrate the forward/backward symmetry of transition time in the mean (see Tab. S8) and the entire distribution of transition time (see Fig. S16). In the following subsection we address biased diffusion obtained in the limit of vanishing free energy barriers (B → 0). Let us now consider the limit in which the rotational free energy landscape vanishes U rot = 0 (or B = 0). Introducing for convenience the reduced coordinates x = θ/120 • with l ± = 1 the local potential simplifies to βU ± = ∓β(M × 120 • )x ≡ ∓f x. The splitting probability is still given by Eq. (S108), since it is independent of the the height of the barrier, B, in the form. Using Eqs. (S87) and (S90) we obtain the mean and variance of exit time t exit = e f − 1 f (e f + 1) and σ 2 exit = t 2 exit − ( t exit ) 2 = 2(e 2f − 2f e f − 1) f 3 (e f + 1) 2 , (S110) respectively, where we further inserted the local potential βU ± = ∓f x along with D ± = 1 into the first line of the corresponding auxiliary integrals in Eq. (S78). Memory prevails even in equilibrium. Interestingly, the fluctuations are not only sub-Markov σ exit / t exit ≤ 1 but using Eq. (S110) also satisfy σ exit / t exit ≤ 2/3 ≈ 0.816 with equality holding at equilibrium conditions. In contrast to the results presented in Fig. 3e in the main text (see also previous subsection), the dynamics does not become memory-less when approaching the equilibrium at f = 0. To see this we compare in Fig. S17 the probability densities of the exit time for the biased diffusion for different values of the non-equilibrium driving f (see Fig. S17a) to driven diffusion with 5 k B T barriers (see Fig. S17b) which are adopted from Fig. 3e in the main text. The results displayed in Fig. S17a are obtained as follows. In the case of plain biased diffusion without any free energy barriers one can expand the exit time density via separation of variables 1 yielding where we recall D ± = 1. Notably, at equilibrium (φ + = φ − ) the mean square angular deviation is not affected by the memory in the exit time fluctuations (deviations from memory-less), since according to Eq. (S109), they will enter via the last term therein, which scales as ∝ (φ + − φ − ) 2 (see also following paragraph for more details). Conversely, the memory in the dynamics will become particularly relevant for the mean squared angular deviation once the system is strongly driven φ + φ − , i.e., ln(φ + /φ − ) 2 = M × 120 • /(k B T ) 1 (see Fig. 3f in the main text). where Φ ij = φ loc i|j and T ij = φ loc i|j t loc i|j , which are both entering Eqs. 6 and 7 in the main text. From the total set of states Ω = {1, 2, 3, 4, 5} we consider a pair of target states A = {1, 2} such that the complement becomes Table S9. Mean versus asymptotics. Each "experimental" value is deduced from Nsim = 4 × 10 5 simulated exits from each state generated by the stochastic Milstein scheme with ∆t = 10 −4 . The "theory" values are obtained from a numerically evaluation of the integrals in the first line of Eq. (S78) and inserting them into the results for mean and second moment of the exit time in Eq. (S87) and (S90), respectively (see Supporting note 2 for its derivation). Each experimental value has a relative statistical error of about 1/ √ Nsim ≈ 0.0016. (S114) Inserting Eqs. (S113) and (S114) into Eqs. 6 and 7 in the main text (here Eqs. (S32) and (S33)) yields φ A 1|3 = 0.4, φ A 2|3 = 0.6, t A 1|3 ≈ 8.26004, and t A 2|3 ≈ 6.80104. (S115) A Markov model in which T M ij = φ i|j t exit j would inevitably infer the conditional first passage to state 1 to be erroneously faster than the one to state 2. More precisely a Markov state model with the same local splitting probabilities would yield t A,M 1|3 = 5.91808 and t A,M 2|3 = 9.14299 while the correct values are given by Eq. (S115). Second moment of the exit time can accurately estimate the long-time asymptotics. It has been found that the long time asymptotics of the leaving any state i becomes a single exponential decay ∝ e µ ∞ i t . One can formally show that µ ∞ i = lim n→∞ n t n−1 exit / t n exit . It turns out that n = 1 provides a good estimate, µ i = 2 t exit / t 2 exit ≈ µ ∞ i (see also Ref. 8). To illustrate this here we simulate 400 000 exits on the graph from each state in the network using the Milstein scheme Supporting note 3.A(ii) with time increment ∆t = 10 −4 . The results for the probability density of all local waiting times between all states in the five-state network are shown in Fig. S18b-i. Panels (g)-(i) depict the probability densities on a semi-log-scale. Note that straight lines in Fig. S18g-i correspond to an exponential decay. The thick gray lines are are obtained with the values listed in Tab. S9 (in the main text we used µ 3 = µ for the long-time asymptotics depicted in Fig. 2c therein).
Non-normalized long-time asymptotics -a signature of long transitions. Whenever the transition is slow we observe in Fig. S18g-i the long-time asymptotics of local probability density above the normalized gray line, which can be explained as follows. When the transition is long, the probability density ℘ loc j|i (t) becomes negligibly small on time-scales shorter than the transition time t δt tr j|i . Since ℘ loc j|i (t)/φ loc j|i must be normalized ∞ 0 ℘ loc j|i (t)/φ loc j|i dt = 1 one inevitably requires more weight of the probability density at long times. Note that all lines plotted in Fig. (S18)b-i are probability densities which are normalized to unity. In other words, the blue solid line in Fig. (S18)i is above the gray thick line at long times since it is below the thick gray line at short times.
To summarize this subsection, we showed that a first passage problem any network can be decomposed into its local state-to-state changes (see Fig. S18), which represent local star-like graphs. We used the local kinetics in star-like graphs in Eq. (S113) to infer the kinetics without star-like topology in Eq. (S115). This example illustrates now one can straightforwardly analyze kinetics in larger scale networks according to Supporting note 2 by using the results of all star-like sub-graphs. Notably, higher moments of conditional first passage time can be obtained from Eqs. (S31) and (S34).