Noise-induced polarization switching in complex networks

The combination of bistability and noise is ubiquitous in complex systems, from biology to social interactions, and has important implications for their functioning and resilience. Here we use a simple three-state dynamical process, in which nodes go from one pole to another through an intermediate state, to show that noise can induce polarization switching in bistable systems if dynamical correlations are significant. In large, fully connected networks, where dynamical correlations can be neglected, increasing noise yields a collapse of bistability to an unpolarized configuration where the three possible states of the nodes are equally likely. In contrast, increased noise induces abrupt and irreversible polarization switching in sparsely connected networks. In multiplexes, where each layer can have a different polarization tendency, one layer is dominant and progressively imposes its polarization state on the other, offsetting or promoting the ability of noise to switch its polarization. Overall, we show that the interplay of noise and dynamical correlations can yield discontinuous transitions between extremes, which cannot be explained by a simple mean-field description.

Figure 1

Three-state model for bistability. (a) Unconditional random process. With probability $\varepsilon$, a node is selected and changes its state with probability $1/2$ as shown. (b) Conditional random process. At probability $1-\varepsilon$, transitions are conditional on the state of a selected neighbor, e.g., transition $u\to v$ for a given node is possible with probability $p_3$ if the selected neighbor is in $w$, and no transition takes place otherwise.

Figure 2

Polarization as a function of noise. (a) Symmetric process, $p=1$, on a degree regular random graph with $k=3$ (bold black lines) when initializing in the two extremal states $\overline{w}=1$ and $\overline{u}=1$, respectively, and infinite, fully connected network (gray curves). Solid (dashed) gray lines show stable (unstable) fixed points; vertical orange mark indicates the bifurcation point ${\varepsilon }_c\left(p\right)={(1+3/\sqrt{p})}^{-1}=1/4$ for the fully connected network. (b) Similar to (a) but for $p=0.2$, i.e., for a nonsymmetric dynamics. Dashed vertical line marks the abrupt transition from $\overline{M}<0$ to $\overline{M}>0$. The vertical orange line marks ${\varepsilon }_c\left(0.2\right)\approx 0.13$. Light blue symbols mark the distribution maxima obtained through the tree approximation (Fig. 6) for $k=3$. Arrows mark the path taken by $\overline{M}$ when starting from $\overline{M}=-1$ and $\varepsilon =0$, then first increasing $\varepsilon$ towards unity and then again reducing $\varepsilon$ to zero. (c) Similar to (b) but for scale-free degree distribution with $\alpha =2.4$ and $\langle k\rangle \approx 3$. Hysteresis in (c) is similar as in (b), arrows not shown. System size for simulations: $N=5000$ (see Appendix pp2), in (b) near the transition: $N=20000$.

Figure 3

Phase portraits for fully and sparsely connected systems. (a) Fully connected system with $p=0.2$ and $\varepsilon =0.02$, i.e., in the bistable phase. Gray open, red filled, and red open circles represent initial conditions, stable fixed points, and unstable fixed points, respectively. Colors of curves distinguish trajectories. (b) Similar to (a) but for a degree-regular random graph with $\langle k\rangle =3$. Panels were obtained using simulations with $30000$ nodes.

Figure 4

Dynamics on the nervous system of C. elegans. Electric and chemical synaptic connections [23] (279 nodes). (a) Time-averaged values of $M$ for individual nodes after completion of the transient. Nodes were sorted by increasing $M$ for the simulation of $\varepsilon =0.02$ (orange curve, see legend). This ranking was maintained when plotting the results for other values of $\varepsilon$. Colors (red to purple) correspond to increasing values of $\varepsilon \in \{0.01,\cdots ,0.07\}$. Inset: Several representative time series of network average $\overline{M}\equiv \overline{w}-\overline{u}$ when initializing all nodes at $M=-1$ at $t=0$. Time is measured in units of system updates. (b) Black points indicate the degree of each node. Note the logarithmic vertical axis. The nodes to the very right have degree zero and are therefore not shown on the log scale. Note that degree-zero nodes have no interactions with other nodes, therefore they are not bistable ($w-u\approx 0$ in panel a for these nodes).

Figure 5

Transitions for additional network structures. Similar to Fig. 2 ($N=5000$) but for a Poissonian degree distribution with $\langle k\rangle =3$ (shown in a) as well as a square lattice (shown in b).

Figure 6

Tree approximation. Example for connectivity $k=3$. Bethe lattice where a central node is connected to three nearest neighbors, which themselves are each connected to three nearest neighbors. We distinguish internal and external links, which are indicated by red and black, respectively. Also shown are several configurations of unique four-node states, i.e., those that do not map onto one another by permutations of the individual single-node states.

Figure 7

Effect of clustering. (a) Transitions for varying levels of disorder for $k=4, {\varepsilon }^{*}$ increases with disorder. The fully disordered system is a regular graph of degree $k=4$. System size: $N=2000$. (b) Summary of ${\varepsilon }^{*}$ for the simulations of varying connectivity ($k=4,6,20$) as a function of clustering coefficient.

Figure 8

Dynamics in a two-layer multiplex with variable coupling. (a) Both layers A (dashed lines) and B (solid lines) are initialized at $M=1$ for different values of $\varepsilon$ (see schematic, inset). The plot legend (top right) indicates various values of coupling $c_f$ increasing from 0.005 to 0.64 for colors ranging from red to blue. Vertical arrow indicates the approximate position of ${\varepsilon }^{*}$ for the isolated system B. $N=2000$. (b) Transition for perfect coupling and various combinations of model parameters. Parameters $p^A$ and $p^B$ correspond to the parameter $p$ in the model (Fig. 1) for the two-layer networks $A$ and $B$. Updating a node in one subsystem will also update the node in the other subsystem. Curves obtained for subsystems of $N=2000$ nodes each and degree $k=3$. Mixing two layers has the main effect of averaging the rates of the two subsystems, seen by comparing the cases of $p^a=0.2,p^b=0.4$ to $p^a=0.3,p^b=0.3$.

Figure 9

Dynamics in a two-layer multiplex with alternative couplings. Dashed (solid) lines represent systems A and B, respectively. (a) Similar to Fig. 8 but keeping $c_f=1$ maximal, i.e., all nodes are capable of copying their states. Now $c_r$ is varied as shown in the legend. (b) Similar to Fig. 8 but keeping the product $c_r{c}_f=0.1$ fixed. Parameters of different curves as shown in the legend. (c) Coupling two systems as shown in inset. $c_r=1$. Numbers near lines and in the legend represent the fraction of nodes permanently coupled ($c_f$). System sizes in all panels: $N=2000$.

Figure 10

Finite system. (a) Zero crossing of $\overline{M}$ for $p=0.2$. Dashed black curve shows the analytical solution for $n=2$, where no zero crossing is observed. Dashed blue curve shows the analytical approximation for $n=3$ [Eq. (A2)] (neglecting nonlinear terms in $\varepsilon$). Solid blue curve shows the full numerical solution for $n=3$, where all powers of $\varepsilon$ were retained. The zero crossing for $n=3$ is indicated by the label ${\varepsilon }_0$. (b) $n=3$, the level of noise $\varepsilon$ where a zero crossing for $\overline{M}$ occurs for different values of $p$. Dashed and solid blue curves again show analytical approximation [Eq. (A2)] and numerical solution, respectively.

Figure 11

System size dependence. Simulation for regular graph with $k=3$, but varying system size and $p$. Each curve represents one value of $p$ (marked in legend) and points are simulation results for the transition between $u$-dominant and $w$-dominant states. Lines serve as guide to the eye and connect the points obtained by the simulations. Vertical dashed line indicates $N=5000$, a value we find sufficient for finite size effects to be considered small, unless stated otherwise in the text.

Figure 12

Potential wells. For each value of noise $\varepsilon$ long simulations with $n=59$ nodes and $k\approx 3\ll n$ were carried out. As these systems are comparably small, sufficiently many fluctuations between the quasistable states at negative and positive $M$ occur and allow computation of the probability density function $P(M=w-u)$. Curves ranging from blue to red show the function $-log\left[P\right(w-u\left)\right]$ for a range of $\varepsilon$ (indicated on the right vertical axis). Inspecting the PDFs shows that the minimum at $M<0$ becomes increasingly weak and finally disappears. As $\varepsilon$ increases, the PDFs obtain an overall slant towards positive $M$. (Details are found in Appendix pp3.)

Figure 13

Maxima of probability distribution function. (a) Star topology consisting of four sites, as shown schematically in Fig. 6 as red links. Solid points show the respective local maxima obtained by inspecting the density of states in the tree approximation (Sec. 5). Thin line shows the average value of $w-u$. (b) As in (a) but interacting with surrounding sites, i.e., as in Fig. 6 including the black links. Note the more abrupt decrease of the $u$-dominant state, i.e., maxima for $w$-dominant states persist in noise regions where $u$-dominant states are absent. Note the increase of the average $w-u$ w.r.t. the star topology. (c) Similar to (a) but for a star consisting of five sites. (d) Similar to (c) but for a star consisting of five sites. Note that the maxima at $u>w$ have now disappeared.

Figure 14

Transition point vs degree and noise bias. (a) Difference of fully connected, infinite system bifurcation point ${\varepsilon }_c$ and regular graph transition point ${\varepsilon }^{*}$ for varying values of degree $k$ and different values of $p$. System sizes: 8000 for $p=0.1$ and $p=0.4$, 4000 for $p=0.2$. (b) System with noise bias, i.e., for $u\to v$ and $v\to w$ noisy transitions were increased to $\varepsilon +\delta$ while the opposite direction was decreased to $\varepsilon -\delta$. Values of $\delta$ as indicated in plot. System sizes: $N=5000$, except for $\delta =0$, where $N=5000$ was used.