Uncovering the topology of configuration space networks

The configuration space network (CSN) of a dynamical system is an effective approach to represent the ensemble of configurations sampled during a simulation and their dynamic connectivity. To elucidate the connection between the CSN topology and the underlying free-energy landscape governing the system dynamics and thermodynamics, an analytical solution is provided to explain the heavy tail of the degree distribution, neighbor connectivity, and clustering coefficient. This derivation allows us to understand the universal CSN topology observed in systems ranging from a simple quadratic well to the native state of the beta3s peptide and a two-dimensional lattice heteropolymer. Moreover, CSNs are shown to fall in the general class of complex networks described by the fitness model.

Figure 1

(Color online) Network topology for the quadratic well in $D=2$ dimensions and different values of the parameter $M$. (a) Degree distribution. For clarity a binning has been applied for $M>10$. Inset: Degree distribution for uncorrelated sampling without binning. (b) Average neighbor degree. (c) Clustering coefficient. Blue circles surrounded by black correspond to a random sampling of the energy landscape (uncorrelated case; see text). Red dashed line shows the analytical estimation.

Figure 2

(Color online) Assortativity coefficient for different values of the parameter $M$ in the three systems under study.

Figure 3

(Color online) Distribution of the relaxation times from three different initial configurations to the bottom configuration of the quadratic well in $D=2$ dimensions for $M=\left(\mathrm{a}\right)$ 1 and (b) 1000. The value $r$ indicates the radial distance from the starting node for each of the three curves.

Figure 4

(Color online) Network topology for the beta3s peptide CSN at different values of the parameter $M$. (a) Degree distribution. To reduce noise a logarithmic binning has been applied. The native state of beta3s is displayed in the inset. (b) Average neighbor connectivity. (c) Clustering coefficient.

Figure 5

(Color online) Network topology for the random lattice heteropolymer CSN at different values of the parameter $M$. (a) Degree distribution. The most visited configuration of the heteropolymer is displayed in the inset. (b) Average neighbor connectivity. (c) Clustering coefficient.

Figure 6

(Color online) Relation between the free-energy barrier to the configuration at the bottom of the native state basin $\left(\Delta F_{i\to 1}\right)$, and the configuration free energy $\left(\Delta F_i\right)$ for the most visited nodes of the beta3s network. Empty and full dots represent the $M=1$ and ${10}^5$ cases, respectively.