Information Dimension of Stochastic Processes on Networks: Relating Entropy Production to Spectral Properties

Abstract

We consider discrete stochastic processes, modeled by classical master equations, on networks. The temporal growth of the lack of information about the system is captured by its non-equilibrium entropy, defined via the transition probabilities between different nodes of the network. We derive a relation between the entropy and the spectrum of the master equation’s transfer matrix. Our findings indicate that the temporal growth of the entropy is proportional to the logarithm of time if the spectral density shows scaling. In analogy to chaos theory, the proportionality factor is called (stochastic) information dimension and gives a global characterization of the dynamics on the network. These general results are corroborated by examples of regular and of fractal networks.

Appendix: \(NT_D\) Graph

In this Appendix we extend our results by considering a structure with a non-monotonous density of states $\rho (\lambda )$.

Fig. 4

(color online) a Sketch of an $NT_D$ graph of generation $G=4$ and $k=2$. b Entropy $\overline{S}(t)$ and the leading term of $\overline{S}_\mathrm{mf}(t)$, Eq. (10), for $NT_D$ graph of $k=2$ and $G=3$. c The behavior of $-\ln [\overline{p}(t)]$, $(d_s^{\mathrm {lin}}/2) \ln (t)$ and $(d_s^{\mathrm {br}}/2) \ln (t)$, for $G=6$ and $k=2$

In Fig. 4a we illustrate the construction of the so-called $NT_D$ graph [8,9,10]. At every iteration G to each end-node of these trees k linear chains of length $2^G$ are attached. In this way, the Laplacian spectrum of the $NT_D$ graphs is dominated by the behavior of the linear chains, whose spectral dimension is $d_s^{\mathrm {lin}}=1$. On the other hand, for these trees the states corresponding to the smallest eigenvalues are described by the relaxation of the branches as whole (similarly as for dendrimers [11, 14, 15, 17]). As has been found in Refs. [8, 9], the related spectral dimension is given by $d_s^{\mathrm {br}}=1+\log k/\log 2$. As we proceed to show, both aspects of the spectrum $\{\lambda \}$ are reflected in the temporal growth of the entropy.

First, in Fig. 4b we show that, as for Vicsek fractals, the leading term $-\ln [\bar{p}(t)]$ determines the temporal behavior of the entropy $\bar{S}(t)$ for the $NT_D$ graphs. We observe a scaling of the linear chains, i.e. $\bar{S}(t)\sim (1/2)\ln (t)$. Increasing generation G leads to an appearance of lower and lower $\lambda$’s that get separated from the (continuous) spectrum of the linear chain. This leads to a change in the behavior of the term ${-}\ln [\bar{p}(t)]$ for longer times, see Fig. 4c. Thus, different parts of the spectrum $\{\lambda \}$ translate their behavior to the time-dependent entropy $\bar{S}(t)$.