Random walks with stochastic resetting in complex networks: a discrete time approach

TL;DR

The paper develops a discrete-time stochastic resetting framework for random walks on finite undirected graphs, where resets occur via a renewal process and relocation to nodes is governed by a relocation matrix $\mathbf{R}$. It derives an exact propagator in terms of backward recurrence times and shows that a non-equilibrium steady state (NESS) emerges when the mean resetting interval is finite; it also treats non-Markovian renewal processes, including fat-tailed Sibuya inter-reset times, and analyzes first-hitting statistics to target nodes. The authors provide explicit MFPT expressions and Kemeny constants for Markovian resets on Watts-Strogatz and Barabási-Albert graphs, revealing nontrivial dependences on the fraction of relocation and target nodes and the resetting rate, with resets notably enhancing search efficiency in large-world networks. They further develop a non-Markovian, killing-target framework to study first-hitting dynamics, establish ergodicity conditions, and exhibit non-ergodic regimes via deterministic and T-periodic resets, offering a versatile toolkit for optimizing search processes under resetting in complex networks.

Abstract

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time PDFs and in the light-tailed case we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts-Strogatz and Barabási-Albert random graphs. We show non trivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes and of the resetting rates. It turns out that, in the large-world case of the Watts-Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.

Figures (6)

• Figure 1: (a) Random search dynamics with resets on a realization of the Barabási-Albert random graphs with $N=500$ nodes and attachment parameter $m=4$. Frames (b,c) show the MFPT $\langle T_{ij} \rangle$ of Eq. (\ref{['represntation_Tij']}) as a function of the Bernoulli probability $p$ for different relocation scenarios of the searcher. The sets of r-nodes to which resetting is allowed are randomly selected by an independent Bernoulli trial for each node, generating homogeneously distributed fractions of r-node populations. We plot the MFPT for various fractions ($10\%$--$100\%$) of the r-node populations. The starting node ($i=100$) is marked in blue, the target node ($j=200$) in red. Frame (b): MFPT versus $p$ for uniform reset ($R_r=R$) to the r-nodes). Frame (c): MFPT versus $p$ for preferential resets to the r-nodes depending on the node degree ($R_r \propto K_r$). Frame (d): MFPT of (b) with a larger window. frame (e): MFPT as a function of the r-node fraction for some values of $p$ (same setting of (b)).
• Figure 2: MFPT $\langle T_{100,200}\rangle$ for uniform resetting probabilities $R_r=1/n_R$. The situations of $j=200 \notin {\cal L}_R$ and $j=200 \in {\cal L}_R$ are indicated by dashed and continuous curves, respectively. All parameters and the network are identical as in Fig. \ref{['BA_network']}.
• Figure 3: (a) Random search with resets between the starting node (blue) and the target node (red) in a Watts-Strogatz graph with $N=500$ nodes, attachment parameter $m=2$ (large-world) and rewiring probability $0.7$. The Kemeny constant (Eq. (\ref{['Kemeny_result']})) is pictured in (b) and (c) in cyan color. We plot $\langle T_{ij} \rangle$ (Eq. (\ref{['represntation_Tij']})) with respect to $p$ for the starting node $i=100$ and the target node $j=200$. (b) Same resetting scenarios and color codes as in Fig. \ref{['BA_network']}(b) with uniform resetting probabilities ($R_r=R$) to the r-nodes. (c) Same resetting scenarios and color codes as in Fig. \ref{['BA_network']}(c) with preferential resetting to the r-nodes ($R_r \propto K_r$). In both frames, the t-node $200$ is excluded from ${\cal L}_R$ for one, $10$ and $10\%$ of r-nodes, whereas it is included for all other cases.
• Figure 4: Comparison of the (a) Kemeny constant ${\cal K}(p)$ and (b) search efficiency $E(p)$ of Eq. (\ref{['searcher_efficieny']}) for the dynamics with resets in the Barabási-Albert and Watts-Strogatz graphs. The parameters are those of Figs. \ref{['BA_network']} and \ref{['WS_network']}. The dotted lines denote the CC graph with ${\cal K}_{CC} \approx N=500$ and $E_{CC}(p) \approx 1$ ($N \gg 1$).
• Figure 5: Sibuya resetting rate ${\cal R}_{\alpha}$ of Eq. (\ref{['resettying_sib_rate']}) as a function of $\alpha$ and different values of $t$. The numerical results for $1<t<100$ are coded with the colorbar, whereas the two extremal cases $t=1$ and $t=100$ discontinuous lines are depicted.

Theorems & Definitions (5)

• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.3