Cover time for random walks on arbitrary complex networks

Benjamin F. Maier and Dirk Brockmann

Phys. Rev. E 96, 042307 – Published 18 October 2017

Abstract

We present an analytical method for computing the mean cover time of a discrete-time random walk process on arbitrary, complex networks. The cover time is defined as the time a random walker requires to visit every node in the network at least once. This quantity is particularly important for random search processes and target localization on network structures. Based on the global mean first-passage time of target nodes, we derive a method for computing the cumulative distribution function of the cover time based on first-passage time statistics. Our method is viable for networks on which random walks equilibrate quickly. We show that it can be applied successfully to various model and real-world networks. Our results reveal an intimate link between first-passage and cover time statistics and offer a computationally efficient way for estimating cover times in network-related applications.

Received 6 June 2017
Revised 29 September 2017

DOI:https://doi.org/10.1103/PhysRevE.96.042307

Physics Subject Headings (PhySH)

Network diffusion Network structure Random walks

Statistical Physics & ThermodynamicsNetworks

Authors & Affiliations

Benjamin F. Maier^*

Robert Koch-Institute, Nordufer 20, D-13353 Berlin, Germany and Department of Physics, Humboldt-University of Berlin, Newtonstraße 15, D-12489 Berlin, Germany

Dirk Brockmann

Robert Koch-Institute, Nordufer 20, D-13353 Berlin, Germany and Institute for Theoretical Biology, Humboldt-University of Berlin, Philippstraße 13, D-10115 Berlin, Germany

^*bfmaier@physik.hu-berlin.de

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Vol. 96, Iss. 4 — October 2017

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Images

Figure 1
Illustration of our approach. Each node of the network is viewed as an independent entity that can be visited by the walker starting at the green node. Each target node $v$ has its own first-passage time (FPT) distribution which is asymptotically (for larger times $τ$ ) distributed as $\propto exp (- τ / τ_{v})$ with its global mean FPT $τ_{v}$ . To compute the cover time, we draw one FPT $τ$ from every target's distribution. Then the cover time is given as the maximum time of all drawn FPTs.
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Figure 2
The mean cover time $〈T〉$ of the largest component of (a) Erdős–Rényi (ER) networks, (b) Barabási–Albert (BA) networks, and (c) random $k$ -regular networks. Shown are averages measured from 1000 simulations per data point ( $□$ ) and both estimations of the mean cover time using ( $♦$ ) estimated GMFPTs from the target nodes' degrees Eq. (11) and ( $◯$ ) exact GMFPTs Eq. (12) computed from the unnormalized graph Laplacian's spectrum in (a) and (b). Respectively, we used Eq. (13) for random $k$ -regular networks in (c). Dashed lines are simulation results and a guide to the eye. Further displayed are the asymptotic results derived by Cooper et al. for the ER [13] and BA [14] network model as well as the lower bound Eq. (14) for random $k$ -regular networks.
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Figure 3
Mean cover times of simple discrete-time random walks on the largest component of (a) various real-world networks (data sources and relative errors given in Table 1) and (b) various subway networks (data source and relative errors given in Table 2). The estimated cover times are compared to the measured cover times (from 50 simulations for each data point). The dashed lines represent the ideal case $〈T^{sim}〉 = 〈T^{es / ex}〉$ . Theoretical results are computed from (left) estimated GMFPTs from the target nodes' degrees Eq. (11) and (right) exact GMFPTs computed from the unnormalized graph Laplacian's spectrum Eq. (12).
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Figure 4
Example of our method yielding results with rather large deviations from simulations. (a) Mean cover time for low-dimensional $(d \leq 2)$ lattices as well as lattices in dimension $d = 3$ as ( $□$ ) an average over 1000 simulations for each data point and ( $◯$ ) theoretical result from exact GMFPTs computed from the unnormalized graph Laplacian's spectrum Eq. (12). (b) The relative error is increasing with increasing system size but is comparably lower for $d = 3$ (cubes).
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Figure 5
The relative error of our heuristic is increasing with increasing inverse second smallest eigenvalue $λ_{2}^{- 1}$ of the unnormalized graph Laplacian, indicating that our method produces higher deviations for networks with relatively high relaxation times as per Eq. (4). Networks are marked with the same symbols as in Fig. 3.
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Figure 6
Relative error Eq. (C1) between integrals Eqs. (8) and (10) for all networks investigated in this study besides lattices. For each network we show the relative error for both sets of rates, $τ_{v}^{es}$ and $τ_{v}^{ex}$ . For ER, BA, and random $k$ -regular networks, we built means over the largest components of 100 network realizations for $〈k〉 \in {3, 5, 7, 9}$ and $m \in {3, 5, 7, 9}$ , respectively. The measured relative errors are roughly following the scaling relation Eq. (C2).
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