Analytical results for the distribution of cover times of random walks on random regular graphs

Tishby, Ido; Biham, Ofer; Katzav, Eytan

doi:10.1088/1751-8121/ac3a34

Cited by 3 publications

(4 citation statements)

References 106 publications

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“…is called the scale parameter. This is in agreement with the results of [34]. The location parameter μ is equal to the mode of the Gumbel distribution.…”

Section: Summary and Discussionsupporting

confidence: 92%

“…Yet another important event, which occurs at much longer time scales, is the step at which the RW completes visiting all the nodes in the network. The time at which this happens is called the cover-time, which scales like t ∼ N ln N [34,43]. This means that on average an RW visits each node ln N times before it completes visiting all the nodes in the network at least once.…”

Section: Summary and Discussionmentioning

confidence: 99%

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Analytical results for the distribution of first-passage times of random walks on random regular graphs

2022

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We present analytical results for the distribution of first-passage (FP) times of random walks (RWs) on random regular graphs that consist of N nodes of degree c ⩾ 3. Starting from a random initial node at time t = 0, at each time step t ⩾ 1 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. We calculate the distribution P(T FP = t) of first-passage times from a random initial node i to a random target node j, where j ≠ i. We distinguish between FP trajectories whose backbone follows the shortest path (SPATH) from the initial node i to the target node j and FP trajectories whose backbone does not follow the shortest path (¬SPATH). More precisely, the SPATH trajectories from the initial node i to the target node j are defined as trajectories in which the subnetwork that consists of the nodes and edges along the trajectory is a tree network. Moreover, the shortest path between i and j on this subnetwork is the same as in the whole network. The SPATH scenario is probable mainly when the length ℓ ij of the shortest path between the initial node i and the target node j is small. The analytical results are found to be in very good agreement with the results obtained from computer simulations.

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“…is called the scale parameter. This is in agreement with the results of [34]. The location parameter μ is equal to the mode of the Gumbel distribution.…”

Section: Summary and Discussionsupporting

confidence: 92%

Section: Summary and Discussionmentioning

confidence: 99%

Analytical results for the distribution of first-passage times of random walks on random regular graphs

2022

Self Cite

View full text Add to dashboard Cite

show abstract

“…Yet another important event which occurs at much longer time scales is the step at which the RW completes visiting all the nodes in the network. The time at which this happens is called the cover-time, which scales like t ∼ N ln N [33,39]. This means that on average an RW visits each node ln N times before it completes visiting all the nodes in the network at least once.…”

Section: Discussionmentioning

confidence: 99%

Analytical results for the distribution of first-passage times of random walks on random regular graphs

Tishby¹,

Biham²,

Katzav³

2022

Preprint

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We present analytical results for the distribution of first-passage (FP) times of random walks (RWs) on random regular graphs that consist of N nodes of degree c ≥ 3. Starting from a random initial node at time t = 0, at each time step t ≥ 1 an RW hops into a random neighbor of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. We calculate the distribution P (T FP = t) of first-passage times from a random initial node i to a random target node j, where j = i. We distinguish between FP trajectories that follow the shortest path (SPATH) from the initial node i to the target node j and FP trajectories, which do not follow the shortest path (¬SPATH). More precisely, the SPATH trajectories are defined as trajectories in which the subnetwork that consists of the nodes and edges along the trajectory is a tree network and the distance ℓ ij between i and j on this subnetwork is the same as in the whole network. The SPATH scenario is probable mainly in case that the distance ℓ ij between the initial node i and the target node j is small. The analytical results are found to be in very good agreement with the results obtained from computer simulations.

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“…This strict bound is, nevertheless, much larger than the actual cover time in most networks, yet the precise mean value depends on the degree distribution [58,59]. For instance, in random regular graphs it drops to N log N [60,61]. We will see next that, by combining the self-avoiding constraint with resetting, we can speed up the exploration of the network with respect to the standard RW, the pure SARW and the non-backtracking random walk.…”

Section: Sarws With Resettingmentioning

confidence: 91%

Efficient network exploration by means of resetting self-avoiding random walkers

Colombani,

Bertagnolli,

Artime

2023

J. Phys. Complex.

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The self-avoiding random walk (SARW) is a stochastic process whose state variable avoids returning to previously visited states. This non-Markovian feature has turned SARWs a powerful tool for modeling a plethora of relevant aspects in network science, such as network navigability, robustness and resilience. We analytically characterize self-avoiding random walkers that evolve on complex networks and whose memory suffers stochastic resetting, that is, at each step, with a certain probability, they forget their previous trajectory and start free diffusion anew. Several out-of-equilibrium properties are addressed, such as the time-dependent position of the walker, the time-dependent degree distribution of the non-visited network and the first-passage time distribution, and its moments, to target nodes. We examine these metrics for different resetting parameters and network topologies, both synthetic and empirical, and find a good agreement with simulations in all cases. We also explore the role of resetting on network exploration and report a non-monotonic behavior of the cover time: frequent memory resets induce a global minimum in the cover time, significantly outperforming the well-known case of the pure random walk, while reset events that are too spaced apart become detrimental for the network discovery. Our results provide new insights into the profound interplay between topology and dynamics in complex networks, and shed light on the fundamental properties of SARWs in nontrivial environments.

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Analytical results for the distribution of cover times of random walks on random regular graphs

Cited by 3 publications

References 106 publications

Analytical results for the distribution of first-passage times of random walks on random regular graphs

Analytical results for the distribution of first-passage times of random walks on random regular graphs

Analytical results for the distribution of first-passage times of random walks on random regular graphs

Efficient network exploration by means of resetting self-avoiding random walkers

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