"Spectrally gapped" random walks on networks: a Mean First Passage Time  formula

Bartolucci, Silvia; Caccioli, Fabio; Caravelli, Francesco; Vivo, Pierpaolo

doi:10.21468/scipostphys.11.5.088

Cited by 7 publications

(5 citation statements)

References 76 publications

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“…6 serves as its cumulative distribution function. Usually interest in the mean first passage time stimulates the development of computational methods [29], but Ref. [28] argued that the first passage time distribution has a universal exponential tail with variable decay rate consistent with our Eq.…”

Section: Discussionsupporting

confidence: 54%

Exposure theory for learning complex networks with random walks

Klishin¹,

Bassett²

2022

Preprint

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Random walks are a common model for exploration and discovery of complex networks. While numerous algorithms have been proposed to map out an unknown network, a complementary question arises: in a known network, which nodes and edges are most likely to be discovered by a random walker in finite time? Here we introduce exposure theory, a statistical mechanics framework that predicts the learning of nodes and edges across several types of networks, including weighted and temporal, and show that edge learning follows a universal trajectory. While the learning of individual nodes and edges is noisy, exposure theory produces a highly accurate prediction of aggregate exploration statistics.

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Section: Discussionsupporting

confidence: 54%

Exposure theory for learning complex networks with random walks

Klishin¹,

Bassett²

2022

Preprint

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“…Plugging in P(T FP = t, SPATH| ) from equation (9) and P(SPATH| ) from equation ( 14) into equation (17), it is found that P(T FP = t| , SPATH) = 0 for t < , while for t it is given by…”

Section: The Distribution Of First-passage Times Via Spath Trajectoriesmentioning

confidence: 99%

“…where E[T FP |SPATH] is given by equation ( 48) and E[T FP |¬SPATH] is given by equation ( 65). The mean first passage time T FP was recently calculated for specific network instances and a given pair of initial and target nodes [17,[38][39][40]. These calculations were done using matrix methods that apply to a wide range of network structures including weighted networks.…”

Section: The Overall Distribution Of First-passage Timesmentioning

confidence: 99%

“…This problem was studied extensively on regular lattices [12,13]. The mean first-passage time T FP of an RW on a random network was studied in references [14][15][16][17]. The statistical properties of first passage processes in empirical networks were applied in order to characterize the heterogeneity and correlations in these networks [18].…”

Section: Introductionmentioning

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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“…The mean first-passage time T FP of an RW on a random network was studied in Refs. [14][15][16][17]. However, the distribution P (T FP = t) of first-passage times has not been studied in detail.…”

Section: Introductionmentioning

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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"Spectrally gapped" random walks on networks: a Mean First Passage Time formula

Cited by 7 publications

References 76 publications

Exposure theory for learning complex networks with random walks

Exposure theory for learning complex networks with random walks

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

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