Cover time for random walks on arbitrary complex networks

Maier, Benjamin F.; Brockmann, Dirk

doi:10.1103/physreve.96.042307

Cited by 44 publications

(32 citation statements)

References 32 publications

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“…The properties of random walks on various networks have been studied extensively [31,32,51]. Quite often the asymptotic analysis and the averaged characteristics of diffusive transport and random walk dynamics, such as mean return time and mean first-passage time, are calculated [8,37]. Here we focused on studying the probability distribution of the first-passage time for random walk on complex networks with heterogeneities.…”

Section: Discussionmentioning

confidence: 99%

“…The comparison of regular versus irregular structures allowed us to refine some properties of the FPT densities on complex networks. In particular we analyzed effects of structural and distributional heterogeneities using the first-passage time as one of the key indicators of how fast information diffuses in a given system [37,44]. Heterogeneities are encoded in the generalized transition matrix Q(t) [21], which affects its spectral properties and as the result the dynamical properties of processes on such networks.…”

Section: Discussionmentioning

confidence: 99%

“…[33]. For homogeneous networks these problems have been studied extensively [37,26], for example, using the underlying backward equation approach [8]. However, the analytical approaches, describing random walks on feature-rich (when nodes or links are endowed with some attributes [7]) heterogeneous networks, are lacking [8].…”

Section: Introductionmentioning

confidence: 99%

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Structural and temporal heterogeneities on networks

Tupikina

2019

Appl Netw Sci

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A heterogeneous continuous time random walk is an analytical formalism for studying and modeling diffusion processes in heterogeneous structures on microscopic and macroscopic scales. In this paper we study both analytically and numerically the effects of spatio-temporal heterogeneities onto the diffusive dynamics on different types of networks. We investigate how the distribution of the first passage time is affected by the global topological network properties and heterogeneities in the distributions of the travel times. In particular, we analyze transport properties of random networks and define network measures based on the first-passage characteristics. The heterogeneous continuous time random walk framework has potential applications in biology, social and urban science, search of optimal transport properties, analysis of the effects of heterogeneities or bursts in transportation networks.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Structural and temporal heterogeneities on networks

Tupikina

2019

Appl Netw Sci

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“…If we assume that the random walk relaxes quickly [36] (i.e., that it approaches the equilibrium distribution in a small number of steps t N), it is furthermore possible to find the mean cover time of the random walk analytically as…”

Section: Effective Medium Approximationmentioning

confidence: 99%

Modular hierarchical and power-law small-world networks bear structural optima for minimal first passage times and cover time

Maier

Huepe

Brockmann

2019

Journal of Complex Networks

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Networks that are organized as a hierarchy of modules have been the subject of much research, mainly focusing on algorithms that can extract this community structure from data. The question of why modular hierarchical organizations are so ubiquitous in nature, however, has received less attention. One hypothesis is that modular hierarchical topologies may provide an optimal structure for certain dynamical processes. We revisit a modular hierarchical network model that interpolates, using a single parameter, between two known network topologies: from strong hierarchical modularity to an Erdős-Rényi random connectivity structure. We show that this model displays a similar small-world effect as the Kleinberg model, where the connection probability between nodes decays algebraically with distance. We find that there is an optimal structure, in both models, for which the pair-averaged first passage time (FPT) and mean cover time of a discrete-time random walk are minimal, and provide a heuristic explanation for this effect. Finally, we show that analytic predictions for the pair-averaged FPT based on an effective medium approximation fail to reproduce these minima, which implies that their presence is due to a network structure effect. this locally-informed search process does not necessarily reflect the underlying dynamics of many natural hierarchical systems, which could follow more random, diffusion-like processes.In this paper we study the topological structure of and random walk processes on a class of MH networks with different levels of hierarchical modularity. This class corresponds to a variant of Watts' hierarchical model, in which all hierarchical levels are statistically self-similar and where the amount of hierarchical modularity can be varied using a single structural control parameter, while keeping the mean degree fixed. We will refer to this model as the self-similar modular hierarchical (SSMH) network model. In this model, the control parameter determines the degree of topological non-locality of the system, i.e. the fraction of connections that are made within a module at each level of the hierarchy. We also analyze the class of networks resulting from a variant of a one-dimensional Kleinberg small-world model [27], in which all pairs of nodes including nearest neighbors are linked with power-law probability on the distance between them. The mean degree is also kept constant and the structural control parameter is defined as the exponent of the power-law, thus controlling the level of spatial non-locality in a one-dimensional embedding space. We will refer to this model as the power-law small-world (PLSW) network model. We will show that the SSMH and PLSW models share many properties, although the latter does not have an inherent modular structure.The definition of the SSMH and PLSW models as statistically self-similar systems, each with a single structural control parameter that determines the level of non-locality of the con-arXiv:1808.00240v1 [physics.soc-ph] 1 Aug 2018

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“…The spectral gap is then a natural candidate for predicting density optima as the features of networks that slow down convergence, such as bottlenecks (identified through nodes of high betweenness centrality 45 ), also significantly increase the parallel cover time. Furthermore, single-searcher cover times can be related 23 There is a tight relationship between the spectral gap and the optimal density of searchers for general networks (Fig. 2), quantified through high mutual information (SI Text).…”

Section: General Topologies and The Spectral Gapmentioning

confidence: 99%

Topology-dependent density optima for efficient simultaneous network exploration

2018

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A random search process in a networked environment is governed by the time it takes to visit every node, termed the cover time. Often, a networked process does not proceed in isolation but competes with many instances of itself within the same environment. A key unanswered question is how to optimize this process: How many concurrent searchers can a topology support before the benefits of parallelism are outweighed by competition for space? Here, we introduce the searcher-averaged parallel cover time (APCT) to quantify these economies of scale. We show that the APCT of the networked symmetric exclusion process is optimized at a searcher density that is well predicted by the spectral gap. Furthermore, we find that nonequilibrium processes, realized through the addition of bias, can support significantly increased density optima. Our results suggest alternative hybrid strategies of serial and parallel search for efficient information gathering in social interaction and biological transport networks.

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Cover time for random walks on arbitrary complex networks

Cited by 44 publications

References 32 publications

Structural and temporal heterogeneities on networks

Structural and temporal heterogeneities on networks

Modular hierarchical and power-law small-world networks bear structural optima for minimal first passage times and cover time

Topology-dependent density optima for efficient simultaneous network exploration

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