Localization of the Maximal Entropy Random Walk

Burda, Z.; Duda, Jarek; Luck, J. M.; Waclaw, Bartłomiej

doi:10.1103/physrevlett.102.160602

Cited by 185 publications

(259 citation statements)

References 23 publications

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“…For this purpose, more emphasis must be put into the definition of the reference networks, which for now have been assumed to just be a randomized correspondent of the specific network under study. Another interesting line of inquiry would be to study the dependence of the statistical complexity with respect to the choice of the underlying random walk, such as the maximum entropy random walk [90], as an alternative to the generic one-step random walked used in this paper.…”

Section: Discussionmentioning

confidence: 99%

Mapping and discrimination of networks in the complexity-entropy plane

et al. 2017

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Complex networks are usually characterized in terms of their topological, spatial, or informationtheoretic properties and combinations of the associated metrics are used to discriminate networks into different classes or categories. However, even with the present variety of characteristics at hand it still remains a subject of current research to appropriately quantify a network's complexity and correspondingly discriminate between different types of complex networks, like infrastructure or social networks, on such a basis. Here, we explore the possibility to classify complex networks by means of a statistical complexity measure that has formerly been successfully applied to distinguish different types of chaotic and stochastic time series. It is composed of a network's averaged pernode entropic measure characterizing the network's information content and the associated JensonShannon divergence as a measure of disequilibrium. We study 29 real world networks and show that networks of the same category tend to cluster in distinct areas of the resulting complexity-entropy plane. We demonstrate that within our framework, connectome networks exhibit among the highest complexity while, e.g, transportation and infrastructure networks display significantly lower values. Furthermore, we demonstrate the utility of our framework by applying it to families of random scale-free and Watts-Strogatz model networks. We then show in a second application that the proposed framework is useful to objectively construct threshold-based networks, such as functional climate networks or recurrence networks, by choosing the threshold such that the statistical network complexity is maximized.

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Section: Discussionmentioning

confidence: 99%

Mapping and discrimination of networks in the complexity-entropy plane

et al. 2017

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“…The MERW considered here is a discrete time random walk 49 . At each discrete time step, the walker jumps from its current position i to any of its neighboring nodes j with probability…”

Section: A Formulating the Problemmentioning

confidence: 99%

“…In MERW, the transition probability incorporates the node centrality measured by the eigenvector associated with the largest eigenvalue of adjacent matrix of the graph where the dynamical process takes place. This local transition rate leads to substantial effects on the diffusion behaviors such as stationary distribution 49 and relaxation time 50 . The principle of entropy maximization has found some applications, including optimal sampling algorithm 51 and demographic stability of population 52,53 .…”

Section: Introductionmentioning

confidence: 99%

“…However, sometimes it is more suitable to describe some particular problems by a biased random walk than the unbiased one 46 , since the transition probability depends on not only network topologies but also other properties relevant to the diffusion dynamics 47 . Among numerous biased random walks, maximal entropy random walk (MERW) 48 , maximizes the entropy of paths, has been studied recently 49 . In MERW, the transition probability incorporates the node centrality measured by the eigenvector associated with the largest eigenvalue of adjacent matrix of the graph where the dynamical process takes place.…”

Section: Introductionmentioning

confidence: 99%

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Maximal entropy random walk improves efficiency of trapping in dendrimers

Peng

Zhang

2014

The Journal of Chemical Physics

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We use maximal entropy random walk (MERW) to study the trapping problem in dendrimers modeled by Cayley trees with a deep trap fixed at the central node. We derive an explicit expression for the mean first passage time from any node to the trap, as well as an exact formula for the average trapping time (ATT), which is the average of the source-to-trap mean first passage time over all non-trap starting nodes. Based on the obtained closed-form solution for ATT, we further deduce an upper bound for the leading behavior of ATT, which is the fourth power of ln N , where N is the system size. This upper bound is much smaller than the ATT of trapping depicted by unbiased random walk in Cayley trees, the leading scaling of which is a linear function of N . These results show that MERW can substantially enhance the efficiency of trapping performed in dendrimers.

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“…Entropy measures provide a natural way of quantifying information; in fact, this measurements have been applied successfully in complex networks before [17][18][19][20][21]. In particular, these methods were applied recently to quantify information in the problem of targeted signaling or navigability [2,22,23] and in the complementary problem of efficient diffusion in a network [24,25]. This is the approach we adopted in our work to quantify the difficulty of choosing the appropriate paths that connect the source with the target.…”

Section: Introductionmentioning

confidence: 99%

Smart random walkers: The cost of knowing the path

Perotti

Billoni

2012

Phys. Rev. E

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In this work we study the problem of targeting signals in networks using entropy information measurements to quantify the cost of targeting. We introduce a penalization rule that imposes a restriction on the long paths and therefore focuses the signal to the target. By this scheme we go continuously from fully random walkers to walkers biased to the target. We found that the optimal degree of penalization is mainly determined by the topology of the network. By analyzing several examples, we have found that a small amount of penalization reduces considerably the typical walk length, and from this we conclude that a network can be efficiently navigated with restricted information.

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Localization of the Maximal Entropy Random Walk

Cited by 185 publications

References 23 publications

Mapping and discrimination of networks in the complexity-entropy plane

Mapping and discrimination of networks in the complexity-entropy plane

Maximal entropy random walk improves efficiency of trapping in dendrimers

Smart random walkers: The cost of knowing the path

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