Classes of random walks on temporal networks with competing timescales

Petit, Julien; Lambiotte, Renaud; Carletti, Timotéo

doi:10.1007/s41109-019-0204-6

Cited by 10 publications

(8 citation statements)

References 36 publications

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“…Many variations of this fundamental process have since then been considered. These include more sophisticated dynamical implementations, which allow to targeting the walks towards nodes with given structural features [36], let them interact at the nodes of the network [37], investigate non-linear transition probabilities [38] and crowded conditions [39], consider the temporal [40][41][42] or multilayer [43,44] dimensions of the edges under different network topologies.…”

Section: Introductionmentioning

confidence: 99%

Random walks on hypergraphs

Battiston²,

et al. 2020

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In the last twenty years network science has proven its strength in modelling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Yet, in many relevant cases, interactions are not pairwise but involve larger sets of nodes, at a time. These systems are thus better described in the framework of hypergraphs, whose hyperedges effectively account for multibody interactions. We hereby propose a new class of random walks defined on such higher-order structures, and grounded on a microscopic physical model where multi-body proximity is associated to highly probable exchanges among agents belonging to the same hyperedge. We provide an analytical characterisation of the process, deriving a general solution for the stationary distribution of the walkers. The dynamics is ultimately driven by a generalised random walk Laplace operator that reduces to the standard random walk Laplacian when all the hyperedges have size 2 and are thus meant to describe pairwise couplings. We illustrate our results on synthetic models for which we have a full control of the high-order structures, and real-world networks where higher-order interactions are at play. As a first application of the method, we compare the behaviour of random walkers on hypergraphs to that of traditional random walkers on the corresponding projected networks, drawing interesting conclusions on node rankings in collaboration networks. As a second application, we show how information derived from the random walk on hypergraphs can be successfully used for classification tasks involving objects with several features, each one represented by a hyperedge. Taken together, our work contributes to unveiling the effect of higher-order interactions on diffusive processes in higher-order networks, shading light on mechanisms at the hearth of biased information spreading in complex networked systems.

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Section: Introductionmentioning

confidence: 99%

Random walks on hypergraphs

Battiston²,

et al. 2020

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“…The technique of encoding relationships between vertices by a scaled sum over random walks of all lengths is a heuristic that dates back to the earliest work in this field (Katz, 1953) as well as more recent proposals for vertex similarity (Leicht et al, 2006). In this work, we add to this approach by recognizing that the columns and rows of matrix K Equation (2) are diffusion patterns that can be understood through the relationship of K to the Markov transition operator of a random walk process known as active, node-centric continuous time random walk (Petit et al, 2019;Masuda et al, 2017). In this process, a random walker at vertex i takes a step to vertex j with probability A [i,j] ∀μ A [i,μ] and that event occurs after a waiting time t which is an exponentially distributed random variable.…”

Section: Relation Of This Methods To Notions Of Diffusion On Network 321 Relation To Markov Processesmentioning

confidence: 99%

“…In this process, a random walker at vertex i takes a step to vertex j with probability A [i,j] ∀μ A [i,μ] and that event occurs after a waiting time t which is an exponentially distributed random variable. Equation ( 5) is known as the master equation of this continuous time random walk (Petit et al, 2019;Angstmann et al, 2013) where the (n × 1) function of time y is the probability distribution on vertex set V, sometimes referred to as the residence probabilities (Petit et al, 2019):…”

Section: Relation Of This Methods To Notions Of Diffusion On Network 321 Relation To Markov Processesmentioning

confidence: 99%

“…Using our experimental framework, we assess the behavior of our diffusion profile similarity as it relates to changes in basic structured network model parameters. Specifically, our similarity method assigns each vertex to the diffusion pattern it generates in a model of diffusion that has been explored in the literature as heat kernel pagerank (Chung, 2007) of a vertex in a graph and also through unbiased and continuous time random walks (Delvenne et al, 2013;Masuda et al, 2017;Petit et al, 2019;Angstmann et al, 2013) beginning at a vertex. Beginning from several of the principles in these prior works, we adapt the solution to the relevant ordinary differential equation to model diffusion on a network in a way that detects both the properties of the neighborhood of vertices in a community and layered or parallel structures in a network within the diffusion patterns into and out of the vertex.…”

Section: Introductionmentioning

confidence: 99%

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Diffusion profile embedding as a basis for graph vertex similarity

Payne¹,

Fuller

Spirou³

et al. 2021

Net Sci

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We describe here a notion of diffusion similarity, a method for defining similarity between vertices in a given graph using the properties of random walks on the graph to model the relationships between vertices. Using the approach of graph vertex embedding, we characterize a vertex vi by considering two types of diffusion patterns: the ways in which random walks emanate from the vertex vi to the remaining graph and how they converge to the vertex vi from the graph. We define the similarity of two vertices vi and vj as the average of the cosine similarity of the vectors characterizing vi and vj. We obtain these vectors by modifying the solution to a differential equation describing a type of continuous time random walk.This method can be applied to any dataset that can be assigned a graph structure that is weighted or unweighted, directed or undirected. It can be used to represent similarity of vertices within community structures of a network while at the same time representing similarity of vertices within layered substructures (e.g., bipartite subgraphs) of the network. To validate the performance of our method, we apply it to synthetic data as well as the neural connectome of the C. elegans worm and a connectome of neurons in the mouse retina. A tool developed to characterize the accuracy of the similarity values in detecting community structures, the uncertainty index, is introduced in this paper as a measure of the quality of similarity methods.

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“…Guillen-Perez and Cano [23] examined REMMs that are currently used in terms of mobility, positioning, and distribution models for flying ad hoc networks (FANET). Petit et al [24] conducted an experiment where they examined the status of the RW method in general stochastic temporal networks that allow its permanent interactions. They investigated the situations provided by the RW method based on the structure of the network that was created.…”

Section: Related Workmentioning

confidence: 99%

Novel random models of entity mobility models and performance analysis ofrandom entity mobility models

Bilgin¹

2020

Turk J Elec Eng & Comp Sci

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It has become possible to collect data from geographically large areas with smart devices that are prevalently used today. Sensors that are integrated into smart devices make it possible for these devices to receive and transmit data wirelessly. The most important problem of this model that is known as mobile crowd sensing and that allows inferences on the data obtained from its users is lack of data. The main reason for this problem is the lack of sufficient usage of the sensors on devices by the user. To increase the amount of data collected, while users may be incentivized in various ways, the amount and accuracy of the collected data may be increased by developing random entity mobility models (REMMs). In this study, two new models (random point and random journey) were proposed as alternatives to existing REMMs. In the experiment environment that was created to measure the performances of the proposed models, their performances were compared to those that are currently used prevalently (random waypoint (RWP), random walk (RW), and random direction (RD)). In the experiment environment, the performances were compared in terms of three different metrics (visiting rates of nodes, rates of reaching the basis, and the number of messages they carried to the basis). The greatest increase in differently sized areas and at different numbers of nodes in the RP model in terms of rates of reaching the basis was 2.6% compared to RWP, 7% compared to RW, and 46.34% compared to RD, while these values for the number of nodes that were visited were 3% compared to RWP, 1.5% compared to RW, and 17.67% compared to RD. In the same conditions in terms of the metric on the number of messages, the model collected 1465.4, 2933.46, and 7260.12 more messages than those in respectively RWP, RW, and RD. The greatest increase in differently sized areas and at different numbers of nodes in the RJ model in terms of reaching the basis was 1% compared to RWP, 3.5% compared to RW, and 25% compared to RD, while these values for the number of nodes that were visited were 0.75% compared to RWP, 2% compared to RW, and 21.4% compared to RD. In the same conditions in terms of the metric on the number of messages, the model collected 1109.56, 1534.26, and 4488.5 more messages than those in RWP, RW, and RD, respectively.

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Classes of random walks on temporal networks with competing timescales

Cited by 10 publications

References 36 publications

Random walks on hypergraphs

Random walks on hypergraphs

Diffusion profile embedding as a basis for graph vertex similarity

Novel random models of entity mobility models and performance analysis ofrandom entity mobility models

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