Random Walks on Hypergraphs with Edge-Dependent Vertex Weights

Chitra, Uthsav; Raphael, Benjamin J.

doi:10.48550/arxiv.1905.08287

Cited by 4 publications

(8 citation statements)

References 27 publications

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“…By invoking Theorem 4 in [56], we can finally conclude that our process is equivalent to a random walk on a weighted projected network, where the weights of the link ij is given by k H ij , that is the weights scale extensively with the region of influence of the nodes, namely the size of the hyperedge they belong to. It is indeed quite remarkable that a properly weighted binary network encapsulates the higher order information, as stemming for the corresponding hypergraph representation.…”

Section: Modelmentioning

confidence: 93%

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: Modelmentioning

confidence: 93%

Random walks on hypergraphs

Battiston²,

et al. 2020

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“…In this subsection, we consider a setting where pairwise link statuses and hyperlink statuses are conditional dependent given latent factors Z, i.e., ρ i 1 i 2 •••im > 0 and the hyperlink statuses are generated based on both Z and the pairwise link clique (10). We investigate the performance comparisons between the augmented joint linkembedding method (Aug JLE) and other link-embedding methods using observed networks only.…”

Section: Study 3: Link Prediction Under the Conditional Dependent Modelmentioning

confidence: 99%

“…Existing works on hyperlink modeling are also considered in node classification and community detection. For instance, [10,30,1] suggest methods based on hyperlink expansions or random walks to reconstruct hyperlinks from pairwise links. These methods use a principle of generating hyperlinks based on pre-specified relations among hyperlinks and pairwise links, while treating hyperlinks as a subgraph with a certain configuration such as a fully-connected clique or starshaped subgroup of nodes.…”

Section: Introductionmentioning

confidence: 99%

High-order joint embedding for multi-level link prediction

Yuan¹,

Qu²

2021

Preprint

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Link prediction infers potential links from observed networks, and is one of the essential problems in network analyses. In contrast to traditional graph representation modeling which only predicts two-way pairwise relations, we propose a novel tensor-based joint network embedding approach on simultaneously encoding pairwise links and hyperlinks onto a latent space, which captures the dependency between pairwise and multi-way links in inferring potential unobserved hyperlinks. The major advantage of the proposed embedding procedure is that it incorporates both the pairwise relationships and subgroup-wise structure among nodes to capture richer network information. In addition, the proposed method introduces a hierarchical dependency among links to infer potential hyperlinks, and leads to better link prediction. In theory we establish the estimation consistency for the proposed embedding approach, and provide a faster convergence rate compared to link prediction utilizing pairwise links or hyperlinks only. Numerical studies on both simulation settings and Facebook ego-networks indicate that the proposed method improves both hyperlink and pairwise link prediction accuracy compared to existing link prediction algorithms.

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“…It is worth emphasising that the dynamics defined on this weighted network is equivalent [45] to the dynamics on the corresponding hypergraph. This observation allows us to transport existing tools targeted to networks' analysis to the realm where nodes are made to interact via hypergraphs.…”

Section: Hypergraphsmentioning

confidence: 99%

Dynamical systems on Hypergraphs

Carletti¹,

Fanelli²

2020

Preprint

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Networks are a widely used and efficient paradigm to model real-world systems where basic units interact pairwise. Many body interactions are often at play, and cannot be modelled by resorting to binary exchanges. In this work, we consider a general class of dynamical systems anchored on hypergraphs. Hyperedges of arbitrary size ideally encircle individual units so as to account for multiple, simultaneous interactions. These latter are mediated by a combinatorial Laplacian, that is here introduced and characterised. The formalism of the Master Stability Function is adapted to the present setting. Turing patterns and the synchronisation of non linear (regular and chaotic) oscillators are studied, for a general class of systems evolving on hypergraphs. The response to externally imposed perturbations bears the imprint of the higher order nature of the interactions. I. INTRODUCTION.Network science [1,2] has proved successful in describing many real-world systems [3-5], which, despite inherent differences, share common structural features. Even more interestingly, dynamical processes and hosting networks are indissolubly entangled with the ensuing patterns that reflect in fact the complex topology of the supports to which they are anchored [6][7][8].Networks constitute abstract frameworks, where pairwise interactions among generic agents, represented by nodes, are schematised by edges. Stated simply, two agents are connected if they interact. Hence by their very first definition, networks encode for binary relationships among units. This descriptive framework is sufficiently accurate in many cases of interest, although several examples exist of systems for which it holds true just as a first order approximation [9,10]. The relevance of high-orders structures has been indeed emphasised in the context of functional brain networks [11,12], in applications to protein interaction networks [13], to the study of ecological communities [14] and co-authorship networks [15,16].Starting from this observation, higher-order models have been developed so as to capture the many body interactions among interacting units. The most notable examples are simplicial complexes [17][18][19] and hypergraphs [20][21][22], non trivial mathematical generalisations of ordinary networks that are currently attracting a lot of interest. The concept of simplicial complexes has been for instance invoked to address problems in epidemic spreading [23,24] or synchronisation phenomena [25,26]. Our work is positioned in the framework of hypergraphs, a domain of investigation which is still in its infancy. In this respect, we mention applications to social contagion model [27,28], to the modelling of random walks [16] and to the study of synchronisation [29,30] and diffusion [28].Hypergraphs constitute indeed a very flexible paradigm: an arbitrary number of agents are allowed to interact, thus extending beyond the limit of binary interactions of conventional network models. On the other hand, hypergraphs define a leap forward as compared to simplicial com...

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Random Walks on Hypergraphs with Edge-Dependent Vertex Weights

Cited by 4 publications

References 27 publications

Random walks on hypergraphs

Random walks on hypergraphs

High-order joint embedding for multi-level link prediction

Dynamical systems on Hypergraphs

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