Random walks on hypergraphs

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 multi-body 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

Random Walks on Hypergraphs with Edge-Dependent Vertex Weights

Hypergraphs are used in machine learning to model higher-order relationships in data. While spectral methods for graphs are well-established, spectral theory for hypergraphs remains an active area of research. In this paper, we use random walks to develop a spectral theory for hypergraphs with edge-dependent vertex weights: hypergraphs where every vertex $v$ has a weight $\gamma_e(v)$ for each incident hyperedge $e$ that describes the contribution of $v$ to the hyperedge $e$. We derive a random walk-based hypergraph Laplacian, and bound the mixing time of random walks on such hypergraphs. Moreover, we give conditions under which random walks on such hypergraphs are equivalent to random walks on graphs. As a corollary, we show that current machine learning methods that rely on Laplacians derived from random walks on hypergraphs with edge-independent vertex weights do not utilize higher-order relationships in the data. Finally, we demonstrate the advantages of hypergraphs with edge-dependent vertex weights on ranking applications using real-world datasets.Comment: Accepted to ICML 201

Hyperlink prediction via local random walks and Jensen-Shannon divergence

Many real-world systems involving higher-order interactions can be modeled by hypergraphs, where vertices represent the systemic units and hyperedges describe the interactions among them. In this paper, we focus on the problem of hyperlink prediction which aims at inferring missing hyperlinks based on observed hyperlinks. We propose three similarity indices for hyperlink prediction based on local random walks and Jensen-Shannon divergence. Numerical experiments show that the proposed indices outperform the state-of-the-art methods on a broad range of datasets.Comment: IoP Latex, 15 pages, 1 figure

On some building blocks of hypergraphs: units, twin-units, regular, co-regular, and symmetric sets

Here, we introduce and investigate different building blocks, named units, twin units, regular sets, symmetric sets, and co-regular sets in a hypergraph. Our work shows that the presence of these building blocks leaves certain traces in the spectrum and the corresponding eigenspaces of the connectivity operators associated with the hypergraph. We also show that, conversely, some specific footprints in the spectrum and in the corresponding eigenvectors retrace the presence of some of these building blocks in the hypergraph. The hypergraph remains invariant under the permutations among the vertices in some building blocks. These vertices behave similarly, in random walks on the hypergraph and play an important role in hypergraph automorphisms. Identifying similar vertices in certain building blocks results in a smaller hypergraph that contains some spectral information of the original hypergraph. The number of specific building blocks provides an upper bound of the chromatic number of the hypergraph. A pseudo metric is introduced to measure distances between vertices in the hypergraph by using one of the building blocks. Here, we use the concept of general connectivity operators of a hypergraph for our spectral study.Comment: Title is changed and some new section adde