The celebrated Cheeger's Inequality [AM85, Alo86] establishes a bound on the edge expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph.In this paper we introduce a new hypergraph Laplacian operator generalizing the Laplacian matrix of graphs. In particular, the operator is induced by a diffusion process on the hypergraph, such that within each hyperedge, measure flows from vertices having maximum weighted measure to those having minimum. Since the operator is non-linear, we have to exploit other properties of the diffusion process to recover a spectral property concerning the "second eigenvalue" of the resulting Laplacian. Moreover, we show that higher order spectral properties cannot hold in general using the current framework.We consider a stochastic diffusion process, in which each vertex also experiences Brownian noise from outside the system. We show a relationship between the second eigenvalue and the convergence behavior of the process.We show that various hypergraph parameters like multi-way expansion and diameter can be bounded using this operator's spectral properties. Since higher order spectral properties do not hold for the Laplacian operator, we instead use the concept of procedural minimizers to consider higher order Cheeger-like inequalities. For any k ∈ N, we give a polynomial time algorithm to compute an O(log r)-approximation to the k-th procedural minimizer, where r is the maximum cardinality of a hyperedge. We show that this approximation factor is optimal under the SSE hypothesis (introduced by [RS10]) for constant values of k.Moreover, using the factor preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs. * A preliminary version of this paper appeared in STOC 2015 [Lou15] and the current paper is the result of a merge with [CTZ15].

The celebrated Cheeger's Inequality [AM85, Alo86] establishes a bound on the expansion of a graph via its spectrum. This inequality is central to a rich spectral theory of graphs, based on studying the eigenvalues and eigenvectors of the adjacency matrix (and other related matrices) of graphs. It has remained open to define a suitable spectral model for hypergraphs whose spectra can be used to estimate various combinatorial properties of the hypergraph.In this paper we introduce a new hypergraph Laplacian operator (generalizing the Laplacian matrix of graphs) and study its spectra. We prove a Cheeger-type inequality for hypergraphs, relating the second smallest eigenvalue of this operator to the expansion of the hypergraph. We bound other hypergraph expansion parameters via higher eigenvalues of this operator. We give bounds on the diameter of the hypergraph as a function of the second smallest eigenvalue of the Laplacian operator. The Markov process underlying the Laplacian operator can be viewed as a dispersion process on the vertices of the hypergraph that can be used to model rumour spreading in networks, brownian motion, etc., and might be of independent interest. We bound the Mixing-time of this process as a function of the second smallest eigenvalue of the Laplacian operator. All these results are generalizations of the corresponding results for graphs.We show that there can be no linear operator for hypergraphs whose spectra captures hypergraph expansion in a Cheeger-like manner. Our Laplacian operator is non-linear and thus computing its eigenvalues exactly is intractable. For any k ∈ 0 , we give a polynomial time algorithm to compute an approximation to the k th smallest eigenvalue of the operator . We show that this approximation factor is optimal under the SSE hypothesis (introduced by [RS10]) for constant values of k.We give a O log k log r log log k -approximation algorithm for the general sparsest cut in hypergraphs, where k is the number of "demands" in the instance and r is the size of the largest hyperedge.Finally, using the factor preserving reduction from vertex expansion in graphs to hypergraph expansion, we show that all our results for hypergraphs extend to vertex expansion in graphs.

Link prediction in simple graphs is a fundamental problem in which new links between vertices are predicted based on the observed structure of the graph. However, in many real-world applications, there is need to model relationships among vertices which go beyond pairwise associations. For example, in a chemical reaction, relationship among the reactants and products is inherently higherorder. Additionally, there is need to represent the direction from reactants to products. Hypergraphs provide a natural way to represent such complex higher-order relationships. Graph Convolutional Networks (GCN) have recently emerged as a powerful deep learning-based approach for link prediction over simple graphs. However, their suitability for link prediction in hypergraphs is underexplored-we fill this gap in this paper and propose Neural Hyperlink Predictor (NHP). NHP adapts GCNs for link prediction in hypergraphs. We propose two variants of NHP-NHP-U and NHP-D-for link prediction over undirected and directed hypergraphs, respectively. To the best of our knowledge, NHP-D is the first ever method for link prediction over directed hypergraphs. An important feature of NHP is that it can also be used for hyperlinks in which dissimilar vertices interact (e.g. acids reacting with bases). Another attractive feature of NHP is that it can be used to predict unseen hyperlinks at test time (inductive hyperlink prediction). Through extensive experiments on multiple real-world datasets, we show NHP's effectiveness. CCS CONCEPTS • Computing methodologies → Neural networks; Unsupervised learning.

We study the complexity of approximating the vertex expansion of graphs G = (V, E), defined asWe give a simple polynomial-time algorithm for finding a subset with vertex expansion O φ V log d where d is the maximum degree of the graph. Our main result is an asymptotically matching lower bound: under the Small Set Expansion (SSE) hypothesis, it is hard to find a subset with expansion less than C φ V log d for an absolute constant C. In particular, this implies for all constant ε > 0, it is SSE-hard to distinguish whether the vertex expansion < ε or at least an absolute constant. The analogous threshold for edge expansion is √ φ with no dependence on the degree (Here φ denotes the optimal edge expansion). Thus our results suggest that vertex expansion is harder to approximate than edge expansion. In particular, while Cheeger's algorithm can certify constant edge expansion, it is SSE-hard to certify constant vertex expansion in graphs.Our proof is via a reduction from the Unique Games instance obtained from the SSE hypothesis to the vertex expansion problem. It involves the definition of a smoother intermediate problem we call Balanced Analytic Vertex Expansion which is representative of both the vertex expansion and the conductance of the graph. Both reductions (from the UGC instance to this problem and from this problem to vertex expansion) use novel proof ideas.