Networks beyond pairwise interactions: Structure and dynamics

Authored by Dr. Safiya U. Noble, an assistant professor at the University of Southern California Annenberg School of Communication, this text is one of the preeminent works that explicitly addresse...

Algorithms of Oppression: How Search Engines Reinforce Racism

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zeroIt has been conjectured that an analogous characterization holds for higher multiplicities: There are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut We resolve this conjecture positively Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into Rk, and then apply geometric considerations to the embeddingWe also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size a n/k and λk, the kth smallest eigenvalue of the normalized Laplacian, where n is the number of vertices In particular, we show that in every graph there are at least k/2 disjoint sets (one of which will have size at most 2n/k), each having expansion at most O(√λk log k) Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result The √log k bound is tight, up to constant factors, for the “noisy hypercube” graphs

Multiway Spectral Partitioning and Higher-Order Cheeger Inequalities

Subexponential time approximation algorithms are presented for the Unique Games and Small-Set Expansion problems. Specifically, for some absolute constant c, the following two algorithms are presented.(1) An exp(kne)-time algorithm that, given as input a k-alphabet unique game on n variables that has an assignment satisfying 1-ec fraction of its constraints, outputs an assignment satisfying 1-e fraction of the constraints.(2) An exp(ne/δ)-time algorithm that, given as input an n-vertex regular graph that has a set S of δn vertices with edge expansion at most ec, outputs a set S' of at most δ n vertices with edge expansion at most e.subexponential algorithm is also presented with improved approximation to Max Cut, Sparsest Cut, and Vertex Cover on some interesting subclasses of instances. These instances are graphs with low threshold rank, an interesting new graph parameter highlighted by this work.Khot's Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for Unique Games. While the results here stop short of refuting the UGC, they do suggest that Unique Games are significantly easier than NP-hard problems such as Max 3-Sat, Max 3-Lin, Label Cover, and more, which are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio.Of special interest in these algorithms is a new notion of graph decomposition that may have other applications. Namely, it is shown for every e >0 and every regular n-vertex graph G, by changing at most δ fraction of G's edges, one can break G into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most ne eigenvalues larger than 1-η, where η depends polynomially on e. The subexponential algorithm combines this decomposition with previous algorithms for Unique Games on graphs with few large eigenvalues [Kolla and Tulsiani 2007; Kolla 2010].

https://www.boazbarak.org/Papers/ssesubexp.pdf

Subexponential Algorithms for Unique Games and Related Problems

Farooq Qureshi, FRCS Anderson Walk Smithfield. SA 5114 so that the full thickness of the vas is not destroyed. The distal end is fulgurated at the cut surface and the sheath of the vas closed over it with a single suture. This creation of a fascial barrier between the ends prevents spontaneous rejoining, and also facilitates reversal later on if required. Ligatures or metal clips should not be used on the vas, whether absorbable or unabsorbable. These cause necrosis and sloughing of the vessel, and it becomes permanently fibrosed.

Smoking and health.

In many real-world network datasets such as co-authorship, co-citation, email communication, etc., relationships are complex and go beyond pairwise. Hypergraphs provide a flexible and natural modeling tool to model such complex relationships. The obvious existence of such complex relationships in many real-world networks naturaly motivates the problem of learning with hypergraphs. A popular learning paradigm is hypergraph-based semi-supervised learning (SSL) where the goal is to assign labels to initially unlabeled vertices in a hypergraph. Motivated by the fact that a graph convolutional network (GCN) has been effective for graph-based SSL, we propose HyperGCN, a novel GCN for SSL on attributed hypergraphs. Additionally, we show how HyperGCN can be used as a learning-based approach for combinatorial optimisation on NP-hard hypergraph problems. We demonstrate HyperGCN's effectiveness through detailed experimentation on real-world hypergraphs.

HyperGCN: A New Method of Training Graph Convolutional Networks on Hypergraphs

Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset S such that its expansion (a.k.a. conductance) is bounded as follows: [ φ(S) def= (w(S,bar{S}))/(min set(w(S), w(bar(S)))) ≤ √(2 λ2) ] where w is the total edge weight of a subset or a cut and λ2 is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer k ∈ [n], there exist ck disjoint subsets S1, ..., Sck, such that [ maxi φ(Si) ≤ C √(λk log k) ] where λk is the kth smallest eigenvalue of the normalized Laplacian and c0 are suitable absolute constants. Our proof is via a polynomial-time algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any k, there is a subset S whose weight is at most a O(1/k) fraction of the total weight and φ(S) ≤ C √(λk log k). Both results are the best possible up to constant factors. The underlying algorithmic problem, namely finding k subsets such that the maximum expansion is minimized, besides extending sparse cuts to more than one subset, appears to be a natural clustering problem in its own right.

Many sparse cuts via higher eigenvalues

In many real-world network datasets such as co-authorship, co-citation, email communication, etc., relationships are complex and go beyond pairwise. Hypergraphs provide a flexible and natural modeling tool to model such complex relationships. The obvious existence of such complex relationships in many real-world networks naturaly motivates the problem of learning with hypergraphs. A popular learning paradigm is hypergraph-based semi-supervised learning (SSL) where the goal is to assign labels to initially unlabeled vertices in a hypergraph. Motivated by the fact that a graph convolutional network (GCN) has been effective for graph-based SSL, we propose HyperGCN, a novel GCN for SSL on attributed hypergraphs. Additionally, we show how HyperGCN can be used as a learning-based approach for combinatorial optimisation on NP-hard hypergraph problems. We demonstrate HyperGCN's effectiveness through detailed experimentation on real-world hypergraphs. We have made HyperGCN's source code available to foster reproducible research.

/pdf/hypergcn-a-new-method-for-training-graph-convolutional-421fhuqapv.pdf

HyperGCN: A New Method For Training Graph Convolutional Networks on Hypergraphs

The celebrated Cheeger’s Inequality (Alon and Milman 1985; Alon 1986) 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 article, 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 nonlinear, we have to exploit other properties of the diffusion process to recover the Cheeger’s Inequality that relates hyperedge expansion with the “second eigenvalue” of the resulting Laplacian. However, we show that higher-order spectral properties cannot hold in general using the current framework.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 kth 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 Raghavendra and Steurer (2010)) 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.

Spectral Properties of Hypergraph Laplacian and Approximation Algorithms

The celebrated Cheeger's Inequality [AM85,a86] 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. Our operator can be viewed as the gradient operator applied to a certain natural quadratic form for hypergraphs. We show that various hypergraph parameters (for e.g. expansion, diameter, etc) can be bounded using this operator's eigenvalues. We study the heat diffusion process associated with this Laplacian operator, and bound its parameters in terms of its spectra. All our 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, we give a polynomial time algorithm to compute an approximation to the kth 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.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.

Anand Louis

Papers

HyperGCN: A New Method of Training Graph Convolutional Networks on Hypergraphs

Many sparse cuts via higher eigenvalues

HyperGCN: A New Method For Training Graph Convolutional Networks on Hypergraphs

Spectral Properties of Hypergraph Laplacian and Approximation Algorithms

Hypergraph Markov Operators, Eigenvalues and Approximation Algorithms