14 research outputs found
Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clustering
We study p-Laplacians and spectral clustering for a recently proposed
hypergraph model that incorporates edge-dependent vertex weights (EDVW). These
weights can reflect different importance of vertices within a hyperedge, thus
conferring the hypergraph model higher expressivity and flexibility. By
constructing submodular EDVW-based splitting functions, we convert hypergraphs
with EDVW into submodular hypergraphs for which the spectral theory is better
developed. In this way, existing concepts and theorems such as p-Laplacians and
Cheeger inequalities proposed under the submodular hypergraph setting can be
directly extended to hypergraphs with EDVW. For submodular hypergraphs with
EDVW-based splitting functions, we propose an efficient algorithm to compute
the eigenvector associated with the second smallest eigenvalue of the
hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the
vertices, achieving higher clustering accuracy than traditional spectral
clustering based on the 2-Laplacian. More broadly, the proposed algorithm works
for all submodular hypergraphs that are graph reducible. Numerical experiments
using real-world data demonstrate the effectiveness of combining spectral
clustering based on the 1-Laplacian and EDVW
Core-periphery detection in hypergraphs
Core-periphery detection is a key task in exploratory network analysis where
one aims to find a core, a set of nodes well-connected internally and with the
periphery, and a periphery, a set of nodes connected only (or mostly) with the
core. In this work we propose a model of core-periphery for higher-order
networks modeled as hypergraphs and we propose a method for computing a
core-score vector that quantifies how close each node is to the core. In
particular, we show that this method solves the corresponding non-convex
core-periphery optimization problem globally to an arbitrary precision. This
method turns out to coincide with the computation of the Perron eigenvector of
a nonlinear hypergraph operator, suitably defined in term of the incidence
matrix of the hypergraph, generalizing recently proposed centrality models for
hypergraphs. We perform several experiments on synthetic and real-world
hypergraphs showing that the proposed method outperforms alternative
core-periphery detection algorithms, in particular those obtained by
transferring established graph methods to the hypergraph setting via clique
expansion
Local Hypergraph Clustering using Capacity Releasing Diffusion
Local graph clustering is an important machine learning task that aims to
find a well-connected cluster near a set of seed nodes. Recent results have
revealed that incorporating higher order information significantly enhances the
results of graph clustering techniques. The majority of existing research in
this area focuses on spectral graph theory-based techniques. However, an
alternative perspective on local graph clustering arises from using max-flow
and min-cut on the objectives, which offer distinctly different guarantees. For
instance, a new method called capacity releasing diffusion (CRD) was recently
proposed and shown to preserve local structure around the seeds better than
spectral methods. The method was also the first local clustering technique that
is not subject to the quadratic Cheeger inequality by assuming a good cluster
near the seed nodes. In this paper, we propose a local hypergraph clustering
technique called hypergraph CRD (HG-CRD) by extending the CRD process to
cluster based on higher order patterns, encoded as hyperedges of a hypergraph.
Moreover, we theoretically show that HG-CRD gives results about a quantity
called motif conductance, rather than a biased version used in previous
experiments. Experimental results on synthetic datasets and real world graphs
show that HG-CRD enhances the clustering quality.Comment: 18 pages, 6 figure
Equivariant Hypergraph Diffusion Neural Operators
Hypergraph neural networks (HNNs) using neural networks to encode hypergraphs
provide a promising way to model higher-order relations in data and further
solve relevant prediction tasks built upon such higher-order relations.
However, higher-order relations in practice contain complex patterns and are
often highly irregular. So, it is often challenging to design an HNN that
suffices to express those relations while keeping computational efficiency.
Inspired by hypergraph diffusion algorithms, this work proposes a new HNN
architecture named ED-HNN, which provably represents any continuous equivariant
hypergraph diffusion operators that can model a wide range of higher-order
relations. ED-HNN can be implemented efficiently by combining star expansions
of hypergraphs with standard message passing neural networks. ED-HNN further
shows great superiority in processing heterophilic hypergraphs and constructing
deep models. We evaluate ED-HNN for node classification on nine real-world
hypergraph datasets. ED-HNN uniformly outperforms the best baselines over these
nine datasets and achieves more than 2\% in prediction accuracy over
four datasets therein.Comment: Code: https://github.com/Graph-COM/ED-HN
Generalizing p-Laplacian: spectral hypergraph theory and a partitioning algorithm
For hypergraph clustering, various methods have been proposed to defne hypergraph
p-Laplacians in the literature. This work proposes a general framework for an abstract class
of hypergraph p-Laplacians from a diferential-geometric view. This class includes previously proposed hypergraph p-Laplacians and also includes previously unstudied novel
generalizations. For this abstract class, we extend current spectral theory by providing
an extension of nodal domain theory for the eigenvectors of our hypergraph p-Laplacian.
We use this nodal domain theory to provide bounds on the eigenvalues via a higher-order
Cheeger inequality. Following our extension of spectral theory, we propose a novel hypergraph partitioning algorithm for our generalized p-Laplacian. Our empirical study shows
that our algorithm outperforms spectral methods based on existing p-Laplacians
A Generalized Ratio Cut Objective in Graphs and Efficient Algorithm for Solving the Localized Generalized Expansion Ratio Problem
In graph clustering, ratio cut objectives represent the ratio between the connectivity of the subgraph and some notation of the graph properties including size and density. These ratio cut objectives are widely studied and used for many tasks including graph partitioning. Specifically, the standard expansion ratio measures the ratio between subgraph’s connections to the rest of the graph and the subgraph size. This thesis introduces a generalized version of the standard expansion ratio objective and studies a localized variant of the expansion ratio problem by presenting its connections to existing problems and numerical results on existing problems.
The generalized version of the expansion ratio concerned in this thesis replaces the subgraph size with a convex function of the set size, generalizing more than one existing objective functions. The localized variant of the expansion ratio problem removes the constraint of subgraph size while restricting the resulting subgraph to a given seed set of nodes. While the original expansion problem is NP-hard to solve, this thesis introduces a polynomial-time algorithm for the novel localized variant of the problem. By varying the convex function and tuning parameters, numerical experiments show solving this new problem with the generalized objective allows one implicitly control the size of the resulting subgraph