397 research outputs found
SIS epidemic propagation on hypergraphs
Mathematical modeling of epidemic propagation on networks is extended to
hypergraphs in order to account for both the community structure and the
nonlinear dependence of the infection pressure on the number of infected
neighbours. The exact master equations of the propagation process are derived
for an arbitrary hypergraph given by its incidence matrix. Based on these,
moment closure approximation and mean-field models are introduced and compared
to individual-based stochastic simulations. The simulation algorithm, developed
for networks, is extended to hypergraphs. The effects of hypergraph structure
and the model parameters are investigated via individual-based simulation
results
Alignment and integration of complex networks by hypergraph-based spectral clustering
Complex networks possess a rich, multi-scale structure reflecting the
dynamical and functional organization of the systems they model. Often there is
a need to analyze multiple networks simultaneously, to model a system by more
than one type of interaction or to go beyond simple pairwise interactions, but
currently there is a lack of theoretical and computational methods to address
these problems. Here we introduce a framework for clustering and community
detection in such systems using hypergraph representations. Our main result is
a generalization of the Perron-Frobenius theorem from which we derive spectral
clustering algorithms for directed and undirected hypergraphs. We illustrate
our approach with applications for local and global alignment of
protein-protein interaction networks between multiple species, for tripartite
community detection in folksonomies, and for detecting clusters of overlapping
regulatory pathways in directed networks.Comment: 16 pages, 5 figures; revised version with minor corrections and
figures printed in two-column format for better readability; algorithm
implementation and supplementary information available at Google code at
http://schype.googlecode.co
Hypergraph Learning with Line Expansion
Previous hypergraph expansions are solely carried out on either vertex level
or hyperedge level, thereby missing the symmetric nature of data co-occurrence,
and resulting in information loss. To address the problem, this paper treats
vertices and hyperedges equally and proposes a new hypergraph formulation named
the \emph{line expansion (LE)} for hypergraphs learning. The new expansion
bijectively induces a homogeneous structure from the hypergraph by treating
vertex-hyperedge pairs as "line nodes". By reducing the hypergraph to a simple
graph, the proposed \emph{line expansion} makes existing graph learning
algorithms compatible with the higher-order structure and has been proven as a
unifying framework for various hypergraph expansions. We evaluate the proposed
line expansion on five hypergraph datasets, the results show that our method
beats SOTA baselines by a significant margin
A framework to generate hypergraphs with community structure
In recent years hypergraphs have emerged as a powerful tool to study systems
with multi-body interactions which cannot be trivially reduced to pairs. While
highly structured methods to generate synthetic data have proved fundamental
for the standardized evaluation of algorithms and the statistical study of
real-world networked data, these are scarcely available in the context of
hypergraphs. Here we propose a flexible and efficient framework for the
generation of hypergraphs with many nodes and large hyperedges, which allows
specifying general community structures and tune different local statistics. We
illustrate how to use our model to sample synthetic data with desired features
(assortative or disassortative communities, mixed or hard community
assignments, etc.), analyze community detection algorithms, and generate
hypergraphs structurally similar to real-world data. Overcoming previous
limitations on the generation of synthetic hypergraphs, our work constitutes a
substantial advancement in the statistical modeling of higher-order systems.Comment: 18 pages, 8 figures, revised versio
How Transitive Are Real-World Group Interactions? -- Measurement and Reproduction
Many real-world interactions (e.g., researcher collaborations and email
communication) occur among multiple entities. These group interactions are
naturally modeled as hypergraphs. In graphs, transitivity is helpful to
understand the connections between node pairs sharing a neighbor, and it has
extensive applications in various domains. Hypergraphs, an extension of graphs,
are designed to represent group relations. However, to the best of our
knowledge, there has been no examination regarding the transitivity of
real-world group interactions. In this work, we investigate the transitivity of
group interactions in real-world hypergraphs. We first suggest intuitive axioms
as necessary characteristics of hypergraph transitivity measures. Then, we
propose a principled hypergraph transitivity measure HyperTrans, which
satisfies all the proposed axioms, with a fast computation algorithm
Fast-HyperTrans. After that, we analyze the transitivity patterns in real-world
hypergraphs distinguished from those in random hypergraphs. Lastly, we propose
a scalable hypergraph generator THera. It reproduces the observed transitivity
patterns by leveraging community structures, which are pervasive in real-world
hypergraphs. Our code and datasets are available at
https://github.com/kswoo97/hypertrans.Comment: To be published in KDD 2023. 12 pages, 7 figures, and 11 table
Encapsulation structure and dynamics in hypergraphs
Hypergraphs have emerged as a powerful modeling framework to represent systems with multiway interactions, that is systems where interactions may involve an arbitrary number of agents. Here we explore the properties of real-world hypergraphs, focusing on the encapsulation of their hyperedges, which is the extent that smaller hyperedges are subsets of larger hyperedges. Building on the concept of line graphs, our measures quantify the relations existing between hyperedges of different sizes and, as a byproduct, the compatibility of the data with a simplicial complex representation–whose encapsulation would be maximum. We then turn to the impact of the observed structural patterns on diffusive dynamics, focusing on a variant of threshold models, called encapsulation dynamics, and demonstrate that non-random patterns can accelerate the spreading in the system
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