32 research outputs found
Estrada index of hypergraphs via eigenvalues of tensors
A uniform hypergraph is corresponding to an adjacency tensor
. We define an Estrada index of by using
all the eigenvalues of as
. The bounds for the Estrada indices of uniform
hypergraphs are given. And we characterize the Estrada indices of -uniform
hypergraphs whose spectra of the adjacency tensors are -symmetric.
Specially, we characterize the Estrada indices of uniform hyperstars
Fast Parallel Tensor Times Same Vector for Hypergraphs
Hypergraphs are a popular paradigm to represent complex real-world networks
exhibiting multi-way relationships of varying sizes. Mining centrality in
hypergraphs via symmetric adjacency tensors has only recently become
computationally feasible for large and complex datasets. To enable scalable
computation of these and related hypergraph analytics, here we focus on the
Sparse Symmetric Tensor Times Same Vector (STTVc) operation. We introduce
the Compound Compressed Sparse Symmetric (CCSS) format, an extension of the
compact CSS format for hypergraphs of varying hyperedge sizes and present a
shared-memory parallel algorithm to compute STTVc. We experimentally show
STTVc computation using the CCSS format achieves better performance than
the naive baseline, and is subsequently more performant for hypergraph
-eigenvector centrality
HPRA: Hyperedge Prediction using Resource Allocation
Many real-world systems involve higher-order interactions and thus demand
complex models such as hypergraphs. For instance, a research article could have
multiple collaborating authors, and therefore the co-authorship network is best
represented as a hypergraph. In this work, we focus on the problem of hyperedge
prediction. This problem has immense applications in multiple domains, such as
predicting new collaborations in social networks, discovering new chemical
reactions in metabolic networks, etc. Despite having significant importance,
the problem of hyperedge prediction hasn't received adequate attention, mainly
because of its inherent complexity. In a graph with nodes the number of
potential edges is , whereas in a hypergraph, the number of
potential hyperedges is . To avoid searching through such a
huge space, current methods restrain the original problem in the following two
ways. One class of algorithms assume the hypergraphs to be -uniform.
However, many real-world systems are not confined only to have interactions
involving components. Thus, these algorithms are not suitable for many
real-world applications. The second class of algorithms requires a candidate
set of hyperedges from which the potential hyperedges are chosen. In the
absence of domain knowledge, the candidate set can have
possible hyperedges, which makes this problem intractable. We propose HPRA -
Hyperedge Prediction using Resource Allocation, the first of its kind
algorithm, which overcomes these issues and predicts hyperedges of any
cardinality without using any candidate hyperedge set. HPRA is a
similarity-based method working on the principles of the resource allocation
process. In addition to recovering missing hyperedges, we demonstrate that HPRA
can predict future hyperedges in a wide range of hypergraphs.Comment: Accepted at WebSci'2
Detecting informative higher-order interactions in statistically validated hypergraphs
Recent empirical evidence has shown that in many real-world systems, successfully represented as networks, interactions are not limited to dyads, but often involve three or more agents at a time. These data are better described by hypergraphs, where hyperlinks encode higher-order interactions among a group of nodes. In spite of the extensive literature on networks, detecting informative hyperlinks in real world hypergraphs is still an open problem. Here we propose an analytic approach to filter hypergraphs by identifying those hyperlinks that are over-expressed with respect to a random null hypothesis, and represent the most relevant higher-order connections. We apply our method to a class of synthetic benchmarks and to several datasets, showing that the method highlights hyperlinks that are more informative than those extracted with pairwise approaches. Our method provides a first way, to the best of our knowledge, to obtain statistically validated hypergraphs, separating informative connections from noisy ones
Vector Centrality in Hypergraphs
Identifying the most influential nodes in networked systems is of vital
importance to optimize their function and control. Several scalar metrics have
been proposed to that effect, but the recent shift in focus towards network
structures which go beyond a simple collection of dyadic interactions has
rendered them void of performance guarantees. We here introduce a new measure
of node's centrality, which is no longer a scalar value, but a vector with
dimension one lower than the highest order of interaction in a hypergraph. Such
a vectorial measure is linked to the eigenvector centrality for networks
containing only dyadic interactions, but it has a significant added value in
all other situations where interactions occur at higher-orders. In particular,
it is able to unveil different roles which may be played by the same node at
different orders of interactions -- information that is otherwise impossible to
retrieve by single scalar measures. We demonstrate the efficacy of our measure
with applications to synthetic networks and to three real world hypergraphs,
and compare our results with those obtained by applying other scalar measures
of centrality proposed in the literature.Comment: 10 pages, 9 figure
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
Accelerating the Computation of Tensor -eigenvalues
Efficient solvers for tensor eigenvalue problems are important tools for the
analysis of higher-order data sets. Here we introduce, analyze and demonstrate
an extrapolation method to accelerate the widely used shifted symmetric higher
order power method for tensor -eigenvalue problems. We analyze the
asymptotic convergence of the method, determining the range of extrapolation
parameters that induce acceleration, as well as the parameter that gives the
optimal convergence rate. We then introduce an automated method to dynamically
approximate the optimal parameter, and demonstrate it's efficiency when the
base iteration is run with either static or adaptively set shifts. Our
numerical results on both even and odd order tensors demonstrate the theory and
show we achieve our theoretically predicted acceleration.Comment: 22 pages, 8 figures, 4 table