957 research outputs found
Distance-Dependent Kronecker Graphs for Modeling Social Networks
This paper focuses on a generalization of stochastic
Kronecker graphs, introducing a Kronecker-like operator and
defining a family of generator matrices H dependent on distances
between nodes in a specified graph embedding. We prove
that any lattice-based network model with sufficiently small
distance-dependent connection probability will have a Poisson
degree distribution and provide a general framework to prove
searchability for such a network. Using this framework, we focus
on a specific example of an expanding hypercube and discuss
the similarities and differences of such a model with recently
proposed network models based on a hidden metric space. We
also prove that a greedy forwarding algorithm can find very short
paths of length O((log log n)^2) on the hypercube with n nodes,
demonstrating that distance-dependent Kronecker graphs can
generate searchable network models
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
Hyperbolic Geometry of Complex Networks
We develop a geometric framework to study the structure and function of
complex networks. We assume that hyperbolic geometry underlies these networks,
and we show that with this assumption, heterogeneous degree distributions and
strong clustering in complex networks emerge naturally as simple reflections of
the negative curvature and metric property of the underlying hyperbolic
geometry. Conversely, we show that if a network has some metric structure, and
if the network degree distribution is heterogeneous, then the network has an
effective hyperbolic geometry underneath. We then establish a mapping between
our geometric framework and statistical mechanics of complex networks. This
mapping interprets edges in a network as non-interacting fermions whose
energies are hyperbolic distances between nodes, while the auxiliary fields
coupled to edges are linear functions of these energies or distances. The
geometric network ensemble subsumes the standard configuration model and
classical random graphs as two limiting cases with degenerate geometric
structures. Finally, we show that targeted transport processes without global
topology knowledge, made possible by our geometric framework, are maximally
efficient, according to all efficiency measures, in networks with strongest
heterogeneity and clustering, and that this efficiency is remarkably robust
with respect to even catastrophic disturbances and damages to the network
structure
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