646 research outputs found
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
Transversals of subtree hypergraphs and the source location problem in hypergraphs
A hypergraph is a subtree hypergraph if there is a tree~ on~ such that each hyperedge of~ induces a subtree of~. Since the number of edges of a subtree hypergraph can be exponential in , one can not always expect to be able to find a minimum size transversal in time polynomial in~. In this paper, we show that if it is possible to decide if a set of vertices is a transversal in time~ (\,where \,), then it is possible to find a minimum size transversal in~. This result provides a polynomial algorithm for the Source Location Problem\,: a set of -sources for a digraph is a subset~ of~ such that for any there are~ arc-disjoint paths that each join a vertex of~ to~ and~ arc-disjoint paths that each join~ to~. The Source Location Problem is to find a minimum size set of -sources. We show that this is a case of finding a transversal of a subtree hypergraph, and that in this case~ is polynomial
Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets
An identifying code of a (di)graph is a dominating subset of the
vertices of such that all distinct vertices of have distinct
(in)neighbourhoods within . In this paper, we classify all finite digraphs
which only admit their whole vertex set in any identifying code. We also
classify all such infinite oriented graphs. Furthermore, by relating this
concept to a well known theorem of A. Bondy on set systems we classify the
extremal cases for this theorem
Grassmann Integral Representation for Spanning Hyperforests
Given a hypergraph G, we introduce a Grassmann algebra over the vertex set,
and show that a class of Grassmann integrals permits an expansion in terms of
spanning hyperforests. Special cases provide the generating functions for
rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All
these results are generalizations of Kirchhoff's matrix-tree theorem.
Furthermore, we show that the class of integrals describing unrooted spanning
(hyper)forests is induced by a theory with an underlying OSP(1|2)
supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J.
Phys.
GRACE as a unifying approach to graph-transformation-based specification1 1This work was partially supported by the ESPRIT Working Group Applications of Graph Transformation (APPLIGRAPH) and the EC TMR Network GETGRATS (General Theory of Graph Transformation Systems).
AbstractIn this paper, we sketch some basic ideas and features of the graph-transformation-based specification language GRACE. The aim of GRACE is to support the modeling of a wide spectrum of graph and graphical processes in a structured and uniform way including visualization and verification
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