726 research outputs found
Small-worlds: How and why
We investigate small-world networks from the point of view of their origin.
While the characteristics of small-world networks are now fairly well
understood, there is as yet no work on what drives the emergence of such a
network architecture. In situations such as neural or transportation networks,
where a physical distance between the nodes of the network exists, we study
whether the small-world topology arises as a consequence of a tradeoff between
maximal connectivity and minimal wiring. Using simulated annealing, we study
the properties of a randomly rewired network as the relative tradeoff between
wiring and connectivity is varied. When the network seeks to minimize wiring, a
regular graph results. At the other extreme, when connectivity is maximized, a
near random network is obtained. In the intermediate regime, a small-world
network is formed. However, unlike the model of Watts and Strogatz (Nature {\bf
393}, 440 (1998)), we find an alternate route to small-world behaviour through
the formation of hubs, small clusters where one vertex is connected to a large
number of neighbours.Comment: 20 pages, latex, 9 figure
GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs
We present a prototype of a software tool for exploration of multiple
combinatorial optimisation problems in large real-world and synthetic complex
networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial
Explorer), provides a unified framework for scalable computation and
presentation of high-quality suboptimal solutions and bounds for a number of
widely studied combinatorial optimisation problems. Efficient representation
and applicability to large-scale graphs and complex networks are particularly
considered in its design. The problems currently supported include maximum
clique, graph colouring, maximum independent set, minimum vertex clique
covering, minimum dominating set, as well as the longest simple cycle problem.
Suboptimal solutions and intervals for optimal objective values are estimated
using scalable heuristics. The tool is designed with extensibility in mind,
with the view of further problems and both new fast and high-performance
heuristics to be added in the future. GraphCombEx has already been successfully
used as a support tool in a number of recent research studies using
combinatorial optimisation to analyse complex networks, indicating its promise
as a research software tool
Cliques in rank-1 random graphs: the role of inhomogeneity
We study the asymptotic behavior of the clique number in rank-1 inhomogeneous
random graphs, where edge probabilities between vertices are roughly
proportional to the product of their vertex weights. We show that the clique
number is concentrated on at most two consecutive integers, for which we
provide an expression. Interestingly, the order of the clique number is
primarily determined by the overall edge density, with the inhomogeneity only
affecting multiplicative constants or adding at most a
multiplicative factor. For sparse enough graphs the clique number is always
bounded and the effect of inhomogeneity completely vanishes.Comment: 29 page
Random subcube intersection graphs I: cliques and covering
We study random subcube intersection graphs, that is, graphs obtained by
selecting a random collection of subcubes of a fixed hypercube to serve
as the vertices of the graph, and setting an edge between a pair of subcubes if
their intersection is non-empty. Our motivation for considering such graphs is
to model `random compatibility' between vertices in a large network. For both
of the models considered in this paper, we determine the thresholds for
covering the underlying hypercube and for the appearance of s-cliques. In
addition we pose some open problems.Comment: 38 pages, 1 figur
Testing bounded arboricity
In this paper we consider the problem of testing whether a graph has bounded
arboricity. The family of graphs with bounded arboricity includes, among
others, bounded-degree graphs, all minor-closed graph classes (e.g. planar
graphs, graphs with bounded treewidth) and randomly generated preferential
attachment graphs. Graphs with bounded arboricity have been studied extensively
in the past, in particular since for many problems they allow for much more
efficient algorithms and/or better approximation ratios.
We present a tolerant tester in the sparse-graphs model. The sparse-graphs
model allows access to degree queries and neighbor queries, and the distance is
defined with respect to the actual number of edges. More specifically, our
algorithm distinguishes between graphs that are -close to having
arboricity and graphs that -far from having
arboricity , where is an absolute small constant. The query
complexity and running time of the algorithm are
where denotes
the number of vertices and denotes the number of edges. In terms of the
dependence on and this bound is optimal up to poly-logarithmic factors
since queries are necessary (and .
We leave it as an open question whether the dependence on can be
improved from quasi-polynomial to polynomial. Our techniques include an
efficient local simulation for approximating the outcome of a global (almost)
forest-decomposition algorithm as well as a tailored procedure of edge
sampling
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