312,623 research outputs found
Spectral Graph Forge: Graph Generation Targeting Modularity
Community structure is an important property that captures inhomogeneities
common in large networks, and modularity is one of the most widely used metrics
for such community structure. In this paper, we introduce a principled
methodology, the Spectral Graph Forge, for generating random graphs that
preserves community structure from a real network of interest, in terms of
modularity. Our approach leverages the fact that the spectral structure of
matrix representations of a graph encodes global information about community
structure. The Spectral Graph Forge uses a low-rank approximation of the
modularity matrix to generate synthetic graphs that match a target modularity
within user-selectable degree of accuracy, while allowing other aspects of
structure to vary. We show that the Spectral Graph Forge outperforms
state-of-the-art techniques in terms of accuracy in targeting the modularity
and randomness of the realizations, while also preserving other local
structural properties and node attributes. We discuss extensions of the
Spectral Graph Forge to target other properties beyond modularity, and its
applications to anonymization
Finding local community structure in networks
Although the inference of global community structure in networks has recently
become a topic of great interest in the physics community, all such algorithms
require that the graph be completely known. Here, we define both a measure of
local community structure and an algorithm that infers the hierarchy of
communities that enclose a given vertex by exploring the graph one vertex at a
time. This algorithm runs in time O(d*k^2) for general graphs when is the
mean degree and k is the number of vertices to be explored. For graphs where
exploring a new vertex is time-consuming, the running time is linear, O(k). We
show that on computer-generated graphs this technique compares favorably to
algorithms that require global knowledge. We also use this algorithm to extract
meaningful local clustering information in the large recommender network of an
online retailer and show the existence of mesoscopic structure.Comment: 7 pages, 6 figure
Community detection algorithms: a comparative analysis
Uncovering the community structure exhibited by real networks is a crucial
step towards an understanding of complex systems that goes beyond the local
organization of their constituents. Many algorithms have been proposed so far,
but none of them has been subjected to strict tests to evaluate their
performance. Most of the sporadic tests performed so far involved small
networks with known community structure and/or artificial graphs with a
simplified structure, which is very uncommon in real systems. Here we test
several methods against a recently introduced class of benchmark graphs, with
heterogeneous distributions of degree and community size. The methods are also
tested against the benchmark by Girvan and Newman and on random graphs. As a
result of our analysis, three recent algorithms introduced by Rosvall and
Bergstrom, Blondel et al. and Ronhovde and Nussinov, respectively, have an
excellent performance, with the additional advantage of low computational
complexity, which enables one to analyze large systems.Comment: 12 pages, 8 figures. The software to compute the values of our
general normalized mutual information is available at
http://santo.fortunato.googlepages.com/inthepress
Benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities
Many complex networks display a mesoscopic structure with groups of nodes
sharing many links with the other nodes in their group and comparatively few
with nodes of different groups. This feature is known as community structure
and encodes precious information about the organization and the function of the
nodes. Many algorithms have been proposed but it is not yet clear how they
should be tested. Recently we have proposed a general class of undirected and
unweighted benchmark graphs, with heterogenous distributions of node degree and
community size. An increasing attention has been recently devoted to develop
algorithms able to consider the direction and the weight of the links, which
require suitable benchmark graphs for testing. In this paper we extend the
basic ideas behind our previous benchmark to generate directed and weighted
networks with built-in community structure. We also consider the possibility
that nodes belong to more communities, a feature occurring in real systems,
like, e. g., social networks. As a practical application, we show how
modularity optimization performs on our new benchmark.Comment: 9 pages, 13 figures. Final version published in Physical Review E.
The code to create the benchmark graphs can be freely downloaded from
http://santo.fortunato.googlepages.com/inthepress
Community Structure in Industrial SAT Instances
Modern SAT solvers have experienced a remarkable progress on solving
industrial instances. Most of the techniques have been developed after an
intensive experimental process. It is believed that these techniques exploit
the underlying structure of industrial instances. However, there are few works
trying to exactly characterize the main features of this structure.
The research community on complex networks has developed techniques of
analysis and algorithms to study real-world graphs that can be used by the SAT
community. Recently, there have been some attempts to analyze the structure of
industrial SAT instances in terms of complex networks, with the aim of
explaining the success of SAT solving techniques, and possibly improving them.
In this paper, inspired by the results on complex networks, we study the
community structure, or modularity, of industrial SAT instances. In a graph
with clear community structure, or high modularity, we can find a partition of
its nodes into communities such that most edges connect variables of the same
community. In our analysis, we represent SAT instances as graphs, and we show
that most application benchmarks are characterized by a high modularity. On the
contrary, random SAT instances are closer to the classical Erd\"os-R\'enyi
random graph model, where no structure can be observed. We also analyze how
this structure evolves by the effects of the execution of a CDCL SAT solver. In
particular, we use the community structure to detect that new clauses learned
by the solver during the search contribute to destroy the original structure of
the formula. This is, learned clauses tend to contain variables of distinct
communities
Post-Processing Hierarchical Community Structures: Quality Improvements and Multi-scale View
Dense sub-graphs of sparse graphs (communities), which appear in most
real-world complex networks, play an important role in many contexts. Most
existing community detection algorithms produce a hierarchical structure of
community and seek a partition into communities that optimizes a given quality
function. We propose new methods to improve the results of any of these
algorithms. First we show how to optimize a general class of additive quality
functions (containing the modularity, the performance, and a new similarity
based quality function we propose) over a larger set of partitions than the
classical methods. Moreover, we define new multi-scale quality functions which
make it possible to detect the different scales at which meaningful community
structures appear, while classical approaches find only one partition.Comment: 12 Pages, 4 figure
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