18,087 research outputs found
Finding community structure in networks using the eigenvectors of matrices
We consider the problem of detecting communities or modules in networks,
groups of vertices with a higher-than-average density of edges connecting them.
Previous work indicates that a robust approach to this problem is the
maximization of the benefit function known as "modularity" over possible
divisions of a network. Here we show that this maximization process can be
written in terms of the eigenspectrum of a matrix we call the modularity
matrix, which plays a role in community detection similar to that played by the
graph Laplacian in graph partitioning calculations. This result leads us to a
number of possible algorithms for detecting community structure, as well as
several other results, including a spectral measure of bipartite structure in
networks and a new centrality measure that identifies those vertices that
occupy central positions within the communities to which they belong. The
algorithms and measures proposed are illustrated with applications to a variety
of real-world complex networks.Comment: 22 pages, 8 figures, minor corrections in this versio
Modularity and community structure in networks
Many networks of interest in the sciences, including a variety of social and
biological networks, are found to divide naturally into communities or modules.
The problem of detecting and characterizing this community structure has
attracted considerable recent attention. One of the most sensitive detection
methods is optimization of the quality function known as "modularity" over the
possible divisions of a network, but direct application of this method using,
for instance, simulated annealing is computationally costly. Here we show that
the modularity can be reformulated in terms of the eigenvectors of a new
characteristic matrix for the network, which we call the modularity matrix, and
that this reformulation leads to a spectral algorithm for community detection
that returns results of better quality than competing methods in noticeably
shorter running times. We demonstrate the algorithm with applications to
several network data sets.Comment: 7 pages, 3 figure
Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths
We introduce several novel and computationally efficient methods for
detecting "core--periphery structure" in networks. Core--periphery structure is
a type of mesoscale structure that includes densely-connected core vertices and
sparsely-connected peripheral vertices. Core vertices tend to be well-connected
both among themselves and to peripheral vertices, which tend not to be
well-connected to other vertices. Our first method, which is based on
transportation in networks, aggregates information from many geodesic paths in
a network and yields a score for each vertex that reflects the likelihood that
a vertex is a core vertex. Our second method is based on a low-rank
approximation of a network's adjacency matrix, which can often be expressed as
a tensor-product matrix. Our third approach uses the bottom eigenvector of the
random-walk Laplacian to infer a coreness score and a classification into core
and peripheral vertices. We also design an objective function to (1) help
classify vertices into core or peripheral vertices and (2) provide a
goodness-of-fit criterion for classifications into core versus peripheral
vertices. To examine the performance of our methods, we apply our algorithms to
both synthetically-generated networks and a variety of networks constructed
from real-world data sets.Comment: This article is part of EJAM's December 2016 special issue on
"Network Analysis and Modelling" (available at
https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/issue/journal-ejm-volume-27-issue-6/D245C89CABF55DBF573BB412F7651ADB
Detecting Multiple Communities Using Quantum Annealing on the D-Wave System
A very important problem in combinatorial optimization is partitioning a
network into communities of densely connected nodes; where the connectivity
between nodes inside a particular community is large compared to the
connectivity between nodes belonging to different ones. This problem is known
as community detection, and has become very important in various fields of
science including chemistry, biology and social sciences. The problem of
community detection is a twofold problem that consists of determining the
number of communities and, at the same time, finding those communities. This
drastically increases the solution space for heuristics to work on, compared to
traditional graph partitioning problems. In many of the scientific domains in
which graphs are used, there is the need to have the ability to partition a
graph into communities with the ``highest quality'' possible since the presence
of even small isolated communities can become crucial to explain a particular
phenomenon. We have explored community detection using the power of quantum
annealers, and in particular the D-Wave 2X and 2000Q machines. It turns out
that the problem of detecting at most two communities naturally fits into the
architecture of a quantum annealer with almost no need of reformulation. This
paper addresses a systematic study of detecting two or more communities in a
network using a quantum annealer
Considerations about multistep community detection
The problem and implications of community detection in networks have raised a
huge attention, for its important applications in both natural and social
sciences. A number of algorithms has been developed to solve this problem,
addressing either speed optimization or the quality of the partitions
calculated. In this paper we propose a multi-step procedure bridging the
fastest, but less accurate algorithms (coarse clustering), with the slowest,
most effective ones (refinement). By adopting heuristic ranking of the nodes,
and classifying a fraction of them as `critical', a refinement step can be
restricted to this subset of the network, thus saving computational time.
Preliminary numerical results are discussed, showing improvement of the final
partition.Comment: 12 page
Overlapping Community Detection Optimization and Nash Equilibrium
Community detection using both graphs and social networks is the focus of
many algorithms. Recent methods aimed at optimizing the so-called modularity
function proceed by maximizing relations within communities while minimizing
inter-community relations.
However, given the NP-completeness of the problem, these algorithms are
heuristics that do not guarantee an optimum. In this paper, we introduce a new
algorithm along with a function that takes an approximate solution and modifies
it in order to reach an optimum. This reassignment function is considered a
'potential function' and becomes a necessary condition to asserting that the
computed optimum is indeed a Nash Equilibrium. We also use this function to
simultaneously show partitioning and overlapping communities, two detection and
visualization modes of great value in revealing interesting features of a
social network. Our approach is successfully illustrated through several
experiments on either real unipartite, multipartite or directed graphs of
medium and large-sized datasets.Comment: Submitted to KD
- …