61 research outputs found
Experiments on comparing graph clusterins
A promising approach to compare graph clusterings is based on
using measurements for calculating the distance. Existing
measures either use the structure of clusterings or
quality--based aspects. Each approach suffers from critical
drawbacks. We introduce a new approach combining both aspects
and leading to better results for comparing graph clusterings.
An experimental evaluation of existing and new measures shows
that the significant drawbacks of existing techniques are not
only theoretical in nature and proves that the results of our
new measures are more coherent with intuition
Combinatorial approach to Modularity
Communities are clusters of nodes with a higher than average density of
internal connections. Their detection is of great relevance to better
understand the structure and hierarchies present in a network. Modularity has
become a standard tool in the area of community detection, providing at the
same time a way to evaluate partitions and, by maximizing it, a method to find
communities. In this work, we study the modularity from a combinatorial point
of view. Our analysis (as the modularity definition) relies on the use of the
configurational model, a technique that given a graph produces a series of
randomized copies keeping the degree sequence invariant. We develop an approach
that enumerates the null model partitions and can be used to calculate the
probability distribution function of the modularity. Our theory allows for a
deep inquiry of several interesting features characterizing modularity such as
its resolution limit and the statistics of the partitions that maximize it.
Additionally, the study of the probability of extremes of the modularity in the
random graph partitions opens the way for a definition of the statistical
significance of network partitions.Comment: 8 pages, 4 figure
A new paradigm for complex network visualization
We propose a new layout paradigm for drawing a nested
decomposition of a large network. The visualization supports the
recognition of abstract features of the decomposition, while
drawing all elements. In order to support the visual analysis
that focuses on the dependencies of the individual parts of the
decomposition, we use an annulus as the general underlying
shape. This method has been evaluated using real world data and
offers surprising readability
Size reduction of complex networks preserving modularity
The ubiquity of modular structure in real-world complex networks is being the
focus of attention in many trials to understand the interplay between network
topology and functionality. The best approaches to the identification of
modular structure are based on the optimization of a quality function known as
modularity. However this optimization is a hard task provided that the
computational complexity of the problem is in the NP-hard class. Here we
propose an exact method for reducing the size of weighted (directed and
undirected) complex networks while maintaining invariant its modularity. This
size reduction allows the heuristic algorithms that optimize modularity for a
better exploration of the modularity landscape. We compare the modularity
obtained in several real complex-networks by using the Extremal Optimization
algorithm, before and after the size reduction, showing the improvement
obtained. We speculate that the proposed analytical size reduction could be
extended to an exact coarse graining of the network in the scope of real-space
renormalization.Comment: 14 pages, 2 figure
Enhance the Efficiency of Heuristic Algorithm for Maximizing Modularity Q
Modularity Q is an important function for identifying community structure in
complex networks. In this paper, we prove that the modularity maximization
problem is equivalent to a nonconvex quadratic programming problem. This result
provide us a simple way to improve the efficiency of heuristic algorithms for
maximizing modularity Q. Many numerical results demonstrate that it is very
effective.Comment: 9 pages, 3 figure
Modularity clustering is force-directed layout
Two natural and widely used representations for the community structure of
networks are clusterings, which partition the vertex set into disjoint subsets,
and layouts, which assign the vertices to positions in a metric space. This
paper unifies prominent characterizations of layout quality and clustering
quality, by showing that energy models of pairwise attraction and repulsion
subsume Newman and Girvan's modularity measure. Layouts with optimal energy are
relaxations of, and are thus consistent with, clusterings with optimal
modularity, which is of practical relevance because both representations are
complementary and often used together.Comment: 9 pages, 7 figures, see http://code.google.com/p/linloglayout/ for
downloading the graph clustering and layout softwar
Vertex and edge covers with clustering properties: complexity and algorithms
We consider the concepts of a t-total vertex cover and a t-total edge cover (t≥1), which generalise the notions of a vertex cover and an edge cover, respectively. A t-total vertex (respectively edge) cover of a connected graph G is a vertex (edge) cover S of G such that each connected component of the subgraph of G induced by S has at least t vertices (edges). These definitions are motivated by combining the concepts of clustering and covering in graphs. Moreover they yield a spectrum of parameters that essentially range from a vertex cover to a connected vertex cover (in the vertex case) and from an edge cover to a spanning tree (in the edge case). For various values of t, we present NP-completeness and approximability results (both upper and lower bounds) and FTP algorithms for problems concerned with finding the minimum size of a t-total vertex cover, t-total edge cover and connected vertex cover, in particular improving on a previous FTP algorithm for the latter problem
Analysis of the autonomous system network and of overlay networks using visualization
Taking the physical Internet at the Autonomous System (AS) level
as an instance of a complex network, and Gnutella as a popular
peer-to-peer application running on top of it, we investigated
the correlation of overlay networks with their underlying
topology using visualization. We find that while overlay
networks create arbitrary topologies, they differ from randomly
generated networks, and there is a correlation with the
underlying network. In addition, we successfully validated the
applicability of our visualization technique for AS topologies
by comparing Routeviews data sets with DIMES data sets, and by
analyzing the temporal evolution in the Routeviews data sets
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