1,469 research outputs found
Mixing patterns and community structure in networks
Common experience suggests that many networks might possess community
structure - division of vertices into groups, with a higher density of edges
within groups than between them. Here we describe a new computer algorithm that
detects structure of this kind. We apply the algorithm to a number of
real-world networks and show that they do indeed possess non-trivial community
structure. We suggest a possible explanation for this structure in the
mechanism of assortative mixing, which is the preferential association of
network vertices with others that are like them in some way. We show by
simulation that this mechanism can indeed account for community structure. We
also look in detail at one particular example of assortative mixing, namely
mixing by vertex degree, in which vertices with similar degree prefer to be
connected to one another. We propose a measure for mixing of this type which we
apply to a variety of networks, and also discuss the implications for network
structure and the formation of a giant component in assortatively mixed
networks.Comment: 21 pages, 9 postscript figures, 2 table
Optimal design, robustness, and risk aversion
Highly optimized tolerance is a model of optimization in engineered systems,
which gives rise to power-law distributions of failure events in such systems.
The archetypal example is the highly optimized forest fire model. Here we give
an analytic solution for this model which explains the origin of the power
laws. We also generalize the model to incorporate risk aversion, which results
in truncation of the tails of the power law so that the probability of
disastrously large events is dramatically lowered, giving the system more
robustness.Comment: 11 pages, 2 figure
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
Vulnerability and Protection of Critical Infrastructures
Critical infrastructure networks are a key ingredient of modern society. We
discuss a general method to spot the critical components of a critical
infrastructure network, i.e. the nodes and the links fundamental to the perfect
functioning of the network. Such nodes, and not the most connected ones, are
the targets to protect from terrorist attacks. The method, used as an
improvement analysis, can also help to better shape a planned expansion of the
network.Comment: 4 pages, 1 figure, 3 table
Maps of random walks on complex networks reveal community structure
To comprehend the multipartite organization of large-scale biological and
social systems, we introduce a new information theoretic approach that reveals
community structure in weighted and directed networks. The method decomposes a
network into modules by optimally compressing a description of information
flows on the network. The result is a map that both simplifies and highlights
the regularities in the structure and their relationships. We illustrate the
method by making a map of scientific communication as captured in the citation
patterns of more than 6000 journals. We discover a multicentric organization
with fields that vary dramatically in size and degree of integration into the
network of science. Along the backbone of the network -- including physics,
chemistry, molecular biology, and medicine -- information flows
bidirectionally, but the map reveals a directional pattern of citation from the
applied fields to the basic sciences.Comment: 7 pages and 4 figures plus supporting material. For associated source
code, see http://www.tp.umu.se/~rosvall
Multiscale Dynamics in Communities of Phase Oscillators
We investigate the dynamics of systems of many coupled phase oscillators with
het- erogeneous frequencies. We suppose that the oscillators occur in M groups.
Each oscillator is connected to other oscillators in its group with
"attractive" coupling, such that the coupling promotes synchronization within
the group. The coupling between oscillators in different groups is "repulsive";
i.e., their oscillation phases repel. To address this problem, we reduce the
governing equations to a lower-dimensional form via the ansatz of Ott and
Antonsen . We first consider the symmetric case where all group parameters are
the same, and the attractive and repulsive coupling are also the same for each
of the M groups. We find a manifold L of neutrally stable equilibria, and we
show that all other equilibria are unstable. For M \geq 3, L has dimension M -
2, and for M = 2 it has dimension 1. To address the general asymmetric case, we
then introduce small deviations from symmetry in the group and coupling param-
eters. Doing a slow/fast timescale analysis, we obtain slow time evolution
equations for the motion of the M groups on the manifold L. We use these
equations to study the dynamics of the groups and compare the results with
numerical simulations.Comment: 29 pages, 6 figure
Mixture models and exploratory analysis in networks
Networks are widely used in the biological, physical, and social sciences as
a concise mathematical representation of the topology of systems of interacting
components. Understanding the structure of these networks is one of the
outstanding challenges in the study of complex systems. Here we describe a
general technique for detecting structural features in large-scale network data
which works by dividing the nodes of a network into classes such that the
members of each class have similar patterns of connection to other nodes. Using
the machinery of probabilistic mixture models and the expectation-maximization
algorithm, we show that it is possible to detect, without prior knowledge of
what we are looking for, a very broad range of types of structure in networks.
We give a number of examples demonstrating how the method can be used to shed
light on the properties of real-world networks, including social and
information networks.Comment: 8 pages, 4 figures, two new examples in this version plus minor
correction
Resolution limit in community detection
Detecting community structure is fundamental to clarify the link between
structure and function in complex networks and is used for practical
applications in many disciplines. A successful method relies on the
optimization of a quantity called modularity [Newman and Girvan, Phys. Rev. E
69, 026113 (2004)], which is a quality index of a partition of a network into
communities. We find that modularity optimization may fail to identify modules
smaller than a scale which depends on the total number L of links of the
network and on the degree of interconnectedness of the modules, even in cases
where modules are unambiguously defined. The probability that a module conceals
well-defined substructures is the highest if the number of links internal to
the module is of the order of \sqrt{2L} or smaller. We discuss the practical
consequences of this result by analyzing partitions obtained through modularity
optimization in artificial and real networks.Comment: 8 pages, 3 figures. Clarification of definition of community in
Section II + minor revision
Evidential Communities for Complex Networks
Community detection is of great importance for understand-ing graph structure
in social networks. The communities in real-world networks are often
overlapped, i.e. some nodes may be a member of multiple clusters. How to
uncover the overlapping communities/clusters in a complex network is a general
problem in data mining of network data sets. In this paper, a novel algorithm
to identify overlapping communi-ties in complex networks by a combination of an
evidential modularity function, a spectral mapping method and evidential
c-means clustering is devised. Experimental results indicate that this
detection approach can take advantage of the theory of belief functions, and
preforms good both at detecting community structure and determining the
appropri-ate number of clusters. Moreover, the credal partition obtained by the
proposed method could give us a deeper insight into the graph structure
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