4,436 research outputs found
Community detection in networks via nonlinear modularity eigenvectors
Revealing a community structure in a network or dataset is a central problem
arising in many scientific areas. The modularity function is an established
measure quantifying the quality of a community, being identified as a set of
nodes having high modularity. In our terminology, a set of nodes with positive
modularity is called a \textit{module} and a set that maximizes is thus
called \textit{leading module}. Finding a leading module in a network is an
important task, however the dimension of real-world problems makes the
maximization of unfeasible. This poses the need of approximation techniques
which are typically based on a linear relaxation of , induced by the
spectrum of the modularity matrix . In this work we propose a nonlinear
relaxation which is instead based on the spectrum of a nonlinear modularity
operator . We show that extremal eigenvalues of
provide an exact relaxation of the modularity measure , however at the price
of being more challenging to be computed than those of . Thus we extend the
work made on nonlinear Laplacians, by proposing a computational scheme, named
\textit{generalized RatioDCA}, to address such extremal eigenvalues. We show
monotonic ascent and convergence of the method. We finally apply the new method
to several synthetic and real-world data sets, showing both effectiveness of
the model and performance of the method
A framework for community detection in heterogeneous multi-relational networks
There has been a surge of interest in community detection in homogeneous
single-relational networks which contain only one type of nodes and edges.
However, many real-world systems are naturally described as heterogeneous
multi-relational networks which contain multiple types of nodes and edges. In
this paper, we propose a new method for detecting communities in such networks.
Our method is based on optimizing the composite modularity, which is a new
modularity proposed for evaluating partitions of a heterogeneous
multi-relational network into communities. Our method is parameter-free,
scalable, and suitable for various networks with general structure. We
demonstrate that it outperforms the state-of-the-art techniques in detecting
pre-planted communities in synthetic networks. Applied to a real-world Digg
network, it successfully detects meaningful communities.Comment: 27 pages, 10 figure
Generalized Markov stability of network communities
We address the problem of community detection in networks by introducing a
general definition of Markov stability, based on the difference between the
probability fluxes of a Markov chain on the network at different time scales.
The specific implementation of the quality function and the resulting optimal
community structure thus become dependent both on the type of Markov process
and on the specific Markov times considered. For instance, if we use a natural
Markov chain dynamics and discount its stationary distribution -- that is, we
take as reference process the dynamics at infinite time -- we obtain the
standard formulation of the Markov stability. Notably, the possibility to use
finite-time transition probabilities to define the reference process naturally
allows detecting communities at different resolutions, without the need to
consider a continuous-time Markov chain in the small time limit. The main
advantage of our general formulation of Markov stability based on dynamical
flows is that we work with lumped Markov chains on network partitions, having
the same stationary distribution of the original process. In this way the form
of the quality function becomes invariant under partitioning, leading to a
self-consistent definition of community structures at different aggregation
scales
Post-processing partitions to identify domains of modularity optimization
We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP)
algorithm to prune and prioritize different network community structures
identified across multiple runs of possibly various computational heuristics.
Given a set of partitions, CHAMP identifies the domain of modularity
optimization for each partition ---i.e., the parameter-space domain where it
has the largest modularity relative to the input set---discarding partitions
with empty domains to obtain the subset of partitions that are "admissible"
candidate community structures that remain potentially optimal over indicated
parameter domains. Importantly, CHAMP can be used for multi-dimensional
parameter spaces, such as those for multilayer networks where one includes a
resolution parameter and interlayer coupling. Using the results from CHAMP, a
user can more appropriately select robust community structures by observing the
sizes of domains of optimization and the pairwise comparisons between
partitions in the admissible subset. We demonstrate the utility of CHAMP with
several example networks. In these examples, CHAMP focuses attention onto
pruned subsets of admissible partitions that are 20-to-1785 times smaller than
the sets of unique partitions obtained by community detection heuristics that
were input into CHAMP.Comment: http://www.mdpi.com/1999-4893/10/3/9
Community detection in temporal multilayer networks, with an application to correlation networks
Networks are a convenient way to represent complex systems of interacting
entities. Many networks contain "communities" of nodes that are more densely
connected to each other than to nodes in the rest of the network. In this
paper, we investigate the detection of communities in temporal networks
represented as multilayer networks. As a focal example, we study time-dependent
financial-asset correlation networks. We first argue that the use of the
"modularity" quality function---which is defined by comparing edge weights in
an observed network to expected edge weights in a "null network"---is
application-dependent. We differentiate between "null networks" and "null
models" in our discussion of modularity maximization, and we highlight that the
same null network can correspond to different null models. We then investigate
a multilayer modularity-maximization problem to identify communities in
temporal networks. Our multilayer analysis only depends on the form of the
maximization problem and not on the specific quality function that one chooses.
We introduce a diagnostic to measure \emph{persistence} of community structure
in a multilayer network partition. We prove several results that describe how
the multilayer maximization problem measures a trade-off between static
community structure within layers and larger values of persistence across
layers. We also discuss some computational issues that the popular "Louvain"
heuristic faces with temporal multilayer networks and suggest ways to mitigate
them.Comment: 42 pages, many figures, final accepted version before typesettin
Router-level community structure of the Internet Autonomous Systems
The Internet is composed of routing devices connected between them and
organized into independent administrative entities: the Autonomous Systems. The
existence of different types of Autonomous Systems (like large connectivity
providers, Internet Service Providers or universities) together with
geographical and economical constraints, turns the Internet into a complex
modular and hierarchical network. This organization is reflected in many
properties of the Internet topology, like its high degree of clustering and its
robustness.
In this work, we study the modular structure of the Internet router-level
graph in order to assess to what extent the Autonomous Systems satisfy some of
the known notions of community structure. We show that the modular structure of
the Internet is much richer than what can be captured by the current community
detection methods, which are severely affected by resolution limits and by the
heterogeneity of the Autonomous Systems. Here we overcome this issue by using a
multiresolution detection algorithm combined with a small sample of nodes. We
also discuss recent work on community structure in the light of our results
Enhancing community detection using a network weighting strategy
A community within a network is a group of vertices densely connected to each
other but less connected to the vertices outside. The problem of detecting
communities in large networks plays a key role in a wide range of research
areas, e.g. Computer Science, Biology and Sociology. Most of the existing
algorithms to find communities count on the topological features of the network
and often do not scale well on large, real-life instances.
In this article we propose a strategy to enhance existing community detection
algorithms by adding a pre-processing step in which edges are weighted
according to their centrality w.r.t. the network topology. In our approach, the
centrality of an edge reflects its contribute to making arbitrary graph
tranversals, i.e., spreading messages over the network, as short as possible.
Our strategy is able to effectively complements information about network
topology and it can be used as an additional tool to enhance community
detection. The computation of edge centralities is carried out by performing
multiple random walks of bounded length on the network. Our method makes the
computation of edge centralities feasible also on large-scale networks. It has
been tested in conjunction with three state-of-the-art community detection
algorithms, namely the Louvain method, COPRA and OSLOM. Experimental results
show that our method raises the accuracy of existing algorithms both on
synthetic and real-life datasets.Comment: 28 pages, 2 figure
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