8,088 research outputs found
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
Detecting Core-Periphery Structures by Surprise
Detecting the presence of mesoscale structures in complex networks is of
primary importance. This is especially true for financial networks, whose
structural organization deeply affects their resilience to events like default
cascades, shocks propagation, etc. Several methods have been proposed, so far,
to detect communities, i.e. groups of nodes whose connectivity is significantly
large. Communities, however do not represent the only kind of mesoscale
structures characterizing real-world networks: other examples are provided by
bow-tie structures, core-periphery structures and bipartite structures. Here we
propose a novel method to detect statistically-signifcant bimodular structures,
i.e. either bipartite or core-periphery ones. It is based on a modification of
the surprise, recently proposed for detecting communities. Our variant allows
for bimodular nodes partitions to be revealed, by letting links to be placed
either 1) within the core part and between the core and the periphery parts or
2) just between the (empty) layers of a bipartite network. From a technical
point of view, this is achieved by employing a multinomial hypergeometric
distribution instead of the traditional (binomial) hypergeometric one; as in
the latter case, this allows a p-value to be assigned to any given
(bi)partition of the nodes. To illustrate the performance of our method, we
report the results of its application to several real-world networks, including
social, economic and financial ones.Comment: 11 pages, 10 figures. Python code freely available at
https://github.com/jeroenvldj/bimodular_surpris
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