11,733 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
Model selection and hypothesis testing for large-scale network models with overlapping groups
The effort to understand network systems in increasing detail has resulted in
a diversity of methods designed to extract their large-scale structure from
data. Unfortunately, many of these methods yield diverging descriptions of the
same network, making both the comparison and understanding of their results a
difficult challenge. A possible solution to this outstanding issue is to shift
the focus away from ad hoc methods and move towards more principled approaches
based on statistical inference of generative models. As a result, we face
instead the more well-defined task of selecting between competing generative
processes, which can be done under a unified probabilistic framework. Here, we
consider the comparison between a variety of generative models including
features such as degree correction, where nodes with arbitrary degrees can
belong to the same group, and community overlap, where nodes are allowed to
belong to more than one group. Because such model variants possess an
increasing number of parameters, they become prone to overfitting. In this
work, we present a method of model selection based on the minimum description
length criterion and posterior odds ratios that is capable of fully accounting
for the increased degrees of freedom of the larger models, and selects the best
one according to the statistical evidence available in the data. In applying
this method to many empirical unweighted networks from different fields, we
observe that community overlap is very often not supported by statistical
evidence and is selected as a better model only for a minority of them. On the
other hand, we find that degree correction tends to be almost universally
favored by the available data, implying that intrinsic node proprieties (as
opposed to group properties) are often an essential ingredient of network
formation.Comment: 20 pages,7 figures, 1 tabl
A framework for the construction of generative models for mesoscale structure in multilayer networks
Multilayer networks allow one to represent diverse and coupled connectivity patterns—such as time-dependence, multiple subsystems, or both—that arise in many applications and which are difficult or awkward to incorporate into standard network representations. In the study of multilayer networks, it is important to investigate mesoscale (i.e., intermediate-scale) structures, such as dense sets of nodes known as communities, to discover network features that are not apparent at the microscale or the macroscale. The ill-defined nature of mesoscale structure and its ubiquity in empirical networks make it crucial to develop generative models that can produce the features that one encounters in empirical networks. Key purposes of such models include generating synthetic networks with empirical properties of interest, benchmarking mesoscale-detection methods and algorithms, and inferring structure in empirical multilayer networks. In this paper, we introduce a framework for the construction of generative models for mesoscale structures in multilayer networks. Our framework provides a standardized set of generative models, together with an associated set of principles from which they are derived, for studies of mesoscale structures in multilayer networks. It unifies and generalizes many existing models for mesoscale structures in fully ordered (e.g., temporal) and unordered (e.g., multiplex) multilayer networks. One can also use it to construct generative models for mesoscale structures in partially ordered multilayer networks (e.g., networks that are both temporal and multiplex). Our framework has the ability to produce many features of empirical multilayer networks, and it explicitly incorporates a user-specified dependency structure between layers. We discuss the parameters and properties of our framework, and we illustrate examples of its use with benchmark models for community-detection methods and algorithms in multilayer networks
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