30,310 research outputs found
Sequence mixed graphs
A mixed graph can be seen as a type of digraph containing some edges (or two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and literated line digraphs. These structures are proven to be useful in the problem of constructing dense graphs or digraphs, and this is related to the degree/diameter problem. Thus, our generalized approach gives rise to graphs that have also good ratio order/diameter. Moreover, we propose a general method for obtaining a sequence mixed diagraph by identifying some vertices of certain iterated line digraph. As a consequence, some results about distance-related parameters (mainly, the diameter and the average distance) of sequence mixed graphs are presented.Postprint (author's final draft
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
Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution
Even though power-law or close-to-power-law degree distributions are
ubiquitously observed in a great variety of large real networks, the
mathematically satisfactory treatment of random power-law graphs satisfying
basic statistical requirements of realism is still lacking. These requirements
are: sparsity, exchangeability, projectivity, and unbiasedness. The last
requirement states that entropy of the graph ensemble must be maximized under
the degree distribution constraints. Here we prove that the hypersoft
configuration model (HSCM), belonging to the class of random graphs with latent
hyperparameters, also known as inhomogeneous random graphs or -random
graphs, is an ensemble of random power-law graphs that are sparse, unbiased,
and either exchangeable or projective. The proof of their unbiasedness relies
on generalized graphons, and on mapping the problem of maximization of the
normalized Gibbs entropy of a random graph ensemble, to the graphon entropy
maximization problem, showing that the two entropies converge to each other in
the large-graph limit
Sparse Maximum-Entropy Random Graphs with a Given Power-Law Degree Distribution
Even though power-law or close-to-power-law degree distributions are
ubiquitously observed in a great variety of large real networks, the
mathematically satisfactory treatment of random power-law graphs satisfying
basic statistical requirements of realism is still lacking. These requirements
are: sparsity, exchangeability, projectivity, and unbiasedness. The last
requirement states that entropy of the graph ensemble must be maximized under
the degree distribution constraints. Here we prove that the hypersoft
configuration model (HSCM), belonging to the class of random graphs with latent
hyperparameters, also known as inhomogeneous random graphs or -random
graphs, is an ensemble of random power-law graphs that are sparse, unbiased,
and either exchangeable or projective. The proof of their unbiasedness relies
on generalized graphons, and on mapping the problem of maximization of the
normalized Gibbs entropy of a random graph ensemble, to the graphon entropy
maximization problem, showing that the two entropies converge to each other in
the large-graph limit
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