136 research outputs found
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
Projective, Sparse, and Learnable Latent Position Network Models
When modeling network data using a latent position model, it is typical to
assume that the nodes' positions are independently and identically distributed.
However, this assumption implies the average node degree grows linearly with
the number of nodes, which is inappropriate when the graph is thought to be
sparse. We propose an alternative assumption---that the latent positions are
generated according to a Poisson point process---and show that it is compatible
with various levels of sparsity. Unlike other notions of sparse latent position
models in the literature, our framework also defines a projective sequence of
probability models, thus ensuring consistency of statistical inference across
networks of different sizes. We establish conditions for consistent estimation
of the latent positions, and compare our results to existing frameworks for
modeling sparse networks.Comment: 51 pages, 2 figure
On sparsity, power-law and clustering properties of graphex processes
This paper investigates properties of the class of graphs based on
exchangeable point processes. We provide asymptotic expressions for the number
of edges, number of nodes and degree distributions, identifying four regimes:
(i) a dense regime, (ii) a sparse almost dense regime, (iii) a sparse regime
with power-law behaviour, and (iv) an almost extremely sparse regime. We show
that under mild assumptions, both the global and local clustering coefficients
converge to constants which may or may not be the same. We also derive a
central limit theorem for the number of nodes. Finally, we propose a class of
models within this framework where one can separately control the latent
structure and the global sparsity/power-law properties of the graph
- …