2,103 research outputs found
Convergence of rank based degree-degree correlations in random directed networks
We introduce, and analyze, three measures for degree-degree dependencies,
also called degree assortativity, in directed random graphs, based on
Spearman's rho and Kendall's tau. We proof statistical consistency of these
measures in general random graphs and show that the directed configuration
model can serve as a null model for our degree-degree dependency measures.
Based on these results we argue that the measures we introduce should be
preferred over Pearson's correlation coefficients, when studying degree-degree
dependencies, since the latter has several issues in the case of large networks
with scale-free degree distributions
Phase transitions for scaling of structural correlations in directed networks
Analysis of degree-degree dependencies in complex networks, and their impact
on processes on networks requires null models, i.e. models that generate
uncorrelated scale-free networks. Most models to date however show structural
negative dependencies, caused by finite size effects. We analyze the behavior
of these structural negative degree-degree dependencies, using rank based
correlation measures, in the directed Erased Configuration Model. We obtain
expressions for the scaling as a function of the exponents of the
distributions. Moreover, we show that this scaling undergoes a phase
transition, where one region exhibits scaling related to the natural cut-off of
the network while another region has scaling similar to the structural cut-off
for uncorrelated networks. By establishing the speed of convergence of these
structural dependencies we are able to asses statistical significance of
degree-degree dependencies on finite complex networks when compared to networks
generated by the directed Erased Configuration Model
Limit theorems for assortativity and clustering in null models for scale-free networks
An important problem in modeling networks is how to generate a randomly
sampled graph with given degrees. A popular model is the configuration model, a
network with assigned degrees and random connections. The erased configuration
model is obtained when self-loops and multiple edges in the configuration model
are removed. We prove an upper bound for the number of such erased edges for
regularly-varying degree distributions with infinite variance, and use this
result to prove central limit theorems for Pearson's correlation coefficient
and the clustering coefficient in the erased configuration model. Our results
explain the structural correlations in the erased configuration model and show
that removing edges leads to different scaling of the clustering coefficient.
We then prove that for the rank-1 inhomogeneous random graph, another null
model that creates scale-free simple networks, the results for Pearson's
correlation coefficient as well as for the clustering coefficient are similar
to the results for the erased configuration model
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
Average nearest neighbor degrees in scale-free networks
The average nearest neighbor degree (ANND) of a node of degree is widely
used to measure dependencies between degrees of neighbor nodes in a network. We
formally analyze ANND in undirected random graphs when the graph size tends to
infinity. The limiting behavior of ANND depends on the variance of the degree
distribution. When the variance is finite, the ANND has a deterministic limit.
When the variance is infinite, the ANND scales with the size of the graph, and
we prove a corresponding central limit theorem in the configuration model (CM,
a network with random connections). As ANND proved uninformative in the
infinite variance scenario, we propose an alternative measure, the average
nearest neighbor rank (ANNR). We prove that ANNR converges to a deterministic
function whenever the degree distribution has finite mean. We then consider the
erased configuration model (ECM), where self-loops and multiple edges are
removed, and investigate the well-known `structural negative correlations', or
`finite-size effects', that arise in simple graphs, such as ECM, because large
nodes can only have a limited number of large neighbors. Interestingly, we
prove that for any fixed , ANNR in ECM converges to the same limit as in CM.
However, numerical experiments show that finite-size effects occur when
scales with
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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