Correlation between clustering and degree in affiliation networks

We are interested in the probability that two randomly selected neighbors of a random vertex of degree (at least) $k$ are adjacent. We evaluate this probability for a power law random intersection graph, where each vertex is prescribed a collection of attributes and two vertices are adjacent whenever they share a common attribute. We show that the probability obeys the scaling $k^{-\delta}$ as $k\to+\infty$. Our results are mathematically rigorous. The parameter $0\le \delta\le 1$ is determined by the tail indices of power law random weights defining the links between vertices and attributes

Moment-based parameter estimation in binomial random intersection graph models

Binomial random intersection graphs can be used as parsimonious statistical models of large and sparse networks, with one parameter for the average degree and another for transitivity, the tendency of neighbours of a node to be connected. This paper discusses the estimation of these parameters from a single observed instance of the graph, using moment estimators based on observed degrees and frequencies of 2-stars and triangles. The observed data set is assumed to be a subgraph induced by a set of $n_0$ nodes sampled from the full set of $n$ nodes. We prove the consistency of the proposed estimators by showing that the relative estimation error is small with high probability for $n_0 \gg n^{2/3} \gg 1$. As a byproduct, our analysis confirms that the empirical transitivity coefficient of the graph is with high probability close to the theoretical clustering coefficient of the model.Comment: 15 pages, 6 figure

Parameter estimators of random intersection graphs with thinned communities

This paper studies a statistical network model generated by a large number of randomly sized overlapping communities, where any pair of nodes sharing a community is linked with probability $q$ via the community. In the special case with $q=1$ the model reduces to a random intersection graph which is known to generate high levels of transitivity also in the sparse context. The parameter $q$ adds a degree of freedom and leads to a parsimonious and analytically tractable network model with tunable density, transitivity, and degree fluctuations. We prove that the parameters of this model can be consistently estimated in the large and sparse limiting regime using moment estimators based on partially observed densities of links, 2-stars, and triangles.Comment: 15 page

Complexity of determining the irregular chromatic index of a graph

A graph G is locally irregular if adjacent vertices of G have different degrees. A k-edge colouring φ of G is locally irregular if each of the k colours of φ induces a locally irregular subgraph of G. The irregular chromatic index χ ′ irr (G) of G is the least number of colours used by a locally irregular edge colouring of G (if any). We show that determining whether χ ′ irr (G) = 2 is NP-complete, even when G is assumed to be a planar graph with maximum degree at most 6