Alignment strength and correlation for graphs

When two graphs have a correlated Bernoulli distribution, we prove that the alignment strength of their natural bijection strongly converges to a novel measure of graph correlation ϱT that neatly combines intergraph with intragraph distribution parameters. Within broad families of the random graph parameter settings, we illustrate that exact graph matching runtime and also matchability are both functions of ϱT, with thresholding behavior starkly illustrated in matchability

Driving interconnected networks to supercriticality

Networks in the real world do not exist as isolated entities, but they are often part of more complicated structures composed of many interconnected network layers. Recent studies have shown that such mutual dependence makes real networked systems potentially exposed to atypical structural and dynamical behaviors, and thus there is a urgent necessity to better understand the mechanisms at the basis of these anomalies. Previous research has mainly focused on the emergence of atypical properties in relation with the moments of the intra- and inter-layer degree distributions. In this paper, we show that an additional ingredient plays a fundamental role for the possible scenario that an interconnected network can face: the correlation between intra- and inter-layer degrees. For sufficiently high amounts of correlation, an interconnected network can be tuned, by varying the moments of the intra- and inter-layer degree distributions, in distinct topological and dynamical regimes. When instead the correlation between intra- and inter-layer degrees is lower than a critical value, the system enters in a supercricritical regime where dynamical and topological phases are not longer distinguishable.Comment: 7 pages, 4 figures + Supplementary Informatio