Diameter, connectivity, and phase transition of the uniform random intersection graph

Abstract

a b s t r a c tWe study properties of the uniform random intersection graph model G (n, m, d). We find asymptotic estimates on the diameter of the largest connected component of the graph near the phase transition and connectivity thresholds. Moreover we manage to prove an asymptotically tight bound for the connectivity and phase transition thresholds for all possible ranges of d, which has not been obtained before. The main motivation of our research is the usage of the random intersection graph model in the studies of wireless sensor networks.

Hyperbolicity, Degeneracy, and Expansion of Random Intersection Graphs

“…On the other hand, we showed that the metric structure of random intersection graphs is not tree-like for any value of α: the hyperbolicity (and treelength) grows at least logarithmically in n. While we only determine a lower bound for the hyperbolicity, we believe this bound to be the correct order of magnitude since the diameter (a natural upper bound for the hyperbolicity) of a similar model of random intersection graphs was shown to be O(log n) [40]. Our experimental results support this hypothesis: the ratio of hyperbolicity to diameter seems to converge to a constant.…”

“…also upper bounds). This would require that the the diameter of connected components is also logarithmic in n, which has been shown for a similar model [40].…”

Recent Progress in Complex Network Analysis: Models of Random Intersection Graphs

“…The phase transition in the component size of an active random intersection graph has been studied in Behrisch (2007), Bloznelis (2010b), Bloznelis et al (2009), Godehardt et al (2007), and Rybarczyk (2011a). The effect of the clustering property on the phase transition in the component size and on the epidemic spread has been studied in Lagerås and Lindholm (2008), Bloznelis (2010c), Britton et al (2008) respectively.…”

“…Also note that G 1 .n; m; p/ is G.n; m; p/ as defined in Karoński et al (1999). The active intersection graph G s .n; m; ı x /, where ı x is the probability distribution putting mass 1 on a positive integer x (i.e., all random sets are of the same, non-random size x) has attracted particular attention in the literature (Blackburn and Gerke 2009;Bloznelis and Łuczak 2013;Eschenauer and Gligor 2002;Godehardt and Jaworski 2003;Nikoletseas et al 2011;Rybarczyk 2011a;Yagan and Makowski 2009) as it provides a convenient model of a secure wireless network. It is called the uniform random intersection graph and denoted G s .n; m; x/.…”

Zero-One Law for Connectivity in Superposition of Random Key Graphs on Random Geometric Graphs

“…In [6], Rybarczyk gave asymptotic tight bounds for the thresholds of the connectivity, phase transition, and diameter of the largest connected component in RKGs for all ranges of .…”