One, Two and Three Times log n/n for Paths in a Complete Graph with Random Weights
Combinatorics, Probability and Computing
The diameter of sparse random graphs
Random Structures & Algorithms
On tree census and the giant component in sparse random graphs
Random Structures & Algorithms
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In this paper we explore first passage percolation (FPP) on the ErdÅ聭s-Rényi random graph Gn(pn), where we assign independent random weights, having an exponential distribution with rate 1, to the edges. In the sparse regime, i.e., when npn â聠聮 λ 1, we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to (λ log n)/(λ-1). Furthermore, we prove that the minimal weight centred by (log n)/(λ-1) converges in distribution. We also investigate the dense regime, where npn â聠聮 ∞. We find that although the base graph is ultra-small (meaning that graph distances between uniformly chosen vertices are o(log n)), attaching random edge weights changes the geometry of the network completely. Indeed, the hopcount Hn satisfies the universality property that whatever the value of pn, Hn/log n â聠聮 1 in probability and, more precisely, (Hn-βn log n)/, where βn = λn/(λn-1), has a limiting standard normal distribution. The constant βn can be replaced by 1 precisely when λn â聣芦 , a case that has appeared in the literature (under stronger conditions on λn) in [4, 13]. We also find lower bounds for the maximum, over all pairs of vertices, of the optimal weight and hopcount. This paper continues the investigation of FPP initiated in [4] and [5]. Compared to the setting on the configuration model studied in [5], the proofs presented here are much simpler due to a direct relation between FPP on the ErdÅ聭s-Rényi random graph and thinned continuous-time branching processes.