Universality for first passage percolation on sparse random graphs

The Newman-Watts model is given by taking a cycle graph of n vertices and then adding each possible edge (i, j), |i − j| = 1 mod n with probability ρ/n for some ρ > 0 constant. In this paper we add i.i.d. exponential edge weights to this graph, and investigate typical distances in the corresponding random metric space given by the least weight paths between vertices. We show that typical distances grow as 1 λ log n for a λ > 0 and determine the distribution of smaller order terms in terms of limits of branching process random variables. We prove that the number of edges along the shortest weight path follows a Central Limit Theorem, and show that in a corresponding epidemic spread model the fraction of infected vertices follows a deterministic curve with a random shift.

Section: Remark 14supporting

confidence: 53%

Section: Universality Classmentioning

confidence: 99%

See 1 more Smart Citation

First Passage Percolation on the Newman–Watts Small World Model

Komjáthy

Vadon

2016

“…FPP on the configuration model with finite mean degrees for exponential edge-weights is treated in [11], with finite variance degrees (i.e., power law exponent at least 3) and arbitrary edge-weight distributions in [13], and for infinite variance degrees (power-law exponent ∈ (2, 3)) for a class of edge-weights [8].…”

Section: Introductionmentioning

confidence: 99%

Weighted Distances in Scale-Free Configuration Models

Adriaans

Komjáthy

2018

In this paper we study first-passage percolation in the configuration model with empirical degree distribution that follows a power-law with exponent τ ∈ (2, 3). We assign independent and identically distributed (i.i.d.) weights to the edges of the graph. We investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical distances. When the underlying age-dependent branching process approximating the local neighborhoods of vertices is found to produce infinitely many individuals in finite time-called explosive branching process-Baroni, Hofstad and the second author showed in Baroni et al. (J Appl Probab 54(1):146-164, 2017) that typical distances converge in distribution to a bounded random variable. The order of magnitude of typical distances remained open for the τ ∈ (2, 3) case when the underlying branching process is not explosive. We close this gap by determining the first order of magnitude of typical distances in this regime for arbitrary, not necessary continuous edge-weight distributions that produce a non-explosive age-dependent branching process with infinite mean power-law offspring distributions. This sequence tends to infinity with the amount of vertices, and, by choosing an appropriate weight distribution, can be tuned to be any growing function that is O(log log n), where n is the number of vertices in the graph. We show that the result remains valid for the the erased configuration model as well, where we delete loops and any second and further edges between two vertices.

“…7 Actually the theorem by Davies in [16] is somewhat more general, since it allows for a slightly larger class of slowly-varying functions than the criterion in (1.5) Nevertheless, the theorem applies for the case described in (1.7)and (1.5) 8 The only reason to denote this quantity by n is to be consistent with the notation in [6].…”

Section: Coupling the Initial Stages Of Bfs To Branching Processesmentioning

confidence: 99%

When is a Scale-Free Graph Ultra-Small?

Hofstad

2017

Self Cite

In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent τ ∈ (2, 3), up to value n β n for some β n (log n) −γ and γ ∈ (0, 1). This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law empirical degree distributions where the (possibly exponential) truncation happens at n β n . These examples are commonly observed in many real-life networks. We show that the graph distance between two uniformly chosen vertices centers around 2 log log(n β n )/| log(τ − 2)| + 1/(β n (3 − τ )), with tight fluctuations. Thus, the graph is an ultrasmall world whenever 1/β n = o(log log n). We determine the distribution of the fluctuations around this value, in particular we prove these form a sequence of tight random variables with distributions that show log log-periodicity, and as a result it is non-converging. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order n f n β n , where f n ∈ (0, 1) is a random variable that oscillates with n. We decompose shortest paths into three segments, two 'end-segments' starting at each of the two uniformly chosen vertices, and a middle segment. The two end-segments of any shortest path have length log log(n β n )/| log(τ − 2)|+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length 1/(β n (3 − τ ))+tight, and it contains only vertices with degree at least of order n (1− f n )β n , thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least n β n , and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees.