When is a Scale-Free Graph Ultra-Small?

Abstract

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.

Diameter in ultra‐small scale‐free random graphs

“…power‐law degrees with τ ∈ (2,3), the difference Δ n between the typical distance and the asymptotic behavior does not converge in distribution, even though it is tight (ie, for every ϵ > 0 there is M < ∞ such that for all n ∈ ℕ). These results have been significantly improved in van der Hofstad and Komjáthym .…”

“…They prove that for τ ∈ (2, 3) typical distances are of order log log n, while for τ > 3 is of order log n, and it presents the explicit constants of asymptotic growth. Van der Hofstad, Hooghiemstra and Znamensky [HHZ07b] shows for τ > 2 and when vertices of degree 1 or 2 are present, that with high probability the diameter of CM n is bounded from below by a constant times log n, while when τ ∈ (2, 3) and the minimal degree is 3, the diameter is bounded from above by a constant times log log n. In [HK17], Van der Hofstad and Komjáthy investigate typical distances for configuration models and τ ∈ (2, 3) in great generality, extending the results in [HHZ07b] beyond the setting of i.i.d. degrees.…”

Linking the mixing times of random walks on static and dynamic random graphs

“…We expect Theorem 1.5(B) to fail without Condition 3.6. Namely, when the degree distribution has infinite variance, graph distances are of smaller order than log n, and in fact are of order log log n under an appropriate powerlaw assumption on the empirical degree distribution [15,25,27,28]. In the latter setting, for r n = c log n we expect near-to-global rewiring to behave similarly as global-to-global rewiring.…”