Critical Window for Connectivity in the Configuration Model

Abstract

We identify the asymptotic probability of a configuration model CM n (d) to produce a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which the asymptotic number of simple connected graphs with a prescribed degree sequence follows.

Uniform random colored complexes

“…Crucially, is a very special case of the (directed) configuration model, and general results by Cooper and Frieze , combined with the recent ones of Federico and van der Hofstad , allow us to obtain the following asymptotic results for the connected components of :…”

“…To obtain the number of connected components of , we thus have to determine its number of cycles. To do so, we adapt a result from on the undirected configuration model, to the directed case:…”

Mixing times of random walks on dynamic configuration models

“…4. Theorem 1.7 can be compared to the analogous result for the static configuration model only when P n (D n ≥ 3) = 1 for all n ∈ N. In fact, only under the latter condition does the probability of having a connected graph tend to one (see Luczak [21], Federico and van der Hofstad [12]). If (R3) holds, then on the dynamic graph the walk mixes on the whole of H, while on the static graph it mixes on the subset of H corresponding to the giant component.…”

Tight Fluctuations of Weight-Distances in Random Graphs with Infinite-Variance Degrees

“…We assume that F (1) = 0, so that min i∈[n] d i ≥ 2 a.s. The condition on the minimal degree guarantees that almost all the vertices of the graph lie in the same connected component (see [3, Proposition 2.1]), or, equivalently, the giant component has size n(1 − o(1)) (see [8,Theorem 2.2]). All edges are equipped with i.i.d.…”