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Critical Window for Connectivity in the Configuration Model 3 CRITICAL WINDOW FOR CONNECTIVITY IN THE CONFIGURATION MODEL LORENZO FEDERICO AND REMCO VAN DER HOFSTAD Abstract. We identify the asymptotic probability of a configuration model CMn(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 con- dition 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. Keywords: configuration model, connectivity threshold, degree sequence MSC 2010: 05A16, 05C40, 60C05 1. Introduction In this paper we consider the configuration model CMn(d) on n vertices with a prescribed degree sequence d = (d1,d2, ..., dn). We investigate the condition on d for CMn(d) to be with high probability connected or disconnected in the limit as n , and we analyse the behaviour of the model in the critical window for connectivity→ ∞ (i.e., when the asymptotic probability of producing a connected graph is in the interval (0, 1)). Given a vertex v [n]= 1, 2, ..., n , we call d its degree. ∈ { } v The configuration model is constructed by assigning dv half-edges to each vertex v, after which the half-edges are paired randomly: first we pick two half-edges at arXiv:1603.03254v1 [math.PR] 10 Mar 2016 random and create an edge out of them, then we pick two half-edges at random from the set of remaining half-edges and pair them into an edge, etc. We assume the total degree v∈[n] dv to be even. The construction can give rise to self-loops and multiple edges between vertices, but these imperfections are relatively rare when n is large; see [4, 6P, 7]. We define the random variable Dn as the degree of a vertex chosen uniformly at random from the vertex set [n]. We call Ni the set of all vertices of degree i and ni its cardinality. The configuration model is known to have a phase transition for the Date: October 14, 2018. 1 2 LORENZOFEDERICOANDREMCOVANDERHOFSTAD existence of a giant component with critical point at E[Dn(Dn 1)] νn = − =1 E[Dn] (see e.g., [11] or [8]). When ν ν > 1, there is a (unique) giant component n → Cmax containing a positive proportion of the vertices, while when νn ν 1, the maximal connected component contains a vanishing proportion of the→ vertices.≤ Assuming that the second moment of Dn remains uniformly bounded, the subcritical behaviour was analysed by Janson in [5]. In this paper, we focus on the connectivity transition of the configuration model. Let us first describe the history of this problem. Wormald [13] showed that for k 3 a random k-regular graph on n vertices is with high probability k-connected as≥n (see also [3]). TomaszLuczak [10] proved that also if the graph is not → ∞ regular, but dv k for every v [n], then CMn(d) in with high probability k- connected, and found≥ the asymptotic∈ probability to have a connected graph when dv 2 and the graph is simple. ActuallyLuczak’s model was defined in a different way≥ from the configuration model and does not allow for vertices of degree 1, but the results could easily be adapted to the configuration model. We will refine his results, including the case in which there are vertices of degree 1 and we give more precise asymptotics on the size of the complement of the maximal connected component [n] C . We start by introducing some notation. \ max Notation. All limits in this paper are taken as n tends to infinity unless stated otherwise. A sequence of events (An)n≥1 happens with high probability (whp) if d P P(An) 1. For random variables (Xn)n≥1,X, we write Xn X and Xn X to de- note convergence→ in distribution and in probability, respectively.→ For real-valued→ se- quences (an)n≥1,(bn)n≥1, we write an = O(bn) if the sequence (an/bn)n≥1 is bounded; an = o(bn) if an/bn 0; an = Θ(bn) if the sequences (an/bn)n≥1 and (bn/an)n≥1 are both bounded; and→a b if a /b 1. Similarly, for sequences (X ) , (Y ) n ∼ n n n → n n≥1 n n≥1 of random variables, we write Xn = OP(Yn) if the sequence (Xn/Yn)n≥1 is tight; and P Xn = oP(Yn) if Xn/Yn 0. Moreover, Poi(λ) always denotes a Poisson distributed random variable with mean→ λ and Bin(n, p) denotes a random variable with binomial distribution with parameters n and p. 2. Main Results We start by defining the conditions for CMn(d) to be in the connectivity criti- cal window. We define the random variable Dn as the degree of a vertex chosen uniformly at random in [n]. We assume these conditions to hold throughout this paper: Condition 2.1 (Critical window for connectivity). We define a sequence CMn(d) to be in the critical window for connectivity when the following conditions are satisfied: CRITICAL WINDOW FOR CONNECTIVITY IN THE CONFIGURATION MODEL 3 (1) There exists a limiting degree variable D such that D d D; n → (2) n0 =0; (3) lim n /√n = ρ [0, ); n→∞ 1 1 ∈ ∞ (4) limn→∞ n2/n = p2 [0, 1); (5) lim E[D ]= d<∈ . n→∞ n ∞ Under these conditions, we prove our main theorem. In its statement, we write Cmax for the maximal connected component in CMn(d): Theorem 2.2 (Connectivity threshold for the configuration model). Consider CMn(d) in the critical window for connectivity as described in Condition 2.1. Then 1/2 2 d 2p2 ρ1 lim P(CMn(d) is connected) = − exp . (2.1) n→∞ d −2(d 2p ) − 2 Moreover, n C d X, (2.2) −| max| → where X = k k(Ck + Lk), and (Ck, Lk) k≥1 are independent random variables such that P ρ2(2p)k−2 (2p )k L =d Poi 1 2 , C =d Poi 2 . k 2dk−1 k 2kdk Finally, 2 ρ1(2d p2) p2 lim E[n Cmax ]= − + . (2.3) n→∞ −| | 2(d p )2 d 2p − 2 − 2 The convergence in distribution of n Cmax to a proper random variable with finite mean is a stronger result than proved−| byLuczak | [10], who instead proved that C P | max| 1. (2.4) n → Our improvement is achieved by an application of the multivariate method of mo- ments, as well as a careful estimate of the probability that there exists v [n] that is not part of C . We next investigate the boundary cases: ∈ | max| Remark 2.3 (Boundary cases). Our proof also applies to the boundary cases where ρ = ,p =0 or d = . When d< , we obtain 1 ∞ 2 ∞ ∞ 0 when ρ1 = , P(CMn(d) is connected) ∞ (2.5) → (1 when ρ1,p2 =0. When d = , instead ∞ 2 n1 lim P(CMn(d) is connected) = lim exp . (2.6) n→∞ n→∞ − 2ℓn n o 4 LORENZOFEDERICOANDREMCOVANDERHOFSTAD We next investigate how many connected graphs there are with prescribed degrees in the connectivity window defined in Condition 2.1. Assuming also that D has finite second moment the configuration model is simple with non-vanishing probability. Under this additional condition we can prove the following results: Theorem 2.4 (Connectivity conditioned on simplicity and number of connected simple graphs). Consider CMn(d) in the connectivity critical window defined in Condition 2.1. If E[D (D 1)] lim n n − = ν < , n→∞ E[Dn] ∞ then lim P(CMn(d) is connected CMn(d) is simple) n→∞ | d 2p 1/2 ρ2 p2 + dp (2.7) = − 2 exp 1 + 2 2 . d −2(d 2p ) d2 − 2 N C d d Let n ( ) be the number of connected simple graphs with degree distribution . Then (ℓ 1)!! d 2p 1/2 N C (d)= n − − 2 n d ! d i∈[n] i (2.8) ν ν2 ρ2 p2 + dp Q exp 1 + 2 2 (1 + o(1)), × − 2 − 4 − 2(d 2p ) d2 − 2 where ℓn = i∈[n] di denotes the total degree. With theseP results, the connectivity critical window is fully explored. Indeed, we have determined the asymptotic probability for the configuration model to produce a connected graph for all possible choices of the limiting degree distribution under finite mean assumption. What remains is to find the asymptotic of the number of connected simple graphs with degree distribution d when it is above the connectivity 1/2 critical window (i.e., when n1 n ). In this case we should analyse how fast the probability to produce a connected≫ graph vanishes, which is a hard problem. It is also worth noticing that the size of the largest component is very sensitive to the precise way how n2/n 1 (recall that we assume that p2 < 1 in Condition 2.1). We define C (v) as the connected→ component of a uniformly chosen vertex. Indeed, when n2 = n, it is not hard to see that C d C (v) d | max| S; | | T, (2.9) n → n → where S, T are proper random variables that satisfy the relation S =d T [(1 T )S]. P ∨ P − Instead, Cmax /n 0 when n2 = n n1, with n1 , while Cmax /n 1 when n = n |n , with| →n . The latter− two statements→∞ can be proved| | by→ relating it 2 − 4 4 →∞ CRITICAL WINDOW FOR CONNECTIVITY IN THE CONFIGURATION MODEL 5 ′ to the case where n2 = n.
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