The Giant in Random Graphs Is Almost Local

The giant in random graphs is almost local

Remco van der Hofstad Department of Mathematics and Computer Science, Eindhoven University of Technology, e-mail: [email protected] Abstract: Local convergence techniques have become a key methodology to study random graphs in sparse settings where the average degree remains bounded. However, many random graph properties do not directly converge when the random graph converges locally. A notable, and important, random graph property that does not follow from local convergence is the size and uniqueness of the giant component. We provide a simple criterion that guar- antees that local convergence of a random graph implies the convergence of the proportion of vertices in the maximal connected component. We further show that, when this condition holds, the local properties of the giant are also described by the local limit. We apply this novel method to the conﬁguration model as a proof of concept, reproving a result that is well-established. As a side result this proof, we show that the proof also implies the small-world nature of the conﬁguration model.

MSC2020 subject classiﬁcations: 60K37, 05C81, 82C27, 37A25. Keywords and phrases: Random graphs, Local convergence, Giant component, Conﬁgu- ration model.

1. Introduction

Local convergence techniques, as introduced by Aldous and Steele [3] and Benjamini and Schramm [13], have become the main methodology to study random graphs in sparse settings where the average degree remains bounded. Local convergence roughly means that the proportion of vertices whose neighborhoods have a certain shape converges to some limiting value, which is to be considered as a measure on rooted graphs. We refer to [39, Chapter 2] for background on local convergence. The giant component problem has received enormous attention ever since the very first and seminal results by Erd˝osand Rényi on the Erd˝os-Rényi random graph [33], see also [48, 18] for detailed further results for this model. The simplest form of this question is whether there exists a linear-size connected component or not, in many cases a sharp transition occurs depending on a certain graph parameter. See [6, 32, 50, 38, 39], as well as the references therein, for several books on the topic. This paper combines these two threads by investigating the size of the giant component when the random graph converges locally. Consider a sequence of random graphs (Gn)n≥1, where we will simplify the notation by assuming that Gn = (V (Gn),E(Gn)) is such that |V (Gn)| = n. We let

|Cmax| = max |C (v)| (1.1) v∈V (Gn)

arXiv:2103.11733v1 [math.PR] 22 Mar 2021 denotes the maximal cluster size. When the random graph converges locally to some limiting rooted P measure, then one would expect that also |Cmax|/n −→ ζ, where ζ is the survival probability of the local limit. However, while the number of connected components is well behaved in the local topology, the proportion of vertices in the giant is not so nicely behaved. Indeed, since local convergence is all about proportions of vertices whose ﬁnite (but arbitrarily large) neighborhoods converge to a limit, there is an enormous gap between surviving locally and being in the giant component. Thus, local convergence is obviously not enough to imply the convergence of the giant. In this paper, we identify what extra condition is needed to imply this natural implication.

2. Asymptotics and properties of the giant

In this section, we investigate the behavior of the giant component |Cmax| for random graphs that converge locally. In Section 2.1, we introduce the notion of local convergence in probability. 1 2

In Section 2.2, we study the asymptotic of |Cmax| for random graphs that converge locally in probability, and in Section 2.3 we investigate local properties of the giant.

2.1. Local convergence in probability Local convergence was introduced independently by Aldous and Steele in [3] and Benjamini and Schramm in [13]. The purpose of Aldous and Steele in [3] was to describe the local structure of the so-called ‘stochastic mean-field model of distance’, meaning the complete graph with i.i.d. exponential edge weights. This local description allowed Aldous to prove the celebrated ζ(2) limit of the random assignment problem [1]. Benjamini and Schramm in [13] instead used local weak convergence to show that limits of planar graphs are with probability one recurrent. Since its conception, local convergence has proved a key ingredient in random graph theory. In this section, we provide some basics of local convergence. For more detailed discussions, we refer the reader to [22] or [39, Chapter 2]. Let us start with some definitions. A rooted graph is a pair (G, o), where G = (V (G),E(G)) is a graph with vertex set V (G) and edge set E(G), and o ∈ V (G) is a vertex. Further, a rooted or non-rooted graph is called locally finite when each of its vertices has finite degree (though not necessarily uniformly bounded). Two (finite or infinite) graphs G1 = (V (G1),E(G1)) and G2 = (V (G2),E(G2)) are called isomorphic, which we write as G1 ' G2, when there exists a bijection φ: V (G1) 7→ V (G2) such that {u, v} ∈ E(G1) precisely when {φ(u), φ(v)} ∈ E(G2). Similarly, two rooted (finite or infinite) graphs (G1, o1) and (G2, o2) ,wtih Gi = (V (Gi),E(Gi)) for i ∈ {1, 2}, are called isomorphic, abbreviated as (G1, o1) ' (G2, o2), when there exists a bijection φ: V (G1) 7→ V (G2) such that φ(o1) = o2 and {u, v} ∈ E(G1) precisely when {φ(u), φ(v)} ∈ E(G2). These notions can be easily adapted to multi-graphs (which we will need to rely on), for which G = (V (G), (xi,j)i,j∈V (G)), where xi,j denotes the number of edges between i and j, and xi,i the number of self-loops at i. There, instead, 0 0 the isomorphism φ: V (G) 7→ V (G ) is required to satisfy that xi,j = xφ(i),φ(j), where (xi,j)i,j∈V (G) 0 0 and (xi,j)i,j∈V (G0) are the edge multiplicities of G and G respectively. We let G? be the space of (possibly infinite) connected rooted graphs, where we consider two graphs to be equal when they are isomorphic. Thus, we consider G? as the set of equivalence classes of rooted graphs modulo isomorphisms. The space G? of rooted graphs is a nice metric space, with an explicit metric, see [39, Appendix A] for details. (G) For a rooted graph (G, o), we let Br (o) denote the (rooted) subgraph of (G, o) of all vertices at (G) (G) (G) graph distance at most r away from o. Formally, this means that Br (o) = ((V (Br (o)),E(Br (o)), o), where

(G) V (Br (o)) = {u: dG(o, u) ≤ r}, (2.1) (G) E(Br (o)) = {{u, v} ∈ E(G): dG(o, u), dG(o, v) ≤ r}.

Let o ∈ V (Gn) be chosen uniformly at random (u.a.r.) in V (Gn). We say that the graph sequence ? (Gn)n≥1 converges locally in probability to a limit (G, o) ∼ µ, when, for every r ≥ 0 and H ∈ G?, 1 X 1 P (G) ? {B(Gn)(v)'H?} −→ µ(Br (v) ' H ). (2.2) |V (Gn) r v∈V (Gn)

This means that the subgraph proportions in the random graph Gn are close, in probability, to those given by µ. There are related notions of local convergence, such as local weak convergence, where (2.2) is replaced by the expectation of the left-hand side converging to those on the right hand side, and local almost sure convergence, where the convergence in probability in (2.2) is replaced by convergence almost surely. For our purposes, local convergence in probability is the most convenient, for example since it implies that the neighborhoods of two uniformly chosen vertices are close to being independent (see e.g., [39, Corollary 2.18]), which is central in our proof.

2.2. Asymptotics of the giant

Given a random graph sequence Gn that converges in locally probability to (G, o) ∼ µ, one P would expect that |Cmax|/n −→ ζ := µ(|C (o)| = ∞). However, the proportion of vertices in the 2.2 Asymptotics of the giant 3

largest connected component |Cmax|/n is not continuous in the local convergence topology, as it is a global object. In fact, also |C (on)|/n does not converge in distribution when Gn converges locally in probability to (G, o) ∼ µ. However, local convergence still tells us a useful story about the existence of a giant, as well as its size. Indeed, Corollary 2.1 shows that the upper bound is always valid, while Theorem 2.2 shows that a relative simple condition suﬃces to yield the lower bound as well:

Corollary 2.1 (Upper bound on the giant). Let (Gn)n≥1 be a sequence of graphs whose sizes |V (Gn)| = n tend to inﬁnity. Assume that Gn converges locally in probability to (G, o) ∼ µ. Write ζ = µ(|C (o)| = ∞) for the survival probability of the limiting graph (G, o). Then, for every ε > 0 ﬁxed, P(|Cmax| ≤ n(ζ + ε)) → 1. (2.3)

P In particular, Corollary 2.1 implies that |Cmax|/n −→ 0 when ζ = 0, so that there can only be a giant when the local limit has a positive survival probability. Proof. Deﬁne X 1 Z≥k = {|C (v)|≥k}. (2.4)

v∈V (Gn)

Assume that Gn converges locally in probability to (G, o) ∼ µ as deﬁned in (2.2). Since {|C (v)| ≥ (G) k} = {|Bk (v)| ≥ k},with ζ≥k = µ(|C (o)| ≥ k), Z ≥k −→P ζ . (2.5) n ≥k For every k ≥ 1,

{|Cmax| ≥ k} = {Z≥k ≥ k}, (2.6) and, on the event that Z≥k ≥ 1, also |Cmax| ≤ Z≥k. Note that ζ = limk→∞ ζ≥k = µ(|C (o)| = ∞). We take k so large that ζ ≥ ζ≥k − ε/2. Then, for every k ≥ 1, ε > 0, and all n large enough,

P(|Cmax| ≥ n(ζ + ε)) ≤ P(Z≥k ≥ n(ζ + ε)) (2.7)

≤ P(Z≥k ≥ n(ζ≥k + ε/2)) = o(1).

We conclude that while local convergence cannot determine the size of the largest connected component, it can prove an upper bound on |Cmax|. There are many results that extend this to P |Cmax|/n −→ ζ (see Section4 for pointers to the literature), but this is no longer a consequence of local convergence alone. Therefore, in general, more involved arguments must be used. We next prove that one, relatively simple, condition suﬃces. In its statement, and for x, y ∈ V (Gn), we write x ←→/ y when x 6∈ C (y):

Theorem 2.2 (The giant is almost local). Let Gn = (V (Gn),E(Gn)) denote a random graph of size |V (Gn)| = n. Assume that Gn converges locally in probability to (G, o) ∼ µ. Assume further that 1 h i lim lim sup 2 E # (x, y) ∈ V (Gn): |C (x)|, |C (y)| ≥ k, x ←→/ y = 0. (2.8) k→∞ n→∞ n

Then, with Cmax and C(2) denoting the largest and second largest connected components (with ties broken arbitrarily),

|Cmax| |C(2)| −→P ζ = µ(|C (o)| = ∞), −→P 0. (2.9) n n Theorem 2.2 shows that a relatively mild condition as in (2.8) suffices for the giant to have the expected limit. In fact, we will see that it is necessary and sufficient. We now start with the preparations for the proof of Theorem 2.2. We first note that, by Corollary 2.1, it suffices to prove Theorem 2.2 for ζ > 0, which we assume from now on. 4

We recall that the vector (|C(i)|)i≥1 denotes the cluster sizes ordered in size, from large to small with ties broken arbitrarily, so that C(1) = Cmax. The following lemma gives a useful estimate on the sum of squares of these ordered cluster sizes. In its statement, we write Xn,k = ok,P(1) when

lim lim sup P(|Xn,k| > ε) = 0. (2.10) k→∞ n→∞ Lemma 2.3 (Convergence of sum of squares of cluster sizes). Under the conditions of Theorem 2.2,

1 X 2 P 2 |C(i)| −→ ζ , (2.11) n2 i≥1 and 1 X 21 2 |C(i)| = ζ + ok, (1). (2.12) n2 {|C(i)|≥k} P i≥1 Proof. By local convergence in probability and for any k ≥ 1 ﬁxed (recall (2.5)) 1 1 1 X 1 X 1 P |C(i)| = = Z≥k −→ ζ≥k, (2.13) n {|C(i)|≥k} n {|C (v)|≥k} n i≥1 v∈V (Gn) where we recall that ζ≥k = µ(|C (o)| ≥ k). Thus, since ζ≥k → ζ,

1 X 1 |C(i)| = ζ + ok, (1). (2.14) n {|C(i)|≥k} P i≥1 Further, 1 # (x, y) ∈ V (Gn): |C (x)|, |C (y)| ≥ k, x ←→/ y (2.15) n2 1 X 1 = |C(i)||C(j)| . n2 {|C(i)|,|C(j)|≥k} i,j≥1 i6=j By the Markov inequality, 1 lim lim sup P( 2 # (x, y) ∈ V (Gn): |C (x)|, |C (y)| ≥ k, x ←→/ y ≥ ε) (2.16) k→∞ n→∞ n 1 h i ≤ lim lim sup 2 E # (x, y) ∈ V (Gn): |C (x)|, |C (y)| ≥ k, x ←→/ y = 0, k→∞ n→∞ εn by our main assumption in (2.8). As a result,

1 # (x, y) ∈ V (Gn): |C (x)|, |C (y)| ≥ k, x ←→/ y = ok, (1). (2.17) n2 P We conclude that, by (2.13), 2 1 X 21 1 X 1 |C(i)| = |C(i)| + ok, (1) (2.18) n2 {|C(i)|≥k} n {|C(i)|≥k} P i≥1 i≥1 2 = ζ + ok,P(1). (2.19) This proves (2.12). Finally,

1 X 21 1 X 1 k |C(i)| ≤ k |C(i)| ≤ , (2.20) n2 {|C(i)|

1 X 2 1 X 21 k |C(i)| = |C(i)| + O , (2.21) n2 n2 {|C(i)|≥k} n i≥1 i≥1 2.2 Asymptotics of the giant 5 which, together with (2.12), completes the proof of (2.11).

We are now ready to complete the proof of Theorem 2.2: Proof of Theorem 2.2. By Lemma 2.3,

1 X 21 2 |C(i)| = ζ + ok, (1). (2.22) n2 {|C(i)|≥k} P i≥1 1 Denote xi,n = |C(i)| {|C(i)|≥k}/n. Then, by (2.13) and (2.22), for every ε > 0 and whp,

X X 2 2 2 xi,n ≤ ζ + ε/4, xi,n ≥ ζ − ε . (2.23) i≥1 i≥1

We wish to show that x1,n = ζ + ok,P(1). Clearly x1,n ≤ ζ + ok,P(1). Since x1,n is bounded, it has a subsequence that converges in probability. Now suppose by contradiction that, along such 1 P a subsequence, x1,n = |C(1)| {|C(1)|≥k}/n −→ ak, where ak → a = ζ − δ for some δ > 0, so that x1,n = ζ − δ + ok,P(1). Let us work along this subsequence. Then, again using (2.22),

X 2 X 2 2 2 2 xi,n = xi,n − x1,n = ζ − (ζ − δ) + ok,P(1) = δ(2ζ − δ) + ok,P(1). (2.24) i≥2 i≥1 P Also, i≥2 xi,n = δ + ok,P(1). As a result,

2 2 X X 2 X δ + ok,P(1) = xi,n = xi,n + xi,nxj,n. (2.25) i≥2 i≥2 i,j≥2 : i6=j

Clearly, by our main assumption (2.8),

2 X X X 2 ok,P(1) = xi,nxj,n ≤ xi,n − xi,n, (2.26) i,j≥2 : i6=j i≥1 i≥1 so that combining (2.24)–(2.26) leads to δ2 = δ(2ζ − δ) which gives δ = ζ. In turn, this implies that x1,n = ok,P(1). By (2.12) and xi,n ≤ x1,n, we get ζ = 0, contradicting our assumption that ζ > 0.

Proof of necessity of (2.8) Here we complete the proof by showing that (2.8) is also necessary P for |Cmax|/n −→ ζ to hold. Indeed, (2.15) implies that when (2.8) fails, for every k ≥ 1 and by monotonicity, h 1 X i lim sup | || |1 = κ ≥ κ > 0. (2.27) E 2 C(i) C(j) {|C(i)|,|C(j)|≥k} k n→∞ n i,j≥1 i6=j

(2) Denote the random variable in the expectation in (2.27) by Pk,n, so that there exists a subsequence (n ) for which lim [P (2) ] = κ > 0. Since | |/n is bounded, along a subsequence it l l≥1 l→∞ E k,nl Cmax converges in probability to some limit a. Our task is to show that a < ζ. For this, by bounded convergence and again along the subsequence (nl)l≥1,

2 1 2 1 X 12 a = lim E[|Cmax| ] ≤ lim E[|C(i)| {| |≥k}] (2.28) l→∞ n2 l→∞ n2 C(i) l l i

1 2 (2) 2 = lim [Z≥k − P ] ≤ ζ≥k − κ. n→∞ 2 E k,nl nl

Letting k → ∞ and using that ζ≥k → ζ, we obtain that a < ζ as claimed. 6

2.3. Local properties of the giant We next extend Theorem 2.2 by investigating the structure of the giant. For this, we ﬁrst let vk(Cmax) denote the number of vertices with degree k in the giant component, and we recall that |E(Cmax)| denotes the number of edges in the giant component: Theorem 2.4 (Properties of the giant). Under the assumptions of Theorem 2.2, when ζ = µ(|C (o)| = ∞) > 0,

v`(Cmax) P −→ µ(|C (o)| = ∞, do = `). (2.29) n Further, assume that D = d(Gn) is uniformly integrable. Then, n on

|E(Cmax)| h i −→P 1 d 1 . (2.30) n 2 Eµ o {|C (o)|=∞}

(Gn) Proof. We now deﬁne, for k ≥ 1 and A ⊆ N and with dv the degree of v in Gn, X ZA,≥k = 1 (Gn) . (2.31) {|C (v)|≥k,dv ∈A} v∈[n]

Assume that Gn converges locally in probability to (G, o) ∼ µ. Then, Z A,≥k −→P µ(|C (o)| ≥ k, d(G) ∈ A). (2.32) n o

Since |Cmax| ≥ k whp by Theorem 2.2, we thus obtain, for every A ⊆ N,

1 X ZA,≥k P (G) va(Cmax) ≤ −→ µ(|C (o)| ≥ k, d ∈ A). (2.33) n n o a∈A Applying this to A = {`}c, we obtain that, for all ε > 0,

1 h i (G) lim P |Cmax| − v`(Cmax) ≤ µ(|C (o)| ≥ k, do 6= `) + ε/2 = 1. (2.34) n→∞ n We argue by contradiction. Suppose that, for some `,

v`(Cmax) (G) lim inf P ≤ µ(|C (o)| = ∞, do = `) − ε = κ > 0. (2.35) n→∞ n

Then, along the subsequence (nl)l≥1 that attains the liminf in (2.35), with asymptotic probability κ > 0, and using (2.34) and (2.35),

|Cmax| 1 v`(Cmax) = [|Cmax| − v`(Cmax)] + ≤ µ(|C (o)| = ∞) − ε/2, (2.36) n n n which contradicts Theorem 2.2. We conclude that (2.35) cannot hold, so that (2.29) follows. For (2.30), we note that 1 X |E(Cmax)| = 2 `v`(Cmax). (2.37) `≥1 We divide by n and split the sum over ` in ` ∈ [K] = {1,...,K} and ` > K as

|E(Cmax)| 1 X 1 X = `v`(Cmax) + `v`(Cmax). (2.38) n 2n 2n `∈[K] `>K For the ﬁrst term in (2.38), by (2.29),

1 X P 1 X (G) 2n `v`(Cmax) −→ 2 µ(|C (o)| = ∞, do = `) (2.39) `∈[K] `∈[K]

1 h i = µ do1 (G) . 2 E {|C (o)|=∞,do ∈[K]} 2.4 The ‘giant is almost local’ condition revisited 7

For the second term in (2.38), we bound, with n` the number of vertices in Gn of degree `, 1 1 n 1 X X ` (Gn) `v ( ) ≤ ` = d 1 (G ) | G . (2.40) ` Cmax E on {d n >K} n 2n 2 n 2 on `>K `>K

By uniform integrability of (d(Gn)) , on n≥1

(Gn) lim lim sup d 1 (G ) = 0. (2.41) E on {d n >K} K→∞ n→∞ on As a result, by the Markov inequality and for every ε > 0, there exists a K = K(ε) < ∞ such that (Gn) d 1 (G ) | G > ε → 0. (2.42) P E on n n {don >K} This completes the proof of (2.30).

It is not hard to extend the above analysis to the local convergence in probability of the giant, as well as its complement, as formulated in the following theorem: Theorem 2.5 (Local limit of the giant). Under the assumptions of Theorem 2.2, when ζ = µ(|C (o)| = ∞) > 0, 1 X 1 P (G) (Gn) −→ µ(|C (o)| = ∞,Br (o) ' H?), (2.43) n {Br (v)'H?} v∈Cmax and 1 X 1 P (G) (Gn) −→ µ(|C (o)| < ∞,Br (o) ' H?). (2.44) n {Br (v)'H?} v6∈Cmax Proof. The convergence in (2.44) follows from that in (2.43) combined with the fact that, by assumption, 1 X 1 P (G) (Gn) −→ µ(Br (o) ' H?). (2.45) n {Br (v)'H?} v∈V (Gn) The convergence in (2.43) can be proved as for Theorem 2.4, now using that

1 1 X 1 ZH?,≥k ≡ (Gn) (2.46) n n {|C (v)|≥k,Br (v)∈H?} v∈V (Gn)

P (G) −→ µ(|C (o)| ≥ k, Br (o) ∈ H?),

P and, since |Cmax|/n −→ ζ > 0 by Theorem 2.2,

1 X 1 (Gn) ≤ ZH?,≥k. (2.47) n {Br (v)∈H?} v∈Cmax We leave the details to the reader.

2.4. The ‘giant is almost local’ condition revisited The ‘giant is almost local’ condition (2.8) is sometimes not so convenient to verify, and we now give an alternative form that is often easier to work with:

Lemma 2.6 (Condition (2.8) revisited). Consider (Gn)n≥1 under the conditions of Theorem 2.2. Assume further that there exists r = rk → ∞ such that

(G) (G) µ(|C (o)| ≥ k, |∂Br (o)| < rk) → 0, µ(|C (o)| < k, |∂Br (o)| ≥ rk) → 0. (2.48) Then, the ‘giant is almost local’ condition in (2.8) holds when 1 h i (Gn) (Gn) lim lim sup 2 E # (x, y) ∈ V (Gn): |∂Br (x)|, |∂Br (y)| ≥ r, x ←→/ y = 0. (2.49) r→∞ n→∞ n 8

Proof. Denote Pk = # (x, y) ∈ V (Gn): |C (x)|, |C (y)| ≥ k, x ←→/ y , (2.50)

(2) (Gn) (Gn) Pr = # (x, y) ∈ V (Gn): |∂Br (x)|, |∂Br (y)| ≥ r, x ←→/ y . (2.51) Then, (2) |Pr − Pk | ≤ 2n Z

Therefore, by local convergence in probability, 1 2 |P − P (2)| ≤ Z + Z n2 k r n

Take r = rk as in (2.48). Then, the right hand side vanishes, so that, by Dominated Convergence, also 1 lim lim sup |P − P (2)| = 0. (2.56) 2 E k rk k→∞ n→∞ n We arrive at 1 1 lim lim sup P ≤ lim lim sup P (2) = 0, (2.57) 2 E k 2 E rk k→∞ n→∞ n k→∞ n→∞ n by (2.49) and since rk → ∞ when k → ∞.

The assumption in (2.49) is more convenient than (2.8), as it requires that most pairs of vertices who have many vertices at distance r are connected to one another. In many random graphs, there (Gn) is some weak dependence between Br (x) and the graph outside of it. This is more complicated when dealing with |C (x)| ≥ k. The assumption in (2.48) on the local limit is often easily veriﬁed. For example, for the unimodular branching processes with bounded oﬀspring, to which we will apply it below, we can take rk = k and use that on the event of survival

−r (G) a.s. ν |∂Br (o)| −→ M, (2.58) where (G, o) denotes the unimodular branching process with bounded oﬀspring, and where M > 0 (G) on the event of survival by [38, Theorem 3.9]. Therefore, µ(|C (o)| ≥ k, |∂Bk (o)| < k) → 0 as (G) k → ∞. Further, µ(|C (o)| < k, |∂Bk (o)| ≥ k) = 0 trivially. However, there are examples where (2.48) fails, and then also the equivalence of (2.8) and (2.49) may be false.

3. Application to the configuration model

In this section, we apply our results to the configuration model. An application to Erd˝os-Rényi random graph is provided in [39, Chapter 2]. The giant in the configuration model has a long history. It was first investigated by Molloy and Reed [55, 56] in a setting where the degrees are general. The problem was revisited by Janson and Luczak [49] and Bollobásand Riordan [20], amongst others. This section is organised as follows. We start by introducing the model and stating our results in Section 3.1. We then prove the main result in Section 3.2, and state a consequence on typical graph distances, that is proved along the way, in Section 3.3. Some remaining technical ingredients are deferred to AppendixA. 3.1 Model definition and results 9

3.1. Model definition and results The conﬁguration model is invented by Bollob´as[17] in the context of random regular graphs. It has the nice property that, when conditioned on being simple, it yields a uniform random graph with the prescribed degree distribution. We refer to [38, Chapter 7] for a thorough introduction to the model.

Model definition and assumptions Fix an integer n that will denote the number of vertices in the random graph. Consider a sequence of degrees d = (di)i∈[n]. Without loss of generality, we assume throughout this paper that dj ≥ 1 for all j ∈ [n], since when dj = 0, vertex j is isolated and can be removed from the graph. We assume that the total degree X `n = dj (3.1) j∈[n] is even. To construct the multigraph where vertex j has degree dj for all j ∈ [n], we have n separate vertices and incident to vertex j, we have dj half-edges. We number the half-edges in an arbitrary order from 1 to `n, and start by randomly connecting the first half-edge with one of the `n − 1 remaining half-edges. Once paired, two half-edges form a single edge of the multigraph, and the half-edges are removed from the list of half-edges that need to be paired. We continue the procedure of randomly choosing and pairing the half-edges until all half-edges are connected, and call the resulting graph the configuration model with degree sequence d, abbreviated as CMn(d). A careful reader may worry about the order in which the half-edges are being paired. In fact, this ordering turns out to be irrelevant since the random pairing of half-edges is completely exchangeable. It can even be done in a random fashion, which will be useful when investigating neighborhoods in the configuration model. See e.g., [38, Definition 7.5 and Lemma 7.6] for more details on this exchangeability. We denote the degree of a uniformly chosen vertex o in [n] by Dn = do. The random variable Dn has distribution function Fn given by 1 X F (x) = 1 , (3.2) n n {dj ≤x} j∈[n] which is the empirical distribution of the degrees. Equivalently, P(Dn = k) = nk/n, where nk denotes the number of vertices of degree k. We assume that the vertex degrees satisfy the following regularity conditions: Condition 3.1 (Regularity conditions for vertex degrees). (a) Weak convergence of vertex weight. There exists a distribution function F such that, as n → ∞, d Dn −→ D, (3.3) where Dn and D have distribution functions Fn and F , respectively. Further, we assume that F (0) = 0, i.e., P(D ≥ 1) = 1. (b) Convergence of average vertex degrees. As n → ∞,

E[Dn] → E[D] < ∞, (3.4) where Dn and D have distribution functions Fn and F from part (a), respectively. While Conditions 3.1(a)-(b) may appear to be quite strong, in fact, they are quite close to assuming uniform integrability of (Dn)n≥1. Indeed, when (Dn)n≥1 is uniformly integrable, then there is a subsequence along which Dn converges in distribution as in Condition 3.1(a), and along this subsequence also Condition 3.1(b) holds. We can then apply our results along this subsequence. 10

The giant in the conﬁguration model We now come to our result on the connected components in the conﬁguration model CMn(d), where the degrees d = (di)i∈[n] satisfy Conditions 3.1(a)-(b). For a graph G, we write vk(G) for the number of vertices of degree k in G, and |E(G)| for the number of edges. The main result concerning the size and structure of the largest connected components of CMn(d) is the following:

Theorem 3.2 (Phase transition in CMn(d)). Suppose that Conditions 3.1(a)-(b) hold and consider the random graph CMn(d), letting n → ∞. Assume that p2 = P(D = 2) < 1. Let Cmax and C(2) be the largest and second largest components of CMn(d) (breaking ties arbitrarily). (a) If ν = E[D(D − 1)]/E[D] > 1, then there exist ξ ∈ [0, 1), ζ ∈ (0, 1] such that

P |Cmax|/n −→ ζ,

P k vk(Cmax)/n −→ pk(1 − ξ ) for every k ≥ 0,

P 1 2 |E(Cmax)|/n −→ 2 E[D](1 − ξ ).

Local structure of Cmax Theorem 3.2 is the main result for CMn(d) proved in [49]. The result in [20] only concerns |Cmax|/n. Related results are in [55, 56, 59]. Many of these proofs use an exploration of the connected components in the graph, albeit that the precise explorations can be performed in rather diﬀerent ways in that in [55, 56] it is in discrete time, while in [49] it is in continuous time. Further, [20] use concentration inequalities combined with a sprinkling argument. The present approach is, even in this light, novel, since it also allows one to study the local structure of the giant and non-giant, as stated in Theorem 2.5. It is not clear whether the proof in [49], for example, can be easily extended to include this extra information.

Local convergence of configuration models The fact that the configuration model converges locally in probability is well-established, and is the starting point for the proof of Theorem 3.2 using Theorems 2.2 and 2.4. Dembo and Montanari [27] crucially rely on it in order to identify the limiting pressure for the Ising model on the configuration model. Many alternative proofs exist. We use some of the ingredients in [39, Proof of Theorem 4.1], since these are helpful in the proof of Theorem 3.2 as well. We also refer to the lecture notes by Bordenave [22] for a nice exposition of local convergence proofs for the configuration model. Further, Bordenave and Caputo [23] prove that the neighborhoods in the configuration model satisfy a large deviation principle at speed n in the context where the degrees of the configuration model are bounded. Let us now describe the local limit of CMn(d) subject to Conditions 3.1(a)-(b). The root has offspring distribution (pk)k≥1 where pk = P(D = k) and D is from Condition 3.1(a), while all other ? individuals in the tree have offspring distribution (pk)k≥0 given by

? k + 1 pk = P(D = k + 1). (3.5) E[D]

? The distribution (pk)k≥0 has the interpretation of the forward degree of a uniform edge. The above branching process is a so-called unimodular branching process (recall [2]) with root oﬀspring distribution (pk)k≥1. We refer to Appendix A.1 for more details on local convergence of CMn(d). Having identiﬁed the local limit of CMn(d), to prove Theorem 3.2, we are left to prove the ‘giant component is almost local’ condition in (2.8), which we will do in the form of (2.49). We do this in the next section. 3.2 The ‘giant component is almost local’ proof 11

Reformulation in terms of branching processes We next interpret the results in Theorem 3.2 in terms of our unimodular branching processes. In terms of this, ξ is the extinction probability ? of a branching process with offspring distribution (pk)k≥0, and ζ as the survival probability of the unimodular branching process with root offspring distribution (pk)k≥1. Thus, ζ satisfies

X k ζ = pk(1 − ξ ), (3.6) k≥1 with ξ the smallest solution to X ? k ξ = pkξ . (3.7) k≥0 Clearly, ξ = 1 precisely when ν ≤ 1, where X 1 X ν = kp? = k(k + 1)p = [D(D − 1)]/ [D], (3.8) k [D] k+1 E E k≥0 E k≥0 by (3.5). This explains the condition on ν in Theorem 3.2(a). Further, to understand the asymptotics of vk(Cmax), we note that there are nk ≈ npk vertices with degree k. Each of the k direct neighbors of a vertex of degree k survives with probability close to 1 − ξ, so that the probability that at least one of them survives is close to 1 − ξk. When one of the neighbors of the vertex of degree k survives, the vertex itself is part of the giant component, which explains why P k vk(Cmax)/n −→ pk(1 − ξ ). Finally, an edge consists of two half-edges, and an edge is part of the giant component precisely when one of the vertices incident to it is, which occurs with asymptotic 2 probability 1 − ξ . There are in total `n/2 = nE[Dn]/2 ≈ nE[D]/2 edges, which explains why P 1 2 |E(Cmax)|/n −→ 2 E[D](1 − ξ ). Therefore, all results in Theorem 3.2 have a simple explanation in terms of the branching process approximation of the connected component for CMn(d) of a uniform vertex in [n].

The condition P(D = 2) = p2 < 1 The case p2 = 1, for which ν = 1, is quite exceptional. We give three examples showing that then quite diﬀerent behaviors are possible, see [39, Section 4.2] for more details. Our ﬁrst example is when di = 2 for all i ∈ [n], so we are studying a random 2-regular graph. In this case, the components are cycles and the distribution of cycle lengths in CMn(d) is given by the Ewen’s sampling formula ESF(1/2), see e.g. [8]. This implies that |Cmax|/n converges in distribution to a non-degenerate distribution on [0, 1] and not to any constant [8,

Lemma 5.7]. Moreover, the same is true for |C(2)|/n (and for |C(3)|/n,...), so in this case there are several large components. Our second example with p2 = 1 is obtained by letting n1 → ∞ be such that n1/n → 0, and n2 = n − n1. In this case, components can either be cycles, or strings of vertices with degree 2 terminated with two vertices with degree 1, for which |Cmax| = oP(n). Our third example with p2 = 1 is obtained by instead letting n4 → ∞ such that n4/n → 0, and n2 = n − n4, in |Cmax| = n − oP(n). We conclude that the case where p2 = P(D = 2) = 1 is quite sensitive to the precise properties of the degree structure that are not captured by the limiting distribution (pk)k≥1 only. In the sequel, we thus ignore the case where p2 = 1.

3.2. The ‘giant component is almost local’ proof In this section, we prove Theorem 3.2 using the ‘giant is almost local’ results in Theorems 2.2 and 2.4. We start by setting the stage.

Setting the stage for the proof of Theorem 3.2 Theorem 3.2(b) follows directly from Corollary 2.1 combined with the local convergence in probability discussed below Theorem 3.2 and the fact that, for ν ≤ 1, the unimodular Galton-Watson tree with root oﬀspring distribution (pk)k≥0 given by pk = P(D = k) dies out. Theorem 3.2(a) follows from Theorem 2.4, together with the facts that, for the unimodular Galton-Watson tree with root oﬀspring distribution (pk)k≥0 given by pk = P(D = k), k µ(|C (o)| = ∞, do = `) = pk(1 − ξ ), (3.9) 12 and h i 1 2 Eµ do {|C (o)|=∞} = E[D](1 − ξ ). (3.10)

Thus, it suﬃces to check the assumptions to Theorem 2.4. The uniform integrability of Dn follows from Condition 3.1(a)-(b). For the conditions in Theorem 2.2, the local convergence in probability is discussed below Theorem 3.2 , so we are left to proving the crucial hypothesis in (2.8), which the remainder of the proof will do.

Before turning to the proof of (2.8), we apply the degree-truncation technique from Theorem A.1 with b large, to which we refer for details. By Condition 3.1(a)-(b), we can choose b so large that ? 1 X 1 εn P(Dn > b) = dv {dv >b} ≤ . (3.11) `n `n v∈[n] 0 This implies that CMn0 (d ) has at most (1 + ε)n vertices, and the at most εn exploded vertices 0 all have degree 1, while the vertices in [n] have degree dv ≤ b. As a result, it suﬃces to prove 0 Theorem 3.2(b) for CMn0 (d ) instead. As a result, in the remainder of this proof, we will assume 0 0 that dmax ≤ b is uniformly bounded. We will thus apply Theorem 2.4 to CMn0 (d ). We denote its 0 0 0 parameters as (pk)k≥1, ξ and ζ , respectively, and note that, for ε as in (3.11), when ε & 0, 0 0 0 pk → pk, ξ → ξ, ζ → ζ, (3.12) 0 0 so that it suﬃces to prove Theorem 2.4 for CMn0 (d ). Thus, from now on, we work with CMn0 (d ), 0 and we let Gn = CMn0 (d ).

0 To start with our proof of (2.8), applied to CMn0 (d ), we ﬁrst use the alternative formulation from Lemma 2.6, and note that (2.48) holds for for the unimodular Galton-Watson tree with root oﬀspring distribution (pk)k≥0 given by pk = P(D = k). Then, note that, with o1, o2 ∈ [n] chosen independently and uniformly at random, 1 h i #(x, y) ∈ V (G ): |∂B(Gn)(x)|, |∂B(Gn)(y)| ≥ r, x ←→/ y (3.13) n2 E n r r (Gn) (Gn) = P(|∂Br (o1)|, |∂Br (o2)| ≥ r, o1 ←→/ o2). Thus, our main aim is to show that

(Gn) (Gn) lim lim sup P(|∂Br (o1)|, |∂Br (o2)| ≥ r, o1 ←→/ o2) = 0. (3.14) r→∞ n→∞ This is what we will focus on from now on.

Coupling to n-dependent branching process We next apply the sharp coupling result stated in Lemma A.2 in Appendix A.3, in particular using the two extensions in Remark A.3. See Appendix A.3 for details. √ Take an arbitrary mn = o( n), then Remark A.3 shows that whp we can perfectly couple (B(Gn)(o )) , where k 1 k≤kn (Gn) kn = inf k : |Bk (o1)| ≥ mn , (3.15) to a unimodular branching processes (BP(1)) with root oﬀspring distribution ( (D = k)) . k k≤kn P n k≥1 (Gn) Since all degree are bounded, |B (o1)| ≤ (1 + b)m = Θ(m ). Let Cn(1) denote the event kn n n that this perfect coupling happens, so that

C (1) = (|∂B(Gn)(o )|) = (|BP(1)|) , and (C (1)) = 1 − o(1). (3.16) n k 1 k≤kn k k≤kn P n

(1) Here |BPk is the size of the kth generation in our n-dependent branching process. We extend the above coupling to also deal with vertex o2, for which we explore a little further. For this, we start by deﬁning the necessary notation. Let

¯ (Gn) kn = inf k : |Bk (o2)| ≥ m¯ n , (3.17) 3.2 The ‘giant component is almost local’ proof 13

(Gn) and, again since all degrees are bounded, |B (o2)| ≤ (1 + b)m ¯ n = Θ(m ¯ n). Further, for δ > 0, we k¯n let (Gn) (2) 2 1+δ ¯ Cn(2) = |∂Bk (o2)| − |BPk | ≤ (m ¯ n/`n) ∀k ∈ [kn] . (3.18) Here (BP(2)) is again an n-dependent unimodular branching process independent of (BP(1)) . k k≥0√ k k≥0 With mn = o( n), we will later pickm ¯ n such that mnm¯ n n to reach our conclusion. The following lemma shows that also Cn(2) occurs whp: 0 2 Lemma 3.3 (Coupling beyond Lemma A.2). Fix CMn0 (d ) and let m¯ n/`n → ∞. Then, for every δ > 0, P(Cn(2)) = 1 − o(1). (3.19) For the proof, we refer to Appendix A.3.

We now deﬁne the successful coupling event Cn to be

Cn = Cn(1) ∩ Cn(2), so that P(Cn) = 1 − o(1). (3.20)

Branching process neighborhood growth The previous step relates the graph exploration (1) (2) process to two independent n-dependent unimodular branching process (BPk , BPk )k≥1. In this (i) (i) step, we investigate the growth of these branching processes. Denote b0 = |BPr |, which we assume (Gn) (Gn) to be at least r (and which is true on the event Cn ∩ {|∂Br (o1)|, |∂Br (o2)| ≥ r}). Let 0 1 X 0 0 νn = dv(dv − 1), (3.21) `n v∈[n0] 0 be the expected forward degree of a uniform half-edge in CMn0 (d ), which equals the expected (i) oﬀspring of the branching processes (BPk )k≥0. Deﬁne

(i) (i) ¯(i) Dn = bk ≤ |BPr+k| ≤ bk ∀i ∈ [2], k ≥ 1 , (3.22)

(i) ¯(i) (i) ¯(i) (i) 1 where (bk )k≤kn−r and (bk )k≤kn−r satisfy the recursions b0 = b0 = b0 , while, for some α ∈ ( 2 , 1),

(i) 0 (i) ¯(i) α ¯(i) 0 ¯(i) ¯(i) α bk+1 = νnbk − (bk ) , bk+1 = νnbk + (bk ) . (3.23)

(i) ¯(i) The following lemma investigates the asymptotics of (bk )k≥1 and (bk )k≥1: (i) ¯(i) 0 Lemma 3.4 (Asymptotics of bk and bk ). Assume that limn→∞ νn = ν > 1, and assume that (i) (i) b0 = BPr ≥ r. Then, there exists an A = Ar > 1 such that, for all k ≥ 1,

¯(i) (i) 0 k (i) (i) 0 k bk ≤ Ab0 (νn) , bk ≥ b0 (νn) /A. (3.24)

¯(i) (i) 0 k Proof. First, obviously, bk ≥ b0 (νn) . Thus,

¯(i) 0 ¯(i) ¯(i) α 0 ¯(i) ¯(i) α−1 bk+1 = νnbk + (bk ) = νnbk (1 + (bk ) ) (3.25) 0 ¯(i) −(1−α) 0 −(1−α)k ≤ νnbk 1 + r (νn) ,

By iteration, this implies the upper bound with A replaced by A¯r given by

¯ Y −(1−α) 0 −(1−α)k Ar = 1 + r (νn) < ∞. (3.26) k≥0

¯(i) ¯ (i) 0 k For the lower bound, we use that bk ≤ Arb0 (νn) to obtain

(i) 0 (i) ¯α (i) α 0 αk bk+1 ≥ νnbk − Ar (b0 ) (νn) . (3.27) We now use induction to show that (i) (i) 0 k bk ≥ akb0 (νn) , (3.28) 14

where a0 = 1 and ¯α 1−α 0 (α−1)k−1 ak+1 = ak − Ar r (νn) . (3.29) (i) (i) The initialization follows, since b0 = b0 and a0 = 1. To advance the induction hypothesis, we substitute the induction hypothesis to obtain that

(i) (i) 0 k+1 ¯α (i) α 0 αk bk+1 ≥ akb0 (νn) − Ar (b0 ) (νn) (3.30) (i) 0 k+1 ¯α (i) α−1 0 (α−1)k = b0 (νn) ak + Ar (b0 ) (νn) (i) 0 k+1 ¯α α−1 0 (α−1)k−1 (i) 0 k+1 ≥ b0 (νn) ak + Ar r (νn) = ak+1b0 (νn) , by (3.29). Finally, ak & a ≡ 1/Ar, where Ar < ∞ for r large enough, so that the claim follows ¯ with A = Ar = max{Ar,Ar}.

The following lemma shows that Dn occurs whp when ﬁrst n → ∞ followed by r → ∞:

(i) (i) Lemma 3.5 (Dn occurs whp). Assume that b0 = |BPr | ≥ r. Then

c Dn,k ∩ Dn,k−1 (3.36) (1) 0 (1) ¯(1) α (2) 0 (2) ¯(2) α ⊆ |BPr+k| − νn|BPr+k−1| ≥ (bk−1) ∪ |BPr+k| − νn|BPr+k−1| ≥ (bk−1) .

By the Chebychev inequality, conditionally on Dn,k−1, (i) 0 (i) ¯(i) α P |BPr+k| − νn|BPr+k−1| ≥ (bk−1) | Dn,k−1 (3.37) Var(|BP(i) | | |BP(i) |) σ2 |BP(i) | ≤ r+k r+k−1 ≤ n r+k−1 ≤ σ2 (¯b(i) )1−2α, ¯(i) 2α ¯(i) 2α n k−1 (bk−1) (bk−1) 2 where σn is the variance of the oﬀspring distribution given by

2 1 X 0 0 2 0 2 σn = dv(dv − 1) − (νn) , (3.38) `n v∈[n0] which is uniformly bounded. Thus,

c 2 ¯(i) 1−2α P(Dn,k ∩ Dn,k−1) ≤ σn(bk−1) , (3.39) and we conclude that ∞ c 2 X ¯(1) 1−2α ¯(2) 1−2α P(Dn) ≤ σn (bk−1) + (bk−1) . (3.40) k=1 2 2 The claim now follows from Lemma 3.4 and the fact that σn ≤ b(b − 1) remains uniformly bounded. 3.3 Small-world nature of conﬁguration model 15

Completion of the proof Recall (3.14). Let Gn = Cn ∩ Dn be the good event. By (3.20) and Lemma 3.5, (Gn) (Gn) c lim lim sup P(|∂Br (o1)|, |∂Br (o2)| ≥ r, o1 ←→/ o2; Gn) = 0, (3.41) r→∞ n→∞

(Gn) (Gn) so that it suﬃces to investigate P(|∂Br (o1)|, |∂Br (o2)| ≥ r, o1 ←→/ o2; Gn). On Gn (recall (3.18)), (Gn) (2) 2 1+δ |∂B (o2)| − |BP | ≥ −(m ¯ /`n) . (3.42) k¯n k¯n n

Further, on Gn (recall (3.16)), (Gn) (1) |∂B (o1)| − |BP | = 0. (3.43) kn kn (Gn) (Gn) There are two possibilities: either B (o1) ∩ B¯ (o2) 6= , in which case o1 ←→ o2, so this does kn kn ∅ (Gn) (Gn) (Gn) (Gn) not contribute to (|∂B (o1)|, |∂B (o2)| ≥ r, o1 ←→/ o2; Gn); or B (o1) ∩ B¯ (o2) = . P r r kn kn ∅ 2 (Gn) In this case, by Lemma 3.4 and whenm ¯ /`n → ∞ suﬃciently slowly, |∂B (o1)| = Θ(m ) n kn n (Gn) (Gn) and |∂B¯ (o2)| = Θ(m ¯ n). Thus, the probability that ∂B (o1) is not directly connected to kn kn (Gn) ∂B (o2) vanishes when m m¯ n n. As a result, as n → ∞, k¯n n

(Gn) (Gn) P(|∂Br (o1)|, |∂Br (o2)| ≥ r, o1 ←→/ o2; Gn) = o(1), (3.44) as required.

3.3. Small-world nature of configuration model

In this section, we use the above proof to study the typical distances in CMn(d). Such distances have attracted considerable attention, see e.g., [29, 30, 40, 41, 42]. Here, we partially reprove a result from [40]:

Theorem 3.6 (Typical distances in CMn(d) for ﬁnite-variance degrees). Consider the conﬁgura- 2 2 tion model CMn(d) subject to Conditions 3.1(a)-(b) and where E[Dn] → E[D ] < ∞ with ν ∈ (1, ∞). (3.45)

Then, conditionally on distCMn(d)(o1, o2) < ∞,

P distCMn(d)(o1, o2)/ log n −→ 1/ log ν. (3.46) Proof. The proof in (3.41)–(3.44) also shows that, whp as ﬁrst n → ∞ followed by r → ∞,

¯ log n distCM (d0)(o , o ) ≤ k + k + 1 = (1 + o (1)). (3.47) n0 1 2 n n 0 P log νn 0 ¯ 0 Indeed, Lemma 3.4 implies that kn = (1 + oP(1)) log mn/ log νn and kn = (1 + oP(1)) logm ¯ n/log νn on the event Gn = Cn ∩ Dn, so that ¯ 0 0 kn + kn + 1 = (1 + oP(1)) log (mnm¯ n)/log νn = (1 + oP(1)) log n/log νn (3.48) 0 0 when mnm¯ n/n → ∞ slowly enough. The giants Cmax in CMn0 (d ) and Cmax in CMn(d) satisfy 0 that |Cmax∆Cmax| ≤ εn whp for b suﬃciently large, where ∆ denotes the symmetric diﬀerence 0 between sets. Further, νn − ε ≤ νn ≤ νn, so that, for every ε > 0,

P(distCMn(d)(o1, o2)/ log n ≤ (1 + ε)/ log ν) = 1 − o(1). (3.49) The reverse inequality applies more generally, and follows from the fact that

(Gn) (distCM (d0)(o , o ) ≤ k) = [|B (o )|/n], (3.50) P n0 1 2 E k 1 together with the fact that (Gn) 0 k E[|Bk (o1)|] ≤ (νn) (3.51) by [47, Lemma 5.1]. Thus, 0 −ε (distCM (d0)(o , o ) ≤ (1 − ε) log n/log ν k) ≤ n = o(1), (3.52) P n0 1 2 n as required. 16

4. Discussion and open problems

In this section, we discuss our proof and results, and state some open problems.

Minimal conditions for the proof Inspection of the proof of Corollary 2.1 and Theorem 2.2 P shows that Z≥k/n −→ ζ≥k, with ζ≥k & ζ suﬃces to obtain Theorem 2.2 (see in particular (2.5) and (2.13)). Theorem 2.4 requires a little more (see (2.32)), while only Theorem 2.5 requires the full local convergence in probability. The use of Z≥k to study connected components (particularly close to criticality) has a long history, and is implicitly present in [26], while being formally introduced in the high-dimensional percolation context in [24], see [25, 44, 45] for applications to percolation on the n-cube. [37] used it to study the critical behavior for randk-1 inhomogeneous random graphs, and [38] for the phase transition on the Erd˝os-R´enyi random graph.

Related models We use the configuration model as a proof of concept for the method in this paper. We believe that our approach is quite versatile. For example, in [39, Section 2.5], the method is performed for the Erd˝os-Rényi random graph. That proof can easily be adapted to finite-type inhomogeneous random graphs as studied in the seminal paper by Bollobás, Janson and Riordan [19]. In turn, such an approach would also yield the type distribution of the giant, as it was identified in greater generality in [51, Lemma 4.10].

Local convergence There are many models for which local convergence has been established, and thus our results might be applied. For inhomogeneous random graphs, local convergence was not established explicitly in the seminal paper by Bollobás,Janson and Riordan [19], but the methodology is very related and does allow for a simple proof of it (see [39, Chapter 3] for a formal proof, and [27] for a proof for Erd˝os-Rényi random graph). For the configuration model, and the related uniform random graph with prescribed degrees, it was proved explicitly by Dembo and Montanari [27], see also [22, 28] as well as [39, Chapter 4] for related proofs. Berger, Borgs, Chayes and Saberi [14] establish local convergence for preferential attachment models (see also [39, Chapter 5] for some extensions). A related preferential attachment model with conditionally independent edges is treated by Dereich and Mörters[31], who again do not formally prove local convergence, but state a highly related result. Random intersection graphs are treated by [53], see also [43]. In most of these models, the size of the giant is established, though mostly in ways that are quite different from the current proof (see [39] for an extensive overview). It would be of interest to see whether the current proof simplifies some of these analyses.

Limiting properties that follow from local convergence As stated in the introduction, local convergence is a versatile tool to prove various properties of random graphs. The number of connected components, the clustering coefficient, local neighborhoods, and edge-neighborhoods are all local quantities (see [39, Chapter 2] for many examples of local quantities). In some cases, some extra local conditions (typically on the degree distribution) need to be made in order to be able to use such results. Some less obvious properties also converge when the graph converges locally. An important and early example is the Ising model partition function [27]. Also the PageRank distribution is local [36]. A property that is almost local is the density of the densest subgraph in a random graph, as shown by Anantharam and Salez [7]. Lyons [54] shows that the exponential growth rate of the number of spanning trees of a finite connected graph can be computed through the local limit. See also [60] for weighted spanning subgraphs, [35] for maximum-eight independent sets, and [15] for the limiting spectral distribution of the graph adjacency matrix. Our approach shows that while the proportion of vertices in the giant is not a local property, it is almost local in the sense that one extra condition does guarantee that the proportion of vertices in the giant is what one would expect it to be on the basis of the local limit. It would be interesting to investigate whether the idea of ‘almost local’ properties can be extended to other properties as well. For example, while Theorem 3.6 suggests that also logarithmic distances are an almost local property, the statement in Theorem 3.6 follows from the proof of Theorem 3.2, rather than from a general ‘almost local’ result as in Theorem 2.2. It would be interesting to explore this A Further ingredients for configuration model 17 further, to see whether also related typical distance results might be proved in a similar way (see [29, 30, 40, 41, 42] for such related results).

Percolation on finite graphs Another area where our results may be useful is percolation on finite graphs. There, it is often not even clear how to precisely define the critical value or critical behavior. We refer to Janson and Warnke [52] and Nachmias and Peres [58] for extensive discussions on the topic. Percolation on random graphs has also attracted substantial attention, see for example Janson [46] or Fountoulakis [34] for the derivation of the limiting percolation threshold for the configuration model. This is related to the locality of the percolation threshold as investigated by Benjamini, Nachmias and Peres [12] in the context of d-regular graphs that have large girth, and inspired by a question by Oded Schramm in 2008. Indeed, Schramm conjectured that the critical percolation threshold on a converging sequence of infinite graphs Gn should converge to that of the graph limit. In a transitive setting, local convergence can be used also on infinite graphs (as every vertex is basically the same, thus skipping the necessity of drawing a root uniformly at random). See also [5, 10, 57] for related work on sharp threshold for the existence of a giant in the context of expanders. Recently, Alimohammadi, Borgs and Saberi [4] brought the discussion significantly forward, by showing that the percolation critical value is is indeed local for expanders. Further, interestingly, they applied their results to the Barabási-Albert preferential attachment model [9], in the version of Bollobáset al. in [21], to show that pc = 0 and identify the giant for all p > 0. The proof relies on a sprinkling argument. It would be interesting to study the relation between our central assumption in (2.8) and (Gn)n≥1 being a sequence of expander graphs. Further, it would be very interesting to see whether local convergence has consequences for the critical behavior of percolation on locally convergent graph sequences.

Appendix A: Further ingredients for configuration model

In this section, we provide some technical ingredients that are useful in the proof of Theorem 3.2. This section is organised as follows. In Section A.1, we give some more details of local convergence in probability for the conﬁguration model. In Section A.2, we discuss a convenient degree-truncation argument that shows that it suﬃces to prove Theorem 3.2 for uniformly bounded degrees. In Section A.3, we present an explicit coupling of the graph exploration process for CMn(d) and an n-dependent unimodular branching process.

A.1. Local convergence of the configuration model As mentioned already below Theorem 3.2, there are several ways in which local convergence in probability, as formalized in (2.2), can be proved. Such proofs always consist of two steps. In the ﬁrst, we investigate the expected number of vertices whose r-neighborhood is isomorphic to a speciﬁc tree, in the second, we prove that this number if highly concentrated. Convergence of the mean follows from the coupling result in Lemma A.2 below (see the discussion right below it). Concentration can be proved using martingale techniques (by pairing edges one by one and using Azuma-Hoefding), see [27] or [22]. Alternatively, one can use a second moment method and results similar to Lemma A.2. See [39, Section 4.1] for such an approach.

A.2. A useful degree-truncation argument for the configuration model In this section, we present a useful degree-truncation argument that allows us to go from a conﬁguration model with degrees satisfying Conditions 3.1(a)-(b), to degrees that are uniformly bounded. This result makes the proof of the ‘giant is almost local’ condition in (2.49) simpler, and is interesting in its own right:

Theorem A.1 (Degree truncation for configuration models). Consider CMn(d) with general de- 0 grees. Fix b ≥ 1. There exists a related configuration model CMn0 (d ) that is coupled to CMn(d) and satisfies that 18

0 0 (a) the degrees in CMn0 (d ) are truncated versions of those in CMn(d), i.e., dv = (dv ∧ b) for 0 0 v ∈ [n], and dv = 1 for v ∈ [n ] \ [n]; 0 P 0 P (b) the total degree in CMn0 (d ) is the same as that in CMn(d), i.e., v∈[n0] dv = v∈[n] dv; 0 (c) for all u, v ∈ [n], if u and v are connected in CMn0 (d ), then so are u and v in CMn(d).

Proof. The proof relies on an ‘explosion’ or ‘fragmentation’ of the vertices [n] in CMn(d) inspired by [46]. Label the half-edges from 1 to `n. We go through the vertices v ∈ [n] one by one. When 0 dv ≤ b, we do nothing. When dv > b, then we let dv = b, and we keep the b half-edges with the lowest labels. The remaining dv −b half-edges are exploded from vertex v, in that they are incident 0 to vertices of degree 1 in CMn0 (d ), and are given vertex labels above n. We give the exploded half-edges the remaining labels of the half-edges incident to v. Thus, the half-edges receive labels 0 both in CMn(d) as well as in CMn0 (d ), and the labels of the half-edges incident to v ∈ [n] in 0 + P CMn0 (d ) are a subset of those in CMn(d). In total, we thus create an extra n = v∈[n](dv − b)+ ‘exploded’ vertices of degree 1, and n0 = n + n+. 0 We then pair the half-edges randomly, in the same way in CMn(d) and in CMn0 (d ). This means that when the half-edge with label x is paired to the half-edge with label y in CMn(d), the half-edge 0 with label x is also paired to the half-edge with label y in CMn0 (d ), for all x, y ∈ [`n]. We now check parts (a)-(c). Obviously parts (a) and (b) follow. For part (c), we note that all 0 created vertices have degree 1. Further, for vertices u, v ∈ [n], if there exists a path in CMn0 (d ) connecting them, then the intermediate vertices have degree at least 2, so that they cannot cor- respond to exploded vertices. Thus, the same path of paired half-edges also exists in CMn(d), so that also u and v are connected in CMn(d).

A.3. Coupling of configuration neighborhoods and branching processes We next relate the neighborhood in a random graph to a certain n-dependent unimodular branching process where the root has oﬀspring distribution Dn. Such a coupling has previously appeared in [16]. Since the branching process is unimodular, all other individuals have oﬀspring ? distribution Dn − 1, where

? k P(Dn = k) = P(Dn = k), k ∈ N, (A.1) E[Dn] is the size-biased distribution of Dn. Denote this branching process by (BPn(t))t∈N0 . Here, BPn(t) denotes the branching process when it contains precisely t vertices, and we explore it in the breadth- d ? d ? first order. Clearly, by Conditions 3.1(a)-(b), Dn −→ D and Dn −→ D , which implies that d BPn(t) −→ BP(t) for every t finite, where BP(t) is the restriction of the unimodular branching process BP with root offspring distribution (pk)k≥1 for which pk = P(D = k) to its first t individuals.

We let (Gn(t))t∈N0 denote the graph exploration process from a uniformly chosen vertex o ∈ [n]. Here Gn(t) is the exploration up to t vertices, in the breadth-ﬁrst manner. In particular, (Gn) from (Gn(t))t∈N0 we can retrieve (Br (o))r∈N0 . The following lemma proves that we can couple the graph exploration to the branching process in such a way that (Gn(t))0≤t≤mn is equal to

(BPn(t))0≤t≤mn whenever mn → ∞ arbitrarily slowly. In the statement, we write (Gbn(t), BPc n(t))t∈N0 for the coupling of (Gn(t))0≤t≤mn and (BPn(t))0≤t≤mn : Lemma A.2 (Coupling graph exploration and branching process). Subject to Conditions 3.1(a)-

(b), there exists a coupling (Gbn(t), BPc n(t))t∈N0 of (Gn(t))0≤t≤mn and (BPn(t))0≤t≤mn such that P (Gbn(t))0≤t≤mn 6= (BPc n(t))0≤t≤mn = o(1), (A.2) whenever mn → ∞ arbitrarily slowly. Lemma A.2 also implies that the proportion of vertices whose r neighborhood is isomorphic to a specific tree converges to the probability that the unimodular branching process with root offspring distribution (pk)k≥1, which is a crucial ingredient in local convergence. A.3 Coupling of configuration neighborhoods and branching processes 19

Remark A.3 (Extensions). Here we discuss some extensions of Lemma A.2. First, in its proof, p we will see that any mn = o( n/dmax) is allowed. Here dmax = maxi∈[n] di is the maximal vertex degree in the graph. Secondly, Lemma A.2 can easily be extended to deal with the explorations p from two sources (o1, o2), where we can still take mn = o( n/dmax), and the two branching (1) processes to which we couple the exploration from two sources, denoted by (BPc n (t))0≤t≤mn and (2) (BPc n (t))0≤t≤mn , are i.i.d.

Proof. We let the offspring of the root of the branching process Dbn be equal to do, which is the number of neighbors of the vertex o ∈ [n] that is chosen uniformly at random. By construction, Dbn = do, so that also Gbn(1) = BPc n(1). We next explain how to jointly construct (Gbn(t), BPc n(t))0≤t≤m given that we have already constructed (Gbn(t), BPc n(t))0≤t≤m−1. To obtain (Gbn(t)0≤t≤m, we take the first unpaired half-edge xm. For the configuration model, this half-edge needs to be paired to a uniform half-edge that has not been paired so far. We draw a uniform half-edge ym from the collection of all half-edges, independently of the past, and we let the (m − 1)st individual in (BPc n(t))0≤t≤m−1 have precisely dUm − 1 children, where Um ? is the vertex incident to ym. Note that dU − 1 has the same distribution as D − 1 and, by m n construction, the collection dUt − 1 t≥0 is i.i.d. When ym is still free, i.e., has not yet been paired in (Gbn(t))0≤t≤m−1, then we also let xm be paired to ym for (Gbn(t))0≤t≤m, and we have constructed (Gbn(t))0≤t≤m. However, a problem arises when ym has already been paired in (Gbn(t))0≤t≤m−1, in 0 0 which case we draw a uniform unpaired half-edge ym and pair xm to ym instead. Clearly, this might give rise to a difference between (Gbn(t))t≤m and (BPc n(t))0≤t≤m. We now provide bounds on the probability that an error occurs before time mn. There are two sources of differences between (Gbn(t))t≥0 and (BPc n(t))t≥0:

Half-edge re-use. In the above coupling ym had already been paired and is being re-used in the 0 branching process, and we need to redraw ym; Vertex re-use. In the above coupling, ym is a half-edge that has not yet been paired in (Gbn(t))0≤t≤m−1, but it is incident to a half-edge that has already been paired in (Gbn(t))0≤t≤m−1. In particular, the vertex Um to which it is incident has already appeared in (Gbn(t))0≤t≤m−1 and it is being re-used in the branching process. In this case, a copy of the vertex appears in (BPc n(t))0≤t≤m, while a cycle appears in (Gbn(t))0≤t≤m. We continue by providing a bound on both contributions:

Half-edge re-use Up to time m − 1, exactly 2m − 1 half-edges are forbidden to be used by (Gbn(t))t≤m. The probability that the half-edge ym equals one of these two half-edges is at most 2m − 1 . (A.3) `n

Hence the expected number of half-edge re-uses before time mn is at most m Xn 2m − 1 m2 = n = o(1), (A.4) ` ` m=1 n n √ when mn = o( n).

Vertex re-use The probability that vertex i is chosen in the mth draw is equal to di/`n. The probability that vertex i is drawn twice before time mn is at most 2 mn(mn − 1) di 2 . (A.5) 2 `n

By the union bound, the probability that a vertex re-use has occurred before time mn is at most 2 mn(mn − 1) X di 2 dmax ≤ mn = o(1), (A.6) 2`n `n `n i∈[n] 20

p by Condition 3.1(a)-(b) and when mn = o( n/dmax). This completes the coupling part of Lemma A.2, including the bound on mn as formulated in Remark A.3. It is straightforward to check that the exploration can be performed from the two sources (o1, o2) independently, thus establishing the requested coupling to two independent n-dependent branching processes claimed in Remark A.3.

Finally, we adapt the above argument to prove Lemma 3.3: 2 1+δ Proof of Lemma 3.3.. Deﬁne an = (m ¯ n/`n) where δ > 0. We apply a ﬁrst-moment method and bound

c 1 h (Gn) (2) i P(Cn(2) ) ≤ E |Bk¯ (o2)| − |BPk¯ | (A.7) an n n h i 1 (Gn) (2) (Gn) (2) = E |Bk¯ (o2)| − |BPk¯ | + + |Bk¯ (o2)| − |BPk¯ | − an n n n n 1 h i ≤ E |Gbn(mn)| − |BPc n(mn)| + + |Gbn(mn)| − |BPc n(mn)| − , an where the notation is as in Lemma A.2 and |Gbn(mn)| and |BPc n(mn)| denote the number of half- edges and individuals found up to the mnth step of the exploration starting from o2, and mn = (b + 1)m ¯ n, while x+ = max{0, x} and x− = max{0, −x}. To bound these expectations, we adapt the proof of Lemma A.2 to our setting. We start with the ﬁrst term in (A.7), for which we use the exploration up to sizem ¯ n used in the proof of Lemma A.2. We note that the only way that |Gbn(t + 1)| − |Gbn(t)| can be larger than |BPc n(t + 1)| − |BPc n(t)| is when a half-edge re-use occurs, and then the redraw in |Gbn(t + 1)| − |Gbn(t)| is larger than then original draw of |BPc n(t + 1)| − |BPc n(t)|. Since all degrees are bounded by b,

|Gbn(m ¯ n)| − |BPc n(m ¯ n)| ≤ b#{half-edge re-uses up to time mn}. (A.8) Thus, by (A.4), 2 h i mn E |Gbn(m ¯ n)| − |BPc n(m ¯ n)| + ≤ b . (A.9) `n

|BPc n(m ¯ n)| − |Gbn(m ¯ n)| ≤ b#{half-edge and vertex re-uses up to time mn}. (A.10) Thus, by (A.4) and (A.6),

2 h i 2 mn E |Gbn(m ¯ n)| − |BPc n(m ¯ n)| + ≤ (b + b ) . (A.11) `n We conclude that 2 δ c 1 2 mn `n P(Cn(2) ) ≤ (2b + b ) = O 2 = o(1), (A.12) an `n mn by assumption and the fact that mn = (b + 1)m ¯ n.

Acknowledgements The work in this paper was supported by the Netherlands Organisation for Scientiﬁc Research (NWO) through Gravitation-grant NETWORKS-024.002.003. I thank Joel Spencer for proposing the ‘easiest solution’ to the giant component problem for the Erd˝os-R´enyi random graph, which inspired this paper. I thank Shankar Bhamidi for useful and inspiring discussions, Souvik Dhara for literature references, and Christian Borgs for suggestions on local convergence and encouragement. REFERENCES 21

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