Scaling limits for critical inhomogeneous random graphs with finite third moments

We identify the scaling limits for the sizes of the largest components at criticality for inhomogeneous random graphs when the degree exponent $\tau$ satisfies $\tau>4$. We see that the sizes of the (rescaled) components converge to the excursion lengths of an inhomogeneous Brownian motion, extending results of \cite{Aldo97}. We rely heavily on martingale convergence techniques, and concentration properties of (super)martingales. This paper is part of a programme to study the critical behavior in inhomogeneous random graphs of so-called rank-1 initiated in \cite{Hofs09a}.


Introduction 1.Model
We start by describing the model considered in this paper.While there are many variants available in the literature, the most convenient for our purposes model is the model often referred to as Poissonian graph process or Norros-Reittu model [20].See Section 1.3 below for consequences for other models.To define the model, we consider the vertex set [n] := {1, 2, . . ., n}, and attach an edge with probability p ij between vertices i and j, where and Different edges are independent.
Below, we shall formulate general conditions on the weight sequence w = (w 1 , . . ., w n ), and for now formulate two main examples.The first key example arises when we take w to be an i.i.d.sequence of random variables with distribution function F satisfying The second key example (which is also studied in [14]) arises when we let the weight sequence w = (w i ) n i=1 be defined by where F (x) is a distribution function satisfying that, with W a random variable with distribution function F satisfying (1.3), and where [1 − F ] −1 is the generalized inverse function of 1 − F defined, for u ∈ (0, 1), by [1 − F ] −1 (u) = inf{s : [1 − F ](s) ≤ u}. (1.5) By convention, we set [1 − F ] −1 (1) = 0. Write Then, by [5], the random graphs we consider are subcritical when ν < 1 and supercritical when ν > 1.Indeed, when ν > 1, then there is one giant component of size Θ(n) while all other components are of smaller size o P (n), while when ν ≤ 1 the largest connected component has size o P (n).Thus, the critical value of the model is ν = 1.Here, and throughout this paper, we use the following standard notation.We write f (n) = O(g(n)) for functions f, g ≥ 0 and n → ∞ if there exists a constant C > 0 such that f (n) ≤ Cg(n) in the limit, and . Furthermore, we write f = Θ(g) if f = O(g) and g = O(f ).We write O P (b n ) for a sequence of random variables X n for which |X n |/b n is tight as n → ∞, o P (b n ) for a sequence of random variables X n for which |X n |/b n P −→ 0 as n → ∞.Finally, we write that a sequence of events (E n ) n≥1 occurs with high probability (whp) when P(E n ) → 1.
We shall write G 0 n (w) to be the graph constructed via the above procedure, while, for any fixed t ∈ R, we shall write G t n (w) when we use the weight sequence (1 + tn −1/3 )w, for which the probability that i and j are neighbors equals 1 − exp −(1 + tn −1/3 )w i w j /l n .In this setting we take n so large that 1 + tn −1/3 > 0.
We now formulate the general conditions on the weight sequence w.In Section 3, we shall verify that these conditions are satisfied for i.i.d.weights with finite third moment, as well as for the choice in (1.4).We assume the following three conditions on the weight sequence w: (a) Maximal weight bound.We assume that the maximal weight is o(n 1/3 ), i.e., max i∈ [n] w i = o(n 1/3 ).
(1.7) (b) Weak convergence of weight distribution.We assume that the weight of a uniform vertex converges in distribution to some distribution function F , i.e., let V n ∈ [n] be a uniform vertex.Then we assume that for some limiting random variable W with distribution function F .Condition (1.8) is equivalent to the statement that, for every x which is a continuity point of x → F (x), we have (1.9) (c) Convergence of first three moments.We assume that . (1.12) Note that condition (a) follows from conditions (b) and (c), as we prove around (2.41) below.When w is random, for example in the case where (w i ) n i=1 are i.i.d.random variables with finite third moment, then we need the estimates in conditions (a), (b) and (c) to hold in probability.
We shall simply refer to the above three conditions as conditions (a), (b) and (c).Note that (1.10) and (1.11) in condition (c) also imply that Before we write our main result we shall need one more construct.For fixed t ∈ R consider the inhomogeneous Brownian motion (W t (s)) s≥0 with where B is standard Brownian motion, and that has drift t − s at time s.We want to consider this process restricted to be non-negative, which is why we introduce the reflected process It [1] it is shown that the excursions of W t from 0 can be ranked in increasing order as, say, γ . .denote the sizes of the components in G t n (w) arranged in increasing order.Define l 2 to be the set of infinite sequences x = (x i ) ∞ i=1 with x 1 ≥ x 2 ≥ . . .≥ 0 and ∞ i=1 x 2 i < ∞, and define the l 2 metric by Then, our main result is as follows: Theorem 1.1 (The critical behavior).Assume that the weight sequence w satisfies conditions (a), (b) and (c).Then, as n → ∞, in distribution and with respect to the l 2 topology.
Theorem 1.1 extends the work of Aldous [1], who identified the scaling limit of the largest connected components in the Erdős-Rényi random graph.Indeed, he proved for the critical Erdős-Rényi random graph with p = (1 + tn −1/3 )/n that the ordered connected components are given by (γ i (t)) i≥1 , i.e., the ordered excursions of the reflected process in (1.15).Hence, Aldous' result corresponds to Theorem 1.1 with µ = σ 3 = 1.The sequence γ * i (t) i≥1 is in fact the sequence of ordered excursions of the reflected version of the process which reduces to the process in (1.14) again when µ = σ 3 = 1.
We next investigate the two key examples, and show that conditions (a), (b) and (c) indeed hold in this case: , where the result in Theorem 1.1 is conjectured in the case where w is as in (1.4), where F is a distribution function of a random variable W with E[W 3+ε ] < ∞ for some ε > 0. The current result implies that E[W 3 ] is a sufficient condition for this result to hold, and we believe this condition also to be necessary (as the constant E[W 3 ] also appears in our results, see (1.18) and (1.19)).Note, however, that in [14, Conjecture 1.6], the constant in front of −s 2 /2 in (1.19) is erroneously taken as 1, while it should be

Overview of the proof
In this section, we give an overview of the proof of Theorem 1.1.After having set the stage for the proof, we shall provide a heuristic that indicates how our main result comes about.We start by describing the cluster exploration: Cluster exploration.The proof involves two key ingredients: • The exploration of components via breath-first search; and • The labeling of vertices in a size-biased order of their weights w.
More precisely, we shall explore components and simultaneously construct the graph G t n (w) in the following manner: First, for all ordered pairs of vertices (i, j), let V (i, j) be exponential random variables with rate 1 + tn −1/3 w j /l n random variables.Choose vertex v(1) with probability proportional to w, so that (1.20) The children of v( 1) are all the vertices j such that Suppose v(1) has c(1) children.Label these as v(2), v(3), . . .v(c(1) + 1) in increasing order of their V (v(1), •) values.Now move on to v(2) and explore all of its children (say c(2) of them) and label them as before.Note that when we explore the children of v(2), its potential children cannot include the vertices that we have already identified.More precisely, the children of v(2) consists of the set {v / ∈ {v(1), . . .v(c(1) + 1)} : and so on.Once we finish exploring one component, we move onto the next component by choosing the starting vertex in a size-biased manner amongst remaining vertices and start exploring its component.It is obvious that this constructs all the components of our graph G t n (w).
Write the breadth-first walk associated to this exploration process as for i = 1, . . ., n. Suppose C * (i) is the size of the ith component explored in this manner (here we write C * (i) to distinguish this from C i n (t), the ith largest component).Then these can be easily recovered from the above walk by the following prescription: For j ≥ 0, write η(j) as the stopping time η(j) = min{i : Z n (i) = −j}. (1.23) Further, Z n (η(j)) = −j, Z n (i) ≥ −j for all η(j) < i < η(j + 1). (1.25) Recall that we started with vertices labeled 1, 2, . . ., n with corresponding weights w = (w 1 , w 2 , . . ., w n ).
The size-biased order v * (1), v * (2), . . ., v * (n) is a random reordering of the above vertex set where v(1) = i with probability equal to w i /l n .Then, given v * (1), we have that v * (2) = j ∈ [n] \ {v * (1)} with probability proportional to w j and so on.By construction and the properties of the exponential random variables, we have the following representation, which lies at the heart of our analysis: , . . ., v(n) in the above construction of the breadth-first exploration process is the size-biased ordering v * (1), v * (2), . . ., v * (n) of the vertex set [n] with weights proportional to w.
Proof.The first vertex v( 1) is chosen from [n] via the size-biased distribution.Suppose it has no neighbors.Then, by construction, the next vertex is chosen via the size-biased distribution amongst all remaining vertices.If vertex 1 does have neighbors, then we shall use the following construction.For j ≥ 2 choose τ 1j exponentially distributed with rate (1 + tn −1/3 )w j /l n .Rearrange the vertices in increasing order of their τ 1j values (so that v ′ (2) is the vertex with the smallest τ 1j value, v ′ (3) is the vertex with the second smallest value and so on).Note that by the properties of the exponential distribution Similarly, given the value of v(2), and so on.Thus the above gives us a size-biased ordering of the vertex sex [n] \ {v(1)}.Suppose c(1) of the exponential random variables are less than w 1 .Then set v(j) = v ′ (j) for 2 ≤ j ≤ c(1) + 1 and discard all the other labels.This gives us the first c(1) + 1 values of our size-biased ordering.
Once we are done with v(1), let the potentially unexplored neighbors of v(2) be and, again, for j in U 2 , we let τ 2j be exponential with rate (1 + tn −1/3 )w j /l n and proceed as above.
Proceeding this way, it is clear that at the end, the random ordering v(1), v(2), . . ., v(n) that we obtain is a size-biased random ordering of the vertex set [n].This proves the lemma.
Heuristic derivation of Theorem 1.1.We next provide a heuristic that explains the limiting process in (1.19).Note that by our assumptions on the weight sequence, for the graph G t n (w) where (1.30) In the remainder of the proof, wherever we need p ij , we shall use p * ij instead, which shall simplify the calculations and exposition.
Recall the cluster exploration described above, and, in particular, Lemma 1.3.We explore the cluster one vertex at a time, in breadth-first search.We choose v(1) according to w, i.e., P(v(1) = j) = w j /l n .We say that a vertex is explored when its neighbors have been investigated, and unexplored when it has been found to be part of the cluster found so far, but its neighbors have not been investigated yet.Finally, we say that a vertex is neutral, when it has not been considered at all.Thus, in our cluster exploration, as long as there are unexplored vertices, we explore the vertices (v(i)) i≥1 in the order of appearance.When there are no unexplored vertices left, then we draw (size-biased) from the neutral vertices.Then, Lemma 1.3 states that (v(i)) n i=1 is a size-biased reordering of [n].Let c(i) denote the number of neutral neighbors of v(i), and denote the process The clusters of our random graph are found in between successive times in which (Z n (l)) l∈[n] reaches a new minimum.Now, Theorem 1.1 follows from the fact that Zn (s) = n −1/3 Z n (⌊n 2/3 s⌋) weakly converges to (W t * (s)) s≥0 defined as in (1.19).General techniques from [1] show that this also implies that the ordered excursions between successive minima of ( Zn (s)) s≥0 also converge to the ones of (W t * (s)) s≥0 .These ordered excursions were denoted by γ * 1 (t) > γ * 2 (t) > . ... Using Brownian scaling, it can be seen that with W t defined in (1.15).Hence, from the relation (1.31) it immediately follows that which then proves Theorem 1.1.
We complete the sketch of proof by giving a heuristic argument that indeed Zn (s) = n −1/3 Z n (⌊n 2/3 s⌋) weakly converges to (W t * (s)) s≥0 .For this, we investigate c(i), the number of neutral neighbors of v(i).Throughout this paper, we shall denote wj = w j (1+tn −1/3 ), so that the G t n (w) has weights w = ( wj ) j∈[n] .We note that since p ij in (1.1) is quite small, the number of neighbors of a vertex j is close to Poi( wj ), where Poi(λ) denotes a Poisson random variable with mean λ.Thus, the number of neutral neighbors is close to the total number of neighbors minus the active neighbors, i.e., wv(i) wv(j) ln is, conditionally on (v(j)) i j=1 , the expected number of edges between v(i) and (v(j) i−1 j=1 .We conclude that the increase of the process Z n (l) equals The change in Z n (l) is not stationary, and decreases on the average as l increases, due to two reasons.First of all, the number of neutral vertices decreases (as is apparent from the sum which is subtracted in (1.38)), and the law of wv(l) becomes stochastically smaller as l increases.The latter can be understood by noting that )l n /n, and, by Cauchy-Schwarz, , and the inequality becomes strict when Var(W ) > 0. We now study these two effects in more detail.
The random variable Poi( wv(i) ) − 1 has asymptotic mean However, since we sum Θ(n 2/3 ) contributions, and we multiply by n −1/3 , we need to be rather precise and compute error terms up to order n −1/3 in the above computation.We shall do this rather precisely now, by conditioning on (v(j)) i−1 j=1 .Indeed, ), these terms are indeed both of order n −1/3 , and shall thus contribute to the scaling limit of (Z n (l)) l≥0 .
The variance of Poi( wv(i) ) is approximately equal to ). Summing the above over i = 1, . . ., sn 2/3 and multiplying by n −1/3 intuitively explains that where we write σ 2 = E[W 3 ]/E[W ] and we let (B(s)) s≥0 denote a standard Brownian motion.Note that when Var(W ) > 0, then > 1 and the constant in front of s 2 is negative.We shall make the limit in (1.43) precise by using a martingale functional central limit theorem.
The second term in (1.38) turns out to be well-concentrated around its mean, so that, in this heuristic, we shall replace it by its mean.The concentration shall be proved using concentration techniques on appropriate supermartingales.This leads us to compute the last asymptotic equality again following from the fact that the random variable involved is concentrated.
We conclude that . (1.45) Subtracting (1.45) from (1.43), these computations suggest, informally, that as required.Note the cancelation of the terms s 2 2E[W ] in (1.43) and (1.45), where they appear with an opposite sign.Our proof will make this analysis precise.

Discussion
Our results are generalizations of the critical behavior of Erdős-Rényi random graphs, which have received tremendous attention over the past decades.We refer to [1], [4], [17] and the references therein.Properties of the limiting distribution of the largest component γ 1 (t) can be found in [21], which, together with the recent local limit theorems in [15], give excellent control over the joint tail behavior of several of the largest connected components.
Comparison to results of Aldous.We have already discussed the relation between Theorem 1.1 and the results of Aldous on the largest connected components in the Erdős-Rényi random graph.However, Theorem 1.1 is related to another result of Aldous [1, Proposition 4], which is less well known, and which investigates a kind of Norros-Reittu model (see [20]) for which the ordered weights of the clusters are determined.Here, the weight of a set of vertices A ⊆ [n] is defined by wA = a∈A w a .Indeed, Aldous defines an inhomogeneous random graph where the edge probability is equal to and assumes that the pair (q, (x i ) n i=1 ) satisfies the following scaling relation: x 2 i . (1.48) When we pick then these assumptions are very similar to conditions (a)-(c).However, the asymptotics of q in (1.48) is replaced with where we recall that γ i (t) ı≥1 is the scaling limit of the ordered component sizes in the Erdős-Rényi random graph with parameter p = (1 + tn −1/3 )/n.Now, and one would expect that wC i n (t) ∼ C i n (t), which is consistent with (1.46) and (1.32).
Related models.The model studied here is asymptotically equivalent to many related models appearing in the literature, for example to the random graph with prescribed expected degrees that has been studied intensively by Chung and Lu (see [7,8,9,10,11]).This model corresponds to the rank-1 case of the general inhomogeneous random graphs studied in [5].Here and the generalized random graph [6], for which See [14, Section 2], which in turn is based on [16], for more details on the asymptotic equivalence of such inhomogeneous random graphs.Further, Nachmias and Peres [19] recently proved similar scaling limits for critical percolation on random regular graphs.
Alternative approach by Turova.Turova [22] recently obtained results for a setting that is similar to ours.Turova takes the edge probabilities to be p ij = min{x i x j /n, 1}, and assumes that (x i ) n i=1 are i.i.d.random variables with E[X 3 ] < ∞.This setting follows from ours by taking Naturally, the critical point changes in Turova's setting, and is equal to First versions of the paper [22] and this paper were uploaded almost simultaneously on the ArXiv.Comparing the two papers gives interesting insights in how to deal with the inherent size-biased orderings in two rather different ways.Turova applies discrete martingale techniques in the spirit of Martin-Löf's [18] work on diffusion approximations for critical epidemics, while our approach is more along the lines of the original paper of Aldous [1], relying on concentration techniques and supermartingales (see Lemma (2.2)).Further, our result is slightly more general that the one in [22].In fact, our discussions with Turova inspired us to extend our setting to one that includes i.i.d.weights (Turova's original setting).We should also mention that Turova's first identification of the scaling limit was missing a factor E[X 3 ] in the drift term (which was corrected in a later version).This factor arises from rather subtle effects of the sized-biased orderings.

The necessity of conditions (a)-(c).
The conditions (a)-(c) provide conditions under which we prove convergence.One may wonder whether these conditions are merely sufficient, or also necessary.Condition (b) gives stability of the weight structure, which implies that the local neighborhoods in our random graphs locally converge to appropriate branching processes.The latter is a strengthening of the assumption that our random graphs are sparse, and is a natural condition to start with.We believe that, given that condition (b) holds, conditions (c) and (a) are necessary.Indeed, Aldous and Limic give several examples where the scaling of the largest critical cluster is n 2/3 with a different scaling limit when w 1 n 1/3 → c 1 > 0 (see [2, Proof of lemma 8, p. 10]).Therefore, for Theorem 1.1 to hold (with the prescribed scaling limit in terms of ordered Brownian excursions), condition (a) seems to be necessary.Since conditions (b) and (c) imply condition (a), it follows that if we assume condition (b), then we need the other two conditions for our main result to hold.This answers [1, Open problem (2), p. 851].
Inhomogeneous random graphs with infinite third moments.In the present setting, when it is assumed that E[W 3 ] < ∞, the scaling limit turns out to be a scaled version of the scaling limit for the Erdős-Rényi random graph as identified in [1].In [3], we have recently studied the case where E[W 3 ] = ∞, for which the critical behavior turns out to be fundamentally different.Indeed, when W has a power law with exponent τ ∈ (3, 4), the clusters have asymptotic size n τ −2 τ −1 (see [14]).The scaling limit itself turns out to be a so-called 'thinned' Lévy process, that consists of infinitely many Poisson processes of which only the first event is counted, which already appeared in [2] in the context of random graphs having n 2/3 critical behavior.Moreover, we prove in [3] that the vertex i is in the largest connected component with non-vanishing probability as n → ∞, which implies that the highest weight vertices characterize the largest components ('power to the wealthy').This is in sharp contrast to the present setting, where the probability that vertex 1 (with the largest weight) is in the largest component is negligible, and instead the largest connected component is an extreme value event arising from many trials with roughly equal probability ('power to the masses').

Weak convergence of cluster exploration
In this section, we shall study the scaling limit of the cluster exploration studied in Section 1.2 above.The main result in this paper is the following theorem: Theorem 2.1 (Weak convergence of cluster exploration).Assume that the weight sequence w satisfies conditions (a), (b) and (c).Consider the breadth-first walk Z n (•) of (1.25) exploring the components of the random graph G t n (w).Define Zn (s) = n −1/3 Z n (⌊n 2/3 s⌋). (2.1) where W t * is the process defined in (1.19), in the sense of convergence in the J 1 Skorohod topology on the space of right-continuous left-limited functions on R + .Assume this theorem for the time being and let us show how this immediately proves Theorem 1.1.Comparing (1.15) and (1.25), Theorem 2.1 suggests that also the excursions of Zn beyond past minima arranged in increasing order converge to the corresponding excursions of W t * beyond past minima arranged in increasing order.See Aldous [1,Section 3.3] for a proof of this fact.Therefore, Theorem 2.1 implies Theorem 1.1.The remainder of this paper is devoted to the proof of Theorem 2.1.
Proof of Theorem 2.1.We shall make use of a martingale central limit theorem.From Equation (1.29) we had and we shall use the above as an equality for the rest of the proof as this shall simplify exposition.It is quite easy to show that the error made is negligible in the limit.
Recall from (1.22) that Then, we decompose where with F i the natural filtration of Z n .Then, clearly, {M n (k)} n k=0 is a martingale.For a process {S k } n k=0 , we further write Sn (u) = n −1/3 S n (⌊un 2/3 ⌋). (2.7) Furthermore, let Then, by the martingale central limit theorem ([13, Theorem 7.1.4]),Theorem 2.1 follows when the following three conditions hold: Indeed, the last two equations, by [13,Theorem 7.1.4]imply that the process Mn (s) = n −1/3 M n (n 2/3 s) satisfies the asymptotics Mn where as before B is standard Brownian motion, while (2.9) gives the drift term in (1.19) and this completes the proof.
We note that, since M n (k) is a discrete martingale, where ∆ n is the maximal degree in the graph.It is not hard to see that, by condition (a), wi = o(n 1/3 ), so that E(sup This proves (2.11).We continue with (2.9) and (2.10), for which we first analyse c(i).In the course of the proof, we shall make use of the following lemma, which lies at the core of the argument: Lemma 2.2 (Sums over sized-biased orderings).As n → ∞, for all u > 0, Proof.We start by proving (2.15), for which we write (2.17) We shall use a randomization trick introduced by Aldous [1].Indeed, let T j be a sequence of independent exponential random variables with rate w j /l n and define Note that by the properties of the exponential random variables, if we rank the vertices according to the order in which they arrive then they appear in size-biased order.More precisely, for any v, where N (t) := #{j : T j ≤ t}. (2.20) As a result, when N (2tn 2/3 ) ≥ tn 2/3 whp, we have that We shall prove that both terms converge to zero in probability.We start with the second, for which we use that the process is a supermartingale, since as required.Therefore, (2.24) Using the fact that 1 − e −x ≤ x − x 2 /2, we obtain that, also using that Similarly, by the independence of Observe that (2.29) also immediately proves that, whp, N (2tn 2/3 ) ≥ tn 2/3 .To deal with Hn (v), we define where and note that Y 1 (u) is a supermartingale.Indeed, writing F t to be the natural filtration of the above process, we have, for s < t and letting (2.32) Now using the inequality 1 − e −x ≤ x for x ∈ [0, 1] we get that as required.Again we can easily compute, using condition (a), that Therefore, (2.27) completes the proof of (2.15).The proof of (2.16) is a little easier.We denote Then, we compute explicitly
(2.41) Equation (2.41) in particular implies that w 3 1 = o(n), so that conditions (b) and (c) also imply condition (a).By the weak convergence stated in condition (b),

.42)
As a result, we have that

.43)
The latter converges to 0 when K → ∞, since E[W 3 ] < ∞.We finally show that (2.41) implies that For this, we note that, for each K, when we first let n → ∞, followed by K → ∞.We conclude that, uniformly for i ≤ sn 2/3 ,
To complete the proof of Theorem 2.1, we proceed to investigate c(i).By construction, we have that, conditionally on V i , c where I ij are (conditionally) independent indicators with for all j ∈ V i .Furthermore, when we condition on F i−1 , we know V i−1 , and we have that, for all j ∈ V i−1 , gives us all we need to know to compute conditional expectations involving c(i) given F i−1 .Now we start to prove (2.9), for which we note that

.49)
Then we split By condition (a), the last term is bounded by O P (w 2 1 /l n ) = o P (n −1/3 ) and is therefore an error term.We continue to compute For the first sum, we use the Chebycheff inequality to obtain 3.2 Verification of conditions for weights as in (1.4) Here we check conditions (b) and (c) for the case that w = (w 1 , . . ., w n ) where w i is chosen as in (1.4).We shall frequently make use of the fact that (1.3) implies that 1 − F (x) = o(x −3 ) as x → ∞, which, in turn implies that (see e.g., [12, (B.9)]), as u ↓ 0, which proves all necessary bounds for condition (c) at once.

Corollary 1 . 2 (
Theorem 1.1 holds for key examples).Conditions (a), (b) and (c) are satisfied in the case where w is either as in (1.4) where F is a distribution function of a random variable W with E[W 3 ] < ∞, or when w consists of i.i.d.copies of a random variable W with E[W 3 ] < ∞.Theorem 1.1, in conjunction with Corollary 1.2, proves [14, Conjecture 1.6] .40) Indeed, we shall show that conditions (b) and (c) imply that lim