Heavy-tailed configuration models at criticality
We study the critical behavior of the component sizes for the configuration model when the tail of the degree distribution of a randomly chosen vertex is a regularly-varying function with exponent , where . The component sizes are shown to be of the order for some slowly-varying function . We show that the re-scaled ordered component sizes converge in distribution to the ordered excursions of a thinned Lévy process. This proves that the scaling limits for the component sizes for these heavy-tailed configuration models are in a different universality class compared to the Erdős-Rényi random graphs. Also the joint re-scaled vector of ordered component sizes and their surplus edges is shown to have a distributional limit under a strong topology. Our proof resolves a conjecture by Joseph, Ann. Appl. Probab. (2014) about the scaling limits of uniform simple graphs with i.i.d degrees in the critical window, and sheds light on the relation between the scaling limits obtained by Joseph and this paper, which appear to be quite different. Further, we use percolation to study the evolution of the component sizes and the surplus edges within the critical scaling window, which is shown to converge in finite dimension to the augmented multiplicative coalescent process introduced by Bhamidi et. al., Probab. Theory Related Fields (2014). The main results of this paper are proved under rather general assumptions on the vertex degrees. We also discuss how these assumptions are satisfied by some of the frameworks that have been studied previously.
Most random graph models posses a phase-transition property: there is a model-dependent parameter and a critical value such that whenever , the largest component of the graph contains a positive proportion of vertices with high probability (w.h.p) and when , the largest component is of smaller order than the size of the graph w.h.p The random graph is called critical when . The study of critical random graphs started in the 1990s with the works of Bollobás , Łuczak , Janson et al.  and Aldous  for Erdős-Rényi random graphs. A large body of subsequent work in [36, 37, 30, 13, 10, 20, 43] showed that the behavior of a wide array of random graphs at criticality is universal in the sense that certain graph properties do not depend on the precise description of the model. One of these universal features is that the scaling limit of the large component sizes, for many graph models, is identical to that of the Erdős-Rényi random graphs. All these universality results are obtained under the common assumption that the degree distribution is light-tailed, i.e., the asymptotic degree distribution has sufficiently large moments. For critical configuration models, a finite third-moment condition proves to be crucial . However, empirical studies of real-world networks from various fields including physics, biology, computer science [7, 3, 32, 39, 21, 40] show that the degree distribution is heavy-tailed and of power-law type. A first work towards understanding the critical behavior in the heavy-tailed network models is , which showed that, for rank-one inhomogeneous random graphs, when the weight distribution follows a power-law with exponent , the component sizes and the scaling limits turn out to be quite different from that of the Erdős-Rényi random graph. This revealed an entirely new universality class for the phase transition of heavy-tailed random graphs. In this paper, we study the configuration model with heavy-tailed power-law degrees. The configuration model is the canonical model for generating a random multi-graph with a prescribed degree sequence. This model was introduced by Bollobás  to generate a uniform simple -regular graph on vertices, when is even. The idea was later generalized to general degree sequences by Molloy and Reed  and others. We will denote the multi-graph generated by the configuration model on the vertex set with the degree sequence by . The configuration model, conditioned on simplicity, yields a uniform simple graph with the same degree sequence, which explains its popularity.
Our main contribution.
Let be the degree of a uniformly chosen vertex, independently of the random graph . The main goal of this paper is to obtain various asymptotic results for the component sizes of when for some and a slowly-varying function (here denotes an unspecified approximation that will be defined in more detail below). In fact, under a general set of assumptions (see Assumptions 1 and 2), we prove the following:
The largest connected components are of the order and the width of the scaling window is of the order for some slowly-varying function .
The joint distribution of the re-scaled component sizes and the surplus edges converges in distribution to a suitable limiting random vector under a strong topology. It turns out that the scaling limits for the re-scaled ordered component sizes can be described in terms of the ordered excursions of a certain thinned Lévy process that only depends on the asymptotics of the high-degree vertices, which is also the case in . Further, the scaling limits for the surplus edges can be described by Poisson random variables with the parameters being the areas under the excursions of the thinned Lévy process.
The results hold conditioned on the graph being simple, thus solving [30, Conjecture 8.5] in this case.
The scaling limits also hold for the graphs obtained by performing critical percolation on a supercritical graph. The percolation clusters can be coupled in a natural way using the Harris coupling. This enables us to study the evolution of the component sizes and the surplus edges as a dynamic process in the critical window. The evolution of the component sizes and surplus edges is shown to converge to a version of the augmented multiplicative coalescent process that was first introduced in . In fact, our results imply that there exists a version of the augmented multiplicative coalescent process whose one-dimensional distribution can be described by the excursions of a thinned Lévy process and a Poisson process with the intensity being proportional to the thinned Lévy process, which is also novel.
Thus, this paper provides a detailed understanding about the critical component sizes and surplus edges for the heavy-tailed graphs in the critical window. Before stating our main results precisely, we introduce some notation and concepts.
1.1 The model
Consider vertices labeled by and a non-increasing sequence of degrees such that is even. For notational convenience, we suppress the dependence of the degree sequence on . The configuration model on vertices having degree sequence is constructed as follows:
Equip vertex with stubs, or half-edges. Two half-edges create an edge once they are paired. Therefore, initially we have half-edges. Pick any one half-edge and pair it with a uniformly chosen half-edge from the remaining unpaired half-edges and keep repeating the above procedure until all the unpaired half-edges are exhausted.
Note that the graph constructed by the above procedure may contain self-loops or multiple edges. It can be shown that conditionally on being simple, the law of such graphs is uniform over all possible simple graphs with degree sequence [41, Proposition 7.13]. It was further shown in  that, under very general assumptions, the asymptotic probability of the graph being simple is positive.
1.2 Definition and notation
We use the standard notation of , and to denote convergence in probability and in distribution, respectively. We often use the Bachmann–Landau notation , for large asymptotics of real numbers. The topology needed for the convergence in distribution will always be specified unless it is clear from the context. The notation will be used to say that . We say that a sequence of events occurs with high probability (w.h.p) with respect to the probability measures when . Define when is tight; when whp; if both and . For a random variable and a distribution , we write to denote that has distribution . Denote
with the -norm metric . Let denote the product topology of and with denoting the sequences on endowed with the product topology. Define also
with the metric
Further, let be given by
Let denote the -fold product space of . For any , will denote the element of obtained by suitably ordering the coordinates of .
We often use the boldface notation for the process , unless stated otherwise. will denote the space of càdlàg functions from a locally compact second countable Hausdorff space to the metric space equipped with Skorohod -topology. (resp. ) simply denotes the case (resp. ) with . Consider a decreasing sequence . Denote by where independently, and denotes the exponential distribution with rate . Consider the process
for some and define the reflected version of by
The process of the form (1.5) was termed thinned Lévy processes in  (see also [2, 44]), since the summands are thinned versions of Poisson processes. For any function , define . is the subset of consisting of functions with positive jumps only. Note that is continuous when . An excursion of a function is an interval such that
Excursions of a function are defined similarly. We will use to denote an excursion, as well as the length of the excursion to simplify notation.
Also, define the counting process to be the Poisson process that has intensity at time conditional on . Formally, is characterized as the counting process for which
is a martingale. We use the notation to denote the number of marks in the interval .
Finally, we define a Markov process on , called the augmented multiplicative coalescent (AMC) process. Think of a collection of particles in a system with describing their masses and describing an additional attribute at time . Let be constants. The evolution of the system takes place according to the following rule at time :
For , at rate , the and component merge and create a new component of mass and attribute .
For any , at rate , increases to .
Of course, at each event time, the indices are re-organized to give a proper element of . This process was first introduced in  to study the joint behavior of the component sizes and the surplus edges over the critical window. In , the authors extensively study the properties of the standard version of AMC, i.e., the case and showed in [10, Theorem 3.1] that this is a (nearly) Feller process, a property that will play a crucial rule in the final part of this paper.
1.3 Main results for critical configuration models
Throughout this paper we will use the shorthand notation
where and is a slowly-varying function. We state our results under the following assumptions:
Fix . Let be a degree sequence such that the following conditions hold:
(High-degree vertices) For any fixed ,
(Moment assumptions) Let denote the degree of a vertex chosen uniformly at random from , independently of . Then, , for some integer-valued random variable and
(Critical window) For some ,
Let be the number of vertices of degree-one. Then , which is equivalent to assuming that .
The only previous work to understand the critical behavior of the configuration model with heavy-tailed degrees was by Joseph  where the degrees were assumed to be i.i.d an sample from an exact power-law distribution and the results were obtained for the component sizes of (Theorem 1). We will see that Assumption 1 is satisfied for i.i.d degrees in Section 2.2. Thus, a quenched version of [30, Theorem 8.3] follows from our results. Further, if the degrees are chosen approximately as the weights chosen in , then our results continue to hold. This sheds light on the relation between the scaling limits in  and  (see Remark 11). Moreover, Theorem 3 resolves [30, Conjecture 8.5].
The conclusions of Theorems 1, 2, and 3 hold for more general functionals of the components. Suppose that each vertex has a weight associated to it and let denote the total weight of the component , i.e., . Then, under some regularity conditions on the weight sequence , in Section 8 we will show that the scaling limit for is given by , where the constant is given by . Observe that, for , gives the asymptotic number of vertices of degree in the largest component.
It might not be immediate why we should work with Assumption 1. We will see in Section 2.1 that Assumption 1 is satisfied by the degree sequences in some important and natural cases. The reason to write the assumptions in this form is to make the properties of the degree distribution explicit (e.g. in terms of moment conditions and the asymptotics of the highest degrees) that jointly lead to this universal critical limiting behavior. We explain the significance of Assumption 1 in more detail in Section 3.
1.4 Percolation on heavy-tailed configuration models
Percolation refers to deleting each edge of a graph independently with probability . Consider percolation on a configuration model under the following assumptions:
Let denote the graph obtained through percolation on with bond retention probability . The following result gives the asymptotics for the ordered component sizes and the surplus edges for :
Now, consider a graph satisfying Assumption 2 (i). To any edge between vertices and (if any), associate an independent uniform random . Note that the graph obtained by keeping only those edges satisfying is distributed as . This construction naturally couples the graphs using the same set of uniform random variables. Our next result shows that the evolution of the component sizes and the surplus edges of , as varies, can be described by a version of the augmented multiplicative coalescent process described in Section 1.2:
Theorem 5, in fact, shows that there exists a version of the AMC process whose distribution at each fixed can be described by the excursions of a thinned Lévy process and an associated Poisson process. This did not appear in [10, 19], since the scaling limits in their settings were described in terms of the excursions of a Brownian motion with parabolic drift.
The additional assumption in Theorem 5 about the asymtotics is required only in one place for Proposition 24 and the rest of the proof works under Assumption 2 only. That is why we have separated this assumption from the set of conditions in Assumption 2. It is worthwhile mentioning that the condition is not stringent at all, e.g., we will see that this condition is satisfied under the two widely studied set ups in Section 2.1.
As we will see in Section 10, the proof of Theorem 5 can be extended to more general functionals of the components. For example, the evolution of the number of degree vertices along with the surplus edges can also be described by an AMC process. The key idea here is that these component functionals become approximately proportional to the component sizes in the critical window and thus the scaling limit for the component functionals becomes a constant multiple of the scaling limit for the component sizes.
2 Important examples
2.1 Power-law degrees with small perturbation
As discussed in the introduction, our main goal is to obtain results for the critical configuration model satisfying for some . In this section, we consider such an example and show that the conditions of Assumption 1 are satisfied. Thus, the results in Section 1.3 hold for in the following set-up that is closely related to the model studied in  for rank-1 inhomogeneous random graphs.
Fix . Suppose that is the distribution function of a discrete non-negative random variable such that
where is a slowly-varying function so that the tail of the distribution is decaying like a regularly-varying function. Recall that the inverse of a locally bounded non-increasing function is defined as . Therefore, using [15, Theorem 1.5.12],
where is another slowly-varying function. Note that [15, Theorem 1.5.12] is stated for positive exponents only. Since our exponent is negative, the asymptotics in (2.2) holds for . Suppose that the random variable is such that
Define the degree sequence by taking the degree of the vertex to be
where the ’s are non-negative integers satisfying the asymptotic equivalence
Note that, by (2.6), . Also, since is non-increasing
From our choice of , and therefore . By [15, Lemma 1.3.2], . Moreover, and . Thus, the proof follows by first taking and then . ∎
Firstly, Lemma 6 guarantees the convergence of the second moment of the degree sequence. However, (2.10) is more about obtaining sharper asymptotics for . We use similar arguments as in [14, Lemma 2.2]. Denote . Note that , so it is enough to verify that
Fix any . We have
Now by (2.4), . Therefore,
Again, using (2.4),
where the last equality follows using the fact that is a slowly-varying function. Note that the error term in (2.15) satisfies for each fixed . Again,
Using Euler-Maclaurin summation [23, Page 333] it can be seen that is finite which completes the proof. ∎
Note that if we add approximately ( is a constant) ones in the degree sequence given in (2.4), then we end up with another configuration model for which with . Similarly, deleting ones from the degree sequence increases the new value. This gives an obvious way to perturb the degree sequence in such a way that the configuration model is in different locations within the critical scaling window. In our proofs, we will only use the precise asymptotics of the high degree vertices. Thus, a small (suitable) perturbation in the degrees of the low degree vertices does not change the scaling behavior fundamentally, except for a change in the location inside the scaling window.
2.2 Random degrees sampled from a power-law distribution
We now consider the set-up discussed in . Let be i.i.d samples from a distribution , where is defined in (2.1). Therefore, the asymptotic relation in (2.2) holds. Consider the random degree sequence where , being the order statistic of . We show that satisfies Assumption 1 almost surely under a suitable coupling. We use a coupling from [18, Section 13.6]. Let be an i.i.d sequence of unit rate exponential random variables and let . Let
It can be checked that and therefore, we will ignore the bar in the subsequent notation. Note that, by the stong law of large numbers, . Thus, for each fixed , . Using (2.2), we see that satisfies Assumption 1 (i) almost surely under this coupling with . The first two conditions of Assumption 1 (ii) are trivially satisfied by almost surely using the strong law of large numbers. To see the third condition, we first claim that
To see (2.20), note that has a Gamma distribution with shape parameter and scale parameter 1. Thus, for ,
where is the Gamma function and the last equality follows from Stirling’s approximation. Therefore,
and (2.20) follows. Now, using the fact that , we can use arguments identical to (2.9) to show that on the event . Thus, we have shown that the third condition of Assumption 1 (ii) holds almost surely.
Therefore, the results in Section 1.3 hold conditionally on the degree sequence if we assume the degrees to be i.i.d samples from a distribution of the form (2.1). For the percolation results, notice that the additional condition in Theorem 5 is a direct consequence of the convergence rates of sums of i.i.d sequence of random variables [31, Corollary 3.22].
Let us recall the limiting object obtained in [30, Theorem 8.1] and compare this with the limiting object , defined in (1.5) with given by (2.24). We will prove an analogue of [30, Theorem 8.1] in Theorem 8. Although we use a different exploration process from , the fact that the component sizes are huge compared to the number of cycles in a component, one can prove Theorem 8 for the exploration process in  also. This will indirectly imply that Joseph’s limiting object obeys the law of , averaged out over the -values. This is counter intuitive, given the vastly different descriptions of the two processes; for example our process does not have independent increments. We do not have a direct way to prove the above mentioned claim.
Assumptions on the degree distribution. Let us now briefly explain the significance of Assumption 1. Unlike the finite third-moment case , the high-degree vertices dictate the scaling limit in Theorem 1 and therefore it is essential to fix their asymptotics through Assumption 1 (i). Assumption 1 (iii) defines the critical window of the phase transition and Assumption 1 (iv) is reminiscent of the fact that a configuration model with negligibly small amount of degree-one vertices is always supercritical. Assumption 1 (ii) states the finiteness of the first two moments of the degree distribution and fixes the asymptotic order of the third-moment. The order of the third-moment is crucial in our case. The derivation of the scaling limits for the components sizes is based on the analysis of a walk which encodes the information about the component sizes in terms of the excursions above its past minima [4, 37, 20, 14, 13]. Now, the increment distribution turns out to be the size-biased distribution with the sizes being the degrees. Therefore, the third-moment assumption controls the variance of the increment distribution. Another viewpoint is that the components can be locally approximated by a branching process with the variance of the same order as the third-moment of the degree distribution. Thus Assumption 1 (ii) controls the order of the survival probability of , which is intimately related to the asymptotic size of the largest components.
Connecting the barely subcritical and supercritical regimes. The barely subcritical (supercritical) regime corresponds to the case when for some () and . Janson  showed that the size of the largest cluster for a subcritical configuration model (i.e., the case and ) is (see [24, Remark 1.4]). In , we show that this is indeed the case for the entire barely subcritical regime, i.e., the size of the largest cluster is . In the barely supercritical case, the giant component can be locally approximated by a branching process having variance of the order