Abstract
One of the classical questions in random graph theory is to understand the asymptotics of subgraph counts. In inhomogeneous random graph, this question has not been well studied. In this study, we investigate the asymptotic distribution of -cliques in a sparse inhomogeneous random graph. Under mild conditions, we prove that the number of -cliques converges in law to the standard normal distribution.
1. Introduction
The random graph theory was founded by Erdös–Rényi [1]. The well-known Erdös–Rényi random graph is an undirected graph on vertices’ (nodes) set , where any two nodes form an edge independently with probability . In random graph theory, one of the classical questions concerns the asymptotics of the number of subgraphs [2–7]. Much attention has been paid to the limiting distribution of subgraph counts. In a dense Erdös–Rényi graph, the subgraph count converges in law to the standard normal distribution under some conditions [3]. This result was proven to be true for sparse random graph [2]. Analogue results exist for the number of strictly balanced subgraph or short cycles in random regular graphs [4–6, 8].
In practice, a lot of real networks display the inhomogeneity property, that is, the vertex degrees vary a lot. In many cases, the degree follows a power law [9]. To accommodate the inhomogenity, the inhomogeneous random graph has recently been introduced [10]. It is natural to study the asymptotics of subgraphs in an inhomogeneous random graph. Some of the first results concern the asymptotic clique or cycle number in special inhomogeneous random graph [9, 11, 12, 23, 24]. For example, the authors gave the upper bound and lower bound of the average number of cycles in [12]. The authors of [9, 13] obtained the asymptotic order of large cliques in scale-free random graph, and Janson [14] studied the order of the largest component. In [12], the order of expected number of cycles and cliques in a random graph were given. Hu and Dong [15] derived the limiting distribution of the number of edges in a generalized random graph, and Liu and Dong [16] derived the asymptotic distribution of the number of triangles. However, to our knowledge, the limiting distribution of the number of cliques is unknown. In this study, we study this problem in sparse inhomogeneous random graph and derive its asymptotic distribution.
2. The Model and Main Result
In this section, we introduce the model and present the main result. The inhomogeneous random graph is defined as follows: given a positive array , every pair of nodes in are joined as an edge with probability independently. The adjacency matrix of a graph is a symmetric -matrix with zeros on its diagonal and if is an edge, otherwise. For , the adjacency matrix is a symmetric random matrix, with elements following independent Bernoulli distributions, that is,and is independent of if . By symmetry, for and for . This model was introduced in [10], and it contains the models in [9, 11, 18] as a special case. If , the random graph is the homogeneous Erdös–Rényi model, where nodes form an edge with probability [1].
In graph , an -clique is a subgraph of vertices such that any two distinct vertices are adjacent. Given the adjacency matrix and vertices , the node set forms an -clique if and only if
Then, the total number of -clique in is
For the random graph , is a sum of dependent random variables. The asymptotic distribution of is given in the following theorem.
Theorem 1. Let be the number of -cliques in the random inhomogeneous graph with expectation and variance . Suppose is a fixed integer, for some ,for some positive constants , and
Then,where ‘’ represents convergence in distribution.
According to Theorem 1, the scaled and centered number of -cycle in converges in law to the standard normal distribution. Note that Theorem 1 only holds for sparse random graph. The first equation in (4) implies and the second one requires the average degree tends to infinity. Our proof relies on the martingale central limit theorem.
The conditions on seem to be very restrictive. In fact, there are many models which satisfy these conditions. Here, we provide some of the examples that satisfy the conditions of Theorem 1. Consider the rank one model with
, . Suppose satisfies (4). Straightforward computation yields
Hence, (5) holds and Theorem 1 applies to this model. The average degree of each vertex is given by
The degree is approximately times of , that is, . Hence, this model is essentially different from the homogeneous Erdös–Rényi model. Furthermore, suppose , satisfying (4) for . Then, for some constant ,
This model is a special case in [17], where the authors studied asymptotic behavior of the extreme eigenvalues.
Another interesting model is the power-law graphs [22]:
Rewrite the probabilities aswhere and . If , then for some constant . For some generic positive constant ,
3. Proof of Main Result
In this section, we prove Theorem 1. For convenience, denote if for some constants and ; if . The proof relies on the following result.
Proposition 1 (see [20]). Suppose that, for every and , the random variables are a martingale difference sequence relative to an arbitrary filtration . If (I) in probability, (II) in probability, for every , then in distribution.
Now, we prove Theorem 1. Let , , and
Denote , the complement set of in . By Taylor expansion, for generic constants , , it followswhere
Next, we show the first term in (15) is the leading term. Note that if , by the independence of and , it follows
Then, it s easy to obtain
For convenience, let
Note that under the assumption of Theorem 1. To get the order of , let and ; then,which implies . For , let and . Then,from which it yields . For , any involves all the vertices . Hence, by a similar proof of (20) and (21), it follows that . For , any involves at least distinct vertices; then, . When , contains at least 3 distinct vertices; then, .
We claim the following hold:
Firstly, according to the first equation in (4), (22) and (24) hold trivially. Let ; then, by (22). Equations (23) and (25) hold if
Simple algebra yields
Hence, . Then, the first equation in (4) implies (22)–(25).
As a result, by (15), one hasand . Therefore, to prove Theorem 1, it suffices to prove
In the following, we use Proposition 1 to prove (29). To this end, define for , where and
Let and . Then,which implies is a Martingale difference.
Check condition in Proposition 1. It suffices to show the following:
For (30), direct computation yields
For (32), let . Then,
Let and . Then,
Now, (32) follows from the following calculation:
Hence, in probability.
Check condition in Proposition 1. By Cauchy–Schwarz inequality and Markov inequality, for any and generic constant , it follows thatif . Hence, . By Proposition 1, the proof is complete.[19, 21]
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.