Regularity lemmas for clustering graphs

Abstract

For a graph G with a positive clustering coefficient C, it is proved that for any positive constant ϵ, the vertex set of G can be partitioned into finitely many parts, say $S_1,S_2,\dots ,S_m$, such that all but an ϵ fraction of the triangles in G are contained in the projections of tripartite subgraphs induced by $(S_i,S_j,S_k)$ which are ϵ-Δ-regular, where the size m of the partition depends only on ϵ and C. The notion of ϵ-Δ-regular, which is a variation of ϵ-regular for the original regularity lemma, concerns triangle density instead of edge density. Several generalizations and variations of the regularity lemma for clustering graphs are derived.

Introduction

One of the celebrated results of Szemerédi [19] is the so-called regularity lemma which asserts that for any graph on n vertices, the vertex set can be partitioned into finitely many parts so that almost all but $ϵn^2$ edges are contained in the union of bipartite subgraphs between pairs of the parts that are random-like under the notion of ϵ-regular. A bipartite graph is said to be ϵ-regular, if the edge density on any induced sub-bipartite graph on at least ϵn vertices differs from the edge density of the bipartite graph by at most ϵ. The regularity lemma has been a powerful tool in graph theory with numerous applications [11], [14], [17] because any graph (with more than $ϵn^2$ edges) can be approximated by a finite graph in the sense that each vertex of the finite graph can be replaced by a subset of vertices and the bipartite subgraphs between any two subsets are quasirandom.

A major deficiency of the regularity lemma is the fact that it is useful only for graphs with a positive edge density since the error bound of approximation is of order $ϵn^2$. There have been numerous attempts for possible extensions of the regularity lemma to sparse graphs, mostly with either additional assumptions [13] or weakened conditions [9], [18].

In this paper, we give a regularity lemma for clustering graphs without any restriction on edge density. We note that many information networks and social network graphs contain a large number of triangles and thus have nontrivial clustering coefficients [16], [20]. Such a clustering effect is one of the main characteristics of the so-called “small world phenomenon” that appear in a variety of real world graphs [15]. There are many research papers concerning finding dense subgraphs [2], [3] or partitioning into dense clique-like subgraphs [12] for such small-world graphs.

In this paper, we focus on graphs with nontrivial clustering coefficients (or triangle density). Let $t_{\begin{array}{c}\hfill \hfill \\ \hfill G\hfill \end{array}}$ denote the number of triangles in G and $p_{\begin{array}{c}\hfill \hfill \\ \hfill G\hfill \end{array}}$ denote the number of paths of two edges. The clustering coefficient $C_{\begin{array}{c}\hfill \hfill \\ \hfill G\hfill \end{array}}$ is defined to be (see [16])

$$C_{\begin{array}{c}\hfill \hfill \\ \hfill G\hfill \end{array}}=\frac{3t_{\begin{array}{c}\hfill \hfill \\ \hfill G\hfill \end{array}}}{p_{\begin{array}{c}\hfill \hfill \\ \hfill G\hfill \end{array}}}.$$

If $p_{\begin{array}{c}\hfill \hfill \\ \hfill G\hfill \end{array}}=0$, we define $C_{\begin{array}{c}\hfill \hfill \\ \hfill G\hfill \end{array}}=0$. We say G is a clustering graph if its clustering coefficient $C_{\begin{array}{c}\hfill \hfill \\ \hfill G\hfill \end{array}}$ is a positive constant independent of the number of vertices of G.

Theorem 1

For any $ϵ>0$ and any graph G with clustering coefficient C, the vertex set of G can be partitioned into $S_1,S_2,\dots ,S_m$ for some m depending only on ϵ and C, such that all but $ϵt_G$ triangles in G are contained in the projections of tripartite subgraphs with vertex set $(S_i,S_j,S_k)$ that are ϵ-Δ-regular.

The detailed definitions of various terms above will be given in Section 2. The proof of the regularity lemma for clustering graphs are quite similar to the previous proofs for the original regularity lemma [4], [14], [19] except for using an index function involving clustering coefficients. In Section 3 we give a proof of the regularity lemma for tripartite graphs with nontrivial clustering coefficient. The proof is self-contained and relatively short. In Section 4 we then consider a strong version of ϵ-Δ-regular for tripartite graphs. In Section 5 we give a proof of Theorem 1 and a weighted version of the regularity lemma both of which are straightforward applications of the regularity lemma for tripartite graphs with nontrivial clustering coefficients. In Section 6, we consider several generalizations of the regularity lemma. We will give a regularity for graphs which is dense in 4-cycles and, in general, graphs which contain a relatively large number of any specified graph (in comparison with its subgraphs). Some remarks and problems are mentioned in Section 7.