Large subgraphs in pseudo-random graphs

1. (i)

   In Section 2 we apply Theorem 1.1 to exponential random graph models (ergms) in the high temperature regime (Section 2.1), random geometric graphs (Section 2.2), conditioned Erdős-Rényi random graphs (Section 2.3), and random regular graphs (Section 2.4). Section 2.5 describes a network model where Theorem 1.1 is optimal. Proofs of all these results can be found in Section 8.

2. (ii)

   Section 3 discusses the relevance of our results, connections to existing literature as well as possible extensions and future directions to pursue.

3. (iii)

   In Section 4 we provide an outline of our proof technique. We start this section by stating Proposition 4.1, which is the sub-case of Theorem 1.1 for $p=1/2$. For clarity of presentation, we only explain the idea behind the proof of Proposition 4.1. Along with the proof idea we introduce necessary notation and definitions.

4. (iv)

   We prove Proposition 4.1 in Section 5. We state the lemmas needed to prove Proposition 4.1. Proofs of these lemmas are deferred to Section 6.

5. (v)

   In Section 7 we provide a detailed outline about how one can extend the ideas from the proof of Proposition 4.1 to prove Theorem 1.1. Proof of Theorem 1.4 can also be found in Section 7. Section 8 contains proofs for the applications our main result.

The exponential random graph model (ergm) is one of the most widely used models in areas such as sociology and political science since it gives a flexible method for capturing reciprocity (or clustering behavior) observed in most real world social networks. The last few years have seen major breakthroughs in rigorously understanding the behavior of ergm in the large network $n\to\infty$ limt (see [5, 15] and the references therein). It has been shown that in the high temperature regime (which we precisely define in Assumption 2.1), these models converge in the so-called cut-distance, a notion of distance between graphs established in [7], [8], to the same limit as an Erdős-Rényi random graph as the number of vertices increase, where the edge connectivity probability of the Erdős-Rényi random graph is determined explicitly by a function of the parameters (see [15]). In particular this implies that the number of induced isomorphic copies of any subgraph on fixed number of vertices are asymptotically same as in an Erdős-Rényi random graph in that high temperature regime (see [7, Theorem 2.6]). In this section we strengthen the above result to show that in the high temperature regime the number of induced isomorphic copies of subgraphs of growing size in an ergm are approximately same with that of an Erdős-Rényi graph, with appropriate edge connectivity probability.

We start with a precise formulation of the model. We will stick to the simplest specification of this model which incorporates edges and triangles although we believe that the results below carry over for any model in the ferromagnetic case as long as one is in the high temperature regime. Let $\Omega:=\{0,1\}^{{n\choose 2}}$ and note that any simple graph on $n$ vertices can be represented by a tuple $\bm{x}:=(x_{ij})_{1\leqslant i<j\leqslant n}\in\Omega$. Here $x_{ij}=1$ if vertices $i,j$ are connected by an edge and $x_{ij}=0$ otherwise. On this space, we define the Hamiltonian $H$ as follows.

Note that $E(\bm{x}):=\sum_{1\leqslant i<j\leqslant n}x_{ij}$ is just the number of edges in $\bm{x}$ whilst $T(\bm{x}):=\sum_{1\leqslant i<j<k\leqslant n}x_{ij}x_{jk}x_{ik}$ is the number of triangles. Thus the above Hamiltonian has a simpler interpretation as

For simplicity we write $\bm{\beta}:=(\beta,\gamma)$ for the parameters of the model, and consider the probability measure,

In the sequel, to ease notation, we will suppress $n$ and write the above, often as $p_{\bm{\beta}}$. Now for later use we define the function $\varphi_{\bm{\beta}}:[0,1]\to[0,1]$ via,

Below we state the assumption on the ergm with which we work in this paper.

Now we are ready to state the main result regarding ergm.

Theorem 2.2 follows from Theorem 1.1 once we establish that assumptions (A1)-(A2) are satisfied by ${\bm{X}^{\scriptscriptstyle(n)}}$ almost surely. We defer the proof to Section 8.

In this section we consider thresholded graphs obtained from high-dimensional random geometric graphs. Roughly in a geometric graph the vertices are points in some metric space and two vertices are connected if they are within a specified threshold distance. These models have drawn wide interests from different branches of science such as machine learning, statistics etc. It is of interest to study these graphs when the dimension of the underlying metric space is growing with the number of vertices. In the context of applications, one major are of applications of this model is neuroscience, while it is impossible to give even a representative sample of related papers, a starting point would be [11, 35, 24] and the references therein. Paraphrasing from [11, Page 187] (with parts (c) and (d) most relevant for this paper):

“Structural and functional brain networks can be explored using graph theory through the following four steps: ”

1. (a)

   Define the network nodes. These could be defined as electroencephalography or multielectrode-array electrodes, or as anatomically defined regions of histological, MRI or diffusion tensor imaging data.

2. (b)

   Estimate a continuous measure of association between nodes. This could be the spectral coherence or Granger causality measures between two magnetoencephalography sensors, $\ldots$ or the inter-regional correlations in cortical thickness or volume MRI measurements estimated in groups of subjects.

3. (c)

   Generate an association matrix by compiling all pairwise associations between nodes and (usually) apply a threshold to each element of this matrix to produce a binary adjacency matrix or undirected graph.

4. (d)

   Calculate the network parameters of interest in this graphical model of a brain network and compare them to the equivalent parameters of a population of random networks.”

As a test-bed, we study the simplest possible setting for such questions, first studied rigorously in [21] and [10], and study the behavior of the graph as dimension is growing with the number of vertices. To describe the model, we closely follow [21]. Write $\mathscr{S}_{d-1}:=\{\bm{x}\in\mathbb{R}^{d}:\|\bm{x}\|_{2}=1\}$ for the unit sphere in $\mathbb{R}^{d}$, where $\|\cdot\|_{2}$ denotes the usual Euclidean metric. Suppose $\{\bm{X}_{i}:i\in[n]\}$ be $n$ points chosen independently and uniformly distributed on $\mathscr{S}_{d-1}$. Fix $p\in(0,1)$. We will use these points to construct a graph with vertex set $[n]$ as follows: For $i\neq j\in[n]$ say that vertex $i$ is connected to $j$ if and only if

Here $\langle\cdot,\cdot\rangle$ is the usual inner product operation on $\mathbb{R}^{d}$ and $t_{p,d}$ is a constant chosen such that

Call the resulting graph ${\sf G}(n,d,p)$. See Figure 2 for a pictorial representation.

Maximal clique behavior in this model was studied in [21]. Further it was shown in [21] that the total variation distance between ${\sf G}(n,d,p)$ and ${\sf G}(n,p)$ goes to zero as $d\rightarrow\infty$ while $n$ is fixed. It was further shown in [10] that the total variation distance between ${\sf G}(n,d,p)$ and ${\sf G}(n,p)$ goes to zero if $d$ grows faster than $n^{3}$ while the distance goes to $1$ if $d/n^{3}\rightarrow 0$. In this section we will consider weaker notions of convergence of graphs and establish that random geometric graphs look similar to Erdős-Rényi when $d$ is growing as slow as $(\log n)^{c}$ for some $c>1$. Below we work with following assumption on random geometric graphs.

We split our result on ${\sf G}(d,n,p)$ into two parts. In first part we consider induced isomorphic copies of any fixed graph.

Theorem 2.4 shows that the number of induced isomorphic copies of any fixed graph is asymptotically same as that of an Erdős-Rényi random graph. Therefore, from [7, Theorem 2.6] it follows that ${\sf G}(d,n,p)$ and ${\sf G}(n,p)$ are arbitrarily close in cut-metric as soon as $d$ is poly-logarithmic in $n$. Theorem 2.4 is actually a consequence of the following stronger result. This result shows that the number of copies of “large subgraphs” are asymptotically same as that of Erdős-Rényi random graph. However, the size of the subgraphs one can consider depends on how fast $d$ “grows” compared to $n$. More precisely, we have the following result.

Finding maximal cliques in graphs is a fundamental problem in computational complexity. One natural direction of research is average case analysis; more precisely studying this problem in the context of the Erdős-Rényi random graph. Here it is known that the maximal clique in ${\sf G}(n,1/2)$ is $(2+o(1))\log_{2}\hskip-2.0ptn$. Simple greedy algorithms have been formulated to find cliques of size $\log_{2}\hskip-2.0ptn$ in polynomial time but no algorithms are known for finding cliques closer to the maximal size of $(2-\varepsilon)\log_{2}\hskip-2.0ptn$ with $\varepsilon$ small, in polynomial time. The situation is different when one places (hides) a clique of large size in the random graph, the so-called hidden clique problem especially when the size of the hidden clique is of size $\kappa\sqrt{n}$ for some absolute constant $\kappa$. Here a number of polynomial algorithms have been formulated to discover the maximal clique. For example [3] proposed a spectral algorithm that finds a hidden clique of size $\sqrt{cn\log_{2}n}$ in polynomial time. In [20] the authors devised an almost linear time non-spectral algorithm to find a clique of size $\sqrt{n/e}$. Dekel, Gurel-Gurevich, and Peres in [19] describe the “most important” open problem in this area as devising an algorithm (or proving the lack of existence thereof) that finds a hidden clique of size $o(\sqrt{n})$ in polynomial time. Motivated by these questions we investigate the following question,

How does an Erdős-Rényi random graph look like when the largest clique is of size ${n^{1/2-\varepsilon}}$?

We check that the graph is strongly pseudo random graph even when the largest clique is of size ${n^{1/2-\varepsilon}}$ for some $\varepsilon>0$.

For positive integers $n$ and $d:=d(n)$, the random $d$-regular graph ${\sf G}(n,d)$ is a random graph which is chosen uniformly among all regular graphs on $n$ vertices of degree $d$. One key difference between ${\sf G}(n,d)$ and ${\sf G}(n,p)$ is that the edges in ${\sf G}(n,d)$ are not independent of each other. However, using different techniques researchers have been able to study different properties of ${\sf G}(n,d)$. For more details, we refer the reader to [6, 31]. Here we study the number of induced isomorphic copies of large subgraph in ${\sf G}(n,d)$. We obtain the following result. Before stating the result let us recall $a_{n}=\Theta(b_{n})$ implies that theres exists constant $c_{1}$ and $c_{2}$ such that $c_{1}a_{n}\leq b_{n}\leq c_{2}a_{n}$, for all large values of $n$.

The proof of Theorem 2.10 is a direct application of Theorem 1.1 once we establish that ${\sf G}(n,d)$ satisfies conditions (A1)-(A2). However, this is already known in the literature.

Proof of Theorem 2.10 is immediate from Theorem 2.11.

Now as mentioned above we provide the example which shows that the result Theorem 1.1 is optimal. This corresponds to a graph on a vector space over the binary field. For any odd integer $k$ this graph ${\sf G}_{n}^{b}$ is a graph on $n$ vertices where $n=2^{k-1}-1$ and “$b$” is to emphasize “binary” which we now elucidate. The vertices of ${\sf G}_{n}^{b}$ are binary tuples of length $k$ with odd number of ones except the vector of all ones. Note that every vertex $v$ has also a vector representation. We draw an edge between two vertices $u$ and $v$ if and only if $\langle u,v\rangle=1$, where the multiplication and the addition are the corresponding operations in the binary field. Now we are ready to state the main result on ${\sf G}_{n}^{b}$.

Proof of Theorem 2.14(i) is a direct application of Theorem 1.1. We only need to verify that assumptions (A1)-(A2) are satisfied by the graph sequence $\{{\sf G}_{n}^{b}\}$ with $p=1/2$ and $\delta\leq 1/2$. Theorem 2.14(ii)-(iii) have deeper implications in the context of pseudo-random graphs. Theorem 2.14(ii) shows that if in (1.1) we restrict ourselves to some subsets of $\mathcal{G}(r)$ then one may be able to improve the conclusion of Theorem 1.1. Whereas, Theorem 2.14(iii) shows that one cannot expect any such improvement to hold uniformly for all subgraphs in $\mathcal{G}(r)$ beyond the $\frac{1}{2}\log_{2}\hskip-2.0ptn$ barrier, as predicted by Theorem 1.1. Therefore, Theorem 1.1 gives us an optimal result under the assumptions (A1)-(A2). Theorem 2.14(ii) further shows that even if we restrict our attention to some subset of $\mathcal{G}(r)$ any improvement must not go beyond $r=\log_{2}\hskip-2.0ptn$.

In the context of pseudo-random graphs results similar to (1.1) first appeared in the work of Thomason [37]. To explain his result, let us define his notion of Jumbled graph.

In [37, Theorem 2.10] it was shown that for $(p,\alpha)$-jumbled graphs, with $p\leq 1/2$, (1.1) holds whenever $s$ is such that $p^{s}n\gg 42\alpha s^{2}$. In [28] it was established that $\alpha$ must be $\Omega(\sqrt{np})$ (recall $a_{n}=\Omega(b_{n})$ implies that $\liminf_{n}a_{n}/b_{n}\geq c>0$). When $\alpha=\Theta(\sqrt{np})$, then we must have $p^{s}\sqrt{n}\gg 1$ for (1.1) to hold for $(p,\alpha)$-jumbled graphs. Note that, when $\delta,p\leq 1/2$, under the same assumption on $s$ we establish Theorem 1.1.

Pseudo-random graphs satisfying assumptions (A1)-(A2) and jumbled graphs are not very far from each other. For example, if ${\sf G}_{n}$ satisfies (A1)-(A2), then one can check that ${\sf G}_{n}$ is a $(p,O(n^{({1+\delta})/{2}}))$-jumbled graph (see [37, Theorem 1.1]). On the other hand, if ${\sf G}_{n}$ is a $(p,n^{\delta^{\prime}})$-jumbled graph then all but few vertices satisfy assumption (A1) for any $\delta>\delta^{\prime}$ (see [37, Lemma 2.1]). So our assumptions (A1)-(A2) and the condition (3.1) are somewhat comparable.

The advantage with (A1)-(A2) is that given any graph one can easily check those two conditions. On other hand, given any large graph, it is almost impossible to check the condition (3.1) for all $U\subset[n]$. For example, condition (3.1) would be very hard to establish for ergms, random geometric graphs, Erdős-Rényi random graphs conditioned on the existence of a large clique, and even random $d$-regular graphs. We have already seen in Section 2 that for these graphs one can use Theorem 1.1 and deduce that the number of slowly growing subgraphs in those graphs are approximately same as that of an Erdős-Rényi random graph with an appropriate edge connectivity probability.

As mentioned above, if (A1)-(A2) are satisfied by graph ${\sf G}_{n}$, then it is also jumbled graph with certain parameters. Therefore. one can try to apply [37, Theorem 2.10] to obtain a result similar to ours. However, the application of [37, Theorem 2.10] yields a sub-optimal result in the following sense. Note that both random $d$-regular graphs and exponential random graph models satisfy assumptions (A1)-(A2) with $\delta=1/2$ (see Theorem 2.11 and Theorem 8.1). Therefore, applying Theorem 1.1 one would obtain that the number of induced isomorphic copies of subgraphs of size $\frac{1}{2}\log_{2}\hskip-2.0ptn$ (for ease of explanation, let us consider only the case $d/n=p^{*}=1/2$ in Theorem 2.10 and Theorem 2.2) are asymptotically same as that of an ${\sf G}(n,\frac{1}{2})$. However, application of [37, Theorem 1.1] and [37, Theorem 2.10] implies that the same holds for $s\leq\frac{1}{4}\log_{2}\hskip-2.0ptn$. Therefore Theorem 1.1 improves the result of [37].

We now turn our attention to the class of pseudo-random graphs defined by Chung, Graham, and Wilson [16].

One of our main motivations for this work was the notion introduced by Chung, Graham, and Wilson [16]. We state the main theorem in their paper that relates various graph parameters and any of them can be taken as a definition of pseudo-randomness. Before stating that result recall that for a sequence of reals $\{a_{n}\}$ one writes $a_{n}=o(1)$ to imply that $a_{n}\rightarrow 0$ as $n\rightarrow\infty$. We only state parts of the main theorem of [16] that are relevant in the context of our result.

Although [16, Theorem 1] was proved only for $p=1/2$ one can easily adapt their technieues to easily extend the result for any $p\in(0,1)$. In this paper one of our main aims is to understand the role of concentration inequalities and rates of convergence in the above conditions. That is, if we specify the rates of convergence in (3.3) and/or in (3.4), then a natural question is whether one can accommodate a slowly growing $r$ in (3.2). Motivated by this observation we began this work. Note that our assumptions (A1)-(A2) are similar to $P_{7}$ above.

Investigating the proof of Theorem 1.1 one can easily convince oneself that we do not need the maximum over all vertices (and pair of vertices) in (A1)-(A2). The proof goes through as long as the assertions (A1)-(A2) hold on the average. Indeed, using Markov’s inequality, and increasing $\delta$ by some multiple of $\frac{\log n}{\log\log n}$, one can establish that (A1)-(A2) holds for all but few vertices. This does not change the conclusion Theorem 1.1 and it only increases the constant $C_{0}^{\prime}$. For clarity of presentation we do not pursue this generalization and work with the assumption that (A1)-(A2) holds for all vertices (and pair of vertices).

In the context of quasi-random graphs attempts have been made to prove that (3.2) holds even one allows $r$ to go to infinity. In [18] Chung and Graham showed that if a subgraph ${\sf H}_{s}$ is not an induced subgraph of ${\sf G}_{n}$ then one must have a set $S$ of size $n/2$ such that $e(S)$ deviates from $\frac{n^{2}}{16}$ by at least $2^{-2(s^{2}+27)n^{2}}$. This implies that in a quasi-random graph ${\sf G}_{n}$ at least one induced copy of every subgraph of size $O(\sqrt{\log n})$ must be present.

One may also want to check upon plugging in the rates of convergence from (A1)-(A2) and following the steps in the proof of [16, Theorem 1] whether it is possible to show (3.2) for subgraphs of growing size. It can indeed be done. However, the arguments break down at $r=O(\sqrt{\log n})$. Note that this is much weaker than Theorem 1.1. To understand why the argument stops at $O(\sqrt{\log n})$ we need discuss the key idea behind the proof of [16, Theorem 1] which is deferred to Section 4.

Next we direct our attention to Alon’s $(n,d,\lambda)$ graph.

A graph is called an $(n,d,\lambda)$-graph if it is a $d$-regular graph on $n$ vertices whose second largest absolute eigenvalue is $\lambda$. A number of properties of such graphs have been studied (see [32] and the references therein). For such graphs the number of induced isomorphic copies of large subgraphs is also well understood (see [32, Section 4.4]). From [32, Theorem 4.1] it follows that $(n,d,\lambda)$-graphs contain the correct number of cliques (and independent sets) of size $s$ if $s$ satisfies the condition $n\gg\lambda(n/d)^{s}$. Thus, to apply [32, Theorem 4.1] one needs a good bound on the second largest absolute eigenvalue $\lambda$. For nice graph ensembles it is believed that one should have $\lambda=\Theta(\sqrt{d})$. However, this has been established only in few examples. For example, when $d=\Theta(n)$ for a random $d$-regular this has been established very recently in [39]. Any bound on $\lambda$ of the form $O(d^{\frac{1}{2}+\eta})$, for some $\eta>0$, yield a sub-optimal result when applied to [32, Theorem 4.10]. On the other hand, our method being a non-spectral technique, does not require any bound on $\lambda$ and the conditions (A1)-(A2) are much easier to check.

We have already seen that Theorem 1.1 is optimal in the sense that one cannot improve the conclusion of Theorem 1.1 without adding further assumptions. One can consider two possible directions for improvements. The first of those is to have more conditions on the graph sequence. For example, one may assume controls on the number of common neighbors of three and four vertices. This indeed improves the conclusion of Theorem 1.1 when $\delta\leq 1/2$. This is seen in Theorem 1.4.

Another direction we are currently pursuing is extending Theorem 1.1 to the setting of models which incorporate community structure. Here the comparative model is not the Erdős-Rényi random graph but the stochastic block model. Community detection and clustering on networks have spawned an enormous literature over the last decade, see the survey [30] and the references therein. One can easily extend the ideas of Theorem 1.1 to count the number of induced isomorphic copies of large subgraphs in a stochastic block model. To keep the clarity of the presentation of the current work we defer this generalization to a separate paper.