Thresholds in random motif graphs

TL;DR

This paper generalizes the Erdős-Rényi model to include fixed motifs, establishing thresholds for key properties like connectivity and Hamiltonicity, and providing hitting time results in the random motif graph process.

Contribution

It introduces the binomial random motif graph model and determines thresholds for several fundamental graph properties, extending classical results to motif-based random graphs.

Findings

Thresholds for connectivity, Hamiltonicity, and perfect matching are established.

Hitting time results for the emergence of these properties are proved.

The model generalizes Erdős-Rényi graphs by incorporating fixed motifs.

Abstract

We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph model in which random instances of a fixed motif are added independently. The binomial random motif graph $G(H,n,p)$ is the random (multi)graph obtained by adding an instance of a fixed graph $H$ on each of the copies of $H$ in the complete graph on $n$ vertices, independently with probability $p$. We establish that every monotone property has a threshold in this model, and determine the thresholds for connectivity, Hamiltonicity, the existence of a perfect matching, and subgraph appearance. Moreover, in the first three cases we give the analogous hitting time results; with high probability, the first graph in the random motif graph process that has minimum degree one (or two) is connected and contains a perfect matching (or Hamiltonian respectively).

Equations

$${\delta }_d(H):=\lceil d/\delta (H)\rceil -1\text{ and }{p}_d^{\pm }(H):=\frac{\ln n+{\delta }_d(H)\ln \ln n\pm x(n)}{m_1(H)},$$

$$\underset{n\to \infty }{\lim }\mathrm{Pr}\left[G(H,n,p)\in \mathcal{P}\right]=\{\begin{array}{ll}0& \text{ if }p/p^{\ast }\to 0,\\ 1& \text{ if }p/p^{\ast }\to \infty ,\end{array}$$

$$\underset{n\to \infty }{\lim }\mathrm{Pr}\left[G(H,n,p)\text{ is connected }\right]=\{\begin{array}{ll}0& p\le p_1^{-}(H),\\ 1& p\ge p_1^{+}(H).\end{array}$$

$$\underset{n\to \infty }{\lim }\mathrm{Pr}\left[G(H,n,p)\text{ has a perfect matching }\right]=\{\begin{array}{ll}0& p\le p_1^{-}(H),\\ 1& p\ge p_1^{+}(H).\end{array}$$

$$\underset{n\to \infty }{\lim }\mathrm{Pr}\left[G(H,n,p)\text{ is Hamiltonian }\right]=\{\begin{array}{ll}0& p\le p_2^{-}(H),\\ 1& p\ge p_2^{+}(H).\end{array}$$

$$\underset{n\to \infty }{\lim }\mathrm{Pr}\left[S\subseteq \bar{G}(H,n,m)\right]=\{\begin{array}{ll}0& m\ll n^{v-\bar{\gamma }}\\ 1& m\gg n^{v-\bar{\gamma }}.\end{array}$$

$$\bar{\gamma }=\{\begin{array}{ll}v-1+1/\beta & \mathrm{exc}(S)=0,\\ v-1& \mathrm{exc}(S)=1,\\ v-2+\eta & \mathrm{exc}(S)\ge 2.\end{array}$$

$$\operatorname{\sf Pr}\left[G(H,n,p)\in\mathcal{P}\right]=\sum_{i=0}^{m_{0}(H)}\sum_{S\in\binom{m_{0}(H)}{i}}p^{i}(1-p)^{\binom{n}{|V(H)|}-i}\mathbb{I}\bigg{(}\bigcup_{j\in S}H_{j}\in\mathcal{P}\bigg{)}$$

$$\mathrm{Pr}\left[G(H,n,p_{1/2})\in \mathcal{P}\right]=\frac{1}{2}.$$

$$\bigcup _{i\in [k]}{G}_i(H,n,p_{1/2})=G(H,n,(1-(1-p_{1/2}{)}^k))\subseteq G(H,n,k{p}_{1/2}).$$

$$\mathrm{Pr}\left[G(H,n,\omega (n)p_{1/2})\in \mathcal{P}\right]$$

$$\ge \underset{k\to \infty }{\lim }1-\mathrm{Pr}\left[G(H,n,k{p}_{1/2})\notin \mathcal{P}\right]$$

$$\ge 1-\underset{k\to \infty }{\lim }\prod _{1=i}^k\mathrm{Pr}\left[G_i(H,n,p_{1/2})\notin \mathcal{P}\right]=1.$$

$$\bigcup _{i\in [k]}{G}_i(H,n,p_{1/2}/\omega (n))$$

$$\subseteq G(H,n,k{p}_{1/2}/\omega (n))\subseteq G(H,n,p_{1/2}).$$

$$\frac{1}{2}=\mathrm{Pr}\left[G(H,n,p_{1/2})\in \mathcal{P}\right]$$

$$\ge \underset{k\to \infty }{\lim }1-\mathrm{Pr}\left[G(H,n,k{p}_{1/2}/\omega (n))\notin \mathcal{P}\right]$$

$$\ge 1-\underset{k\to \infty }{\lim }\prod _{1=i}^k\mathrm{Pr}\left[G_i(H,n,p_{1/2}/\omega (n))\notin \mathcal{P}\right]$$

$$=1-\mathrm{Pr}{\left[G_i(H,n,p_{1/2}/\omega (n))\notin \mathcal{P}\right]}^k.$$

$$\operatorname{\sf Pr}\left[G_{i}(H,n,p_{1/2}/\omega(n))\notin\mathcal{P}\right]\geq\lim_{k\to\infty}\bigg{(}\frac{1}{2}\bigg{)}^{1/k}=1.\qed$$

$$qp\sim r{f}_k(r/n)\cdot \ln n\ge (1+o(1))k\ln n.$$

$$\mathrm{Pr}\left[S_r>0\right]$$

$$\le {\left[C\cdot \frac{n}{r}\cdot r\cdot p^{(r-1)/(r(k-1))}{n}^{-(1+o(1))k/r}\right]}^r$$

$$={\left[C\cdot \mathrm{polylog}n\cdot n^{1/r-(1+o(1))k/r}\right]}^r=o(1).$$

$$\mathrm{Pr}\left[G(H,n,p)\text{ is not connected}\right]$$

$$=\mathrm{Pr}\left[S_1>0\right]+\sum _{r=k}^{n/2}\mathrm{Pr}\left[S_r>0\right]=\mathrm{Pr}\left[S_1>0\right]+o(1),$$

$$E\left[D_0\right]=n(1-p_{-}{)}^{m_1(H)}\sim n{e}^{-p_{-}\cdot m_1(H)}=n{e}^{-\ln n+\ln \ln n}=\ln n.$$

$$E\left[D_0^2\right]=E\left[D_0\right]+n(n-1)(1-p_{-}{)}^{m_2(H)}.$$

$$E\left[D_0^2\right]\le E\left[D_0\right]+E{\left[D_0\right]}^2(1-p_{-}{)}^{-L},$$

$$\mathrm{Var}\left[D_0\right]\le E\left[D_0\right]+E{\left[D_0\right]}^2((1-p_{-}{)}^{-L}-1)\le E\left[D_0\right]+(L+1)p_{-}E{\left[D_0\right]}^2.$$

$$\mathrm{Pr}\left[D_0\ge 2\ln n\right]$$

$$\le (1+o(1))\left(E{\left[D_0\right]}^{-1}+(L+1)p_{-}\right)=o(1).\text{qed}$$

$$m_{-}\le {\tau }_1\le {\tau }_{\mathsf{c}}\le m_{+}.$$

$$\bar{G}(H,n,m_{-})\subseteq \bar{G}(H,n,{\tau }_1)\subseteq \bar{G}(H,n,{\tau }_{\mathsf{c}})\subseteq \bar{G}(H,n,m_{+}),$$

$$\mathrm{Pr}\left[{\tau }_1<{\tau }_{\mathsf{c}}\right]\le o(1)+M\cdot \frac{C_1\left(\genfrac{}{}{0ex}{}{|V_0|}{k}\right)}{m_n(H)-m_{+}}\le o(1)+C_2\cdot \frac{m_n(H)\ln \ln n}{m_1(H)}\cdot \frac{{\ln }^{2}n}{m_n(H)}=o(1).\text{qed}$$

Full text

S. Petti: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1650044

1. Introduction

In the late 1950’s Gilbert [Gil59] and Erdős and Rényi [ER59] introduced two of the most fundamental models for generating random graphs: the binomial random graph $G(n,p)$, generated by independently adding an edge between each pair of vertices in the complete graph on $n$ vertices with probability $p$, and the the uniform random graph $G(n,m)$, which is a uniformly chosen graph from all graphs on $n$ vertices with $m$ edges. Since, the extensive study of these simple constructions has influenced a variety of fields including combinatorics, computer science, and statistical physics (see [FK, Bol, JLR] for surveys).

Detailed analysis of the model has led to the development of plethora of new techniques in probability for analyzing random processes, and the model has been used to verify the existence of structures with certain properties [AS]. In computer science, the model has been used to analyze the performance of algorithms on an “average” case, showing that NP complete problems may be easier random instances.

The rise of data in the form of graphs (e.g. internet connections, biological networks, social networks) has further fueled the study of random graphs. In practice, the comparison of real world networks to the Erdős-Rényi model is a popular technique for highlighting the non-random aspects of a network’s structure [Yeg04, Alo07, Son05, Mil02]. Moreover, the model has inspired many other models which are designed to mirror some characteristic of real-world networks (e.g. Watts-Strogatz graphs have small diameter [WT98], Barabási-Albert preferential attachment graph exhibit a power law degree distribution [BA99]).

In this paper we consider a natural generalization of the Erdős-Rényi model in which random motifs are added rather than random edges. A motif is a fixed small subgraph, such as a triangle. The motifs that are overrepresented in a network are correlated to the function of the network [Yeg04, Alo07, Son05, Mil02]. Analyzing random graphs formed as the union of many instances of a particular motif $H$ will give insight into the structural properties of networks with many copies of the motif $H$.

We define the binomial random motif graph $G(H,n,p)$ as the random (multi)graph obtained by adding an instance of $H$ on each of the $\binom{n}{|V(H)|}\cdot|V(H)|!/\operatorname{aut}(H)$ copies of $H$ in the complete graph on $n$ vertices $K_{n}$, independently with probability $p$. Here by $\operatorname{aut}(H)$ we denote the number of automorphisms of $H$. Note that if $H$ is an edge, then this is exactly $G(n,p)$. Similarly, the uniform random motif graph $\bar{G}(H,n,m)$ is the random (multi)graph obtained by taking the union of $m$ uniformly chosen copies of $H$ in $K_{n}$ without replacement.

Closely related to $\bar{G}(H,n,m)$ is the random motif graph process $\bar{G}_{0}(H,n),\bar{G}_{1}(H,n),\allowbreak...,\allowbreak\bar{G}_{N}(H,n)$. $\bar{G}_{0}(H,n)$ is the empty graph on $n$ vertices and for $0\leq i\leq N=\binom{n}{|V(H)|}/\operatorname{aut}(H)$ the graph $\bar{G}_{i+1}(H,n)$ is generated by adding to $\bar{G}_{i}(H,n)$ a copy of $H$, $H_{i+1}$, chosen uniformly at random from all the copies of $H$ except those in $\{H_{1},H_{2},...,H_{i}\}$ i.e. those that have been added to $\bar{G}_{0}(H,n)$ so far. Clearly $\bar{G}_{m}(H,n)$ has the same law as $\bar{G}(H,n,m)$. In addition, by setting $H$ to be an edge we retrieve the random graph process introduce by Erdős and Rényi [ER60]. By considering the random motif graph process in place of the uniform random motif graph model we can phrase results in a finer way (see for example Theorem 3).

In this work we show that every monotone graph property has a threshold in the binomial random motif graph $G(H,n,p)$. Then we determine the thresholds for connectivity, existence of a perfect matching, Hamiltoncity and subgraph appearance. In the first three cases we also show a hitting time result, according to which w.h.p.111That is, with probability tending to $1$ as $n$ tends to infinity. the first graph in the random motif graph process that has minimum degree one (or two) is connected (or Hamiltonian respectively).

1.1. Notation

Throughout we assume the motif $H$ has no isolated vertices. For an integer $r\geq 0$, denote by $m_{r}(H)$ the number of its copies in $K_{n}$ which intersect the set $[r]$. For an integer $d\geq 0$ we define the quantities $\delta_{d}(H)$ and $p_{d}(H)$ by

[TABLE]

where $x(n)$ is any function of $n$ satisfying $1\ll x(n)\ll\ln{\ln{n}}$. Note that the expected number of added instances of $H$ in $G(H,n,p^{\pm}_{1}(H))$ is $m_{n}(H)\cdot p^{\pm}_{1}(H)$, which only depends on $n$ and on $|V(H)|$.

1.2. Results

A function $p^{*}=p^{*}(n)$ is a threshold for a monotone increasing property $\mathcal{P}$ in the random graph $G(H,n,p)$ if

[TABLE]

as $n\to\infty$. Our first result is a generalization of a theorem by Bollobás and Thomason [BT97].

Theorem 1**.**

Every non-trivial monotone graph property has a threshold.

Given Theorem 1, a natural goal is to find the thresholds for various monotone properties. The remaining results of this paper are dedicated towards this goal; we determine the threshold for connectivity, the existence of a perfect matchings, Hamiltonicity, and subgraph appearance.

A first such result, which generalizes a result in [ER59], shows, in particular, that the expected number of motifs needed to make the random motif graph connected depends only on the number of (non-isolated) vertices of the motif.

Theorem 2**.**

Let $H$ be a fixed graph. Then

[TABLE]

In fact, we show a hitting time result, according to which the hitting time of connectivity equals, w.h.p., the hitting time of minimum degree one. In other words, the random motif graph process becomes connected exactly when the last isolated vertex disappears, with high probability.

Fix an integer $n$ and a graph $H$. Let $\tau_{\mathsf{c}}=\min\{i:\bar{G}_{i}(H,n)\text{ is connected}\}$, and for $d\geq 1$ denote $\tau_{d}=\min\{i:\delta(\bar{G}_{i}(H,n))\geq d\}$.

Theorem 3**.**

Let $H$ be a fixed graph. Then w.h.p. $\tau_{\mathsf{c}}=\tau_{1}$.

We remark that if the motif $H$ is connected, every connectivity related question depends solely on the sets of vertices on which copies of $H$ are added, and not on the way they are put there. Thus, we may model the question as a (binomial or uniform) random $k$-uniform hypergraph, where $k=|V(H)|$. In this case, Theorems 2 and 3 follow immediately from known results about (loose) connectivity in random hypergraphs (see, e.g.,[Poo15]).

In the following two theorems we show that the existence of a perfect matching is also dependent on the number of non-isolated vertices of the motif.

Theorem 4**.**

Let $H$ be a fixed graph, and assume that $n$ is even. Then,

[TABLE]

Let $\tau_{\mathsf{M}}=\min\{i:\bar{G}_{i}(H,n)\text{ has a perfect matching}\}$. The analogue hitting time result is also true.

Theorem 5**.**

Let $H$ be a fixed graph, and assume that $n$ is even. Then w.h.p. $\tau_{\mathsf{M}}=\tau_{1}$.

Theorem 6 establishes that the thresholds for minimum degree $2$ and for Hamiltonicity are the same. Theorem 7 shows the hitting time version of that result.

Theorem 6**.**

Let $H$ be a fixed graph. Then

[TABLE]

Let $\tau_{\mathsf{H}}:=\min\{i:\bar{G}_{i}(H,n)\text{ is Hamiltonian}\}$.

Theorem 7**.**

Let $H$ be a fixed graph. Then w.h.p. $\tau_{\mathsf{H}}=\tau_{2}$.

Next, we describe the threshold for the appearance of a subgraph $S$. If $S$ appears in a random motif graph, then $S$ is a subgraph of some configuration of $b$ copies of $H$ whose union contains $a$ vertices. For such an $(a,b)$ covering of $S$, we call a subset of the covering containing $b^{\prime}$ copies of $H$ whose union contains $a^{\prime}$ vertices an $(a^{\prime},b^{\prime})$ subset. The threshold for the appearance of $S$ depends on $\bar{\gamma}$, the maximum over all covering configurations of the minimum ratio $a^{\prime}/b^{\prime}$ for all subsets of the covering configuration. Definition 15 formally describes $\bar{\gamma}$.

Theorem 8**.**

Let $H$ be a fixed graph, let $S$ be a fixed graph, and set $v=|V(H)|$ and $\bar{\gamma}=\bar{\gamma}(S,H)$. Then

[TABLE]

The number of excess edges of a connected graph $S$, or simply its excess, is defined to be $\operatorname{exc}(S)=|E(S)|-|V(S)|+1$. In particular, trees have excess [math]. We say that $S$ is unicyclic if its excess is $1$, or complex if its excess is at least $2$. The following theorem gives a simple description of $\bar{\gamma}$ when the motif $H$ is a path, which allows us to deduce how the copies of $H$ fit together to form a copy of $S$ at the threshold when $S$ first appears. If $S$ is a tree, a minimal set of edge disjoint copies of $H$ typically forms $S$. If $S$ is complex, each copy of the path $H$ typically contributes a single edge to $S$. If it is unicyclic, it may be formed by any edge disjoint configuration of paths $H$.

Theorem 9**.**

Let $H$ be a path of length $v-1$ and let $S$ be a connected graph. Let $\beta$ be the minimum number of edge-disjoint copies of $H$ whose union contains $S$ as a subgraph. Let $\eta=\min_{X\subseteq S}\frac{|V(X)|}{|E(X)|}$. Then

[TABLE]

In the case where the motif is a long path, this result establishes a connection between the threshold for the appearance of subgraphs in random motif graphs and the threshold for the appearance of subgraphs in the trace of a random walk on the complete graph $K_{n}$ (studied in [KM17]). Let $S$ be a connected graph and $\beta$ be the minimum number of paths in any edge-disjoint decomposition of $S$ into paths. If $H$ is longer than the maximum length path in such a minimum edge-disjoint path decomposition, then the threshold implied by Theorem 9 matches the threshold for the appearance of $S$ in the trace of a random walk on the complete graph [KM17].

This should not come as a surprise; by noticing that when the motif is a long path, the random motif graph model approximates the trace model, in the following sense. One may sequentially “cut” the (lazy) simple random walk into chunks with buffers of length $1$. We delete loops created by the trace of each chunk, and we enforce the condition that the remaining edges span a path of length $\ell$ (which is fixed but large). Hence the trace of each such chunk is an independent copy of a path of length $\ell$. Thus we may couple the trace model and the random motif model such that the trace model will include the random motif model plus some loops plus a small number of buffer edges (which gets smaller as $\ell$ gets larger).

Viewing this analogy this way, we may use Theorems 8 and 9 to reprove the main theorems of [KM17] for the case where the base graph is complete.

2. Existence of thresholds for monotone properties

Proof of Theorem 1.

Assume that $\mathcal{P}$ is a monotone increasing property and let $H_{1},H_{2},\allowbreak...,\allowbreak H_{m_{0}(H)}$ be the copies of $H$ that are spanned by $K_{n}$. Observe that

[TABLE]

is a polynomial in $p$. In addition, since $\mathcal{P}$ is increasing, it is increasing. Therefore we may define $p_{1/2}$ by

[TABLE]

We will show that $p_{1/2}$ is a threshold for $\mathcal{P}$. For two random graphs $G,G^{\prime}$ we write $G\subseteq G^{\prime}$ if $G,G^{\prime}$ can be coupled such that $G$ is a subgraph of $G^{\prime}$.

First let $p=\omega(n)p_{1/2}$ where $\omega(n)\to\infty$ as $n\to\infty$ and let $k\in\mathbb{N}$. Let $G_{i}(H,n,p_{1/2})$ be distributed as a $G(H,n,p_{1/2})$ for $i\in[k]$. Then, by considering the probability of no appearance of a fixed copy of $H$, we have that the graph $\cup_{i\in[k]}G_{i}(H,n,p_{1/2})$ is distributed as $G(H,n,(1-(1-p_{1/2})^{k}))$. Thereafter $1-(1-p_{1/2})^{k}\leq kp_{1/2}$ implies,

[TABLE]

Hence,

[TABLE]

Now assume that $p=p_{1/2}/\omega(n)$ for some $\omega(n)\to\infty$ as $n\to\infty$ and let $k\in\mathbb{N}$. Similarly to before, if we let $G_{i}(H,n,p_{1/2}/\omega(n))$ to be distributed as a $G(H,n,p_{1/2}/\omega(n))$ for $i\in[k]$ then, we have that

[TABLE]

Hence,

[TABLE]

Rearranging the above gives,

[TABLE]

3. Connectivity

Proof of Theorem 2.

If $p\leq p_{1}^{-}(H)$ then by Theorem 19 the minimum degree of $G(H,n,p)$ is w.h.p. [math], hence it is not connected.

Suppose $p\geq p_{1}^{+}(H)$. In fact, for the argument below, we only assume that $p=(\ln{n}\pm o(\ln{n}))/m_{1}(H)$ (and the conclusion will follow by monotonicity). Let $k$ denote the number of vertices of $H$. For $r=1,\ldots,n/2$ denote by $S_{r}$ the number of connected components of size $r$ in $G(H,n,p)$. Note that for $r\geq k$, if a set of cardinality $r$ is a connected component, then there exist $\lceil(r-1)/(k-1)\rceil$ copies of $H$ inside the set which appear in $G(H,n,p)$, and there are no edges between it and its complement, so none of the $q=q_{r}(H)$ copies of $H$ that intersect that set appear. By Lemma 17,

[TABLE]

Let $\eta=k!/\operatorname{aut}(H)$ and suppose $r\geq k$. By Lemma 18 and by the union bound there exist constants $c,c^{\prime},C>0$ depending only on $H$ such that

[TABLE]

It follows that

[TABLE]

but according to Theorem 19 (for $p\geq p_{1}^{+}(H)$), there are no isolated vertices w.h.p., and the result follows. ∎

Note that a consequence of this proof is that for $p=(\ln{n}\pm o(\ln{n}))/m_{1}(H)$, with high probability, every connected component is of cardinality $1$ or at least $n/2$. This means that w.h.p. there exists a unique “giant” component of linear size, and the rest of the vertices are isolated. The next lemma, whose proof uses a simple second moment argument, estimates the number of these isolated vertices for $p_{-}=(\ln{n}-\ln{\ln{n}})/m_{1}(H)$.

Lemma 10**.**

The number of isolated vertices in $G(H,n,p_{-})$ is w.h.p. at most $2\ln{n}$.

Proof.

Let $D_{0}$ be the number of isolated vertices in $G(H,n,p_{-})$. First,

[TABLE]

Moreover,

[TABLE]

Denote $L:=2m_{1}(H)-m_{2}(H)$. Thus

[TABLE]

and since $(1-p_{-})^{-L}-1\sim Lp_{-}$, we have that

[TABLE]

Thus, noting that $Lp_{-}=o(1)$,

[TABLE]

Proof of Theorem 3.

Denote $p_{\pm}=(\ln{n}\pm\ln{\ln{n}})/m_{1}(H)$ and $m_{\pm}=p_{\pm}\cdot m_{n}(H)$. By asymptotic equivalence of the binomial and the uniform models (see, e.g., [JLR]*Section 1.4) we have that w.h.p. $G(H,n,m_{-})$ has a unique giant component, and the rest of the connected components are isolated vertices, whose number is at most $2\ln{n}$. Denote the set of these isolated vertices by $V_{0}$. Together with Theorem 2 we also conclude that w.h.p.

[TABLE]

We may thus couple $\bar{G}(H,n,m_{-})$, $\bar{G}(H,n,\tau_{1})$, $\bar{G}(H,n,\tau_{\mathsf{c}})$ and $\bar{G}(H,n,m^{+})$ such that

[TABLE]

by starting with $\bar{G}(H,n,m_{-})$ and adding $M=m_{+}-m_{-}$ random copies of $H$ to create $\bar{G}(H,n,m_{+})$. Note that if none of these $M$ edges is fully contained in $V_{0}$ (and the coupling succeeds) then $\tau_{1}=\tau_{\mathsf{c}}$. Thus, there exist positive constants $C_{1},C_{2}$ such that,

[TABLE]

4. Hamiltoncity and Perfect Matchings

The proof of Theorems 7 and 5 can be given in parallel, using the same techniques and tools. For clarity though, in this section we focus mainly on proving Theorem 7 and we give a sketch of the proof of Theorem 5 in the appendix.

For proving our Hamiltonicity result we use the standard technique of Posa’s rotations. We define Small to be the vertices of significantly smaller degree than the expected one and we set Large to be the rest of the vertices. We first show that small to medium subsets of Large expand and that the vertices in Small are well spread. This is done in the context of Lemmas 11 and 12, 13 respectively. We use these properties of Small and Large in order to prove all the the ingredients needed to apply the Posa’s rotations, which we gather in Lemma 14.

Let $p_{0}:=(\ln n-2\ln\ln n)/m_{1}(H)$ and recall that $p_{2}^{\pm}=(\ln n+r_{2}\ln\ln n\pm\omega(1))/m_{1}(H)$, $r_{2}=\lfloor 2/\delta(H)-1\rfloor$. W.h.p. (see [FK]) we can couple $G(H,n,p_{0}),G(H,n,p_{2}^{-}),\bar{G}(H,n,\tau_{2})$ and $G(H,n,p_{2}^{+})$ such that

• (i) $G(H,n,p_{0})\subset G(H,n,p_{2}^{-})\subset\bar{G}(H,n,\tau_{2})\subset G(H,n,p_{2}^{+})$ and
• (ii) there are $(1+o(1))(p_{2}^{-}-p_{0})\frac{r!}{\operatorname{aut}(H)}\binom{n}{r}>n\ln\ln n/2r$ copies of $H$ in $G(H,n,p_{2}^{-})$, hence in $\bar{G}(H,n,\tau_{2})$, that are not present in $G(H,n,p_{0})$.

Observe that the above coupling and Theorem 7 imply Theorem 6. In addition a similar coupling and Theorem 5 imply Theorem 4.

We now define the sets Small, Large based on the degrees of the vertices in $G(H,n,p_{0})$. Let \textsc{Large}=\{v\in V:v\text{ intersects at least}\ln\ln n\text{ copies of H in }G(H,n,p_{0})\} and $\textsc{Small}=V\setminus\textsc{Large}$.

Lemma 11**.**

W.h.p. every $S\subset\textsc{Large}$ of size at most $n/30r$ satisfies $|N(S)|\geq 10|S|$.

Lemma 12**.**

W.h.p. for every pair $u,v\in\textsc{Small}$ there do not exist $\ell\leq 6$ copies of $H$ in $G(H,n,p_{2}^{+})$ that span a connected subgraph containing both $u,v$. Hence w.h.p. every pair $u,v\in\textsc{Small}$ is at distance at least 7 in $G(H,n,p_{2}^{+})$.

Lemma 13**.**

W.h.p. for every $v\in V$ there exists at most one copy of $H$ in $G(H,n,p_{2}^{+})$, hence in $\bar{G}(H,n,\tau_{2})$, that intersect both $\{v\}$ and $\textsc{Small}\setminus\{v\}$.

Now we generate $\bar{G}(H,n,\tau_{2})$ as follows. We first generate $G_{0}^{\prime}=G(H,n,p_{0})$. Then we randomly permute the copies of $H$ not appearing in $G_{0}^{\prime}$, let them be $H_{1},H_{2},....$. We also let $S_{0}=\emptyset$. We define the sequences $G_{0}^{\prime},G_{1}^{\prime},...$ and $S_{0},S_{1},...$ in the following way. At step $i\in\mathbb{N}$ we query $H_{i}$ whether it is incident to a vertex in Small. If it is then we set $S_{i}=S_{i-1}$ and $G_{i}^{\prime}=G_{i-1}^{\prime}\cup H_{i}$. Otherwise we set $S_{i}=S_{i-1}\cup\{H_{i}\}$ and $G_{i}^{\prime}=G_{i-1}^{\prime}$. Let $t^{*}=\min\{i:\delta(G_{i}^{\prime})=2\}$ and $S_{t^{*}}=\{H_{i_{1}},H_{i_{2}},...,H_{i_{w}}\}$.

Given the sequence $G_{0}^{\prime},G_{1}^{\prime},...,G_{t^{*}}^{\prime}$ and the set $S_{t^{*}}=\{H_{i_{1}},H_{i_{2}},...,H_{i_{w}}\}$ we define the graph sequence $F_{0},...,F_{w}$ by $F_{0}=G_{t^{*}}^{\prime}$ and $F_{j}=F_{j-1}\cup H_{i_{j}}$ for $1\leq j\leq w$. Observe that $S_{t^{*}}$ consists of all copies of $H$ in $\{H_{1},...,H_{t^{*}}\}$ that have not been added to $G_{0}^{\prime}$, equivalently the copies of $H$ that are not incident to Small. Thus F_{w}=G^{\prime}_{t^{*}}\cup\big{(}\bigcup_{j=1}^{w}H_{i_{j}}\big{)}=G^{\prime}_{0}\cup\big{(}\bigcup_{i=1}^{t^{*}}H_{i}\big{)}=\bar{G}(H,n,\tau_{2}).

Lemma 14**.**

W.h.p. the following hold:

1. i)

$w\geq n\ln\ln n/2r-n$, 2. ii)

every $S\subset V$ of size at most $n/30r$ satisfies $|N(S)|\geq 2|S|$ in $F_{0}$, 3. iii)

$F_{0}$* is connected,* 4. iv)

for every $1\leq j\leq w$, $\epsilon>0$, and every set $Q_{j}$ consisting of $\epsilon n^{2}$ edges not present in $F_{j}$ there exist a constant $C_{\epsilon}>0$ such that the probability that $Q_{j}$ intersects $E(H_{i_{j+1}})$ is at least $C_{\epsilon}$.

We are now ready to apply Posa’s rotations . For that assume that $F_{j}$ is not Hamiltonian and consider a longest path in $F_{j}$, $P_{j}$, $j\geq 0$. Let $x,y$ be the end-vertices of $P_{j}$. Given $yv$ where $v$ is an interior vertex of $P_{j}$ we can obtain a new longest path $P_{j}^{\prime}=x..vy..w$ where $w$ is the neighbor of $v$ on $P_{j}$ between $v$ and $y$. In such a case we say that $P_{j}^{\prime}$ is obtained from $P_{j}$ by a rotation with the end-vertex $x$ being the fixed end-vertex.

Let $\textsc{End}_{j}(x;P_{j})$ be the set of end-vertices of longest paths of $F_{j}$ that can be obtained from $P_{j}$ by a sequence of rotations that keep $x$ as the fixed end-vertex. Thereafter for $z\in\textsc{End}_{j}(x;P_{j})$ let $P_{j}(x,z)$ be a path that has end-vertices $x,z$ and can be obtain form $P_{j}$ by a sequence of rotations that keep $x$ as the fixed end-vertex. Observe that for $z\in\textsc{End}_{j}(x;P_{j})$ and $z^{\prime}\in\textsc{End}_{j}(z;P_{j}(x,z))$ there exists a $z$-$z^{\prime}$ path $P_{z,z^{\prime}}$ of length $|P_{j}|$ that can be obtained from $P_{j}$ via a sequence of Posa rotations. Thus we can conclude that $\{z,z^{\prime}\}$ does not belong to $F_{j}$. Indeed assume that $\{z,z^{\prime}\}\in E(G_{i})$. Then we can close $P_{z,z^{\prime}}$ into a cycle $C_{z,z^{\prime}}$ that is not Hamiltonian. Since $F_{j}$ is connected there is an edge $e$ spanned by $V(C_{z,z^{\prime}})\times V\setminus V(C_{z,z^{\prime}})$. $E(C_{z,z^{\prime}})\cup\{e\}$ spans a path of length $|P_{j}|+2$ contradicting the maximality of $P_{j}$. Similarly if $\{z,z^{\prime}\}\in E(H_{i_{j+1}})$ then $F_{j+1}$ is either Hamiltonian or it contains a path that is longer than $P_{j}$. At the same time it follows (see [FK]*Corollary 6.7) that

[TABLE]

Moreover for every $z\in\textsc{End}_{j}(x;P_{j})$

[TABLE]

As a consequence of Lemma 11, we have that $|\textsc{End}(x,P_{j})|\geq n/30r$ and $|\textsc{End}(z,P_{j}(x,z))|$ $\geq n/30r$ for every $z\in\textsc{End}_{j}(x;P_{j})$. Let $E_{j}=\{\{z,z^{\prime}\}:z\in\textsc{End}_{j}(x;P_{j})\text{ and }z^{\prime}\in\textsc{End}_{j}(z;P_{j}(x,z))\}$. Then $|E_{j}|\geq(n/30r)^{2}/2$.

Now let $Y_{j}$ be the indicator of the event $\{E_{j}\cap E(H_{i_{j+1}})\neq\emptyset\}$ and set $Z=\sum_{j=1}^{w}Y_{i}$. From Lemma 14 iv) we have $\operatorname{\sf Pr}\left[Y_{j}=1\right]\geq C_{\epsilon}$ (here $\epsilon=1/2(30r)^{2}$). In the event that $G_{w}$ is not Hamiltonian, $Z\leq n$ while $Y_{j}$ is a $\mathsf{Bernoulli}(C_{\epsilon})$ random variable for $1\leq j\leq w$ . Since $w\geq n\ln\ln n/2r-n$ we have $\operatorname{\sf Pr}\left[\mathsf{Bin}(w,C_{\epsilon})\leq n\right]=o(1)$. Hence w.h.p. $F_{w}=\bar{G}(H,n,\tau_{2})$ is Hamiltonian and the hitting time for Hamiltonicity equals the hitting time for minimum degree 2.

5. Subgraph appearance

In $G(n,p)$ there is only one way for a specified subgraph to appear on a fixed set of vertices: all the edges in the subgraph must be present. In the case of random motif graphs, there are multiple ways to place motifs so that a specified subgraph appears on a fixed set of vertices. For example, in a random two-path graph, a triangle may appear on $\{1,2,3\}$ if (i) the paths $(1,2,3)$ and $(3,1,z)$ are present or (ii) the paths $(1,2,x)$, $(2,3,y)$ and $(3,1,z)$ are present. In order to pin down the threshold for subgraph appearance, it is necessary to understand the various motif configurations that cause the subgraph to appear and their relative probabilities. The following definition provides the notation to describe such configurations.

Definition 15**.**

Let $V$ be a set of vertices. Let $S$ be a fixed graph on a subset of the vertices of $V$. Let $H_{1},H_{2},\dots H_{b}$ be copies $H$ also defined on subsets of vertices of $V$.

1. (a)

We say $\{H_{1},H_{2},\dots H_{b}\}$ is an $(a,b)$ covering of $S$ if (i) $S\subseteq\bigcup_{j=1}^{b}H_{j}$, (ii) $|V(\bigcup_{j=1}^{b}H_{j})|=a$, and (iii) for each $\ell\in[b]$, $S\not\subseteq\bigcup_{j=1}^{b}H_{j}\setminus H_{\ell}$. 2. (b)

Let $k(a,b)$ be the number of unique configurations of $(a,b)$ coverings, i.e. the number of ways to place $b$ copies of $H$ on $a$ vertices such that conditions (i)-(iii) of (a) hold. Enumerate the possible configurations of $(a,b)$ coverings with values in $[k(a,b)]$. For $i\in[k(a,b)]$, an $(a,b,i)$ covering of $S$ is an $(a,b)$ covering with configuration $i$. 3. (c)

We say the set $\{F_{1},F_{2},\dots F_{b^{\prime}}\}$ (with precisely $b^{\prime}$ elements) is an $(a^{\prime},b^{\prime})$ subset of an $(a,b,i)$ covering $\{H_{1},H_{2},\dots H_{b}\}$ if (i) $\{F_{1},F_{2},\dots F_{b^{\prime}}\}\subseteq\{H_{1},H_{2},\dots H_{b}\}$, and (ii) $|V(\bigcup_{\ell=1}^{b^{\prime}}F_{\ell})|=a^{\prime}$. 4. (d)

Let \mathcal{I}(S,H)=\{(a,b,i)\>|\>\text{there exists an (a,b)$covering of$S$by$H$and$i\in[k(a,b)]}\}. 5. (e)

For $(a,b,i)\in\mathcal{I}(S,H)$, let

[TABLE] 6. (f)

For $(a,b,i)\in\mathcal{I}(S,H)$, let $\gamma(a,b,i)=\min_{(a^{\prime},b^{\prime})\in\mathcal{D}(a,b,i)}\frac{a^{\prime}}{b^{\prime}}$ and denote

[TABLE]

Proof of Theorem 8.

Let $G\sim\bar{G}(H,n,m)$. We say that an instance of the subgraph $S$ in $G$ is an $(a,b,i)$ instance if the placed graphs $H_{1},\dots H_{b}$ that contribute at least one edge to $S$ form an $(a,b,i)$ covering of $S$. Let $X_{i}^{ab}$ denote the number of $(a,b,i)$ instances of $S$ in $G$. Let $Z=\sum_{(a,b,i)\in\mathcal{I}(S,H)}X_{i}^{ab}$ be the total number of instances of the subgraph $S$ in $G$.

First we use the first moment method to show that if $m\ll n^{v-\bar{\gamma}}$, then the probability that $S$ occurs as a subgraph is $o(1)$. It suffices to show that for all $(a,b,i)\in\mathcal{I}(S,H)$, $\operatorname{\sf E}\left[X_{i}^{ab}\right]=o(1)$ since

[TABLE]

and $|\mathcal{I}(S,H)|$ is a constant independent of $n$.

We now compute $\operatorname{\sf E}\left[X_{i}^{ab}\right]$ for a fixed triple $(a,b,i)\in\mathcal{I}(S,H)$. Let $\{F_{1},\dots F_{b^{\prime}}\}$ be an $(a^{\prime},b^{\prime})$ subset of the configuration $(a,b,i)$ with $a^{\prime}/b^{\prime}=\gamma(a,b,i)$. Let $Y$ be the number of instances of $F=\bigcup_{i=1}^{b^{\prime}}F_{b^{\prime}}$ in $G$ formed by the configuration $\{F_{1},\dots F_{b^{\prime}}\}$. Since an $(a,b,i)$ instance of $S$ contains an instance of the configuration $\{F_{1},\dots F_{b^{\prime}}\}$, $X_{i}^{ab}\leq Y$. The number of ways to select $a^{\prime}$ vertices is at most $n^{a^{\prime}}$. The probability that a labeled copy of $H$ is placed on a specified set of vertices is $m/n^{v}$. We compute

[TABLE]

where $c$ is a constant depending only on the number of automorphisms of $S$ and the number of automorphisms of the configuration $\{F_{1},\dots F_{b^{\prime}}\}$. It follows that for $m\ll n^{\bar{\gamma}-v}$, $\operatorname{\sf E}\left[X_{i}^{ab}\right]=o(1)$, as desired.

Next we use the second moment method to show that if $m\gg n^{v-\bar{\gamma}}$ then $S$ appears as a subgraph almost surely. It suffices to show that there exists some $(a,b,i)\in\mathcal{I}(S,H)$ such that $X_{i}^{ab}$ is almost surely positive. Let $(a,b,i)$ be such that $\bar{\gamma}=\gamma(a,b,i)$. We apply Corollary 4.3.5 of [AS] to show that $X_{i}^{ab}$ is almost surely positive. Let $X_{i}^{ab}=\sum_{j}A_{j}$ where $A_{j}$ is an indicator random variable for the event that there is an $(a,b,i)$ instance of $S$ formed by a configuration of $H_{1},H_{2},\dots H_{b}$ each present on a specified set of vertices. Fix $A_{\ell}$, and let

[TABLE]

where $j\sim\ell$ indicates that $A_{j}$ and $A_{\ell}$ are not independent. By 4.3.5 of [AS], if $\operatorname{\sf E}\left[X_{i}^{ab}\right]\to\infty$ and $\Delta^{\ast}=o(\operatorname{\sf E}\left[X_{i}^{ab}\right])$, then $X_{i}^{ab}>0$ almost surely.

First we show that $\operatorname{\sf E}\left[X_{i}^{ab}\right]\to\infty$. We compute as above

[TABLE]

where $c^{\prime}$ is a constant depending only on the number of automorphisms of $S$ and the number of automorphisms of the configuration $\{H_{1},\dots H_{b}\}$. It follows that if $m\gg n^{v-\bar{\gamma}}$ then $\operatorname{\sf E}\left[X_{i}^{ab}\right]\to\infty$.

Finally, we show $\Delta^{\ast}=o(\operatorname{\sf E}\left[X_{i}^{ab}\right])$. Observe that under the assumption $m\gg n^{v-\bar{\gamma}}$,

[TABLE]

6. Conclusion

6.1. The value of the random motif model

The study of random motif graphs has the potential to strengthen the impact of the Erdős-Rényi construction. In the context of analyzing real-world networks with an overrepresented motif, random motif graphs may be a more insightful null hypothesis model to compare against to identify non-random structure. For instance by studying subgraphs counts of random $H$ motif graphs one can determine if some larger motif pattern is a byproduct of having many copies of $H$ or is itself some novel aspect of the network structure. Moreover, it is possible that a random motif graph may be used to establish the existence of a graph with some extremal property of interest. Finally, random motif graphs can be used as an alternate definition of average case for analyzing algorithms under the assumption that the input has some motif structure.

6.2. Future directions: understanding threshold behavior more broadly

We have established that random motif graphs behave similarly to traditional Erdős-Rényi random graphs with respect to thresholds and hitting times for monotone properties. Does similar behavior appear when we consider random graphs formed by randomly adding primitive subgraphs $H$ whose size scales with $n$, the number of vertices of the random graph? Instead of taking $H$ to be a fixed motif, $H$ could be a path, cycle, matching or clique whose size depends on $n$, for example. Some of these cases were in fact studied in several contexts. For example, the union of $d\geq 3$ random perfect matchings is contiguous to the random $d$-regular graph, and is sometimes easier to analyze [Wor99]. Moreover, we can consider the class of models where $H$ itself is chosen from some probability distribution. In several cases, this has been studied as well. For instance, [FKT99] and [FGRV14] consider the case when $H$ is the uniform spanning tree, and [Pet18] considers the case when $H$ is an Erdős-Rényi random graph with constant density and size dependent on $n$. Further study of these models is a first step toward delineating a larger family of random graphs that exhibit Erdős-Rényi like threshold and hitting time behaviors.