Phase transitions in exponential random graphs

Introduction

We will treat a class of models of large, “dense” random graphs, that is, simple graphs on $n$ vertices in which the average number of edges is of order $n^{2}$. More specifically we will consider two-parameter families of exponential random graphs in which dependence between the random edges is defined through certain finite subgraphs $H_{2}$, in imitation of the use of potential energy to provide dependence between particle states in a grand canonical ensemble of statistical physics. Intuitively, the two parameters allow one to adjust the density of edges and the density of subgraphs $H_{2}$, and analyze the extent to which specific values of these densities “interfere” with one another. Exponential random graphs have been widely studied (see F1 , F2 for a range of recent work) since the pioneering work on the independent case by Erdős and Rényi ER . We will concentrate on the phenomenon of phase transitions which can emerge for dependent variables: a sharp, unambiguous partition of parameter ranges separating those values in which changes in parameters lead to smooth changes in the two densities, from those special parameter values where the response in the densities is singular. This subject has attracted enormous interest in mathematics, as well as in various applied disciplines (some references may be found, e.g., in Häggström and Jonasson HJ ). Analyses using mean-field and other uncontrolled approximations (see, e.g., PN1 , PN2 ) have predicted such partitions, but with some qualitative error which we discuss in the last section. There has recently been important progress by Chatterjee and Diaconis CD , including the first rigorous proof of singular dependence on parameters. We will extend their result both in the class of models and parameter values under control and provide an appropriate formalism of phase structure for such models.

Statement of results

We consider the class of models in which the probability of the simple graph $G_{n}$ on $n$ vertices is given by

where $H_{1}$ is an edge, $H_{2}$ is any finite simple graph with $p\geq 2$ edges, $\psi_{n}=\psi_{n}(\beta_{1},\beta_{2})$ is the normalization constant, $t(H,G_{n})$ is the density of graph homomorphisms $H\to G_{n}$

For any allowed $H_{2}$, the pointwise limit

exists and is analytic at all $(\beta_{1},\beta_{2})$ in the upper half-plane ($\beta_{2}\geq 0$) except on a certain continuous curve $\beta_{2}=q(\beta_{1})$ which includes the endpoint

The derivatives $\frac{\partial}{\partial\beta_{1}}\psi_{\infty}$ and $\frac{\partial}{\partial\beta_{2}}\psi_{\infty}$ have (jump) discontinuities across the curve, except at $(\beta_{1}^{c},\beta_{2}^{c})$ where, however, all the second derivatives $\frac{\partial^{2}}{\partial\beta_{1}^{2}}\psi_{\infty}$, $\frac{\partial^{2}}{\partial\beta_{1}\partial\beta_{2}}\psi_{\infty}$ and $\frac{\partial^{2}}{\partial\beta_{2}^{2}}\psi_{\infty}$ diverge.

If the graph $H_{2}$ is a $p$-star, $p\geq 2$, then the pointwise limit $\psi_{\infty}(\beta_{1},\beta_{2})$ exists and is analytic at all $(\beta_{1},\beta_{2})$ in the lower half-plane ($\beta_{2}\leq 0$).

A $p$-star has $p$ edges meeting at a vertex.

For any allowed $H_{2}$, the parameter space $\{(\beta_{1},\beta_{2})\dvtx\break\beta_{2}\geq 0\}$ consists of a single phase with a first order phase transition across the indicated curve and a second order phase transition at the critical point $(\beta_{1}^{c},\beta_{2}^{c})$.

To explain the language of phase transitions in Corollary 2.3 we first give a superficial introduction to the formalism of classical statistical mechanics within $d$-dimensional lattice gas models; for more details see, for instance, M .

Assume each point in a $d$-dimensional cube

is randomly occupied (by one particle) or not occupied, and assume there is a (many-body) potential energy of fixed value $a\neq 0$ associated with every occupied subset of $C$ congruent to a certain $H_{2}\subset C$. The interaction is attractive if $a<0$ and repulsive if $a>0$. We define the probability that the occupied sites in $C$ are precisely $c$ by

the (average) particle density. To model materials in thermal equilibrium, calculations in this formalism normally require that the system size be sufficiently large, and in practice one often resorts to using $n\to\infty$. With this as motivation we tentatively define a “phase” as a set of states (i.e., probability distributions) corresponding to a connected region of the $(\beta,\mu)$ parameter space, which is maximal for the condition that $\lim_{n\to\infty}\frac{\partial^{j+k}}{\partial\beta^{j}\,\partial\mu^{k}}F_{n}(\beta,\mu)$ are analytic in $\beta$ and $\mu$ for all $j,k$. One associates a “phase transition” with singularities which develop in some of these quantities as the system size diverges. Such singularities are sometimes detected in simulations through the variances $\sigma^{2}_{1}(n),\sigma^{2}_{2}(n)$ of the particle and energy densities, $E_{1}/n^{d},E_{2}/n^{d}$, respectively. It is easy to check, for instance, that

and therefore a finite limit for $\frac{\partial^{2}}{\partial\mu^{2}}F_{n}(\beta,\mu)$ as $n\to\infty$ implies $\sigma^{2}_{1}(n)\to 0$ at least as fast as $1/n^{d}$, while the divergence of $\frac{\partial^{2}}{\partial\mu^{2}}F_{n}(\beta,\mu)$ more slowly than $n^{d}$ as $n\to\infty$ implies $\sigma^{2}_{1}(n)\to 0$ more slowly than $1/n^{d}$, and a jump discontinuity in $\frac{\partial}{\partial\mu}F_{n}(\beta,\mu)$ as $n\to\infty$ implies $\sigma^{2}_{1}(n)$ does not go to 0 as $n\to\infty$. An important simplification was proven by Yang and Lee YL who showed that the limiting free energy density $F_{\infty}(\beta,\mu)=\lim_{n\to\infty}F_{n}(\beta,\mu)$ always exists and that certain limits commute:

This implies that phases and phase transitions can be determined from the limiting free energy density, and so a phase is commonly defined (see, e.g., FR ) as a connected region of the $(\beta,\mu)$ parameter space maximal for the condition that $F_{\infty}(\beta,\mu)$ is analytic.

Using the obvious analogues for random graphs, with $\beta_{1}$ playing the role of $-\beta\mu$ and $\beta_{2}$ the role of $-\beta a$ (and therefore positive if and only if the model is “attractive”), $\psi_{n}=\psi_{n}(\beta_{1},\beta_{2})$ plays the key role of the free energy density $F_{n}(\beta,\mu)$. We will show in Theorems 3.9 and 3.10 below that the limiting free energy density $\psi_{\infty}(\beta_{1},\beta_{2})$ exists, and the proof of Theorem 2 by Yang and Lee YL , on the commutation of limits, then goes through without any difficulty in this setting, so we can again define phases and phase transitions through the limiting free energy density, as follows.

A phase is a connected region of the parameter space $\{(\beta_{1},\beta_{2})\}$, maximal for the condition that the limiting free energy density, $\psi_{\infty}(\beta_{1},\beta_{2})$, is analytic. There is a $j$th-order transition at a boundary point of a phase if at least one $j$th-order partial derivative of $\psi_{\infty}(\beta_{1},\beta_{2})$ is discontinuous there, while all lower order derivatives are continuous.

Theorems 2.1 and 2.2 thereby justify our interpretation in Corollary 2.3 that each of our models consists of a single phase with a first order phase transition across the indicated curve, except at the end (or “critical”) point $(\beta^{c}_{1},\beta^{c}_{2})$, where the transition is second order, superficially similar to the transition between liquid and gas in equilibrium materials.

Proofs

Chatterjee and Diaconis have proven that the main object of interest, $\psi_{\infty}(\beta_{1},\beta_{2})$, exists for all $\beta_{1}$ and nonnegative $\beta_{2}$ and is the solution of a certain optimization problem:

Fix one of our models and assume $H_{2}$ has $p\geq 2$ edges. Then for all $(\beta_{1},\beta_{2})$ in the upper half-plane $(\beta_{2}>0)$, the pointwise limit $\psi_{\infty}(\beta_{1},\beta_{2})=\lim_{n\to\infty}\psi_{n}(\beta_{1},\beta_{2})$ exists and

The following detailed analysis of this maximization problem is therefore fundamental to understanding the phase structure of our models, and we now address it.

Fix an integer $p\geq 2$. Consider the maximization problem for

on the interval $$, where$-\infty<\beta_{1}<\infty$and$-\infty<\beta_{2}<\infty$are parameters. Then there is a$V$-shaped region in the$(\beta_{1},\beta_{2})$plane with corner point$(\beta_{1}^{c},\beta_{2}^{c})$such that outside this region,$l(u)$has a unique local maximizer (hence global maximizer)$u^{*}$; whereas inside this region,$l(u)$always has exactly two local maximizers$u_{1}^{*}$and$u_{2}^{*}$. Moreover, for every$\beta_{1}$inside this$V$-shaped region ($\beta_{1}<\beta_{1}^{c}$), there is a unique$\beta_{2}=q(\beta_{1})$such that the two local maximizers of$l(u;\beta_{1},q(\beta_{1}))$are both global maximizers. Furthermore$q$is a continuous and decreasing function of$\beta_{1}$.

By the Lebesgue Differentiation theorem, $q$ being monotone guarantees that it is differentiable almost everywhere. {pf*}Proof of Proposition 3.2 The location of maximizers of $l(u)$ on the interval $$are closely related to properties of its derivatives$l^{\prime}(u)$and$l^{\prime\prime}(u)$,

We first analyze properties of $l^{\prime\prime}(u)$ on the interval $$. Consider instead the function

and the equality holds if and only if $u=\frac{p-1}{p}$. Thus

with the lower bound achieved only at $u=\frac{p-1}{p}$, and $m(u)$ decreases before this minimum and increases after it. This implies that for $\beta_{2}\leq\frac{p^{p-1}}{2(p-1)^{p}}$, $l^{\prime\prime}(u)\leq 0$ on the whole interval $$; whereas for$\beta_{2}>\frac{p^{p-1}}{2(p-1)^{p}}$,$l^{\prime\prime}(u)$will take on both positive and negative values, and we denote the transition points by$u_{1}$and$u_{2}$($u_{1}<\frac{p-1}{p}

Based on properties of $l^{\prime\prime}(u)$, we next analyze properties of $l^{\prime}(u)$ on the interval $$. For$\beta_{2}\leq\frac{p^{p-1}}{2(p-1)^{p}}$,$l^{\prime}(u)$is monotonically decreasing. For$\beta_{2}>\frac{p^{p-1}}{2(p-1)^{p}}$,$l^{\prime}(u)$is decreasing from to$u_{1}$, increasing from$u_{1}$to$u_{2}$and then decreasing again from$u_{2}$to$1$.

Based on properties of $l^{\prime}(u)$ and $l^{\prime\prime}(u)$, we analyze properties of $l(u)$ on the interval $$. Independent of the choice of parameters$\beta_{1}$and$\beta_{2}$,$l(u)$is a bounded continuous function,$l^{\prime}(0)=\infty$and$l^{\prime}(1)=-\infty$, so$l(u)$cannot be maximized at or$1$. For$\beta_{2}\leq\frac{p^{p-1}}{2(p-1)^{p}}$,$l^{\prime}(u)$crosses the$u$-axis only once, going from positive to negative. Thus$l(u)$has a unique local maximizer (hence global maximizer)$u^{*}$. For$\beta_{2}>\frac{p^{p-1}}{2(p-1)^{p}}$, the situation is more complicated and deserves a careful analysis. If$l^{\prime}(u_{1})\geq 0$[resp.,$l^{\prime}(u_{2})\leq 0$],$l(u)$has a unique local maximizer (hence global maximizer) at a point$u^{*}>u_{2}$(resp.,$u^{*}$). If$l^{\prime}(u_{1})<0$, then$l(u)$has two local maximizers$u_{1}^{*}$and$u_{2}^{*}$, with$u_{1}^{*}

Notice that $u_{1}$ and $u_{2}$ are solely determined by the choice of parameter $\beta_{2}>\frac{p^{p-1}}{2(p-1)^{p}}$, and vice versa. By (14),

It is not hard to see that $n(0)=\infty$, $n(1)=\infty$, $n(u)$ is decreasing from to $\frac{p-1}{p}$, then increasing from $\frac{p-1}{p}$ to $1$, and the global minimum value is

This implies in particular that $l^{\prime}(u_{1})\geq 0$ for $\beta_{1}\geq\frac{1}{2}\log(p-1)-\frac{p}{2(p-1)}$. The only possible region in the $(\beta_{1},\beta_{2})$ plane where $l^{\prime}(u_{1})<0<l^{\prime}(u_{2})$ is thus bounded by $\beta_{1}<\frac{1}{2}\log(p-1)-\frac{p}{2(p-1)}$ and $\beta_{2}>\frac{p^{p-1}}{2(p-1)^{p}}$.

We now analyze the behavior of $l^{\prime}(u_{1})$ and $l^{\prime}(u_{2})$ more closely when $\beta_{1}$ and $\beta_{2}$ are chosen from this region. Recall that by construction, $u_{1}<\frac{p-1}{p}<u_{2}$. By monotonicity of $n(u)$ on the intervals $(0,\frac{p-1}{p})$ and $(\frac{p-1}{p},1)$, there exist continuous functions $a(\beta_{1})$ and $b(\beta_{1})$ of $\beta_{1}$, such that $l^{\prime}(u_{1})<0$ for $u_{1}>a(\beta_{1})$ and $l^{\prime}(u_{2})>0$ for $u_{2}>b(\beta_{1})$. $a(\beta_{1})$ is an increasing function of $\beta_{1}$, whereas $b(\beta_{1})$ is a decreasing function, and they satisfy

Also, as $\beta_{1}\rightarrow-\infty$, $a(\beta_{1})\rightarrow 0$ and $b(\beta_{1})\rightarrow 1$. By (14), the restrictions on $u_{1}$ and $u_{2}$ yield restrictions on $\beta_{2}$. We have $l^{\prime}(u_{1})<0$ for $\beta_{2}<m(a(\beta_{1}))$ and $l^{\prime}(u_{2})>0$ for $\beta_{2}>m(b(\beta_{1}))$. Notice that $m(a(\beta_{1}))$ and $m(b(\beta_{1}))$ are both decreasing functions of $\beta_{1}$, and as $\beta_{1}\rightarrow-\infty$, they both grow unbounded. By construction, for every parameter value $(\beta_{1},\beta_{2})$, $l^{\prime}(u_{2})>l^{\prime}(u_{1})$. Also, for fixed $\beta_{1}$, $m(a(\beta_{1}))$ is the value of $\beta_{2}$ for which $l^{\prime}(u_{1})=0$, and $m(b(\beta_{1}))$ is the value for which $l^{\prime}(u_{2})=0$. Thus the curve $m(b(\beta_{1}))$ must lie below the curve $m(a(\beta_{1}))$. And together they generate the bounding curves of the $V$-shaped region in the $(\beta_{1},\beta_{2})$ plane where two local maximizers exist for $l(u)$. It is not hard to see that the corner point is given by $(\beta_{1}^{c},\beta_{2}^{c})=(\frac{1}{2}\log(p-1)-\frac{p}{2(p-1)},\frac{p^{p-1}}{2(p-1)^{p}})$. (See Figures 1–6.)

Fixing an arbitrary $\beta_{1}<\beta_{1}^{c}$, we examine the effect of varying $\beta_{2}$ on the graph of $l^{\prime}(u)$. It is clear from (12) that $l^{\prime}(u)$ shifts upward as $\beta_{2}$ increases. As a result, as $\beta_{2}$ gets large, the positive area bounded by the curve $l^{\prime}(u)$ increases, whereas the negative area decreases. By the fundamental theorem of calculus, the difference between the positive and negative areas is the difference between $l(u_{2}^{*})$ and $l(u_{1}^{*})$, which goes from negative [$l^{\prime}(u_{2})=0$, $u_{1}^{*}$ is the global maximizer] to positive [$l^{\prime}(u_{1})=0$, $u_{2}^{*}$ is the global maximizer] as $\beta_{2}$ goes from $m(b(\beta_{1}))$ to $m(a(\beta_{1}))$. Thus there must be a unique $\beta_{2}\dvtx m(b(\beta_{1}))<\beta_{2}<m(a(\beta_{1}))$ such that $u_{1}^{*}$ and $u_{2}^{*}$ are both global maximizers. We denote this $\beta_{2}$ by $q(\beta_{1})$; see Figures 1 and 6.

By analyzing the graph of $l^{\prime}(u)$, we see that the parameter values of $(\beta_{1},q(\beta_{1}))$ are exactly the ones for which positive and negative areas bounded by $l^{\prime}(u)$ equal each other. An increase in $\beta_{1}$ will induce an upward shift of $l^{\prime}(u)$, which must be balanced by a decrease in $\beta_{2}=q(\beta_{1})$. Similarly, a decrease in $\beta_{1}$ will induce a downward shift of $l^{\prime}(u)$, which must be balanced by an increase in $\beta_{2}=q(\beta_{1})$. This justifies that $q$ is monotonically decreasing in $\beta_{1}$. Furthermore, the continuity of $l^{\prime}(u)$ as a function of $\beta_{1}$ and $\beta_{2}$ implies the continuity of $q$ as a function of $\beta_{1}$.

The transition curve $\beta_{2}=q(\beta_{1})$ displays a universal asymptotic behavior as $\beta_{1}\to-\infty$

So far we have used results from CD but have avoided specific reference to the framework of graph limits, developed by Lovász et al, which was used to prove those results. We now need to refer directly to graph limits; for the notation and an introduction to this material see, for instance, LS or CD .

Let $G_{n}$ be a random graph on $n$ vertices in one of our models. For parameter values of $(\beta_{1},\beta_{2})$ in the upper half-plane $\beta_{2}>-\frac{2}{p(p-1)}$, the behavior of $G_{n}$ in the large $n$ limit is as follows:

where $U$ is the set of maximizers of (11).

The assumptions of Theorems 4.2 or 6.1 in CD are satisfied for parameter values $\beta_{2}>-\frac{2}{p(p-1)}$. By Proposition 3.2, along the curve $(\beta_{1},q(\beta_{1}))$, the maximization problem (11) is solved at two values $u_{1}^{*}$ and $u_{2}^{*}$; whereas off this curve, it is solved at a unique value $u^{*}$. Thus in the large $n$ limit, along the curve $(\beta_{1},q(\beta_{1}))$, $G_{n}$ behaves like an Erdős–Rényi graph $G(n,u)$ ($u$ picked by some distribution from $u_{1}^{*}$ and $u_{2}^{*}$); whereas off this curve, $G_{n}$ is indistinguishable from the Erdős–Rényi graph $G(n,u^{*})$.

Fix any $\beta_{2}>\beta_{2}^{c}$. Let $H$ be an edge, so $t(H,G_{n})$ is the edge density of $G_{n}$. Then there exists a continuous and decreasing function $q^{-1}(\beta_{2})$ such that

Here $u_{1}$ and $u_{2}$ are defined as in the proof of Proposition 3.2: $m(u_{1})=m(u_{2})=\beta_{2}$.

As $\beta_{2}\rightarrow\infty$, $u_{1}\rightarrow 0$ and $u_{2}\rightarrow 1$ and the jump is noticeable even for relatively small values of $\beta_{2}$. {pf*}Proof of Corollary 3.5 As $q(\beta_{1})$ is a continuous and decreasing function of $\beta_{1}$, the inverse function $q^{-1}(\beta_{2})$ exists and is also continuous and decreasing. We examine the effect of varying $\beta_{1}$ on the graph of $l^{\prime}(u)$ [and hence on the global maximizers of $l(u)$]. First note that varying $\beta_{1}$ does not change the shape of $l^{\prime}(u)$. Inside the $V$-shaped region, there are three cases. Recall that $u_{1}^{*}<u_{1}<\frac{p-1}{p}<u_{2}<u_{2}^{*}$. For $\beta_{1}=q^{-1}(\beta_{2})$, positive and negative areas bounded by $l^{\prime}(u)$ equal each other and thus $u_{1}^{*}$ and $u_{2}^{*}$ are both global maximizers. For $\beta_{1}<q^{-1}(\beta_{2})$, the graph of $l^{\prime}(u)$ shifts downward, negative area exceeds positive area and thus $u_{1}^{*}$ is the global maximizer. For $\beta_{1}>q^{-1}(\beta_{2})$, the graph of $l^{\prime}(u)$ shifts upward, positive area exceeds negative area and thus $u_{2}^{*}$ is the global maximizer. Outside the $V$-shaped region, there are two cases. Below the lower bounding curve, $l^{\prime}(u)$ has a unique local maximizer $u^{*}<u_{1}$. Above the upper bounding curve, $l^{\prime}(u)$ has a unique local maximizer $u^{*}>u_{2}$. Our conclusion then follows from Theorem 3.4.

Assume that in one of our models $H_{2}$ is a $p$-star ($p\geq 2$). For all parameter values $(\beta_{1},\beta_{2})$, the behavior of $G_{n}$ in the large $n$ limit is as follows:

where $U$ is the set of maximizers of (11).

This follows from related results in CD . We separate the parameter plane $\{(\beta_{1},\beta_{2})\}$ into upper and lower half-planes. The upper half-plane ($\beta_{2}\geq 0$) satisfies the assumptions of Theorem 4.2, and the lower half-plane ($\beta_{2}\leq 0$) satisfies the assumptions of Theorem 6.4. By similar reasoning as in Theorem 3.4, the rest of the proof follows.

The following is a straightforward application of the standard analytic implicit function theorem, which can be found, for instance, in the text of Krantz and Parks KP , so we omit the proof.

Off the end point $(\beta_{1}^{c},\beta_{2}^{c})$, the local maximizer $u^{*}$ for $l(u;\beta_{1},\beta_{2})$ ($u_{1}^{*}$ and $u_{2}^{*}$ if inside the $V$-shaped region) is an analytic function of the parameters $\beta_{1}$ and $\beta_{2}$.

Off the phase transition curve, $l(u^{*})=\max l(u;\beta_{1},\beta_{2})$ [$l(u_{1}^{*})$ or $l(u_{2}^{*})$ if inside the $V$-shaped region] is an analytic function of the parameters $\beta_{1}$ and $\beta_{2}$.

It is clear that $l(u;\beta_{1},\beta_{2})$ is analytic for $u\in(0,1)$, $\beta_{1}\in(-\infty,\infty)$, and $\beta_{2}\in(-\infty,\infty)$. Outside the $V$-shaped region, $l(u)$ has a unique local maximizer $u^{*}$ in $(0,1)$, which is analytic in $\beta_{1}$ and $\beta_{2}$ by Proposition 3.7. Inside the $V$-shaped region, $l(u)$ has two local maximizers $u_{1}^{*}$ and $u_{2}^{*}$, both have values in $(0,1)$ and are analytic in $\beta_{1}$ and $\beta_{2}$ by Proposition 3.7. Below the phase transition curve, $\max l(u)$ is given by $l(u_{1}^{*})$, which coincides with $l(u^{*})$ along the lower bounding curve. Above the phase transition curve, $\max l(u)$ is given by $l(u_{2}^{*})$, which coincides with $l(u^{*})$ along the upper bounding curve. Our claim follows by realizing that compositions of analytic functions are analytic as long as the domains and ranges match up.

Let $G_{n}$ be a random graph on $n$ vertices in one of our models. The limiting free energy density $\psi_{\infty}=\lim_{n\rightarrow\infty}\psi_{n}$ is an analytic function of the parameters $\beta_{1}$ and $\beta_{2}$ off the phase transition curve in the upper half-plane $\beta_{2}>-\frac{2}{p(p-1)}$.

The assumptions of Theorems 4.1 and 6.1 in CD are satisfied for parameter values $\beta_{2}>-\frac{2}{p(p-1)}$. Our claim then follows from Proposition 3.8.

Assume that in one of our models $H_{2}$ is a $p$-star ($p\geq 2$). The limiting free energy density $\psi_{\infty}=\lim_{n\rightarrow\infty}\psi_{n}$ is an analytic function of the parameters $\beta_{1}$ and $\beta_{2}$ off the phase transition curve.

The assumptions of Theorems 4.1 or 6.4 in CD are satisfied. Our claim again follows from Proposition 3.8.

Let $U,W\dvtx^{2}\rightarrow$ be two symmetric integrable functions. Then for every finite simple graph $F$,

By Corollary 3.5, these two first derivatives $\frac{\partial}{\partial\beta_{1}}\psi_{\infty}$ and $\frac{\partial}{\partial\beta_{2}}\psi_{\infty}$ are discontinuous across the curve (except at the end point). Let us now take a closer look at the behavior of $\psi_{\infty}$ at the critical point. Recall that $l^{\prime}(u;\beta_{1}^{c},\beta_{2}^{c})$ is monotonically decreasing on $$, and the unique zero is achieved at$\frac{p-1}{p}$. Take any$0<\varepsilon<\frac{1}{p}$. Set$\delta=\min\{l^{\prime}(\frac{p-1}{p}-\varepsilon),-l^{\prime}(\frac{p-1}{p}+\varepsilon)\}$. Consider$(\beta_{1},\beta_{2})$so close to$(\beta_{1}^{c},\beta_{2}^{c})$such that$|\beta_{1}-\beta_{1}^{c}|+p|\beta_{2}-\beta_{2}^{c}|<\delta$. For every$u$in$$, we then have$|l^{\prime}(u;\beta_{1},\beta_{2})-l^{\prime}(u;\beta_{1}^{c},\beta_{2}^{c})|<\delta$. It follows that the zeros$u^{*}(\beta_{1},\beta_{2})$($u_{1}^{*}$and$u_{2}^{*}$if inside the$V$-shaped region) must satisfy$|u^{*}-\frac{p-1}{p}|<\varepsilon$, which easily implies the continuity of$\frac{\partial}{\partial\beta_{1}}\psi_{\infty}$and$\frac{\partial}{\partial\beta_{2}}\psi_{\infty}$at$(\beta_{1}^{c},\beta_{2}^{c})$. To see that the transition at the critical point is second-order, we check the second derivatives of$\psi_{\infty}$ in its neighborhood. Off the phase transition curve,

But as was explained in the proof of Proposition 3.2, $l^{\prime\prime}(u^{*})$ converges to zero as $(\beta_{1},\beta_{2})$ approaches $(\beta_{1}^{c},\beta_{2}^{c})$; the desired singularity is thus justified.

Summary

Much of the literature on phase transitions in exponential random graph models uses techniques such as mean-field approximations, which are mathematically uncontrolled. As such they have been useful in discovering interesting behavior, but they can be misleading in detail. For instance, although phase transitions have been discovered in this way for the two-star ($H_{2}$ a two-star) PN1 and edge-triangle ($H_{2}$ a triangle) models PN2 , the approximation leads to an error in the qualitative nature of the transition, attributing phase coexistence to the full $V$-shaped region of Figure 1 rather than just the curve $\beta_{2}=q(\beta_{1})$; in other words it does not distinguish the local maxima in the region from the global maxima.

Chatterjee and Diaconis CD gave the first rigorous proof of singular behavior in an exponential random graph model, the edge-triangle model. Our paper is an extension of this important first step; besides extending the models and parameters under control we have provided a mathematical framework of “phases” which we hope will be useful in motivating future mathematical work in this subject. Our results show that all models with “attraction” ($\beta_{2}>0$) exhibit a transition qualitatively like the gas/liquid transition: a first order transition corresponding to a discontinuity in density, with a second order critical point. In Theorem 7.1 of CD Chatterjee and Diaconis suggest that, quite generally, models with repulsion exhibit a transition qualitatively like the solid/fluid transition, in which one phase has nontrivial structure, as distinguished from the “disordered” Erdős–Rényi graphs, which have independent edges. We have not yet been able to extend our results to this regime, except for $p$-star models with “repulsion” ($\beta_{2}<0$) which we prove do not exhibit a transition. It is an important open problem to determine the qualitative behavior of models based on an $H_{2}$ with chromatic number above 2.

Acknowledgements

It is a pleasure to acknowledge useful discussions with Persi Diaconis and Sourav Chatterjee introducing us to the subject of random graphs and giving us early access to, and explanations of, their preprint CD , as well as suggestions for improving early drafts of this paper.

References