Critical behavior of two-choice rules,
a class of Achlioptas processes

Let $p(s,t)$ be the probability that a randomly chosen vertex belongs to a cluster of size $s$ at time $t$ (in the limit $n\to\infty$). Let $\theta(t)=1-\sum_{s}p(s,t)$, and $\chi_{k}(t)=\sum_{s}s^{k}p(s,t)$ (the value $s=\infty$ is excluded from each of the sums). The exponents $\beta$, $\gamma$, and $\Delta$ are defined by

	$\displaystyle\theta(t)$	$\displaystyle\approx(t-t_{c})^{\beta}\qquad\hbox{as $t\downarrow t_{c}$}$
	$\displaystyle\chi_{1}(t)$	$\displaystyle\approx|t-t_{c}|^{-\gamma}\qquad\hbox{as $t\to t_{c}$}$
	$\displaystyle\hbox{for $k\geq 2$}\quad\chi_{k}(t)/\chi_{k-1}(t)$	$\displaystyle\approx|t-t_{c}|^{-\Delta}\qquad\hbox{as $t\to t_{c}$}$

In the physics literature the meaning of $\approx$ is not precisely defined. It could be something as weak as

$$\frac{\log\theta(t)}{\log(t-t_{c})}\to\beta\qquad\hbox{as $t\downarrow t_{c}$}$$

In order to derive relations between exponents we will suppose $\theta(t)\sim C(t-t_{c})^{\beta}$ where $a(t)\sim b(t)$ means $a(t)/b(t)\to 1$. Our final two exponents concern the behavior near the critical value

$$p(s,t)\approx s^{1-\tau}f(s|t-t_{c}|^{1/\sigma})$$

(1)

where $f$ is a scaling function.

In Section 5 we will compute the critical exponents for the Erdös-Rényi model,

$$\beta=1,\quad\gamma=1,\quad\Delta=2,\quad\tau=5/2,\quad\sigma=1/2$$

(2)

and show that the scaling relationship (1) holds. These results are well-known. We include them for completeness.

The most significant difference between our critical exponents and those for $d$-dimensional percolation is that we do not have a correlation length $\xi(p)$ that gives the spatial size of a typical finite cluster. Thi squantity is often defined in terms of the exponential decay of probability 0 and $x$ are in the same finite cluster

$$\tau^{f}(0,x)=P_{p}(0\leftrightarrow x,|{\cal C}_{0}|<\infty).$$

For example, if $e_{1}$ is the first unit vector

$$\xi(p)=\lim_{n\to\infty}-\frac{1}{n}\log\tau^{f}(0,ne_{1})$$

A simpler approach taken by Kesten [18] is to define the correlation length by

$$\xi(p)=\left(\frac{1}{\chi_{1}(p)}\sum_{y}|y|^{2}P(0\to y;|{\cal C}_{0}|<\infty)\right)^{1/2}$$

and the critical exponent $\nu$ by $\xi(p)\approx|p-p_{c}|^{-\nu}$. Finally on a $d$-dimensional graph we have another exponent $\eta$ called the anomalous dimension

$$P_{cr}(0\to x)\approx|x|^{2-d-\eta}$$

The percolation critical exponents satisfy the following scaling relationships

$$\beta=\frac{2\nu}{\delta+1}\qquad\gamma=2\nu\frac{\delta-1}{\delta+1}\qquad\Delta=2\nu\frac{\delta}{\delta+1}\qquad\eta=\frac{4}{\delta}$$

(3)

Kesten [18] established provided that the exponents $\delta$ and $\nu$ exist.

In the Erdös-Rényi model $p(s,t)$ satisfies

$$\frac{\partial p(s,t)}{\partial t}=s\sum_{u+v=s}p(u,t)p(v,t)-2sp(s,t)$$

(4)

In words if the new edge joins a cluster of size $u$ and $v$ with $u+v=s$ we now have $s$ new vertices that are part of clusters of size $s$, while if a cluster of size $s$ is one of the two that merge then the $s$ vertices in that cluster are no longer part of a cluster of size $s$. If we let $n(s,t)=p(s,t)/s$ be the number of clusters of size s and $K(u,v)=uv$ then (4) becomes the Smoluchowski equation

$$\frac{\partial n(s,t)}{\partial t}=\sum_{u+v=s}K(u,v)n(u,t)n(v,t)-2n(s,t)$$

(5)

For probabilists the best known reference is Aldous [3], but the survey by Leyvraz [19] is more extensive. In addition to results about existence, uniqueness, and asymptotic behavior, there are exact results for $K(u,v)=1$ (Kingman’s coalescent) and $K(u,v)=u+v$ (the additive case).

It has long been known for percolation (see Stauffer [27]) that (1) implies

	$\displaystyle\beta$	$\displaystyle=(\tau-2)/\sigma$		(13)
	$\displaystyle\gamma$	$\displaystyle=(3-\tau)/\sigma$		(14)
	$\displaystyle\Delta$	$\displaystyle=1/\sigma$		(15)

In Section 4.1 we give the proofs, and state the assumptions on $\tau$ and the scaling function $f$ in (1) that are needed for these results to be theorems.

Let $\beta_{\phi}$, $\gamma_{\phi}$, $\delta_{\phi}$, and $\tau_{\phi}$ be the values of the critical exponents when $p(s,t)$ is replaced by $\phi(s,t)$. Since $\phi$ is computed from $p$, we assume that $\phi$ has the scaling property

$$\phi(s,t)\approx s^{\alpha}g(s|t-t_{c}|^{1/\sigma}$$

(16)

Here $\alpha=1-\tau_{\phi}$ and $\sigma_{\phi}=\sigma$. As we explain in Section 4.1 (13)–(15) hold for the exponents with subscript $\phi$, Using (8), (9), (11), and (12) we see that

Using (6) we can show, see Lemma 4 and 6 that if $\Phi=1-\sum_{v}\phi(v,t)$ then

	$\displaystyle\partial\theta(t)/\partial t$	$\displaystyle=2\langle s\rangle_{\phi}\Phi$		(18)
	$\displaystyle\partial\langle s\rangle_{P}/\partial t$	$\displaystyle=2\langle s\rangle^{2}_{\phi}-2\langle s^{2}\rangle_{\phi}\Phi$		(19)

From this it follows that

	$\displaystyle\beta-1$	$\displaystyle=\beta_{\phi}-\gamma_{\phi}\$		(20)
	$\displaystyle-\gamma-1$	$\displaystyle=-2\gamma_{\phi}$		(21)
	$\displaystyle-\gamma-1$	$\displaystyle=\beta_{\phi}-\gamma_{\phi}-\Delta_{\phi}$		(22)

On the second and third lines the two formulas come from $t\uparrow t_{c}$ and $t\downarrow t_{c}$ in (19).

Finally using a calculation from Appendix E of da Costa et al [10] we have the remarkable result which relates the two variable in the scaling relation

$$\sigma=(\tau-1)+2\alpha+2$$

(23)

The proof begins by letting $f(x)=x^{\tau-1}\tilde{f}(x)$, and $g(x)=x^{-\alpha}\tilde{g}(x)$ in order to rewrite the scaling formulas as

	$\displaystyle p(s,t)$	$\displaystyle=s^{1-\tau}f(s\delta^{1/\sigma})=\delta^{(\tau-1)/\sigma}\tilde{f}(s\delta^{1/\sigma})$
	$\displaystyle\phi(s,t)$	$\displaystyle=s^{\alpha}g(s\delta^{1/\sigma})=\delta^{-\alpha/\sigma}\tilde{g}(s\delta^{1/\sigma})$

in order to more easily differentiate with respect to $\delta-|t-t_{c}|$. One then substitutes $\phi(v)=\phi(v)-\phi(s)+\phi(s)$ in (6) of $\partial p(s,t)/\partial t$ so that all of the terms have the large time $s$ in them. When the smoke clears after all of the change of variables, every term on the right0hand side is of order $\delta^{(-2\alpha-2)/\sigma}$ and equating this to the order of the left-hand side gives the result.

Using our scaling relationships we can analyze our examples

Theorem 1 implies that the Min(m) models are in different universlaity classes (if the values of $\Delta$ are the same for $\ell$ and $m$ then $(2\ell-1)\beta_{\ell}=(2m-1)\beta_{m}$ the values of $\tau-2$ must be different). Sabbir and Hassam [25] claim that the sum and the product rule defined in the next section have the same critical exponents, but otherwise it seems that within this class of models, the universality classes are small.

The next result treats our other two models.

While in principle we can study two-edge rules in which the choice functions are different, in practice we can only analyze the case in which the first choice is a randomly chosen vertex,

Generalizing the proofs of Lemmas 4 and 6 in Section 4.2

	$\displaystyle\partial\theta(t)/\partial t$	$\displaystyle=\langle s\rangle_{p}\Phi+\langle s\rangle_{\phi}\theta$
	$\displaystyle\partial\langle s\rangle_{P}/\partial t$	$\displaystyle=2\langle s\rangle_{\phi}\langle s\rangle_{p}-\langle s^{2}\rangle_{p}\Phi-\langle s^{2}\rangle_{\phi}\theta$

From this it follows that

	$\displaystyle\beta-1$	$\displaystyle=\min\{\beta-\gamma_{\phi},\beta_{\phi}-\gamma\}$		(25)
	$\displaystyle-\gamma-1$	$\displaystyle=\gamma_{\phi}-\gamma$		(26)

where in the second equation we have only used the limit $t\uparrow t_{c}$. Generalizing the proof of (23) we can conclude that

$$\sigma=\alpha+2$$

(27)

Combining our results we have

Bohman and Frieze [6] were the first to describe a rule that made percolation come later than in the Erdös-Rényi case. Again $G_{0}=0$ and edges $e_{1},e_{1}^{\prime};e_{2},e_{2}^{\prime};e_{3},e_{3}^{\prime}\ldots$ are chosen at random from the set of all possible edges. If on the $i$th step $e_{i}$ connects two isolated vertices it is added to the graph, otherwise $e_{i}^{\prime}$ is added. Using our convention that the graph after $k$ choices is the sate at time to time $2k/n,$ then as $n\to\infty$ They showed that

As in ordinary percxolation, when $t>t_{c}$ there is a giant component containing a positive fraction of the sites, while for $t<t_{c}$ the cluster size distribution has $P(\kappa\geq s)\leq Ke^{-cs}$ where $K$ and $c$ depend on $t$.

In 2013, Bhamidi, Budhirja and Wang [5] showed that the behavior of the Bohman-Frieze process near the critical values is the same as in the Erdös-Rényi case. To be precise if we consider the behavior at $t_{c}+r/n^{1/3}$ with $-\infty<r<\infty$ then as $n\to\infty$ the system converges to the multiplicative coalecsent in which a cluster of size $x$ and a cluster of size $y$ merge at rate $xy$. See Aldous [2] for the corresponding result for Erdös-Rényi graphs.

Riordan and Warnke [24] have show that “bounded-size rules share (in a strong sense) all the features of the Erdös-Rényi phase transiiton.” Let $L_{j}$ be the size of the $j$th largest component and let

$$S_{r}=\sum_{{\cal C}}|{\cal C}|/n=\sum_{k\geq 1}k^{r-1}N_{k}/n$$

where the first sum is over all components and $N_{k}$ is the number of vertices that belong to components of size $k$. Writing whp (with high probability) for with probaility tending to 1 as $n\to\infty$ they established (see their Theorem 1.1):

In 2009 Achliptas, D’Souza, and Spencer [1] shouted from the pages of Science that they had found “Explosive percolation in networks.” To quote from their paper:

Here we provide conclusive numerical evidence that unbounded size rules can give rise to discontinuous transitions. For concreteness, we present results for the so-called product rule.

In 2011 Riordan and Warnke wrote their own Science paper declaring that “Explosive percolation is continuous.” Here we will follow the version published the next year in the Annals of Applied Probability. [22]. The continuity of the phase transition is proved for a very general set of models

To obtain more detailed results they reduced the collection of models. An $\ell$-vertex rule is said to be merging if whenever ${\cal C}_{1}$ and ${\cal C}_{2}$ are distinct components with $|{\cal C}_{1}|,|{\cal C}_{2}|\geq\epsilon n$ then in the next step we have probability $\geq\epsilon^{\ell}$ of joining ${\cal C}_{1}$ to ${\cal C}_{2}$.

The results in [22] are very general and their proofs are ingenious. However, the proof of their Theorem 1 given on page 1457 is based on an argument by contradiction, and it does not seem possible to extract quantitative result. Our goal here is to prove results about the critical behavior of some of these processes, specifically about the values of critical exponents defined in the next section.

Lemma 1 implies $\beta_{\phi}=m\beta$, $\nu=m$, so using (20)

$$\beta-1=m\beta-\gamma_{\phi}\quad\hbox{or}\quad\gamma_{\phi}=1+(m-1)\beta$$

Using the first equality in (21) $-\gamma-1=-2\gamma_{\phi}$ so

$$\gamma=2\gamma_{\phi}+1=1+2(m-1)\beta$$

To prove the next result we note that Lemma 1 implies

$$\alpha=2m-1-m\tau$$

and using (23) $\sigma=(\tau-1)+2\alpha+2$

	$\displaystyle\sigma$	$\displaystyle=\tau-1+2(2m-1-m\tau)+2$
		$\displaystyle=1+2(2m-1)-(2m-1)\tau=1-(2m-1)(\tau-2)$		(28)

Dividing each side by $\sigma$ and using (13) $(\tau-2)/\sigma=\beta$ gives

$$1/\sigma=1+(2m-1)\beta$$

Rearranging (28) then using the last result twice

$$\tau-2=\frac{1-\sigma}{2m-1}=\frac{(1/\sigma-1)/(2m-1)}{1/\sigma}=\frac{\beta}{1+(2m-1)\beta}$$

which completes the proof.

Since $\phi$ is Min(m), Lemma 1 implies $\beta_{\phi}=m\beta$ so (20) implies $\gamma_{\phi}=1$ and (21) implies $\gamma=1$. Using (13), (14) and (15) we have

$$\Delta=1/\sigma=\beta+\gamma=1+\beta$$

Using (13)

$$\tau-2=\frac{\beta}{1/\sigma}=\frac{\beta}{1+\beta}$$

which completes the proof.

Lemma 1 implies $\beta_{\phi}=\beta$. (26) implies $\gamma_{\phi}=1$. The $\phi$ versions of (13), (14) and (15) imply

$$\Delta=1/\sigma=\beta_{\phi}+\gamma_{\phi}=1+m\beta$$

(27) and (28) imply

$$\sigma=\alpha+2=2m+1-m\tau=1-m(\tau-2)$$

Rearranging gives

$$\tau-2=\frac{1-\sigma}{m}=\frac{(1/\sigma-1)/m}{1/\sigma}=\frac{\beta}{1+m\beta}$$

To get the final result we have to assume that the two quantities on the right hand side of (25) are equal, i.e. $\beta-\gamma_{\phi}=\beta_{\phi}-\gamma$, in order to conclude that

$$\gamma=1+(m-1)\beta$$

To have rigorous result we state explicitly what we assume about $\tau$ and about the scaling function. Throughout we only use the scaling relationship

$$p(s,t)=s^{1-\tau}f(s|t-t_{c}|^{1/\sigma})$$

Since $\phi(s,t)=s^{1-\tau_{\phi}}g(s|t-t_{c}|^{1/\sigma})$ the $\phi$ versions of the scaling relationships follow immediately.

The next result establishes (14) and (15)

While the proof is fresh on the readers mind we will prove the result for the alternate edge rule.