Approximating the Cumulant Generating Function of Triangles in the Erdös–Rényi Random Graph

Abstract

We study the pressure of the “edge-triangle model”, which is equivalent to the cumulant generating function of triangles in the Erdös–Rényi random graph. The investigation involves a population dynamics method on finite graphs of increasing volume, as well as a discretization of the graphon variational problem arising in the infinite volume limit. As a result, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to the one of an equi-bipartite graph.

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Acknowledgements

We acknowledge an enlightening discussion with Remco van der Hofstad. We thank M. Prato and S. Rebegoldi for making available to us their Gradient Projection code.

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Correspondence to Claudio Giberti.

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Communicated by Pierpaolo Vivo.

Appendix: \(n=3\)

Appendix: \(n=3\)

In order to show how the cloning algorithm works, we explicitly compute the cumulant generating function of triangles in the simplest case, that is the graph of size \(n=3\). Since the probability of the unique triangle is \(p^3\), we have

$$\begin{aligned} \mu _{3,p}(\alpha ) = \frac{1}{3} \ln \left\langle \exp \frac{\alpha }{3} T_3(X) \right\rangle ^{ER}_3 = \frac{1}{3} \ln \left[ \mathrm{e}^\frac{\alpha }{3}p^3 +1-p^3 \right] . \end{aligned}$$
(46)

We compute again (46) by applying the dynamics described in Sect. 3 to a family of M clones. The three steps required to construct the edges of the graph \(G_3\) are represented in Fig. 11.

Fig. 11
figure11

Graphical representation of the evolution step, according to the tilted dynamics \(P_\alpha (\cdot , \cdot )\) of cloning scheme. The levels of the tree display the values of the elements of the adjacency matrix \(x_{1,2}, x_{1,3}, x_{2,3}\) during the evolution leading to \(G_3\). M is the initial size of the population. In all levels the expected sizes of the sub-populations with a given configuration of values \(x_{1,2}, x_{1,3}, x_{2,3}\) are reported

The leafs of the tree represent the occupation variables of the three possible edges, \(x_{1,2}, x_{1,3}, x_{2,3}\). In the first step the edge connecting two vertices, say 1 and 2, of each clone is added with probability p. Thus, in the clone population about pM graphs have the edge (1, 2) and \((1-p)M\) graphs have not this edge, see the first level in Fig.11 . In the second step the edge (1, 3) is added, still with probability p, leading to four possible values for the pair \((x_{1,2},x_{1,3})\). Thus, the expected numbers of types are \(Mp^2, Mp(1-p), Mp(1-p), M(1-p)^2\), see level 2 in Fig. 11. Let us observe that in the first two steps, since \(\Delta T(x,y)=0\), the original and tilted transition probabilities coincide: \(P_\alpha (x,y)=P(x,y)\), see (30). The situation changes in the last step. Indeed, the configuration of edges \(x=11\) (which means that \(x_{1,2}=1\) and \(x_{1,3}=1\)) may evolve to \(y=111\), with \(\Delta T(x,y)=1\), or to \(y=110\), in which case \(\Delta T(x,y)=0\). For all the other configurations x we have \(\Delta T(x,y)=0\). Thus, see the definition (30) of the tilted probability:

$$\begin{aligned} P_\alpha (11,111)= \frac{\mathrm{e}^\frac{\alpha }{3}p}{ \mathrm{e}^\frac{\alpha }{3}p+1-p}, \quad P_\alpha (11,110)= \frac{1-p}{ \mathrm{e}^\frac{\alpha }{3}p+1-p}, \end{aligned}$$
(47)

being \(P(11,111)=p\), \(P(11,110)=1-p\) and \(k_\alpha (11)= p\mathrm{e}^\frac{\alpha }{3}+1-p\). Then, the probability of the paths connecting the root \(\phi \) to the leafs 111, respectively 110, will be, :

$$\begin{aligned} {\mathbb {P}}(\phi ,111)= \frac{\mathrm{e}^\frac{\alpha }{3}p^3}{ \mathrm{e}^\frac{\alpha }{3}p+1-p},\quad {\mathbb {P}}(\phi ,110)= \frac{p^2(1-p)}{ \mathrm{e}^\frac{\alpha }{3}p+1-p}. \end{aligned}$$
(48)

The average in (31) can be computed as the sum over the paths from the root to the leafs (equivalently, as the sum over the leafs). Recalling that \(k_\alpha (11)= p\mathrm{e}^\frac{\alpha }{3}+1-p \) and observing also that \(k_\alpha (x)=1\) if \(x\ne 11\), from (31) we have:

$$\begin{aligned} \mu _{3,p}(\alpha )= & {} \frac{1}{3} \ln \left[ {\mathbb {P}}(\phi ,111) k_\alpha (11)+ {\mathbb {P}}(\phi ,110) k_\alpha (11)+ \sum _{x\notin \{111, 110\}}{\mathbb {P}}(\phi ,x ) \right] \end{aligned}$$
(49)
$$\begin{aligned}= & {} \frac{1}{3} \ln \left[ \frac{p^3\mathrm{e}^\frac{\alpha }{3}}{ p\mathrm{e}^\frac{\alpha }{3}+1-p} (p\mathrm{e}^\frac{\alpha }{3}+1-p) + \frac{p^2(1-p)}{ p\mathrm{e}^\frac{\alpha }{3}+1-p} (p\mathrm{e}^\frac{\alpha }{3}+1-p) + (1-p^2) \right] \nonumber \\ \end{aligned}$$
(50)
$$\begin{aligned}= & {} \frac{1}{3} \ln \left[ p^3\mathrm{e}^\frac{\alpha }{3} +1-p^3 \right] , \end{aligned}$$
(51)

The cloning algorithm simulates (51) by producing a final population of expected size

$$\begin{aligned} M_3=M\,{\mathbb {P}}(\phi ,111) k_\alpha (11)+ M\, {\mathbb {P}}(\phi ,110) k_\alpha (11)+\sum _{x\notin \{111, 110 \}} M\, {\mathbb {P}}(\phi , x). \end{aligned}$$

Then, the cumulant generating function is computed from the size of the final population \(M_3\) as

$$\begin{aligned} \mu _{3,p}(\alpha ) = \frac{1}{3} \ln \left[ \frac{M_3}{M} \right] . \end{aligned}$$
(52)

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Giardinà, C., Giberti, C. & Magnanini, E. Approximating the Cumulant Generating Function of Triangles in the Erdös–Rényi Random Graph. J Stat Phys 182, 23 (2021). https://doi.org/10.1007/s10955-021-02707-3

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Keywords

  • Erdös–Rényi random graph
  • Edge-triangle model
  • Rare events simulations
  • Phase transition
  • Graphs limits
  • Ensemble equivalence