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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References
Angeli, L., Grosskinsky, S., Johansen, A. M.: Limit theorems for cloning algorithms. arXiv:1902.00509 (2019)
Angeli, L., Grosskinsky, S., Johansen, A.M., Pizzoferrato, A.: Rare event simulation for stochastic dynamics in continuous time. J. Stat. Phys. 176(5), 1185–1210 (2019)
Aristoff, D., Zhu, L.: Asymptotic structure and singularities in constrained directed graphs. Stoch. Process. Appl. 125(11), 4154–4177 (2015)
Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8(1), 141–148 (1988)
Bhamidi, S., Bresler, G., Sly, A.: Mixing time of Exponential Random Graphs. 2008 49th Annual IEEE Symposium on Foundations of Computer Science. IEEE (2008)
Bhamidi, S., Hannig, J., Lee, C.Y., Nolen, J.: The importance sampling technique for understanding rare events in Erdös–Rényi random graphs. Electron. J. Probab. 20, 1 (2015)
Bollobás, B.: Modern Graph Theory, vol. 184. Springer, New York (2013)
Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Counting graph homomorphisms. Top. Discret. Math. 1, 315–371 (2006)
Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Convergent sequences of dense graphs i: subgraph frequencies, metric properties and testing. Adv. Math. 219(6), 1801–1851 (2008)
Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Convergent sequences of dense graphs ii. Multiway cuts and statistical physics. Ann. Math. 1, 151–219 (2012)
Carollo, F., Pérez-Espigares, C.: Entanglement statistics in Markovian open quantum systems: a matter of mutation and selection. arXiv:1910.13940 (2019)
Chatterjee, S.: Large Deviations for Random Graphs: École D’Été de Probabilités de Saint-Flour XLV-2015, vol. 2197. Springer, New York (2017)
Chatterjee, S., Dembo, A.: Nonlinear large deviations. Adv. Math. 299, 396–450 (2016)
Chatterjee, S., Dey, P.S.: Applications of stein method for concentration inequalities. Ann. Probab. 38(6), 2443–2485 (2010)
Chatterjee, S., Diaconis, P.: Estimating and understanding exponential random Graph models. Ann. Stat. 41(5), 2428–2461 (2013)
Chatterjee, S., Varadhan, S.: The large deviation principle for the Erdös–Rényi random graph. Eur. J. Comb. 32(7), 1000–1017 (2011)
Dembo, A., Lubetzky, E.: A large deviation principle for the Erdös–Rényi uniform random graph. Electron. Commun. Probab. 23, 1 (2018)
Den Hollander, F., Mandjes, M., Roccaverde, A., Starreveld, N.J.: Ensemble equivalence for dense graphs. Electron. J. Probab. 23, 1 (2018)
Giardinà, C., Kurchan, J., Peliti, L.: Direct evaluation of large-deviation functions. Phys. Rev. Lett. 96(12), 120603 (2006)
Giardinà, C., Kurchan, J., Lecomte, V., Tailleur, J.: Simulating rare events in dynamical processes. J. Stat. Phys. 145(4), 787–811 (2011)
Hurtado, P.I., Pérez-Espigares, C., Del Pozo, J., Garrido, P.L.: Thermodynamics of currents in nonequilibrium diffusive systems: theory and simulation. J. Stat. Phys. 154(1–2), 214–264 (2014)
Kenyon, R., Radin, C., Ren, K., Sadun, L.: Multipodal structure and phase transitions in large constrained graphs. J. Stat. Phys. 168(2), 233–258 (2017)
László, L.: Large Networks and Graph Limits, vol. 60. American Mathematical Society, Providence (2012)
László, L., Szegedy, B.: Limits of dense graph sequences. J. Comb. Theory B 96(6), 933–957 (2006)
López, F.A., Coolen, A.C.C.: Imaginary replica analysis of loopy regular random graphs. J. Phys. A 53(6), 065002 (2020)
Lopez, F.A., Coolen, A.C.C.: Transitions in loopy random graphs with fixed degrees and arbitrary degree distributions. arXiv:2008.11002 (2020)
Lubetzky, E., Zhao, Y.: On replica symmetry of large deviations in random graphs. Rand. Struct. Algorithms 47(1), 109–146 (2015)
Metz, F.L., Castillo, I.P.: Condensation of degrees emerging through a first-order phase transition in classical random graphs. Phys. Rev. E 100(1), 012305 (2019)
Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)
Pérez-Espigares, C., Hurtado, P.I.: Sampling rare events across dynamical phase transitions. Chaos Interdiscipl. J. Nonlinear Sci. 29(8), 083106 (2019)
Pikhurko, O., Razborov, A.: Asymptotic structure of graphs with the minimum number of triangles. Comb. Probab. Comput. 26(1), 138–160 (2017)
Radin, C., Sadun, L.: Phase transitions in a complex network. J. Phys. A 46(30), 305002 (2013)
Radin, C., Sadun, L.: Singularities in the entropy of asymptotically large simple graphs. J. Stat. Phys. 158(4), 853–865 (2015)
Touchette, H.: Equivalence and nonequivalence of ensembles: thermodynamic, macrostate, and measure levels. J. Stat. Phys. 159(5), 987–1016 (2015)
Zhao, Y.: On the lower tail variational problem for random graphs. Comb. Probab. Comput. 26(2), 301–320 (2017)
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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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
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.
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:
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, :
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:
The cloning algorithm simulates (51) by producing a final population of expected size
Then, the cumulant generating function is computed from the size of the final population \(M_3\) as
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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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DOI: https://doi.org/10.1007/s10955-021-02707-3