We analyze ensembles of random networks with fixed numbers of edges, triangles, and nodes. In the limit as the number of nodes goes to infinity, this model is known to exhibit sharp phase transitions as the density of edges and triangles is varied. In this paper we study finite size effects in ensembles where the number of nodes is between 30 and 66. Despite the small number of nodes, the phases and phase transitions are clearly visible. Concentrating on 54 nodes and one particular phase transition, we use spectral data to examine the structure of a typical graph in each phase, and see that is it very similar to the graphon that describes the system as n diverges. We further investigate how graphs change their structure from that of one phase to the other under a natural edge flip dynamics in which the number of triangles is either forced downwards or allowed to drift upwards. In each direction, spectral data shows that the change occurs in three stages: a very fast stage of about 100 edge flips, in which the triangle count reaches the targeted value, a slower second stage of several thousand edge flips in which the graph adopts the qualitative features of the new phase, and a very slow third stage, requiring hundreds of thousands of edge flips, in which the fine details of the structure equilibrate.
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Abbe, E.: Community detection and stochastic block models: recent developments. J. Mach. Learn. Res. 18, 6446–6531 (2017)
Borgs, C., Chayes, J., Lovász, L.: Moments of two-variable functions and the uniqueness of graph limits. Geom. Funct. Anal. 19, 1597–1619 (2010)
Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Convergent graph sequences I: subgraph frequencies, metric properties, and testing. Adv. Math. 219, 1801–1851 (2008)
Chatterjee, S., Diaconis, P.: Estimating and understanding exponential random graph models. Ann. Stat. 41, 2428–2461 (2013)
Chatterjee, S., Varadhan, S.R.S.: The large deviation principle for the Erdős–Rényi random graph. Eur. J. Comb. 32, 1000–1017 (2011)
Kenyon, R., Radin, C., Ren, K., Sadun, L.: Multipodal structure and phase transitions in large constrained graphs. J. Stat. Phys. 168, 233–258 (2017)
Kenyon, R., Radin, C., Ren, K., Sadun, L.: Bipodal structure in oversaturated random graphs. Int. Math. Res. Notices 2016, 1009–1044 (2018)
Kenyon, R., Radin, C., Ren, K., Sadun, L.: The phases of large networks with edge and triangle constraints. J. Phys. A 50, 435001 (2017)
Koch, H.: Vertex order in some large constrained random graphs. SIAM J. Math. Anal. 48, 2588–2601 (2016)
Lovász, L.: Large Networks and Graph Limits. American Mathematical Society, Providence (2012)
Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Comb. Theory Ser. B 98, 933–957 (2006)
Lovász, L., Szegedy, B.: Szemerédi’s lemma for the analyst. GAFA 17, 252–270 (2007)
Lovász, L., Szegedy, B.: Finitely forcible graphons. J. Comb. Theory Ser. B 101, 269–301 (2011)
Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)
Radin, C., Sadun, L.: Phase transitions in a complex network. J. Phys. A 46, 305002 (2013)
Radin, C., Sadun, L.: Singularities in the entropy of asymptotically large simple graphs. J. Stat. Phys. 158, 853–865 (2015)
Radin, C., Ren, K., Sadun, L.: The asymptotics of large constrained graphs. J. Phys. A 47, 175001 (2014)
Radin, C., Ren, K., Sadun, L.: A symmetry breaking transition in the edge/triangle network model. Ann. Inst. H. Poincaré D 5, 251–286 (2018)
Radin, C., Swinney, H.: Phases of granular matter. J. Stat. Phys. 175, 542–553 (2019)
Rietz, F., Radin, C., Swinney, H., Schröter, M.: Nucleation in sheared granular matter. Phys. Rev. Lett. 120, 055701 (2018)
Strauss, D.: On a general class of models for interaction. SIAM Rev. 28, 513–527 (1986)
Uhlenbeck, G.E.: In: Cohen, E.G.D. (ed.) Fundamental Problems in Statistical Mechanics II, pp. 16–17. Wiley, New York (1968)
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Communicated by Federico Ricci-Tersenghi.
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Neeman, J., Radin, C. & Sadun, L. Phase Transitions in Finite Random Networks. J Stat Phys (2020). https://doi.org/10.1007/s10955-020-02582-4
- Random graph
- Phase transition
- Finite size effect
- Markov Chain Monte Carlo