## Annals of Applied Probability

### Max $\kappa$-cut and the inhomogeneous Potts spin glass

#### Abstract

We study the asymptotic behavior of the Max $\kappa$-cut on a family of sparse, inhomogeneous random graphs. In the large degree limit, the leading term is a variational problem, involving the ground state of a constrained inhomogeneous Potts spin glass. We derive a Parisi-type formula for the free energy of this model, with possible constraints on the proportions, and derive the limiting ground state energy by a suitable zero temperature limit.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 3 (2018), 1536-1572.

Dates
Revised: July 2017
First available in Project Euclid: 1 June 2018

https://projecteuclid.org/euclid.aoap/1527840026

Digital Object Identifier
doi:10.1214/17-AAP1337

Mathematical Reviews number (MathSciNet)
MR3809471

Zentralblatt MATH identifier
06919732

#### Citation

Jagannath, Aukosh; Ko, Justin; Sen, Subhabrata. Max $\kappa$-cut and the inhomogeneous Potts spin glass. Ann. Appl. Probab. 28 (2018), no. 3, 1536--1572. doi:10.1214/17-AAP1337. https://projecteuclid.org/euclid.aoap/1527840026

#### References

• [1] Aiello, W., Chung, F. and Lu, L. (2001). A random graph model for power law graphs. Exp. Math. 10 53–66.
• [2] Aizenman, M., Sims, R. and Starr, S. L. (2003). Extended variational principle for the Sherrington–Kirkpatrick spin-glass model. Phys. Rev. B 68 214403.
• [3] Albert, R. and Barabási, A.-L. (2002). Statistical mechanics of complex networks. Rev. Modern Phys. 74 47–97.
• [4] Arguin, L.-P. and Aizenman, M. (2009). On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 1080–1113.
• [5] Auffinger, A. and Chen, W.-K. (2017). Parisi formula for the ground state energy in the mixed $p$-spin model. Ann. Probab. 45 4617–4631.
• [6] Auffinger, A. and Chen, W.-K. (2014). Free energy and complexity of spherical bipartite models. J. Stat. Phys. 157 40–59.
• [7] Barra, A., Contucci, P., Mingione, E. and Tantari, D. (2015). Multi-species mean field spin glasses. Rigorous results. Ann. Henri Poincaré 16 691–708.
• [8] Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31 3–122.
• [9] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2008). Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing. Adv. Math. 219 1801–1851.
• [10] Borgs, C., Chayes, J. T., Lovász, L., Sós, V. T. and Vesztergombi, K. (2012). Convergent sequences of dense graphs II. Multiway cuts and statistical physics. Ann. of Math. (2) 176 151–219.
• [11] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence, with a Foreword by Michel Ledoux. Oxford Univ. Press, Oxford.
• [12] Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124 1377–1397.
• [13] Caltagirone, F., Parisi, G. and Rizzo, T. (2012). Dynamical critical exponents for the mean-field Potts glass. Phys. Rev. E 85 051504.
• [14] Chen, W.-K. and Sen, A. (2017). Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed $p$-spin models. Comm. Math. Phys. 350 129–173.
• [15] Chung, F. and Lu, L. (2002). Connected components in random graphs with given expected degree sequences. Ann. Comb. 6 125–145.
• [16] Decelle, A., Krzakala, F., Moore, C. and Zdeborová, L. (2011). Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications. Phys. Rev. E 84 066106.
• [17] Dembo, A., Montanari, A. and Sen, S. (2017). Extremal cuts of sparse random graphs. Ann. Probab. 45 1190–1217.
• [18] Dorogovtsev, S. N. and Mendes, J. F. F. (2002). Evolution of networks. Advances in Physics 51 1079–1187.
• [19] Durrett, R. (2003). Rigorous result for the CHKNS random graph model. In Discrete Random Walks (Paris, 2003). Discrete Math. Theor. Comput. Sci. Proc., AC 95–104. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.
• [20] Elderfield, D. and Sherrington, D. (1983). The curious case of the Potts spin glass. J. Phys. C, Solid State Phys. 16 L497.
• [21] Elderfield, D. and Sherrington, D. (1983). Novel non-ergodicity in the Potts spin glass. J. Phys. C, Solid State Phys. 16 L1169.
• [22] Fu, Y. and Anderson, P. W. (1986). Application of statistical mechanics to NP-complete problems in combinatorial optimisation. J. Phys. A: Math. Gen. 19 1605.
• [23] Ghirlanda, S. and Guerra, F. (1998). General properties of overlap probability distributions in disordered spin systems. towards parisi ultrametricity. J. Phys. A: Math. Gen. 31 9149.
• [24] Guerra, F. (2003). Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys. 233 1–12.
• [25] Holland, P. W., Laskey, K. B. and Leinhardt, S. (1983). Stochastic blockmodels: First steps. Soc. Netw. 5 109–137.
• [26] Jackson, M. O. (2008). Social and Economic Networks. Princeton Univ. Press, Princeton, NJ.
• [27] Jagannath, A. and Tobasco, I. (2017). Low temperature asymptotics of spherical mean field spin glasses. Comm. Math. Phys. 352 979–1017.
• [28] Kahng, A. B., Lienig, J., Markov, I. L. and Hu, J. (2011). VLSI Physical Design: From Graph Partitioning to Timing Closure. Springer, New York.
• [29] Kalikow, S. and Weiss, B. (1988). When are random graphs connected. Israel J. Math. 62 257–268.
• [30] Kirkpatrick, S., Gelatt, C. D. Jr. and Vecchi, M. P. (1983). Optimization by simulated annealing. Science 220 671–680.
• [31] Łuczak, T. (1992). Sparse random graphs with a given degree sequence. In Random Graphs, Vol. 2 (Poznań, 1989). 165–182. Wiley, New York.
• [32] Massoulié, L. (2014). Community detection thresholds and the weak Ramanujan property. In STOC’14—Proceedings of the 2014 ACM Symposium on Theory of Computing 694–703. ACM, New York.
• [33] Mézard, M. and Montanari, A. (2009). Information, Physics, and Computation. Oxford Univ. Press, Oxford.
• [34] Mézard, M., Parisi, G. and Virasoro, M. (1987). Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications 9. World Scientific, Singapore.
• [35] Mossel, E., Neeman, J. and Sly, A. (2013). A proof of the block model threshold conjecture. Preprint. Available at arXiv:1311.4115.
• [36] Nishimori, H. and Stephen, M. J. (1983). Gauge-invariant frustrated Potts spin-glass. Phys. Rev. B 27 5644–5652.
• [37] Norros, I. and Reittu, H. (2006). On a conditionally Poissonian graph process. Adv. in Appl. Probab. 38 59–75.
• [38] Panchenko, D. (2013). Spin glass models from the point of view of spin distributions. Ann. Probab. 41 1315–1361.
• [39] Panchenko, D. (2013). The Parisi ultrametricity conjecture. Ann. of Math. (2) 177 383–393.
• [40] Panchenko, D. (2013). The Sherrington–Kirkpatrick Model. Springer, New York.
• [41] Panchenko, D. (2014). The Parisi formula for mixed $p$-spin models. Ann. Probab. 42 946–958.
• [42] Panchenko, D. (2018). Free energy in the mixed $p$-spin models with vector spins. Ann. Probab. 46 865–896.
• [43] Panchenko, D. (2018). Free energy in the Potts spin glass. Ann. Probab. 46 829–864.
• [44] Panchenko, D. (2015). The free energy in a multi-species Sherrington-Kirkpatrick model. Ann. Probab. 43 3494–3513.
• [45] Poljak, S. and Tuza, Z. (1993). The Max-Cut Problem—a Survey. Acad. Sinica, Taipei.
• [46] Porter, M. A., Onnela, J.-P. and Mucha, P. J. (2009). Communities in networks. Notices Amer. Math. Soc. 56 1082–1097.
• [47] Ruelle, D. (1987). A mathematical reformulation of Derrida’s REM and GREM. Comm. Math. Phys. 108 225–239.
• [48] Sen, S. (2016). Optimization on sparse random hypergraphs and spin glasses. Preprint. Available at arXiv:1606.02365.
• [49] Shepp, L. A. (1989). Connectedness of certain random graphs. Israel J. Math. 67 23–33.
• [50] Sherrington, D. and Kirkpatrick, S. (1975). Solvable model of a spin-glass. Phys. Rev. Lett. 35 1792–1796.
• [51] Söderberg, B. (2002). General formalism for inhomogeneous random graphs. Phys. Rev. E (3) 66 066121, 6.
• [52] Talagrand, M. (2010). Mean Field Models for Spin Glasses: Volume I: Basic Examples 54. Springer, New York.