Abstract
We propose a new method inspired from statistical mechanics for extracting geometric information from undirected binary networks and generating random networks that conform to this geometry. In this method an undirected binary network is perceived as a thermodynamic system with a collection of permuted adjacency matrices as its states. The task of extracting information from the network is then reformulated as a discrete combinatorial optimization problem of searching for its ground state. To solve this problem, we apply multiple ensembles of temperature regulated Markov chains to establish an ultrametric geometry on the network. This geometry is equipped with a tree hierarchy that captures the multiscale community structure of the network. We translate this geometry into a Parisi adjacency matrix, which has a relative low energy level and is in the vicinity of the ground state. The Parisi adjacency matrix is then further optimized by making block permutations subject to the ultrametric geometry. The optimal matrix corresponds to the macrostate of the original network. An ensemble of random networks is then generated such that each of these networks conforms to this macrostate; the corresponding algorithm also provides an estimate of the size of this ensemble. By repeating this procedure at different scales of the ultrametric geometry of the network, it is possible to compute its evolution entropy, i.e. to estimate the evolution of its complexity as we move from a coarse to a fine description of its geometric structure. We demonstrate the performance of this method on simulated as well as real data networks.
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Airoldi, E.M., Blei, D.M., Fienberg, S.E., Xing, E.P.: Mixed membership stochastic blockmodels. J. Mach. Learn. Res. 9, 1981–2014 (2008)
Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)
Anderson, P.W.: More is different. Science 177, 393–396 (1972)
Barabási, A.L.: Scale-free networks: a decade and beyond. Science 325, 412–413 (2009)
Barvinok, A.: On the number of matrices and a random matrix with prescribed row and column sums and 0–1 entries. Adv. Math. 224, 316–339 (2010)
Bascompte, J.: Disentangling the web of life. Science 325, 416–419 (2009)
Bascompte, J.: Structure and dynamics of ecological networks (perspective). Science 329, 765–766 (2010)
Bayati, M., Kim, J.H., Saberi, A.: A sequential algorithm for generating random graphs. Algorithmica 58, 860–910 (2010)
Bianconi, G.: The entropy of randomized network ensembles. Europhys. Lett. 81, 28005 (2008)
Bianconi, G.: Entropy of network ensembles. Phys. Rev. E. 79, 036114 (2009)
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U.: Complex networks: structure and dynamics. Phys. Rep. 424, 175–308 (2006)
Boguñá, M., Krioukov, D.: Navigating ultrasmall worlds in ultrashort time. Phys. Rev. Lett. 102, 058701 (2009)
Boguñá, M., Krioukov, D., Claffy, K.C.: Navigability of complex networks. Nat. Phys. 5, 74–80 (2009)
Boguñá, M., Papadopoulos, F., Krioukov, D.: Sustaining the internet with hyperbolic mapping. Nat. Commun. 1, 62 (2010)
Bollobás, B., Janson, S., Riordan, O.: Sparse random graphs with clustering. Random Struct. Algorithms 38, 269–323 (2011)
Chen, C., Fushing, H.: Multi-scale community geometry in network and its application. Phys. Rev. E. 86, 041120 (2012)
Chen, Y., Diaconis, P., Holmes, S.P., Liu, J.S.: Sequential monte carlo methods for statistical analysis of tables. J. Am. Stat. Assoc. 100, 109–120 (2005)
Croft, D., Madden, J., Franks, D.W., James, R.: Hypothesis testing in animal social networks. J. Trends Ecol. Evol. 26, 502–507 (2011)
Efron, B.: Bootstrap methods: another look at the jackknife. Ann. Stat. 7, 1–26 (1979)
Fallani, F.D.V., Nicosia, V., Latora, V., Chavez, M.: Nonparametric resampling of random walks for spectral network clustering. Phys. Rev. E 89, 012802 (2014)
Fortunato, S.: Community detection in graphs. Phys. Rep. 486, 75–174 (2010)
Fushing, H., McAssey, M.: Time, temperature and data cloud geometry. Phys. Rev. E 82, 061110 (2010)
Fushing, H., Wang, H., Van der Waal, K., McCowan, B., Koehl, P.: Multi-scale clustering by building a robust and self-correcting ultrametric topology on data points. PLoS ONE 8, e56259 (2013)
Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proc. Natl. Acad. Sci. (USA) 99, 7821–7826 (2002)
Goldenberg, A., Zheng, Z.X., Fienberg, S.E., Airoldi, E.M.: A survey of statistical network model. Found. Trends Mach. Learn. 2, 1–117 (2009)
Herbert, S.: The architecture of complexity. Proc. Am. Philos. Soc. 106, 467–482 (1962)
Havlin, S., Cohen, R.: Complex Networks: Structure, Robustness, and Function. Cambridge University Press, Cambridge (2010)
Karrer, B., Newman, M.: Stochastic blockmodels and community structure in network. Phys. Rev. E 83, 016107 (2011)
Kim, J., Vu, V.: Generating random regular graphs. In: Proceedings of ACM Symposium on Theory of Computing (STOC), pp. 213–222 (2003)
Kolacyzk, E.: Statistical Analysis of Network Models. Springer, New York (2009)
Krakhardt, D.: Predicting with networks: non parametric multiple regression analysis of dyadic data. Soc. Netw. 10, 359–381 (1988)
Krause, J., Croft, R., James, R.: Social network theory in the behavioural sciences: potential applications. Behav. Ecol. Sociobiol. 62, 15–27 (2007)
Krioukov, D., Kitsak, M., Sinkovits, R., Rideout, D., Meyer, D., Boguñá, M.: Network cosmology. Sci. Rep. 2, 793 (2012)
Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010)
Krioukov, D., Papadopoulos, F., Vahdat, A., Boguñá, M.: Curvature and temperature of complex networks. Phys. Rev. E 80, 035101 (2009)
Lancichinetti, A., Fortunato, S.: Community detection algorithm: a comparative analysis. Phys. Rev. E. 80, 056117 (2009)
Lancichinetti, A., Fortunato, S., Radicchi, F.: Benchmark graphs for testing community detection algorithms. Phys. Rev. E. 78, 046110 (2008)
Lancichinetti, A., Radicchi, F., Ramasco, J.J., Fortunato, S.: Finding statistically significant communities in networks. PLoS One 6, e18961 (2011)
Manly, B.: A note on the analysis of species co-occurrences. Ecology 76, 1109–1115 (1995)
Manly, B.: Randomization, Bootstrap, and Monte Carlo methods in biology. CRC Press, Boca Raton (2006)
McKay, B.: Asymptotics for symmetric 0–1 matrices with prescribed row sums. Ars. Comb. A. 19, 15–25 (1985)
Molloy, M., Reed, B.: A critical point for random graphs with a given degree sequence. Random Struct. Algorithms 6, 161–179 (1995)
Musmeci, N., Battiston, S., Caldarelli, G., Puliga, M., Gabrielli, A.: Bootstrapping topological properties and systemic risk of complex networks using the fitness model. J. Stat. Phys. 151, 720–734 (2013)
Newman, M.: The structure and function of complex networks. SIAM Rev. 45, 167–246 (2003)
Newman, M.: Detecting community structure in networks. Eur. Phys. J. B 38, 321–330 (2004)
Newman, M.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. (USA) 103, 8577–8582 (2006)
Newman, M.: Random graphs with clustering. Phys. Rev. Lett. 103, 058701 (2009)
Newman, M.: Networks: An Introduction. Oxford University Press, Oxford (2010)
Newman, M.E.J., Barabási, A.L., Watts, D.J.: The Structure and Dynamics of Networks. Princeton Univ. Press, New Jersey (2006)
Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E. 69, 026113 (2004)
Nowicki, K., Snijders, T.A.B.: Estimation and prediction for stochastic blockstructures. J. Am. Stat. Assoc. 96, 1077–1087 (2001)
Parisi, G.: Infinite number of order parameters for spin-glasses. Phys. Rev. Lett. 43, 1754–1756 (1979)
Parisi, G.: A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A 13, L115–L121 (1980)
Proulx, S., Promislow, D., Phillips, P.: Network thinking in ecology and evolution. Trends Ecol. Evol. 20, 345–353 (2005)
Reichardt, J., Alamino, R., Saad, D.: The interplay between microscopic and mesoscopic structures in complex networks. PLoS One 6, e21282 (2011)
Rosvall, M., Bergstrom, C.T.: Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. (USA) 105, 1118–1123 (2007)
Rosvall, M., Bergstrom, C.T.: Mapping change in large networks. PLoS One 5, e8694 (2010)
Sih, A., Hauser, S., McHugh, K.: Social network theory: new insights and issues for behavorial ecologists. Behav. Ecol. Sociobiol. 63, 975–988 (2009)
Snijders, T.A.B., Nowicki, K.: Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classif. 14, 75–100 (1997)
Steger, A., Wormald, N.C.: Generating random regular graphs quickly. Comb. Prob. Comput. 8, 377–396 (1999)
Stephens, P., Buskirk, S., del Rio, C.: Inference in ecology and evolution. Trends Ecol. Evol. 22, 192–197 (2007)
DiCiccio, T.J., Efron, B.: Bootstrap confidence intervals (with discussion). Stat. Sci. 11, 189–228 (1996)
Wainwright, M.J., Jordan, M.I.: Graphical models, exponential families and variational inference. Mach. Learn. 1, 1–305 (2008)
Zachary, W.W.: An information flow model for conflict and fission in small groups. J. Anthropol. Res. 33, 452–473 (1977)
Zhao, Y., Levina, E., Zhu, J.: Community extraction for social networks. Proc. Natl. Acad. Sci. (USA) 108, 7321–7326 (2011)
Acknowledgments
This work was partially supported by National Science Foundation Grant DMS-1007219 (co-funded by Cyber-enabled Discovery and Innovation (CDI) program). P. Koehl acknowledges support from the NIH.
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Fushing, H., Chen, C., Liu, SY. et al. Bootstrapping on Undirected Binary Networks Via Statistical Mechanics. J Stat Phys 156, 823–842 (2014). https://doi.org/10.1007/s10955-014-1043-6
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DOI: https://doi.org/10.1007/s10955-014-1043-6