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Aggregation models on hypergraphs. (English) Zbl 1364.82071

Summary: Following a newly introduced approach by Rasetti and Merelli we investigate the possibility to extract topological information about the space where interacting systems are modelled. From the statistical datum of their observable quantities, like the correlation functions, we show how to reconstruct the activities of their constitutive parts which embed the topological information. The procedure is implemented on a class of polymer models on hypergraphs with hard-core interactions. We show that the model fulfils a set of iterative relations for the partition function that generalise those introduced by Heilmann and Lieb for the monomer-dimer case. After translating those relations into structural identities for the correlation functions we use them to test the precision and the robustness of the inverse problem. Finally the possible presence of a further interaction of peer-to-peer type is considered and a criterion to discover it is identified.

MSC:

82D60 Statistical mechanics of polymers
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
05C90 Applications of graph theory
05C15 Coloring of graphs and hypergraphs
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[1] Rasetti, M.; Merelli, E., (Contucci, Giardina, Advances in Disordered Systems, Random Processes and Some Applications (2016), Cambridge University Press)
[2] Feynman, R. P., Rev. Modern Phys., 20, 2, 367 (1948)
[3] Schwinger, J., Proc. Natl. Acad. Sci., 44, 9, 956-965 (1958)
[4] Symanzik, K., J. Math. Phys., 7, 3, 510-525 (1966)
[5] Chang, T., Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 512-531 (1939)
[6] O’Neil, K. T.; DeGrado, W. F., Science, 250, 4981, 646-651 (1990)
[7] Bordenave, C.; Lelarge, M.; Salez, J., Probab. Theory Related Fields, 157, 1-2, 183-208 (2013)
[8] Karp, R. M.; Sipser, M., (22nd Annual Symposium on Foundations of Computer Science, 1981. 22nd Annual Symposium on Foundations of Computer Science, 1981, SFCS’81 (1981), IEEE), 364-375
[9] Zdeborová, L.; Mézard, M., J. Stat. Mech. Theory Exp., 2006, 05, P05003 (2006)
[10] Barra, A.; Contucci, P.; Sandell, R.; Vernia, C., Integration indicators in immigration phenomena. a statistical mechanics perspective., Tech. Rep. (2013)
[11] Frosini, P., Intelligent Robots and Computer Vision X: Algorithms and Techniques, 122-133 (1992), International Society for Optics and Photonics
[12] Verri, A.; Uras, C.; Frosini, P.; Ferri, M., Biol. Cybernet., 70, 2, 99-107 (1993)
[13] H. Edelsbrunner, D. Letscher, A. Zomorodian, Topological persistence and simplification, in: Proc. 41st IEEE Symp. Found. Comput. Sci., 2000, pp. 454-463.; H. Edelsbrunner, D. Letscher, A. Zomorodian, Topological persistence and simplification, in: Proc. 41st IEEE Symp. Found. Comput. Sci., 2000, pp. 454-463. · Zbl 1011.68152
[14] Edelsbrunner, H.; Harer, J., Contemp. Math., 453, 257-282 (2008)
[15] Petri, G.; Expert, P.; Turkheimer, F.; Carhart-Harris, R.; Nutt, D.; Hellyer, P.; Vaccarino, F., J. R. Soc. Interface, 11, 101, 20140873 (2014)
[16] Heilmann, O. J.; Lieb, E. H., Statistical Mechanics, 45-87 (1972), Springer
[17] Ackley, D. H.; Hinton, G. E.; Sejnowski, T. J., Cogn. Sci., 9, 1, 147-169 (1985)
[18] Bianconi, G., Phys. Rev. E, 87, Article 062806 pp. (2013)
[19] Heilmann, O. J.; Lieb, E. H., Phys. Rev. Lett., 24, 1412-1414 (1970)
[20] Aurell, E.; Ekeberg, M., Phys. Rev. Lett., 108, 9, Article 090201 pp. (2012)
[21] Sessak, V.; Monasson, R., J. Phys. A, 42, 5, Article 055001 pp. (2009)
[22] Roudi, Y.; Tyrcha, J.; Hertz, J., Phys. Rev. E, 79, 5, Article 051915 pp. (2009)
[23] Zuev, K.; Eisenberg, O.; Krioukov, D., J. Phys. A, 48, 46, Article 465002 pp. (2015)
[24] Costa, A.; Farber, M., J. Topol. Anal., 1-31 (2015)
[25] Kahle, M., AMS Contemp. Math., 620, 201-222 (2014)
[26] Courtney, O. T.; Bianconi, G., Phys. Rev. E, 93, Article 062311 pp. (2016)
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