The higher-order spectrum of simplicial complexes: a renormalization group approach. (English) Zbl 1519.82042
Summary: Network topology is a flourishing interdisciplinary subject that is relevant for different disciplines including quantum gravity and brain research. The discrete topological objects that are investigated in network topology are simplicial complexes. Simplicial complexes generalize networks by not only taking pairwise interactions into account, but also taking into account many-body interactions between more than two nodes. Higher-order Laplacians are topological operators that describe higher-order diffusion on simplicial complexes and constitute the natural mathematical objects that capture the interplay between network topology and dynamics. We show that higher-order up and down Laplacians can have a finite spectral dimension, characterizing the long time behaviour of the diffusion process on simplicial complexes that depends on their order \(m\). We provide a renormalization group theory for the calculation of the higher-order spectral dimension of two deterministic models of simplicial complexes: the Apollonian and the pseudo-fractal simplicial complexes. We show that the RG flow is affected by the fixed point at zero mass, which determines the higher-order spectral dimension \(d_{\mathrm{S}}\) of the up-Laplacians of order \(m\) with \(m \ge 0\).
MSC:
| 82B28 | Renormalization group methods in equilibrium statistical mechanics |
| 55U10 | Simplicial sets and complexes in algebraic topology |
Keywords:
simplicial complexes; spectral dimension; higher-order Laplacians; network topology; renormalization groupReferences:
| [1] | Bianconi G 2015 Europhys. Lett.111 56001 · doi:10.1209/0295-5075/111/56001 |
| [2] | Giusti C, Ghrist R and Bassett D S 2016 J. Comput. Neurosci.41 1 · doi:10.1007/s10827-016-0608-6 |
| [3] | Salnikov V, Cassese D and Lambiotte R 2018 Eur. J. Phys.14 014001 |
| [4] | Kahle M 2014 AMS Contemp. Math.620 201 · doi:10.1090/conm/620/12367 |
| [5] | Courtney O T and Bianconi G 2016 Phys. Rev. E 93 062311 · doi:10.1103/physreve.93.062311 |
| [6] | Cohen D, Costa A, Farber M and Kappeler T 2012 Discrete Comput. Geom.47 117 · Zbl 1237.55009 · doi:10.1007/s00454-011-9378-0 |
| [7] | Wu Z, Menichetti G, Rahmede C and Bianconi G 2014 Sci. Rep.5 10073 · doi:10.1038/srep10073 |
| [8] | Bianconi G and Rahmede C 2017 Sci. Rep.7 41974 · doi:10.1038/srep41974 |
| [9] | Bianconi G and Rahmede C 2016 Phys. Rev. E 93 032315 · doi:10.1103/physreve.93.032315 |
| [10] | Mulder D and Bianconi G 2018 J. Stat. Phys.73 783 · Zbl 1476.90072 · doi:10.1007/s10955-018-2115-9 |
| [11] | Fountoulakis N, Iyer T, Mailler C and Sulzbach H 2019 arXiv:1910.12715 |
| [12] | Ghrist R 2014 Elementary Applied Topology (Scotts Valley, CA: Createspace) · Zbl 1427.55001 |
| [13] | Petri P et al 2014 J. R. Soc. Interface11 20140873 · doi:10.1098/rsif.2014.0873 |
| [14] | Tumminello M, Aste T, Di Matteo T and Mantegna R N 2005 Proc. Natl Acad. Sci.102 10421 · doi:10.1073/pnas.0500298102 |
| [15] | Šuvakov M, Andjelković M and Tadić B 2018 Sci. Rep.8 1987 · doi:10.1038/s41598-018-20398-x |
| [16] | Barbarossa S and Sardellitti S 2019 arXiv:1907.11577 |
| [17] | Torres J J and Bianconi G 2020 arXiv:2001.05934 |
| [18] | Millán A P, Torres J J and Bianconi G 2020 Phys. Rev. Lett.124 218301 · doi:10.1103/PhysRevLett.124.218301 |
| [19] | Skardal P S and Arenas A 2019 Phys. Rev. Lett.122 248301 · doi:10.1103/physrevlett.122.248301 |
| [20] | Iacopini I, Petri G, Barrat A and Latora V 2019 Nat. Commun.10 2485 · doi:10.1038/s41467-019-10431-6 |
| [21] | Jhun B, Jo M and Kahng B 2019 arXiv:1910.00375 |
| [22] | Matamalas J T, Gómez S and Arenas A 2019 arXiv:1910.03069 |
| [23] | Chung F R K 1997 Spectral Graph Theory (Providence, RI: American Mathematical Society) 92 · Zbl 0867.05046 |
| [24] | Dorogovtsev S N, Goltsev A V, Mendes J F F and Samukhin A N 2003 Phys. Rev. E 68 046109 · doi:10.1103/physreve.68.046109 |
| [25] | Samukhin A N, Dorogovtsev S N and Mendes J F F 2008 Phys. Rev. E 77 036115 · doi:10.1103/physreve.77.036115 |
| [26] | Wang Y, Yi Y, Xu W and Zhang Z 2020 arXiv:2002.12219 |
| [27] | Rammal R and Toulouse G 1983 J. Phys. Lett.44 1 · doi:10.1051/jphyslet:0198300440206500 |
| [28] | Bianconi G and Dorogovtsev S N 2020 J. Stat. Mech. 014005 · Zbl 1459.05356 · doi:10.1088/1742-5468/ab5d0e |
| [29] | Hwang S, Yun C-K, Lee D-S, Kahng B and Kim D 2010 Phys. Rev. E 82 056110 · doi:10.1103/physreve.82.056110 |
| [30] | Kim D 1984 J. Kor. Phys. Soc.17 3 |
| [31] | Burioni R and Cassi D 1996 Phys. Rev. Lett.76 1091 · doi:10.1103/physrevlett.76.1091 |
| [32] | Burioni R, Cassi D and Vezzani A 1999 Phys. Rev. E 60 1500 · doi:10.1103/physreve.60.1500 |
| [33] | Burioni R, Cassi D, Cecconi F and Vulpiani A 2004 Proteins: Struct., Funct., Bioinf.55 529 · doi:10.1002/prot.20072 |
| [34] | Jonsson T and Wheater J F 1998 Nucl. Phys. B 515 549 · Zbl 0949.82017 · doi:10.1016/s0550-3213(98)00027-3 |
| [35] | Durhuus B, Jonsson T and Wheater J F 2007 J. Stat. Phys.128 1237 · Zbl 1136.82006 · doi:10.1007/s10955-007-9348-3 |
| [36] | Avrachenkov K, Cottatellucci L and Hamidouche M 2019 Int. Conf. on Complex Networks and Their Applications (Cham: Springer) 965 |
| [37] | Bradde S, Caccioli F, Dall’Asta L and Bianconi G 2010 Phys. Rev. Lett.104 218701 · doi:10.1103/physrevlett.104.218701 |
| [38] | Aygün E and Erzan A 2011 J. Phys.: Conf. Ser.319 012007 · doi:10.1088/1742-6596/319/1/012007 |
| [39] | Millán A P, Torres J J and Bianconi G 2018 Sci. Rep.8 9910 · doi:10.1038/s41598-018-28236-w |
| [40] | Millán A P, Torres J J and Bianconi G 2019 Phys. Rev. E 99 022307 · doi:10.1103/physreve.99.022307 |
| [41] | Bradde S and Bialek W 2017 J. Stat. Phys.167 462 · Zbl 1378.82027 · doi:10.1007/s10955-017-1770-6 |
| [42] | Ambjørn J, Jurkiewicz J and Loll R 2005 Phys. Rev. Lett.95 171301 · doi:10.1103/physrevlett.95.171301 |
| [43] | Benedetti D 2009 Phys. Rev. Lett.102 111303 · doi:10.1103/physrevlett.102.111303 |
| [44] | Benedetti D and Henson J 2009 Phys. Rev. D 80 124036 · doi:10.1103/physrevd.80.124036 |
| [45] | Ambjørn J, Jurkiewicz J and Loll R 2005 Phys. Rev. D 72 064014 · doi:10.1103/physrevd.72.064014 |
| [46] | Andrade J S Jr, Herrmann H J, Andrade R F S and Da Silva L R 2005 Phys. Rev. Lett.94 018702 · doi:10.1103/physrevlett.94.018702 |
| [47] | Zhang Z, Rong L and Comellas F 2006 Physica A 364 610 · doi:10.1016/j.physa.2005.09.042 |
| [48] | Graham R L, Lagarias J C, Mallows C L, Wilks A R and Yan C H 2005 Discrete Comput. Geom.34 547 · Zbl 1085.52010 · doi:10.1007/s00454-005-1196-9 |
| [49] | Dorogovtsev S N, Goltsev A V and Mendes J F F 2002 Phys. Rev. E 65 066122 · doi:10.1103/physreve.65.066122 |
| [50] | Bonzom V, Gurau R, Riello A and Rivasseau V 2011 Nucl. Phys. B 853 174 · Zbl 1229.81222 · doi:10.1016/j.nuclphysb.2011.07.022 |
| [51] | Lionni L 2018 Colored Discrete Spaces: Higher Dimensional Combinatorial Maps and Quantum Gravity (Berlin: Springer) · Zbl 1406.81007 · doi:10.1007/978-3-319-96023-4 |
| [52] | Steenbergen J, Klivans C and Mukherjee S 2014 Adv. Appl. Math.56 56 · Zbl 1305.55010 · doi:10.1016/j.aam.2014.01.002 |
| [53] | Parzanchevski O and Rosenthal R 2017 Random Struct. Algorithms50 225 · Zbl 1359.05114 · doi:10.1002/rsa.20657 |
| [54] | Rozenfeld H D, Havlin S and Ben-Avraham D 2007 New J. Phys.9 175 · doi:10.1088/1367-2630/9/6/175 |
| [55] | Rozenfeld H D and Ben-Avraham D 2007 Phys. Rev. E 75 061102 · doi:10.1103/physreve.75.061102 |
| [56] | Boettcher S, Singh V and Ziff R M 2012 Nat. Commun.3 787 · doi:10.1038/ncomms1774 |
| [57] | Boettcher S, Cook J L and Ziff R M 2009 Phys. Rev. E 80 041115 · doi:10.1103/physreve.80.041115 |
| [58] | Auto D M, Moreira A A, Herrmann H J and Andrade J S Jr 2008 Phys. Rev. E 78 066112 · doi:10.1103/physreve.78.066112 |
| [59] | Bianconi G and Ziff R M 2018 Phys. Rev. E 98 052308 · doi:10.1103/physreve.98.052308 |
| [60] | Kryven I, Ziff R M and Bianconi G 2019 Phys. Rev. E 100 022306 · doi:10.1103/physreve.100.022306 |
| [61] | Bianconi G, Kryven I and Ziff R M 2019 Phys. Rev. E 100 062311 · doi:10.1103/physreve.100.062311 |
| [62] | Boettcher S and Brunson C T 2011 Frontiers Physiol.2 102 · doi:10.3389/fphys.2011.00102 |
| [63] | Muhammad A and Egerstedt M 2006 Proc. 17th Int. Symp. on Mathematical Theory of Networks and Systems 1024-38 |
| [64] | Goldberg T E 2002 Combinatorial Laplacians of simplicialcomplexes Senior Thesis Bard College |
| [65] | Horak D and Jost J 2013 Adv. Math.244 303 · Zbl 1290.05103 · doi:10.1016/j.aim.2013.05.007 |
| [66] | Brunekreef J and Reitz M 2020 (in preparation) |
| [67] | Livan G, Novaes M and Vivo P 2018 Introduction to Random Matrices: Theory and Practice (Berlin: Springer) · Zbl 1386.15003 · doi:10.1007/978-3-319-70885-0 |
| [68] | Mehta M L 2004 Random Matrices(Pure and Applied Mathematics) (Amsterdam: Elsevier) · Zbl 1107.15019 |
| [69] | Reitz M and Bianconi G 2020 (in preparation) |
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