When simplicial complexes are used to represent higher-order systems, information regarding when and how interactions happen may be lost. In this paper, we propose the concept of temporal simplicial complexes, in which simplices with timestamps (or temporal simplices) are used to represent interactions, and faces with weights are used to represent relations. Then, we propose a growing model with two rules, face dimension selection (FDS), and preferential attachment. By properly setting the probability parameter vector q in the FDS rule, one can balance network diameter expansion and network centrality, thus attaining more flexibility in the growing process. Our theoretical analysis and simulations that followed show the generalized degree of faces of any dimension follows a power-law distribution, with a scaling component controlled by q. Our work provides a flexible growing model and can be used to study higher-order systems with temporal properties.
REFERENCES
1.
2.
S. N.
Dorogovtsev
and J. F.
Mendes
, “Evolution of networks
,” Adv. Phys.
51
, 1079
–1187
(2002
). 3.
S.
Boccaletti
, V.
Latora
, Y.
Moreno
, M.
Chavez
, and D.-U.
Hwang
, “Complex networks: Structure and dynamics
,” Phys. Rep.
424
, 175
–308
(2006
). 4.
A. R.
Benson
, D. F.
Gleich
, and J.
Leskovec
, “Higher-order organization of complex networks
,” Science
353
, 163
–166
(2016
). 5.
F.
Battiston
, G.
Cencetti
, I.
Iacopini
, V.
Latora
, M.
Lucas
, A.
Patania
, J.-G.
Young
, and G.
Petri
, “Networks beyond pairwise interactions: Structure and dynamics
,” Phys. Rep.
874
, 1
–92
(2020
). 6.
V.
Larivière
, Y.
Gingras
, C. R.
Sugimoto
, and A.
Tsou
, “Team size matters: Collaboration and scientific impact since 1900
,” J. Assoc. Inform. Sci. Technol.
66
, 1323
–1332
(2015
). 7.
J.
Yu
, M.
Ding
, Q.
Wang
, W.
Sun
, and W.
Hu
, “Community sports organization development from a social network evolution perspective—Structures, stages, and stimulus
,” IEEE Trans. Comput. Soc. Syst.
10
(3
), (2021
).8.
A. E.
Sizemore
, C.
Giusti
, A.
Kahn
, J. M.
Vettel
, R. F.
Betzel
, and D. S.
Bassett
, “Cliques and cavities in the human connectome
,” J. Comput. Neurosci.
44
, 115
–145
(2018
). 9.
I.
Iacopini
, G.
Petri
, A.
Barrat
, and V.
Latora
, “Simplicial models of social contagion
,” Nat. Commun.
10
, 2485
(2019
). 10.
G. F.
de Arruda
, G.
Petri
, and Y.
Moreno
, “Social contagion models on hypergraphs
,” Phys. Rev. Res.
2
, 023032
(2020
). 11.
F.
Baccini
, F.
Geraci
, and G.
Bianconi
, “Weighted simplicial complexes and their representation power of higher-order network data and topology
,” Phys. Rev. E
106
, 034319
(2022
). 12.
S.
Boccaletti
, P.
De Lellis
, C.
del Genio
, K.
Alfaro-Bittner
, R.
Criado
, S.
Jalan
, and M.
Romance
, “The structure and dynamics of networks with higher order interactions
,” Phys. Rep.
1018
, 1
–64
(2023
). 13.
S.
Majhi
, M.
Perc
, and D.
Ghosh
, “Dynamics on higher-order networks: A review
,” J. Roy. Soc. Interface
19
, 20220043
(2022
). 14.
F.
Battiston
, E.
Amico
, A.
Barrat
, G.
Bianconi
, G.
Ferraz de Arruda
, B.
Franceschiello
, I.
Iacopini
, S.
Kéfi
, V.
Latora
, Y.
Moreno
et al., “The physics of higher-order interactions in complex systems
,” Nat. Phys.
17
, 1093
–1098
(2021
). 15.
L.
Torres
, A. S.
Blevins
, D.
Bassett
, and T.
Eliassi-Rad
, “The why, how, and when of representations for complex systems
,” SIAM Rev.
63
, 435
–485
(2021
). 16.
G.
Bianconi
, Higher-Order Networks
(Cambridge University Press
, 2021
).17.
M.
Kahle
, “Topology of random simplicial complexes: A survey
,” AMS Contemp. Math.
620
, 201
–222
(2014
). 18.
F.
Callegaro
, F.
Cohen
, C.
De Concini
, E. M.
Feichtner
, G.
Gaiffi
, and M.
Salvetti
, Configuration Spaces: Geometry, Topology and Representation Theory
(Springer
, 2016
), Vol. 14.19.
I.
Iacopini
, G.
Petri
, A.
Barrat
, and V.
Latora
, “Simplicial models of social contagion
,” Nat. Commun.
10
, 2485
(2019
). 20.
O. T.
Courtney
and G.
Bianconi
, “Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes
,” Phys. Rev. E
93
, 062311
(2016
). 21.
J.-G.
Young
, G.
Petri
, F.
Vaccarino
, and A.
Patania
, “Construction of and efficient sampling from the simplicial configuration model
,” Phys. Rev. E
96
, 032312
(2017
). 22.
G.
Bianconi
, I.
Kryven
, and R. M.
Ziff
, “Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
,” Phys. Rev. E
100
, 062311
(2019
). 23.
G.
Petri
and A.
Barrat
, “Simplicial activity driven model
,” Phys. Rev. Lett.
121
, 228301
(2018
). 24.
G.
Bianconi
and C.
Rahmede
, “Complex quantum network manifolds in dimension
are scale-free
,” Sci. Rep.
5
, 13979
(2015
). 25.
G.
Bianconi
and C.
Rahmede
, “Network geometry with flavor: From complexity to quantum geometry
,” Phys. Rev. E
93
, 032315
(2016
). 26.
D.
Mulder
and G.
Bianconi
, “Network geometry and complexity
,” J. Stat. Phys.
173
, 783
–805
(2018
). 27.
G.
Bianconi
and C.
Rahmede
, “Emergent hyperbolic network geometry
,” Sci. Rep.
7
, 41974
(2017
). 28.
O. T.
Courtney
and G.
Bianconi
, “Weighted growing simplicial complexes
,” Phys. Rev. E
95
, 062301
(2017
). 29.
K.
Kovalenko
, I.
Sendiña-Nadal
, N.
Khalil
, A.
Dainiak
, D.
Musatov
, A. M.
Raigorodskii
, K.
Alfaro-Bittner
, B.
Barzel
, and S.
Boccaletti
, “Growing scale-free simplices
,” Commun. Phys.
4
, 43
(2021
). 30.
M. T.
Schaub
, Y.
Zhu
, J.-B.
Seby
, T. M.
Roddenberry
, and S.
Segarra
, “Signal processing on higher-order networks: Livin’on the edge
and beyond
,” Signal Processing
187
, 108149
(2021
). 31.
R.
Ghorbanchian
, J. G.
Restrepo
, J. J.
Torres
, and G.
Bianconi
, “Higher-order simplicial synchronization of coupled topological signals
,” Commun. Phys.
4
, 120
(2021
). 32.
A. P.
Millán
, J. J.
Torres
, and G.
Bianconi
, “Explosive higher-order Kuramoto dynamics on simplicial complexes
,” Phys. Rev. Lett.
124
, 218301
(2020
). 33.
Y.
Lee
, J.
Lee
, S. M.
Oh
, D.
Lee
, and B.
Kahng
, “Homological percolation transitions in growing simplicial complexes
,” Chaos
31
, 041102
(2021
). 34.
A.-L.
Barabási
and R.
Albert
, “Emergence of scaling in random networks
,” Science
286
, 509
–512
(1999
). 35.
G.
Bianconi
, “Emergence of weight-topology correlations in complex scale-free networks
,” Europhys. Lett.
71
, 1029
(2005
). 36.
F. R.
Chung
and L.
Lu
, Complex graphs and networks, CBMS Regional Conference Series in Mathematics 107 (American Mathematical Soc., 2006).37.
38.
A.
Clauset
, C. R.
Shalizi
, and M. E.
Newman
, “Power-law distributions in empirical data
,” SIAM Rev.
51
, 661
–703
(2009
). 39.
V. W.
Berger
and Y.
Zhou
, “Kolmogorov–Smirnov test: Overview
,” in Wiley StatsRef: Statistics Reference Online
(John Wiley & Sons, Ltd.
, 2014
). © 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
