The largest eigenvalue of the matrix describing a network’s contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue, the expansion eigenvalue, for hypergraph dynamical processes. Using a mean-field approach, we derive an approximation to the expansion eigenvalue in terms of the degree sequence for uncorrelated hypergraphs. We introduce a generative model for hypergraphs that includes degree assortativity, and use a perturbation approach to derive an approximation to the expansion eigenvalue for assortative hypergraphs. We define the dynamical assortativity, a dynamically sensible definition of assortativity for uniform hypergraphs, and describe how reducing the dynamical assortativity of hypergraphs through preferential rewiring can extinguish epidemics. We validate our results with both synthetic and empirical datasets.
REFERENCES
1.
M. E. J.
Newman
, “Assortative mixing in networks
,” Phys. Rev. Lett.
89
, 208701
(2002
). 2.
M. E. J.
Newman
, “Mixing patterns in networks
,” Phys. Rev. E
67
, 026126
(2003
). 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.
J. G.
Restrepo
and E.
Ott
, “Mean-field theory of assortative networks of phase oscillators
,” Europhys. Lett.
107
, 60006
(2014
). 5.
M.
Boguñá
and R.
Pastor-Satorras
, “Epidemic spreading in correlated complex networks
,” Phys. Rev. E
66
, 047104
(2002
). 6.
Y.
Moreno
, J. B.
Gómez
, and A. F.
Pacheco
, “Epidemic incidence in correlated complex networks
,” Phys. Rev. E
68
, 035103(R)
(2003
). 7.
M.
Brede
and S.
Sinha
, “Assortative mixing by degree makes a network more unstable,” arXiv:cond-mat/0507710 (2005).8.
Z.
Rong
, X.
Li
, and X.
Wang
, “Roles of mixing patterns in cooperation on a scale-free networked game
,” Phys. Rev. E
76
, 027101
(2007
). 9.
G.
D’Agostino
, A.
Scala
, V.
Zlatić
, and G.
Caldarelli
, “Robustness and assortativity for diffusion-like processes in scale-free networks
,” Europhys. Lett.
97
, 68006
(2012
). 10.
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
). 11.
A. R.
Benson
, D. F.
Gleich
, and D. J.
Higham
, “Higher-order network analysis takes off, fueled by classical ideas and new data,” arXiv:2103.05031 [physics] (2021).12.
S.
Majhi
, M.
Perc
, and D.
Ghosh
, “Dynamics on higher-order networks: A review
,” J. R. Soc. Interface
19
, 011002
(2022
). 13.
J.
Grilli
, G.
Barabás
, M. J.
Michalska-Smith
, and S.
Allesina
, “Higher-order interactions stabilize dynamics in competitive network models
,” Nature
548
, 210
–213
(2017
). 14.
G. F.
de Arruda
, G.
Petri
, and Y.
Moreno
, “Social contagion models on hypergraphs
,” Phys. Rev. Res.
2
, 023032
(2020
). 15.
N. W.
Landry
and J. G.
Restrepo
, “The effect of heterogeneity on hypergraph contagion models
,” Chaos
30
, 103117
(2020
). 16.
I.
Iacopini
, G.
Petri
, A.
Barrat
, and V.
Latora
, “Simplicial models of social contagion
,” Nat. Commun.
10
, 47
(2019
). 17.
P. S.
Skardal
and A.
Arenas
, “Abrupt desynchronization and extensive multistability in globally coupled oscillator simplexes
,” Phys. Rev. Lett.
122
, 248301
(2019
). 18.
G.
St-Onge
, V.
Thibeault
, A.
Allard
, L. J.
Dubé
, and L.
Hébert-Dufresne
, “Social confinement and mesoscopic localization of epidemics on networks
,” Phys. Rev. Lett.
126
, 098301
(2021
). 19.
P. S.
Chodrow
, “Configuration models of random hypergraphs
,” J. Complex Networks
8
, 296
(2020
). 20.
B.
Kamiński
, V.
Poulin
, P.
Prałat
, P.
Szufel
, and F.
Théberge
, “Clustering via hypergraph modularity
,” PLoS One
14
, e0224307
(2019
). 21.
I.
Amburg
, N.
Veldt
, and A.
Benson
, “Clustering in graphs and hypergraphs with categorical edge labels
,” in Proceedings of The Web Conference 2020
, WWW ’20 (Association for Computing Machinery
, New York, USA
, 2020
), pp. 706
–717
. 22.
P.
Chodrow
and A.
Mellor
, “Annotated hypergraphs: Models and applications
,” Appl. Network Sci.
5
, 1
–25
(2020
). 23.
Y.
Wang
, D.
Chakrabarti
, C.
Wang
, and C.
Faloutsos
, “Epidemic spreading in real networks: An eigenvalue viewpoint
,” in 22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings
(IEEE
, 2003
), pp. 25
–34
. 24.
J. G.
Restrepo
, E.
Ott
, and B. R.
Hunt
, “Onset of synchronization in large networks of coupled oscillators
,” Phys. Rev. E
71
, 036151
(2005
). 25.
J. G.
Restrepo
, E.
Ott
, and B. R.
Hunt
, “Weighted percolation on directed networks
,” Phys. Rev. Lett.
100
, 058701
(2008
). 26.
B.
Karrer
, M. E. J.
Newman
, and L.
Zdeborová
, “Percolation on sparse networks
,” Phys. Rev. Lett.
113
, 208702
(2014
). 27.
J. G.
Restrepo
, E.
Ott
, and B. R.
Hunt
, “Approximating the largest eigenvalue of network adjacency matrices
,” Phys. Rev. E
76
, 056119
(2007
). 28.
D. J.
Higham
and H.-L.
de Kergorlay
, “Epidemics on hypergraphs: Spectral thresholds for extinction
,” Proc. R. Soc. A
477
, 20210232
(2021
). 29.
P.
Bonacich
, “Power and centrality: A family of measures
,” Am. J. Sociol.
92
, 1170
–1182
(1987
). 30.
A. R.
Benson
, “Three hypergraph eigenvector centralities
,” SIAM J. Math. Data Sci.
1
, 293
–312
(2019
). 31.
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
). 32.
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
). 33.
H.
Sun
and G.
Bianconi
, “Higher-order percolation processes on multiplex hypergraphs
,” Phys. Rev. E
104
, 034306
(2021
). 34.
A.
Benson
, “Data!” https://www.cs.cornell.edu/∼arb/data/ (2021).35.
J. H.
Fowler
, “Connecting the congress: A study of cosponsorship networks
,” Polit. Anal.
14
, 456
–487
(2006
). 36.
J. H.
Fowler
, “Legislative cosponsorship networks in the US house and senate
,” Social Networks
28
, 454
–465
(2006
). 37.
N.
Veldt
, A. R.
Benson
, and J.
Kleinberg
, “Higher-order homophily is combinatorially impossible,” arXiv:2103.11818 [cs] (2021).38.
N.
Litvak
and R.
van der Hofstad
, “Uncovering disassortativity in large scale-free networks
,” Phys. Rev. E
87
, 022801
(2013
). 39.
R.
van der Hofstad
and N.
Litvak
, “Degree-degree dependencies in random graphs with heavy-tailed degrees
,” Internet Math.
10
, 287
(2014
). 40.
M.
Cinelli
, L.
Peel
, A.
Iovanella
, and J.-C.
Delvenne
, “Network constraints on the mixing patterns of binary node metadata
,” Phys. Rev. E
102
, 062310
(2020
). 41.
L.
Neuhäuser
, A.
Mellor
, and R.
Lambiotte
, “Multibody interactions and nonlinear consensus dynamics on networked systems
,” Phys. Rev. E
101
, 032310
(2020
). 42.
K.
Hayashi
, S. G.
Aksoy
, C. H.
Park
, and H.
Park
, “Hypergraph random walks, laplacians, and clustering
,” in Proceedings of the 29th ACM International Conference on Information & Knowledge Management
, CIKM ’20
(Association for Computing Machinery
, New York, USA
, 2020
), pp. 495
–504
. 43.
J.
Jost
and R.
Mulas
, “Hypergraph Laplace operators for chemical reaction networks
,” Adv. Math.
351
, 870
–896
(2019
). 44.
Á.
Bodó
, G. Y.
Katona
, and P. L.
Simon
, “SIS epidemic propagation on hypergraphs
,” Bull. Math. Biol.
78
, 713
–735
(2016
). 45.
B.
Karrer
and M. E. J.
Newman
, “Message passing approach for general epidemic models
,” Phys. Rev. E
82
, 016101
(2010
). 46.
A.
Kirkley
, G. T.
Cantwell
, and M. E. J.
Newman
, “Belief propagation for networks with loops
,” Sci. Adv.
7
, eabf1211
(2021
). 47.
W.
Wang
, M.
Tang
, H. E.
Stanley
, and L. A.
Braunstein
, “Unification of theoretical approaches for epidemic spreading on complex networks
,” Rep. Prog. Phys.
80
, 036603
(2017
). 48.
A. V.
Goltsev
, S. N.
Dorogovtsev
, and J. F. F.
Mendes
, “Percolation on correlated networks
,” Phys. Rev. E
78
, 051105
(2008
). 49.
J. T.
Matamalas
, S.
Gómez
, and A.
Arenas
, “Abrupt phase transition of epidemic spreading in simplicial complexes
,” Phys. Rev. Res.
2
, 012049(R)
(2020
). 50.
F.
Chung
, L.
Lu
, and V.
Vu
, “Spectra of random graphs with given expected degrees
,” Proc. Natl. Acad. Sci. U.S.A.
100
, 6313
–6318
(2003
). 51.
C.
Castellano
and R.
Pastor-Satorras
, “Relating topological determinants of complex networks to their spectral properties: Structural and dynamical effects
,” Phys. Rev. X
7
, 041024
(2017
). 52.
N.
Landry
, nwlandry/hypergraph-assortativity: v0.7
,” (Zenodo
, 2022
). 53.
G.
Burgio
, A.
Arenas
, S.
Gómez
, and J. T.
Matamalas
, “Network clique cover approximation to analyze complex contagions through group interactions
,” Commun. Phys.
4
, 1
–10
(2021
). © 2022 Author(s). Published under an exclusive license by AIP Publishing.
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