Weighted growing simplicial complexes

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Weighted growing simplicial complexes

Owen T. Courtney and Ginestra Bianconi

Phys. Rev. E 95, 062301 – Published 12 June 2017

Abstract

Simplicial complexes describe collaboration networks, protein interaction networks, and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial complexes are weighted. Here we propose a nonequilibrium model for weighted growing simplicial complexes. The proposed dynamics is able to generate weighted simplicial complexes with a rich interplay between weights and topology emerging not just at the level of nodes and links, but also at the level of faces of higher dimension.

Received 3 March 2017

DOI:https://doi.org/10.1103/PhysRevE.95.062301

Physics Subject Headings (PhySH)

Evolving networks Network evolution Networks & random structures Scale free & inhomogeneous networks Weighted networks

Mean field theory Network Models

Interdisciplinary PhysicsStatistical PhysicsNetworks

Authors & Affiliations

Owen T. Courtney and Ginestra Bianconi

School of Mathematical Sciences, Queen Mary University of London, E1 4NS London, United Kingdom

Article Text

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References

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Issue

Vol. 95, Iss. 6 — June 2017

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Graphical representation of process A [panel (a)] and process B [panel (b)] for a two-dimensional simplicial complex with and starting from a given initial condition.
Skeleton networks of simplicial complexes generated by the model for , and . Node sizes indicate their degrees while link widths indicate their generalized strength. Node and edge colorings indicate community structure calculated according to the Louvain algorithm []. Panel (a) shows the skeleton of a simplicial complex with and , while panel (b) shows the skeleton of a simplicial complex with and .
Generalized degree distributions are shown for simplicial complexes of dimension and flavor [panels (a), (d), and (g)], [panels (b), (e), and (h)], and [panels (c), (f), and (i)], and for faces of dimension [panels (a), (b), and (c)], [panels (d), (e), and (f)], and [panels (g), (h), and (i)]. The results of simulations are shown for , 2, and 3 (blue circles, red squares, and yellow triangles, respectively). The simulated simplicial complexes have nodes, and the results are averaged over 10 simplicial complex realizations.
The average generalized strengths of the -faces arrived in the network at time are shown for simplicial complexes of dimension , and flavor [panels (a), (d), and (g)], [panels (b), (e), and (h)], and [panels (c), (f), and (i)] and for faces of dimension [panels (a), (b), and (c)], [panels (d), (e), and (f)], and [panels (g), (h), and (i)]. The results of simulations are shown for , and (blue circles, red squares, yellow crosses, and purple triangles, respectively). The simulated simplicial complexes have nodes, and the results are averaged over 10 simplicial complex realizations.
The exponents obtained by fitting Eq. () to the data in Fig. are shown vs the predicted exponents given by Eq. () for different -faces. The panels (a), (b), and (c) refer, respectively, to triangles , links , and nodes . The blue stars, red squares, and yellow circles indicate the data obtained, respectively, for the the flavors , 0, and 1.
The average generalized strengths of -faces as a function of their corresponding generalized degree are shown for simplicial complexes of dimension and flavor [panels (a), (d), and (g)], [panels (b), (e), and (h)], and [panels (c), (f), and (i)] and for faces of dimension [panels (a), (b), and (c)], [panels (d), (e), and (f)], and [panels (g), (h), and (i)]. The results of simulations are shown for , and (blue circles, red squares, yellow crosses, and purple triangles, respectively). The simulated simplicial complexes have nodes, and the results are averaged over 10 simplicial complex realizations.

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