Individual heterogeneity generating explosive system network dynamics

Pedro D. Manrique and Neil F. Johnson

Phys. Rev. E 97, 032311 – Published 21 March 2018

Abstract

Individual heterogeneity is a key characteristic of many real-world systems, from organisms to humans. However, its role in determining the system's collective dynamics is not well understood. Here we study how individual heterogeneity impacts the system network dynamics by comparing linking mechanisms that favor similar or dissimilar individuals. We find that this heterogeneity-based evolution drives an unconventional form of explosive network behavior, and it dictates how a polarized population moves toward consensus. Our model shows good agreement with data from both biological and social science domains. We conclude that individual heterogeneity likely plays a key role in the collective development of real-world networks and communities, and it cannot be ignored.

Received 16 December 2017

DOI:https://doi.org/10.1103/PhysRevE.97.032311

Physics Subject Headings (PhySH)

Dynamics of networks Network evolution

Dynamical systems Protein interaction networks Real world networks Social networks

Explosive percolation Network Models

NetworksStatistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Pedro D. Manrique and Neil F. Johnson

Physics Department, University of Miami, Coral Gables, Florida 33126, USA

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Issue

Vol. 97, Iss. 3 — March 2018

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Images

Figure 1
Modeling heterogeneous grouping. (a) The heterogeneity of nodes is modeled by the character ${x}$ , which is a real number defined in the [0,1] interval and randomly assigned to each node from a distribution $q (x)$ . Our model of interacting nodes comprises two mechanisms: M1, which favors linking similar nodes (i.e., close in character), and M2, favoring dissimilar nodes (i.e., distant in character). (b) The model follows a two-step cycle where it first randomly selects a sample of $k$ unconnected nodes and then selects the pair that maximizes the coalescence function $C (S_{i, j})$ that links together. (c) Dynamics of the largest cluster $G_{1}$ as a function of the number of links for both heterogeneous formation mechanisms as well as the random case (i.e., free) and different values of sampling size $k$ . The total population is $N = 10^{4}$ nodes. (d) Variation of the transition point for M1, M2, and the free model with the sampling size $k$ computed at the instant when $G_{1} / N$ and its growth rate exceed the value of the random process at $L / N = 0.5$ . (e) Dependence of the largest gap in the growth of $G_{1}$ for both mechanisms (shadows) and the free model, and several values of sampling. Lines are guides to the eye. For (d) and (e) we show an average over 500 realizations.
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Figure 2
Effect of diversity in grouping. (a) Probability density function of a Beta distribution for several values of the parameter $α$ for the case when $α = β$ . (b)–(f) Grouping evolution as a function of the average character of the largest seven clusters (colored bubbles). The size of the bubble is proportional to the square root of the size, and each panel shows a different $α$ value as shown in each panel. The total population is $N = 10^{3}$ nodes, sampling $k = 10$ , and M1 aggregation rule.
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Figure 3
Heterogeneity in the human protein homology network. (a) Snapshots of the human protein homology network as links are added. The parameter $p$ represents the fraction of added links. The top panel is the early stage for $p = 0.5$ and the bottom panel is the final stage with $p = 1$ . Each panel shows the largest four clusters (i.e., $G_{i}$ for $i = 1, 2, 3, 4$ ). (b) Evolution of the three largest clusters of the protein network shown in (a) (filled symbols) compared to those of a heterogeneous process (empty symbols) with a uniform character distribution and a sampling size of $k = 8$ .
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Figure 4
Explosive grouping in pro-ISIS online groups. (a) Snapshot of online pro-ISIS support groups on the VK platform on January 10, 2015. (b) Evolution of a sample of sharkfin pro-ISIS groups from first detection to shutdown. (c) Examples of explosive behavior in pro-ISIS groups as compared to our heterogeneous model for $k = 5$ (left panel) and $k = 10$ (middle and right panel). For each case, $N$ is set to be the respective group population at the moment of shutdown.
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