Critical tipping point distinguishing two types of transitions in modular network structures

Saray Shai, Dror Y. Kenett, Yoed N. Kenett, Miriam Faust, Simon Dobson, and Shlomo Havlin

Phys. Rev. E 92, 062805 – Published 2 December 2015

Abstract

Modularity is a key organizing principle in real-world large-scale complex networks. The relatively sparse interactions between modules are critical to the functionality of the system and are often the first to fail. We model such failures as site percolation targeting interconnected nodes, those connecting between modules. We find, using percolation theory and simulations, that they lead to a “tipping point” between two distinct regimes. In one regime, removal of interconnected nodes fragments the modules internally and causes the system to collapse. In contrast, in the other regime, while only attacking a small fraction of nodes, the modules remain but become disconnected, breaking the entire system. We show that networks with broader degree distribution might be highly vulnerable to such attacks since only few nodes are needed to interconnect the modules, consequently putting the entire system at high risk. Our model has the potential to shed light on many real-world phenomena, and we briefly consider its implications on recent advances in the understanding of several neurocognitive processes and diseases.

Received 9 April 2015
Revised 22 October 2015

DOI:https://doi.org/10.1103/PhysRevE.92.062805

Authors & Affiliations

Saray Shai^1,2,*, Dror Y. Kenett³, Yoed N. Kenett⁴, Miriam Faust^4,5, Simon Dobson¹, and Shlomo Havlin⁶

¹School of Computer Science, University of St Andrews, St Andrews, Fife KY16 9SX, Scotland, United Kingdom
²Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA
³Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
⁴Gonda Brain Research Center, Bar-Ilan University, 52900 Ramat-Gan, Israel
⁵Department of Psychology, Bar-Ilan University, 52900 Ramat-Gan, Israel
⁶Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel

^*Corresponding author: sshai@live.unc.edu

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Vol. 92, Iss. 6 — December 2015

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Images

Figure 1
Visualization of the model for generating random modular networks. [(a)–(c)] Illustration of the effect of $α$ on the obtained modular network using Gephi [38] on a Erdős-Rényi (ER) network of size $N = 10 000$ with mean degree $k = 8$ divided into $m = 5$ modules. (d) Illustration of the effect of the number of modules $m$ on the obtained network with a number of intermodule links increasing with the number of modules. Interconnected nodes and links are shown in red (light gray).
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Figure 2
Two percolation regimes when attacking interconnected nodes. (a) $p_{c}$ as a function of $m$ calculated for ER networks with $k = 4, α = 100$ . Simulation points obtained from at least 1000 simulation runs of networks of size $N = 600 000$ . Solid lines represent the analytical result obtained in Eqs. (15) and (19). (b) Two-parameter ( $α, m$ ) phase diagram for a fixed mean degree $k = 4$ . The black line corresponds to the critical number of modules $m^{*}$ obtained from Eq. (22). [(c) and (d)] Fraction of nodes in the largest cluster $S$ and second largest cluster $S_{second}$ , respectively, with solid lines representing the analytical result obtained in Eq. (24) and symbols are from simulations.
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Figure 3
Size of modules in the giant component at $S = 0.1$ . Visualization is shown for networks of size $N = 2000$ with mean degree $k = 4$ and $α = 10$ , at the point where the giant component contains 10% of the nodes ( $S = 0.1$ ) [(a) and (c)] for random node removal and [(b) and (d)] for attack on interconnected nodes. [(e) and (f)] Distribution of the number of modules in the giant component and second-largest component at $S = 0.1$ . A module is considered to be part of a component if at least one of its nodes are part of the component. (g) Distribution of the size of modules in the giant component at $S = 0.1$ , normalized by the initial module size. Note that in (g), the size of modules is measured by reconstructing the graph of each module in the giant component and counting its number of nodes in this graph. In other words, interconnected nodes that have been detached from their original module are not considered. Results obtained by at least 1000 simulation runs of networks of size $N = 600 000$ with mean degree $k = 4$ .
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Figure 4
Fragility of modular scale-free networks [(a) and (b)] $p_{c}$ as a function of $m$ calculated for scale-free networks with mean degree $k = 4, α = 100$ , and scaling exponents $λ = 2.5$ and $λ = 2.9$ , respectively. Simulation points obtained from at least 1000 simulation runs of networks of size $N = 600 000$ . Solid lines represent the numerical solution for Eqs. (25, 26, 27). Two-parameter ( $α, m$ ) phase diagram for a fixed mean degree $k = 4$ . The black line in (d) corresponds to the critical number of modules, $m^{*}$ obtained by solving Eqs. (25, 26, 27) numerically and finding the point where the two possible solutions for $p_{c}$ cross. Note that in (c) there is no black line since $m^{*}$ approach to infinite for $λ = 2.5$ .
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Figure 5
Critical concentration of interconnected nodes [(a)–(c)] $p_{c}$ as a function of $k_{inter}$ calculated for (a) ER and [(b) and (c)] SF networks with $m = 10$ and $k_{intra} = 2$ . Simulation points obtained from at least 1000 simulation runs of networks of size $N = 600 000$ . Solid lines represent the analytical result obtained in Eqs. (19), (25, 26, 27). [(d)–(f)] Two-parameter ( $k_{intra}, k_{inter}$ ) phase diagram for a fixed number of modules $m = 10$ . The black line corresponds to the critical mean interdegree $k_{inter}^{*}$ at which the two $p_{c}$ solutions cross.
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