Universal vulnerability in strong modular networks with various degree distributions between inequality and equality

TL;DR

All networks which include typical well-known network structure between them become extremely vulnerable, if a strong modular (or community) structure is added with commonalities of areas, interests, religions, purpose, and so on.

Abstract

Generally, networks are classified into two sides of inequality and equality with respect to the number of links at nodes by the types of degree distributions. One side includes many social, technological, and biological networks which consist of a few nodes with many links, and many nodes with a few links, whereas the other side consists of all nodes with an equal number of links. In comprehensive investigations between them, we have found that, as a more equal network, the tolerance of whole connectivity is stronger without fragmentation against the malfunction of nodes in a wide class of randomized networks. However, we newly find that all networks which include typical well-known network structures between them become extremely vulnerable, if a strong modular (or community) structure is added with commonalities of areas, interests, religions, purpose, and so on. These results will encourage avoiding too dense unions by connecting nodes and taking into account the balanced resource allocation between intra- and inter-links of weak communities. We must reconsider not only efficiency but also tolerance against attacks or disasters, unless no community that is really impossible.

Figures (25)

• Figure 1: Continuously changing degree distributions from (a) power-law as SF networks (red), power-law with exponential cut-off (orange), exponential (yellow), (b) near Poisson as ER random graphs (green), and to narrower ones approaching regular networks (blue, light-blue, and purple). Note that (a) is a log-log plot, while (b) is a semi-log plot.
• Figure 2: Visualization example of SF networks at $\nu = 1$, $m = 2$, and $N = 200$ with modules whose strength is increased by the tentative rewiring rate $w' = 0, 0.5, 0.7, 0.9, 0.95$, and $0.98$ from top-left to bottom-right. Th five colors represent $m_{o} = 5$ modules, and the circle size of node is proportional to its degree.
• Figure 3: Comparison of the areas under the curves represented as the robustness against MB attacks in (a)(b) SF networks at $\nu = 1$ and (c)(d) nearly regular networks at $\nu = -100$ with (a)(c) $m_{o} = 5$ and (b)(d) $200$ modules. The colored lines from red to light-blue correspond to values of $w'$ for increasing the modularity $Q$.
• Figure 4: Rapid decreasing of the vertical axis value: (a)(b) the eigenvalue $\mu_{2}$ of Laplacian matrix and the robustness index $R$ against (c)(d) MB and (e)(f) ID attacks in modular networks measured by the modularity $Q$ of the horizontal axis value. The degree distributions are continuously changed from power-law (as SF networks at $\nu = 1$), power-law with exponential cut-off, exponential, nearly Poisson (as nearly ER random graphs at $\nu = -1$), and narrower ones (approaching regular networks at $\nu < -1$) by varying $\nu$ whose lines are rainbow colored from red to purple. The modularity $Q$ is controlled by varying $w'$ in (a)(c)(e) $m_{o} = 5$ and (b)(d)(f) $200$ modules.
• Figure 5: Scatter plots of the eigenvalue $\mu_{2}$ of Laplacian matrix and the robustness index $R$ against (a) MB, (b) IB, (c) ID attacks in networks with $m_{o} = 5, 10, 20, 50, 100$, and $200$ modules. Rainbow colors for $\nu = 1, 1/2, 0, -1, -3, -5$ and $-100$ correspond to different $P(k)$ interpolated between power-law (as SF networks at $\nu = 1$), power-law with exponential cut-off, exponential, nearly Poisson (as ER random graphs at $\nu = -1$), and narrower ones (approaching regular networks at $\nu < -1$). Lower and upper dashed-oval parts are for $Q > 0.8$ and $Q < 0.8$, respectively, in different $m_{o}$ modules and varying the rewiring rate $w'$.