Fig 1.
Configuration of a modular network before/after rewirings.
(A) Initial configuration of strongly modular network. (B) Rewired modular network to increase the robustness of connectivity. Node colors indicate modules.
Table 1.
Initial numbers of intra- and inter- links.
Fig 2.
The 1st LCC size against MB attacks in rewired networks with d = 9 and the number mo of modules.
(A) mo = 1000, (B) mo = 500, (C) mo = 100, (D) mo = 50, (E) mo = 20, and (F) mo = 5. Color lines represent the rewiring rates w′ on anti-modularization.
Table 2.
Values of the critical fraction qc against three types of attacks in rewired networks with d = 9 and mo = 200.
Fig 3.
Relation between modularity Q and robustness index R against three types of attacks.
(A) RMB against MB, (B) RIB against IB, and (C) RRF against RF. Color lines represent the results for the number mo of modules. Black solid, dotted, and cyan lines represent the robustness after these attacks in non-modular ER, SF networks and random d-regular graphs.
Fig 4.
Relation between modularity Q and six measures of robustness in rewired networks with d = 9.
(A) Average betweenness centrality, (B) reciprocal of network efficiency, (C) average path length, (D) diameter, (E) spectral gap, (F) algebraic connectivity. Color lines represent the results for the numbers mo of modules.
Fig 5.
Average path length L in rewired networks for varying the network size N.
Rewired networks with (A) d = 4, Nm = 10, (B) d = 9, Nm = 20, and (C) d = 19, Nm = 50. Color lines represent a rewiring rate w′ on anti-modularization.
Fig 6.
Disparities of average path length L between the values for rewired modular networks and theoretical value.
Rewired networks with (A) d = 4, Nm = 20, (B) d = 9, Nm = 20, and (C) d = 19, Nm = 20. Color lines represent a rewiring rate w′ on anti-modularization. Black solid and dashed lines indicate numerical and theoretical values for random d-regular graphs without modular structures.
Fig 7.
Degree distribution in rewired networks with d = 9 and mo = 100.
For rewiring rates (A) w = 0.988, (B) w = 0.695, and (C) w = 0.494. The values of w are corresponding to w′ = 0.01, 0.3, 0.5 from Eq (1).
Fig 8.
Process of the modified modularization.
The numbers in brackets denote a sequence of the process. Yellow circles represent modules. Black dotted and Blue solid lines are inter- and intra-links by rewirings, respectively.
Fig 9.
Comparing the robustness against MB attacks in rewired networks on the anti-modularization and modified modularization.
The 1st LCC size against MB attacks in rewired networks with d = 9 and number mo of modules. (A) mo = 100, (B) mo = 20, and (C) mo = 5. Solid and dotted color lines represent the results by rewiring with w′ on anti-modularization and w on modified modularization.