Anti-modularization for both high robustness and efficiency including the optimal case

Although robustness of connectivity and modular structures in networks have been attracted much attentions in complex networks, most researches have focused on those two features in Erdos-Renyi random graphs and Scale-Free networks whose degree distributions follow Poisson and power-law, respectively. This paper investigates the effect of modularity on robustness in a modular d-regular graphs. Our results reveal that high modularity reduces the robustness even from the optimal robustness of a random d-regular graph in the pure effect of degree distributions. Moreover, we find that a low modular d-regular graph exhibits small-world property that average path length is O(logN). These results indicate that low modularity on modular structures leads to coexistence of both high robustness and efficiency of paths.