Recent studies on complex networks show that heuristically enhancing loops decreases the variance of degree distributions and improves the robustness of connectivity. However, the optimal length of loops remains unclear for improving the robustness. Moreover, it has been revealed that networks become extremely vulnerable, when a strong modular (or community) structure is added. To clarify the relations among the length of loops, the variance of degree distributions, and the modularity, we comprehensively investigate the lengths of the shortest loops in networks with/without the modular structure under continuously changing degree distributions. We find that decreasing the variance leads to longer shortest loops, whereas the modular structure gives the inverse effects. In other words, large holes are important for improving the robustness, although it seems contradictory.