Complexity vs. stability in small-world networks

According to the May–Wigner stability theorem, increasing the complexity of a network inevitably leads to its destabilization, such that a small perturbation will be able to disrupt the entire system. One of the principal arguments against this observation is that it is valid only for random networks, and therefore does not apply to real-world networks, which presumably are structured. Here, we examine how the introduction of small-world topological structure into networks affects their stability. Our results indicate that, in structured networks, the parameter values at which the stability–instability transition occurs with increasing complexity is identical to that predicted by the May–Wigner criteria. However, the nature of the transition, as measured by the finite-size scaling exponent, appears to change as the network topology transforms from regular to random, with the small-world regime as the cross-over region. This behavior is related to the localization of the largest eigenvalues along the real axis in the eigenvalue plain with increasing regularity in the network.