Comparing the stability and reliability of networks by spectral analysis

Forty years ago, Gardner and Ashby1, May2, 3 and Daniel and Mackay4 studied network stability by analysing large networks in which species interact at random. They found a negative correlation between system stability and system complexity. However, their works did not address the stability criterion of networks with equal connectance or connectivity, nor they discussed networks with specified topology. Reliability of network is another important quantity that can be either investigated numerically or using the theory of signature5–7, though in general the comparison of the reliability of two large networks with same connectance is difficult. Here we investigate the stability and reliability of a class of simple networks with a specific topology, in the form of twisted-ring. We show how that the eigenvalue spectrum of these simple networks can be used to classify their stability and reliability. In agreement with previous results, we find that the most important factor in determining the stability and reliability is the largest eigenvalue, which reflects the connectance of the network. However, for different networks with the same connectance, their eigenvalue gaps, which reflect the level of symmetry of the networks, provide a guide for the ordering of stability and reliability. These conclusions on the twisted ring networks is conjectured to be general, and has