  The relationship between network topology and system dynamics has significant
implications for unifying our understanding of the interplay among metabolic,
gene-regulatory, and ecosystem network architecures. Here we analyze the
stability and robustness of a large class of dynamics on such networks. We
determine the probability distribution of robustness as a function of network
topology and show that robustness is classified by the number of links between
modules of the network. We also demonstrate that permutation of these modules
is a fundamental symmetry of dynamical robustness. Analysis of these findings
leads to the conclusion that the most robust systems have the most hierarchical
structure. This relationship provides a means by which evolutionary selection
for a purely dynamical phenomenon may shape network architectures across scales
of the biological hierarchy.
