Superelliptical laws for complex networks

All dynamical systems of biological interest--be they food webs, regulation of genes, or contacts between healthy and infectious individuals--have complex network structure. Wigner's semicircular law and Girko's circular law describe the eigenvalues of systems whose structure is a fully connected network. However, these laws fail for systems with complex network structure. Here we show that in these cases the eigenvalues are described by superellipses. We also develop a new method to analytically estimate the dominant eigenvalue of complex networks.