linguistic:voice_change
Resonance in complex networks

A generalization to complex networks of coupled harmonic oscillators shows that such systems have a finite number of natural resonant frequencies, related to the topological structure of the network itself. In particular, such frequencies result related to the eigenvalues of the network's Laplacian matrix. Let A {\displaystyle {\bf {A}}} be the adjacency matrix describing the topological structure of the network and L = K − A {\displaystyle {\bf {L}}={\bf {K}}-{\bf {A}}} the corresponding Laplacian matrix, where K = d i a g { k i } {\displaystyle {\bf {K}}={\rm {diag}}\,\{k_{i}\}} is the diagonal matrix of the degrees of the network's nodes. Then, for a network of classical and identical harmonic oscillators, when a sinusoidal driving force f (t) = F 0 sin ω t {\displaystyle f(t)=F_{0}\sin \omega t} is applied to a specific node, the global resonant frequencies of the network are given by ω i = 1 + μ i {\displaystyle \omega _{i}={\sqrt {1+\mu _{i}}}} where μ i {\displaystyle \mu _{i}} are the eigenvalues of the Laplacian L {\displaystyle {\bf {L}}}.