  Quantifying the relations (e.g., similarity) between complex networks paves
the way for studying the latent information shared across networks. However,
fundamental relation metrics are not well-defined between networks. As a
compromise, prevalent techniques measure network relations in data-driven
manners, which are inapplicable to analytic derivations in physics. To resolve
this issue, we present a theory for obtaining an optimal characterization of
network topological properties. We show that a network can be fully represented
by a Gaussian variable defined by a function of the Laplacian, which
simultaneously satisfies network-topology-dependent smoothness and maximum
entropy properties. Based on it, we can analytically measure diverse relations
between complex networks. As illustrations, we define encoding (e.g.,
information divergence and mutual information), decoding (e.g., Fisher
information), and causality (e.g., Granger causality and conditional mutual
information) between networks. We validate our framework on representative
networks (e.g., random networks, protein structures, and chemical compounds) to
demonstrate that a series of science and engineering challenges (e.g., network
evolution, embedding, and query) can be tackled from a new perspective. An
implementation of our theory is released as a multi-platform toolbox.
