Coarse-Graining and Self-Dissimilarity of Complex Networks

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Abstract
Can complex engineered and biological networks be coarse-grained into smaller and more understandable versions in which each node represents an entire pattern in the original network? To address this, we define coarse-graining units (CGU) as connectivity patterns which can serve as the nodes of a coarse-grained network, and present algorithms to detect them. We show that this approach can systematically reverse-engineer electronic circuits, forming understandable high-level maps from incomprehensible transistor wiring: first, a coarse-grained version in which each node is a gate made of several transistors is established, and then a high-level blueprint in which each node is a circuit-module made of many gates is found. We apply our approach also to a mammalian protein-signaling network, to find a simplified coarse-grained network with three main signaling channels that correspond to cross-interacting MAP-kinase cascades. We find that both biological and electronic networks are 'self-dissimilar', with different network motifs found at each level. This approach can be used to simplify a wide variety of natural and designed networks.

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