Complex biological, technological, and sociological networks can be of very different sizes and connectivities, making it difficult to compare their structures. Here we present an approach to systematically study similarity in the local structure of networks, based on the significance profile (SP) of small subgraphs in the network compared to randomized networks. We find several superfamilies of previously unrelated networks with very similar SPs. One superfamily, including transcription networks of microorganisms, represents "rate-limited" information-processing networks strongly constrained by the response time of their components. A distinct superfamily includes protein signaling, developmental genetic networks, and neuronal wiring. Additional superfamilies include power grids, protein-structure networks and geometric networks, World Wide Web links and social networks, and word-adjacency networks from different languages.
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 use this approach to 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. Then, the coarse-grained network is itself coarse-grained, resulting in a high-level blueprint in which each node is a circuit-module made of multiple gates. 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. The present approach can be used to simplify a wide variety of directed and nondirected, natural and designed networks.
Western harmony is comprised of sequences of chords, which obey grammatical rules. It is of interest to develop a compact representation of the harmonic movement of chord sequences. Here, we apply an approach from analysis of complex networks, known as "network motifs" to define repeating dynamical patterns in musical harmony. We describe each piece as a graph, where the nodes are chords and the directed edges connect chords which occur consecutively in the piece. We detect several patterns, each of which is a walk on this graph, which recur in diverse musical pieces from the Baroque to modern-day popular music. These patterns include cycles of three or four nodes, with up to two mutual edges (edges that point in both directions). Cliques and patterns with more than two mutual edges are rare. Some of these universal patterns of harmony are well known and correspond to basic principles of music theory such as hierarchy and directionality. This approach can be extended to search for recurring patterns in other musical components and to study other dynamical systems that can be represented as walks on graphs.
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