Coarse-graining and self-dissimilarity of complex networks

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 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.

Spontaneous evolution of modularity and network motifs

“…Feed-forward loops, bifans, and diamonds are found in signal transduction and synaptic neuronal networks (7). In signal transduction networks (34) and the neuronal network of C. elegans (39), multilayered feed-forward patterns similar to those in Fig. 5c, are strong network motifs.…”

“…5c, are strong network motifs. An example is multilayered protein kinase cascades, in which families of kinases in each layer activate families of kinases in the next layer (34,40,41).…”

“…We also used this measure to quantitate the modularity of three biological networks. All showed high modularity: The transcription network of the bacterium Escherichia coli (7) had Q m ϭ 0.54, the neuronal synaptic network of the nematode Caenorhabditis elegans (7) had Q m ϭ 0.54, and a human signal transduction interaction network (34) had Q m ϭ 0.58. Modularity measurements results are summarized in Tables 3-5, which are published as supporting information on the PNAS web site.…”

Community detection in graphs

“…18 , but delivers good results for computer generated networks, and meaningful partitions for some real networks, like the world airport network (Barrat et al, 2004), an email exchange network of a Catalan university (Guimerà et al, 2003), a network of electronic circuits (Itzkovitz et al, 2005) and metabolic networks of E. coli .…”

A Faster Algorithm for Detecting Network Motifs

“…2 Some excitement has surrounded the network motif approach with the original paper by Milo et al [15] being cited well over 40 times in some major scientific journals as of June 2005. The analysis of network motifs has led to interesting results (of which we only name a few here), e.g., in the areas of protein-protein interaction prediction [1] and hierarchical network decomposition [7]. The transcriptional network of Escherichia Coli displays motifs to which specific functionalities such as the generation of temporal expression programs or the response to ⋆ Supported by Deutsche Telekom Stiftung and Studienstiftung des deutschen Volkes.…”