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A standard approach to reduce the complexity of very large networks is togroup together sets of nodes into clusters according to some criterion whichreflects certain structural properties of the network. Beyond the well-knownmodularity measures defining communities, there are criteria based on theexistence of similar or identical connection patterns of a node or sets ofnodes to the remainder of the network. A key notion in this context is that ofstructurally equivalent or twin nodes, displaying exactly the same connectionpattern to the remainder of the network.  The first goal of this paper is to extend this idea to subgraphs of arbitraryorder of a given network, by means of the notions of T-twin and F-twinsubgraphs. This is motivated by the need to provide a systematic approach tothe analysis of core-semiperiphery-periphery (CSP) structures, a notion whichsomehow lacks a formal treatment in the literature. The goal is to provide ananalytical framework accommodating and extending the idea that the unique(ideal) core-periphery (CP) structure is a 2-partitioned K2. We provide aformal definition of CSP structures in terms of core eccentricities andperiphery degrees, with semiperiphery vertices acting as intermediaries. TheT-twin and F-twin notions then make it possible to reduce the large number ofresulting structures by identifying isomorphic substructures which share theconnection pattern to the remainder of the graph, paving the way for thedecomposition and enumeration of CSP structures. We compute the resulting CSPstructures up to order six.  We illustrate the scope of our results by analyzing a subnetwork of thenetwork of 1994 metal manufactures trade. Our approach can be further appliedin complex network theory and seems to have many potential extensions.

Twin subgraphs and core-semiperiphery-periphery structures