Twin Subgraphs and Core-Semiperiphery-Periphery Structures

Ricardo Riaza

doi:10.1155/2018/2547270

Abstract

A standard approach to reduce the complexity of very large networks is to group together sets of nodes into clusters according to some criterion which reflects certain structural properties of the network. Beyond the well-known modularity measures defining communities, there are criteria based on the existence of similar or identical connection patterns of a node or sets of nodes to the remainder of the network. A key notion in this context is that of structurally equivalent or twin nodes, displaying exactly the same connection pattern to the remainder of the network. Our first goal is to extend this idea to subgraphs of arbitrary order of a given network, by means of the notions of T-twin and F-twin subgraphs. This research, which leads to graph-theoretic results of independent interest, is motivated by the need to provide a systematic approach to the analysis of core-semiperiphery-periphery (CSP) structures, a notion which is widely used in network theory but that somehow lacks a formal treatment in the literature. The goal is to provide an analytical framework accommodating and extending the idea that the unique (ideal) core-periphery (CP) structure is a 2-partitioned K2, a fact which is here understood to rely on the true and false twin notions for vertices already known in network theory. We provide a formal definition of such CSP structures in terms of core eccentricities and periphery degrees, with semiperiphery vertices acting as intermediaries between both. The T-twin and F-twin notions then make it possible to reduce the large number of resulting structures, paving the way for the decomposition and enumeration of CSP structures. We compute explicitly the resulting CSP structures up to order six. We illustrate the scope of our results by analyzing a subnetwork of the well-known network of metal manufactures trade arising from 1994 world trade statistics.

Highlights

The notion of a core-periphery (CP) structure can be traced back at least to some research on economic and commercial networks developed in the late 1970s and early 1980s [1,2,3], largely emanating from the influential work of Wallerstein on world systems analysis [4]
We say that two disjoint subgraphs H1 and H2 are not adjacent if there is no adjacent pair (u, V) with u ∈ H1 and V ∈ H2; if both subgraphs lie in the same connected component of G, this is equivalent to saying that d(H1, H2) ≥ 2
Many problems related to twin subgraphs and to coresemiperiphery-periphery structures remain open for future study

Summary

Introduction

The notion of a core-periphery (CP) structure can be traced back at least to some research on economic and commercial networks developed in the late 1970s and early 1980s [1,2,3], largely emanating from the influential work of Wallerstein on world systems analysis [4]. Let us clarify that K2 is twin-free only as a 2-partitioned graph; that is, we cannot consider both vertices as (true) twins because they belong to different partition classes; see the beginning of Section 5 When scaling these ideas to define formally core-semiperiphery-periphery (CSP) structures and eventually other structures with more partition classes, one finds the problem that there is no appropriate analog of the twin notions mentioned above for subgraphs with more than one vertex.

Definition and Elementary Properties

T-Twins

Core-Semiperiphery-Periphery Structures

Findings

Concluding Remarks