Global decomposition of networks into multiple cores formed by local hubs

TL;DR

The paper proposes locally defined hub centrality as the basis for a hub-centrality-based edge-pruning decomposition (LED) to uncover multiscale core–periphery structure in networks. By pruning edges with zero hub-centrality product, the authors identify cusp points that separate a backbone from shells, producing onion-like hierarchical layers and an interpretable core–periphery organization. LED is contrasted with the traditional k-core decomposition and shown to reveal finer, locally meaningful substructures, including multiple cores within communities and coarse-grained supernode networks. The approach is validated on real networks and synthetic models (BA and SBM), demonstrating the utility of local information for detecting CP structure and for revealing mesoscale organization with potential dynamical implications.

Abstract

Networks are ubiquitous in various fields, representing systems where nodes and their interconnections constitute their intricate structures. We introduce a network decomposition scheme to reveal multiscale core-periphery structures lurking inside, using the concept of locally defined nodal hub centrality and edge-pruning techniques built upon it. We demonstrate that the hub-centrality-based edge pruning reveals a series of breaking points in network decomposition, which effectively separates a network into its backbone and shell structures. Our local-edge decomposition method iteratively identifies and removes locally least connected nodes, and uncovers an onion-like hierarchical structure as a result. Compared with the conventional $k$-core decomposition method, our method based on relative information residing in local structures exhibits a clear advantage in terms of discovering locally crucial substructures. As an application of the method, we present a scheme to detect multiple core-periphery structures and the decomposition of coarse-grained supernode networks, by combining the method with the network community detection.

Figures (11)

• Figure 1: (a) Relative size $G$ of the giant component as a function of the fraction $p$ of removed edges. The network is a collaboration network in the field of computational geometry. (b)--(c) Fraction $e_0$ of edges with $\mathscr{E}^{h\hbox{-}\mathrm{P}}=0$ and the fraction $n_0$ of nodes with $h=0$ as a function of the cusp point $p_c$ and the giant component size $g$ with zero hub-centrality nodes removed, respectively. Each point represents the outcome of each individual network listed in Table \ref{['tab1']}. The solid lines in (b) and (c) represent $e_0 = p_c$ and $n_0 = 1-g$, respectively.
• Figure 2: (a)--(c) Relative size $G$ of the giant component as a function of the fraction $p$ of removed edges based on the hub-centrality product ($h\hbox{-}\mathrm{P}$). (a) The original collaboration network used in Fig. \ref{['fig1']}. (b) Backbone of the original network. (c) Backbone of the original network's backbone. (d) Relative fraction $g$ of the giant component size with zero hub-centrality nodes removed as a function of the fraction $n_0$ of zero hub-centrality nodes in decomposed networks. The solid line represents $n_0=1-g$, which implies the set of zero hub-centrality nodes $\approx$ the shell at each decomposition level.
• Figure 3: An example network to illustrate the LED decomposition. The nodes filled with the same color belong to the same hierarchical level. The green, blue, and red nodes belong to the lowest, intermediate, and highest levels, respectively. The color and thickness of the edges indicate the decomposition levels as well.
• Figure 4: (a) Relative size $G$ of the giant component as a function of the fraction $p$ of removed edges based on $h\hbox{-}\mathrm{P}$, applied to a single realization of the BA model Barabasi1999 with $10^5$ nodes and the number $m = 4$ of stubs for each newly added node. The black squares indicate the original network. The red circles, blue triangles, and purple inverted triangles represent the results obtained for the primary, secondary, and tertiary backbones, respectively, of the original network. (b) Relative fraction $g$ of the giant component size with zero hub-centrality nodes removed as a function of the fraction $n_0$ of zero hub-centrality nodes in decomposed networks at different levels. The black squares, red circles, and blue triangles represent the cases with $m = 2$, $4$, and $6$, respectively. We only plot the cases where the backbone with the number of nodes $> 1\%$ of that of the original network.
• Figure 5: Fraction of the level (or "shell" in the kCD) relationship between nodes and their neighbors. The colors red, blue, and green indicate that nodes at a given level have neighbors with higher, lower, or the same level, respectively. The regions filled with gray represent the absence of nodes in the corresponding levels. Panel (a) is for the LED, and panel (b) is for the kCD.