Uncovering the hidden core-periphery structure in hyperbolic networks

TL;DR

This study comprehensively explores the presence of an important feature, the core-periphery structure, in the hyperbolic network models, and reveals core-periphery insights applicable to various domains, enhancing network performance and resiliency in transportation and information systems.

Abstract

The hyperbolic network models exhibit very fundamental and essential features, like small-worldness, scale-freeness, high-clustering coefficient, and community structure. In this paper, we comprehensively explore the presence of an important feature, the core-periphery structure, in the hyperbolic network models, which is often exhibited by real-world networks. We focused on well-known hyperbolic models such as popularity-similarity optimization model (PSO) and S1/H2 models and studied core-periphery structures using a well-established method that is based on standard random walk Markov chain model. The observed core-periphery centralization values indicate that the core-periphery structure can be very pronounced under certain conditions. We also validate our findings by statistically testing for the significance of the observed core-periphery structure in the network geometry. This study extends network science and reveals core-periphery insights applicable to various domains, enhancing network performance and resiliency in transportation and information systems.

Figures (13)

• Figure 1: Core-periphery visualisation in hyperbolic networks. (a) Core (cyan) and periphery (red) obtained in a network with $N = 500$ number of nodes, generated by the PSO model with parameters $m = 10$ (corresponding to $<k> = 20$), $\beta = 0.8$ (corresponding to $\gamma = 2.25$) and $T = 0.8$. The layout shows the network in the native disk representation of the two dimensional hyperbolic space of curvature $K =-1$, with the nodes arranged according to their coordinates assigned during the network generation process. (b) Core (cyan) and periphery (red) obtained in a network generated by the $\mathbb{S}^1/\mathbb{H}^2$ model with parameters $N = 500$, $<k> = 20$, $\gamma = 2.25$ and $\alpha = 1.125$, shown in the native disk representation of the hyperbolic plane of curvature $K =-1$
• Figure 2: Core-periphery centralization in the PSO model. We show the core-periphery centralization $C$ as a function of the model parameters $T$ and $\beta$ for networks of size $N = 100$ and the expected average degree: (a). $\langle k \rangle = 4$, (b). $\langle k \rangle = 10$, and (c). $\langle k \rangle = 20$.
• Figure 3: Core-periphery centralization in the PSO model. We show the core-periphery centralization $C$ as a function of the model parameters $T$ and $\beta$ for networks of size $N = 500$ and the expected average degree: (a). $\langle k \rangle = 4$, (b). $\langle k \rangle = 10$, and (c). $\langle k \rangle = 20$.
• Figure 4: Core-periphery centralization in the PSO model. We show the core-periphery centralization $C$ as a function of the model parameters $T$ and $\beta$ for networks of size $N = 1000$ and the expected average degree: (a). $\langle k \rangle = 4$, (b). $\langle k \rangle = 10$, and (c). $\langle k \rangle = 20$.
• Figure 5: Core-periphery centralization in the $\mathbb{S}^1/\mathbb{H}^2$ model. We show the core-periphery centralization $C$ as a function of the model parameters $1/\alpha$ and $1/(\gamma-1)$ for networks of size $N = 100$ and the expected average degree: (a). $\langle k \rangle = 4$, (b). $\langle k \rangle = 10$, and (c). $\langle k \rangle = 20$.