Network shell structure based on hub and non-hub nodes

TL;DR

This paper investigates the shell structure of hub and non-hub nodes in ER and SF networks, deriving analytical expressions for node distributions outside shells and revealing an iterative functional relationship.

Contribution

It introduces a novel analytical framework for understanding shell structures in different network types based on degree distributions and node roles.

Findings

Derived analytical degree distribution outside shells.

Established iterative functional form for shell progression.

Applied framework to ER and SF networks.

Abstract

The shell structure holds significant importance in various domains such as information dissemination, supply chain management, and transportation. This study focuses on investigating the shell structure of hub and non-hub nodes, which play important roles in these domains. Our framework explores the topology of Erd\"{o}s-R\'{e}nyi (ER) and Scale-Free (SF) networks, considering source node selection strategies dependent on the nodes' degrees. We define the shell $l$ in a network as the set of nodes at a distance $l$ from a given node and represent $r_l$ as the fraction of nodes outside shell $l$. Statistical properties of the shells are examined for a selected node, taking into account the node's degree. For a network with a given degree distribution, we analytically derive the degree distribution and average degree of nodes outside shell $l$ as functions of $r_l$. Moreover, we discover that $r_l$ follows an iterative functional form $r_l = \phi(r_{l-1})$, where $\phi$ is expressed in terms of the generating function of the original degree distribution of the network.