Quasistatic Scale-free Networks

TL;DR

This paper introduces a model for scale-free networks on a ring lattice, demonstrating a transition in degree distribution behavior controlled by a parameter, with the network becoming scale-free beyond a critical point.

Contribution

It presents a quasistatic network formation model with a tunable parameter, revealing a phase transition to scale-free degree distributions.

Findings

Degree distribution follows a power law for lpha .

Transition point lpha  marks change in maximum node degree scaling.

Network exhibits scale-free properties for lpha .

Abstract

A network is formed using the $N$ sites of an one-dimensional lattice in the shape of a ring as nodes and each node with the initial degree $k_{in}=2$. $N$ links are then introduced to this network, each link starts from a distinct node, the other end being connected to any other node with degree $k$ randomly selected with an attachment probability proportional to $k^{\alpha}$. Tuning the control parameter $\alpha$ we observe a transition where the average degree of the largest node $<k_m(\alpha,N)>$ changes its variation from $N^0$ to $N$ at a specific transition point of $\alpha_c$. The network is scale-free i.e., the nodal degree distribution has a power law decay for $\alpha \ge \alpha_c$.