Dynamic Scaling, Data-collapse and Self-Similarity in Mediation-Driven Attachment Networks

TL;DR

This paper investigates the dynamic scaling and self-similarity properties of mediation-driven attachment (MDA) networks, revealing a spectrum of scaling exponents and the influence of initial connectivity on network structure, with similarities to BA networks at large m.

Contribution

It introduces the dynamic scaling behavior of MDA networks, showing a range of scaling exponents and the impact of initial conditions, extending understanding beyond BA models.

Findings

MDA networks exhibit dynamic scaling with a variable exponent  1/2 to 1.

Scaling curves differ significantly for small and large initial connectivity m.

MDA and BA networks become similar for large m.

Abstract

Recently, we have shown that if the $i$th node of the Barab\'{a}si-Albert (BA) network is characterized by the generalized degree $q_i(t)=k_i(t)t_i^\beta/m$, where $k_i(t)\sim t^\beta$ and $m$ are its degree at current time $t$ and at birth time $t_i$, then the corresponding distribution function $F(q,t)$ exhibits dynamic scaling. Applying the same idea to our recently proposed mediation-driven attachment (MDA) network, we find that it too exhibits dynamic scaling but, unlike the BA model, the exponent $\beta$ of the MDA model assumes a spectrum of value $1/2\leq \beta \leq 1$. Moreover, we find that the scaling curves for small $m$ are significantly different from those of the larger $m$ and the same is true for the BA networks albeit in a lesser extent. We use the idea of the distribution of inverse harmonic mean (IHM) of the neighbours of each node and show that the number of data points that follow the power-law degree distribution increases as the skewness of the IHM distribution decreases. Finally, we show that both MDA and BA models become almost identical for large $m$.