Degree Distribution, Rank-size Distribution, and Leadership Persistence in Mediation-Driven Attachment Networks

TL;DR

This paper studies a mediation-driven attachment network model, revealing power-law degree distributions, contrasting rank-size distributions with BA networks, and analyzing leadership persistence with power-law decay in leader retention.

Contribution

It introduces the mediation-driven attachment model, showing its degree distribution and leadership persistence properties, and compares these with Barabási-Albert networks.

Findings

Degree distribution follows a power-law with a spectrum of exponents depending on m.

Leadership persistence probability decays as a power-law with a universal exponent in MDA networks.

Contrast between MDA and BA networks in rank-size distribution and leadership dynamics.

Abstract

We investigate the growth of a class of networks in which a new node first picks a mediator at random and connects with $m$ randomly chosen neighbors of the mediator at each time step. We show that degree distribution in such a mediation-driven attachment (MDA) network exhibits power-law $P(k)\sim k^{-\gamma(m)}$ with a spectrum of exponents depending on $m$. To appreciate the contrast between MDA and Barab\'{a}si-Albert (BA) networks, we then discuss their rank-size distribution. To quantify how long a leader, the node with the maximum degree, persists in its leadership as the network evolves, we investigate the leadership persistence probability $F(\tau)$ i.e. the probability that a leader retains its leadership up to time $\tau$. We find that it exhibits a power-law $F(\tau)\sim \tau^{-\theta(m)}$ with persistence exponent $\theta(m) \approx 1.51 \ \forall \ m$ in the MDA networks and $\theta(m) \rightarrow 1.53$ exponentially with $m$ in the BA networks.