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

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#### Abstract

We investigate the growth of a class of networks in which a new node first picks a mediatorat random and connects with m randomly chosen neighbors of the mediator at each timestep. We show that the degree distribution in such a mediation-driven attachment (MDA)network exhibits power-law P(k) ∼ k−γ (m) with a spectrum of exponents dependingon m. To appreciate the contrast between MDA and Barabási–Albert (BA) networks, wethen discuss their rank-size distribution. To quantify how long a leader, the node withthe maximum degree, persists in its leadership as the network evolves, we investigatethe leadership persistence probability F (τ ) i.e. the probability that a leader retains itsleadership up to time τ . We find that it exhibits a power-law F (τ ) ∼ τ−θ (m) withpersistence exponent θ (m) ≈ 1.51 ∀ m in MDA networks and θ (m) → 1.53 exponentiallywith m in BA networks.