Degree distribution, rank-size distribution, and leadership persistence in mediation-driven attachment networks

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 the degree distribution in such a mediation-driven attachment (MDA) network exhibits power-law P(k)∼k−γ(m) with a spectrum of exponents depending on m. To appreciate the contrast between MDA and Barabási-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(τ) i.e. the probability that a leader retains its leadership up to time τ. We find that it exhibits a power-law F(τ)∼τ−θ(m) with persistence exponent θ(m)≈1.51∀m in MDA networks and θ(m)→1.53 exponentially with m in BA networks.