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 degree distribution in such a mediation-driven attachment (MDA) network exhibits power-law $P(k)\sim k^{-\gamma(m)}$withaspectrumofexponentsdependingon$m$. 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 $\theta (m)\approx 1.51\mathrm{\forall }m$ in the MDA networks and $\theta (m)\to 1.53$ exponentially with $m$ in the BA networks.