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) ∼ 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 the MDA networks and θ(m) → 1.53 exponentially with m in the BA networks.

Recently, we have shown that if the ith node of the Barabási-Albert (BA) network is characterized by the generalized degree qi(t) = ki(t)t β i /m, where ki(t) ∼ t β and m are its degree at current time t and at birth time ti, 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 β of the MDA model assumes a spectrum of value 1/2 ≤ β ≤ 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.

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