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 $P(k)\sim k^{-γ(m)}$ 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 $F(τ)\sim τ^{-θ(m)}$ with persistence exponent $θ(m) \approx 1.51 \ \forall \ m$ in the MDA networks and $θ(m) \rightarrow 1.53$ exponentially with $m$ in the BA networks.