  In this paper we study the impact of active participation -- or deliberately
seeking out other agents with an aim to convince them -- on the dynamics of
consensus formation. For this purpose, we propose an adaptive network model in
which two processes shape opinion dynamics at interwoven time-scales: (i)
agents adapt their opinions subject to influence from social network neighbours
who hold opinions within a tolerance interval $\delta$ and (ii) agents rewire
network connections with an aim to maximize their own influence on overall
system opinion. We study this system in both an endogenous setting in which all
agents are subject to influence and also attempt to maximize influence, and in
a setting of exogenous control, in which external agents not subject to
influence adaptively attempt to maximize their influence. In both settings we
find three regimes of stationary opinion configurations: (i) for low $\delta$ a
regime of two evenly balanced radicalized opinion clusters at the extremes of
the opinion space, (ii) for intermediate $\delta$ a 'winner-takes-most' regime
of two unevenly sized radicalized opinion clusters, and (iii) for large
$\delta$ a regime in which very low spread compromise consensus states can be
reached. Comparing to adaptive processes of random and deliberately
spread-reducing rewiring, we demonstrate that in regime (iii) competitive
influence maximization can achieve near-minimal opinion spread within
near-optimal times. Further, we also show that competitive influence maximizing
rewiring can reduce the impact of small influential minorities on consensus
states.
