Adding links wisely: how an influencer seeks for leadership in opinion dynamics?

This paper investigates the problem of leadership development for an external influencer using the Friedkin-Johnsen (FJ) opinion dynamics model, where the influencer is modeled as a fully stubborn agent and leadership is quantified by social power. The influencer seeks to maximize her social power by strategically adding a limited number of links to regular agents. This optimization problem is shown to be equivalent to maximizing the absorbing probability to the influencer in an augmented Markov chain. The resulting objective function is both monotone and submodular, enabling the use of a greedy algorithm to compute an approximate solution. To handle large-scale networks efficiently, a random walk sampling over the Markov chain is employed to reduce computational complexity. Analytical characterizations of the solution are provided for both low and high stubbornness of regular agents. Specific network topologies are also examined: for complete graphs with rank-one weight matrices, the problem reduces to a hyperbolic 0-1 programmming problem, which is solvable in polynomial time; for symmetric ring graphs with circulant weight matrices and uniform agent stubbornness, the optimal strategy involves selecting agents that are sufficiently dispersed across the network. Numerical simulations are presented for illustration.