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We propose a setting for two-phase opinion dynamics in social networks, wherea node's final opinion in the first phase acts as its initial biased opinion inthe second phase. In this setting, we study the problem of two camps aiming tomaximize adoption of their respective opinions, by strategically investing onnodes in the two phases. A node's initial opinion in the second phase naturallyplays a key role in determining the final opinion of that node, and hence alsoof other nodes in the network due to its influence on them. More importantly,this bias also determines the effectiveness of a camp's investment on that nodein the second phase. To formalize this two-phase investment setting, we proposean extension of Friedkin-Johnsen model, and hence formulate the utilityfunctions of the camps. There is a tradeoff while splitting the budget betweenthe two phases. A lower investment in the first phase results in worse initialbiases for the second phase, while a higher investment spares a lower availablebudget for the second phase. We first analyze the non-competitive case whereonly one camp invests, for which we present a polynomial time algorithm fordetermining an optimal way to split the camp's budget between the two phases.We then analyze the case of competing camps, where we show the existence ofNash equilibrium and that it can be computed in polynomial time underreasonable assumptions. We conclude our study with simulations on real-worldnetwork datasets, in order to quantify the effects of the initial biases andthe weightage attributed by nodes to their initial biases, as well as that of acamp deviating from its equilibrium strategy. Our main conclusion is that, ifnodes attribute high weightage to their initial biases, it is advantageous tohave a high investment in the first phase, so as to effectively influence thebiases to be harnessed in the second phase.

A Two Phase Investment Game for Competitive Opinion Dynamics in Social Networks