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Influence estimation in sparse social networks

In this paper we are interested in the estimation of the social influence among n agents that interact in a sparse social network. In particular, we consider the classical Friedkin and Johnsen's model, where agents discuss m ≪ n independent topics, take into account the other opinions but are not completely open-minded, and persistently are influenced by their initial prejudices. By observing the initial and final opinions' profile, we propose a method based on the ℓ0/ℓ1 minimization to infer the topology of the social graph and the strength of the interconnections. Compared to the methods previously introduced in literature, our work does not assume partial knowledge on the social graph and does not consider an optimized placement of stubborn agents injecting inputs that change the terminal behavior of the opinion dynamics. Moreover, the proposed method is suitable for parallel implementation and the influence identification of each agent can be performed independently from the others. Under suitable assumptions on the distribution of the initial prejudices, we derive theoretical conditions that guarantee that the problem is well posed and sufficient requirements on the number of topics under discussion that ensure perfect recovery. Extensive simulations corroborate theoretical results and our findings.