Unlike many complex networks studied in the literature, social networksrarely exhibit unanimous behavior, or consensus. This requires a development ofmathematical models that are sufficiently simple to be examined and capture, atthe same time, the complex behavior of real social groups, where opinions andactions related to them may form clusters of different size. One such model,proposed by Friedkin and Johnsen, extends the idea of conventional consensusalgorithm (also referred to as the iterative opinion pooling) to take intoaccount the actors' prejudices, caused by some exogenous factors and leading todisagreement in the final opinions. In this paper, we offer a novel multidimensional extension, describing theevolution of the agents' opinions on several topics. Unlike the existingmodels, these topics are interdependent, and hence the opinions being formed onthese topics are also mutually dependent. We rigorous examine stabilityproperties of the proposed model, in particular, convergence of the agents'opinions. Although our model assumes synchronous communication among theagents, we show that the same final opinions may be reached "on average" viaasynchronous gossip-based protocols.
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