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One of the main subjects in the field of social networks is to quantifyconflict, disagreement, controversy, and polarization, and some quantitativeindicators have been developed to quantify these concepts. However, directcomputation of these indicators involves the operations of matrix inversion andmultiplication, which make it computationally infeasible for large-scale graphswith millions of nodes. In this paper, by reducing the problem of computingrelevant quantities to evaluating $\ell_2$ norms of some vectors, we present anearly linear time algorithm to estimate all these quantities. Our algorithm isbased on the Laplacian solvers, and has a proved theoretical guarantee of errorfor each quantity. We execute extensive numerical experiments on a variety ofreal networks, which demonstrate that our approximation algorithm is efficientand effective, scalable to large graphs having millions of nodes.

Fast Evaluation for Relevant Quantities of Opinion Dynamics