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Opinion formation cannot be modeled solely as an ideological deduction from a set of principles; rather, repeated social interactions and logic constraints among statements are consequential in the construct of belief systems. We address three basic questions in the analysis of social opinion dynamics: (i) Will a belief system converge? (ii) How long does it take to converge? (iii) Where does it converge? We provide graph-theoretic answers to these questions for a model of opinion dynamics of a belief system with logic constraints. Our results make plain the implicit dependence of the convergence properties of a belief system on the underlying social network and on the set of logic constraints that relate beliefs on different statements. Moreover, we provide an explicit analysis of a variety of commonly used large-scale network models.

The Quality of Service Multicast Tree Problem is a generalization of the Steiner tree problem which appears in the context of multimedia multicast and network design. In this generalization, each node possesses a rate and the cost of an edge with length l in a Steiner tree T connecting the source to non-zero rate nodes is l · re, where re is the maximum node rate in the component of T-{e} that does not contain the source. The best previously known approximation ratios for this problem (based on the best known approximation factor of 1.549 for the Steiner tree problem in networks) are 2.066 for the case of two non-zero rates and 4.212 for the case of an unbounded number of rates. In this paper we give improved approximation algorithms with ratios of 1.960 and 3.802, respectively. When the minimum spanning tree heuristic is used for finding approximate Steiner trees, then the previously best known approximation ratios of 2.667 for two non-zero rates and 5.542 for an unbounded number of rates are reduced to 2.414 and 4.311, respectively.