Fixed-Points of Social Choice: An Axiomatic Approach to Network Communities

We provide the first social choice theory approach to the question of what constitutes a community in a social network. Inspired by social choice theory in voting and other contexts [2], we start from an abstract social network framework, called preference networks [3]; these consist of a finite set of members and a vector giving a total ranking of the members in the set for each of them (representing the preferences of that member). Within this framework, we axiomatically study the formation and structures of communities. Our study naturally involves two complementary approaches. In the first, we apply social choice theory and define communities indirectly by postulating that they are fixed points of a preference aggregation function obeying certain desirable axioms. In the second, we directly postulate desirable axioms for communities without reference to preference aggregation, leading to a natural set of eight community axioms. These two approaches allow us to formulate and analyze community rules. We prove a taxonomy theorem that provides a structural characterization of the family of those community rules that satisfies all eight axioms. The structure is actually quite beautiful: the family satisfying all eight axioms forms a bounded lattice under the natural intersection and union operations of community rules. The taxonomy theorem also gives an explicit characterization of the most comprehensive community rule and the most selective community rule consistent with all community axioms. This structural theorem is complemented with a complexity result: we show that while identifying a community by the selective rule is straightforward, deciding if a subset satisfies the comprehensive rule is coNP-complete. Our studies also shed light on the limitations of defining community rules solely based on preference aggregation. In particular, we show that many aggregation functions lead to communities which violate at least one of our community axioms. These include any aggregation function satisfying Arrow’s independence of irrelevant alternative axiom as well as commonly used aggregation schemes like the Borda count or generalizations thereof. Finally, we give a polynomial-time rule consistent with seven axioms and weakly satisfying the eighth axiom. Supported in part by NSF grants CCF-1111270 and CCF-0964481 and by a Simons Investigator Award from the Simons Foundation.