Graph Aggregation

Graph aggregation is the process of computing a single output graph that constitutes a good compromise between several input graphs, each provided by a different source. One needs to perform graph aggregation in a wide variety of situations, e.g., when applying a voting rule (graphs as preference orders), when consolidating conflicting views regarding the relationships between arguments in a debate (graphs as abstract argumentation frameworks), or when computing a consensus between several alternative clusterings of a given dataset (graphs as equivalence relations). In this paper, we introduce a formal framework for graph aggregation grounded in social choice theory. Our focus is on understanding which properties shared by the individual input graphs will transfer to the output graph returned by a given aggregation rule. We consider both common properties of graphs, such as transitivity and reflexivity, and arbitrary properties expressible in certain fragments of modal logic. Our results establish several connections between the types of properties preserved under aggregation and the choice-theoretic axioms satisfied by the rules used. The most important of these results is a powerful impossibility theorem that generalises Arrow’s seminal result for the aggregation of preference orders to a large collection of different types of graphs. This work refines and extends papers presented at COMSOC-2012 [Endriss and Grandi, 2012] and ECAI-2014 [Endriss and Grandi, 2014b]. We are grateful for the extensive feedback received from several anonymous reviewers as well as from audiences at the SSEAC Workshop on Social Choice and Social Software held in Kiel in 2012, the Dagstuhl Seminar on Computation and Incentives in Social Choice in 2012, the KNAW Academy Colloquium on Dependence Logic held at the Royal Netherlands Academy of Arts and Sciences in Amsterdam in 2014, a course on logical frameworks for multiagent aggregation given at the 26th European Summer School in Logic, Language and Information (ESSLLI-2014) in Tübingen in 2014, the Lorentz Center Workshop on Clusters, Games and Axioms held in Leiden in 2015, and lectures delivered at Sun Yat-Sen University in Guangzhou in 2014 as well as at École Normale Supérieure and Pierre & Marie Curie University in Paris in 2016. This work was partly supported by COST Action IC1205 on Computational Social Choice. It was completed while the first author was hosted at the University of Toulouse in 2015 as well as Paris-Dauphine University, Pierre & Marie Curie University, and the London School of Economics in 2016.

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