- The paper establishes a formal framework for aggregating graphs from multiple sources, analyzing how different aggregation rules affect the preservation of graph properties.
- It investigates axiomatic properties of aggregation rules and provides characterisation theorems for specific rules, showing how desirable conditions like anonymity and monotonicity interact.
- Crucially, it presents impossibility theorems, akin to Arrow's theorem, revealing inherent limitations in preserving certain complex graph properties under common axiomatic conditions.
An Overview of Graph Aggregation in Computational Social Choice Theory
The paper "Graph Aggregation" by Ulle Endriss and Umberto Grandi explores a formal framework for aggregating graphs, drawing inspiration from social choice theory. Graph aggregation refers to constructing a single consensus graph from several input graphs presented by different agents, motivated by scenarios such as voting, debate argument consolidation, and consensus clustering. The paper addresses graph aggregation within a comprehensive framework that examines the preservation of graph properties under aggregation rules, establishing connections to choice-theoretic axioms.
Framework and Definitions
The authors present various graph properties, such as reflexivity, transitivity, symmetry, and completeness, examining how these properties can be preserved under graph aggregation. They clarify that graphs are prevalent in various computer science applications, including decision support systems, social network analysis, and abstract argumentation frameworks. An aggregation rule maps individual graphs into a collective graph, aiming to find a compromise that satisfies desirable properties. The paper illustrates these concepts using examples from preference aggregation, where graphs model preference orders, and other domains like nonmonotonic reasoning and clustering.
Axiomatic Insights and Characterisations
The framework investigates axiomatic properties like independence of irrelevant edges (IIE), independence of irrelevant sources (IIS), neutrality, and the basic economic principle of unanimity. It introduces characterisation theorems for different aggregation rules, notably quota rules and successor-approval rules. These results contribute to understanding how intuitive desiderata like anonymity and monotonicity interact in the context of graph aggregation and highlight the complexity of engineering aggregation rules that meet multiple axiomatic conditions simultaneously.
Impossibility Theorems
Central to the paper are impossibility theorems echoing Arrow's theorem in social choice theory, demonstrating the limitations of aggregating graphs while preserving complex properties like transitivity and completeness. These theorems indicate that aggregation rules, under certain axioms and collective rationality conditions, lead to undesirable outcomes such as dictatorship or oligarchy. This generalisation of impossibility theorems emphasizes the restrictive nature of choice-theoretic conditions and exposes the complexity underlying graph aggregation in a general context.
Theoretical Contributions and Future Implications
The paper underscores the practical implications of these theoretical results by discussing applications in AI - such as preference aggregation with incomplete information, belief merging in nonmonotonic reasoning frameworks, consensus clustering analysis, and argumentation within multiagent systems. These applications reflect the diverse potential uses of graph aggregation methodology in computational settings. The authors highlight directions for future research, proposing investigations into new domains like Semantic Web data integration and theory change in scientific methodologies.
Conclusion
Endriss and Grandi provide a foundational exploration of graph aggregation, grounded in computational social choice theory. By dissecting the interplay between aggregation rules and graph properties, they offer significant insights into the inherent limitations and potential of constructing consensus graphs from distributed information. Their work prompts further exploration into graph aggregation, opening avenues for a broader understanding of aggregative mechanisms in complex data interactions, pivotal in advancing applications in artificial intelligence and beyond.
