Actuarial Applications of Graph Theory and Networks

Graph Theory Basics

A graph consists of vertices (nodes) and edges (connections). Key concepts include degree (number of edges per vertex), paths (sequences of connected vertices), and connected components (maximal connected subgraphs). Directed graphs have edges with orientation. Weighted graphs assign values to edges. The adjacency matrix A has entry A_{ij} = 1 if vertices i and j are connected. Trees are connected acyclic graphs. Graph metrics include diameter (longest shortest path), clustering coefficient (local connectivity), and centrality measures (degree, betweenness, eigenvector centrality).

Insurance and Risk Applications

Network analysis applies to insurance in several ways. Fraud detection uses network graphs to identify rings of connected suspicious claims or providers. Reinsurance networks model cedant-reinsurer relationships and assess systemic risk from interconnectedness. Social network effects influence insurance purchasing behavior and moral hazard. Contagion models study how defaults or catastrophic events propagate through networks of interconnected insurers. Agent and broker networks inform distribution strategy. Exam MAS-I introduces graph theory concepts and their potential applications in actuarial data analysis and risk assessment.