Graph databases have been the subject of significant research and development in the database, data analytics, and applications' communities. Problems such as modularity, centrality, alignment, and clustering have been formalized and solved in various application contexts. In this paper, we focus on databases for applications in which graphs have a spatial basis, which we refer to as rigid graphs. Nodes in such graphs have preferred positions relative to their graph neighbors. Examples of such graphs include abstractions of large biomolecules (e.g., in drug databases), where edges corresponding to chemical bonds have preferred lengths, functional connectomes of the human brain (e.g., the HCP database [13]), where edges corresponding to co-firing regions of the brain have preferred anatomical distances, and mobile device/ sensor communication logs, where edges corresponding to point-to-point communications across devices have distance constraints. When analyzing such networks it is important to consider edge lengths; e.g., when identifying conserved patterns through graph alignment, it is important for conserved edges to have correlated lengths, in addition to topological similarity. Similar considerations exist for clustering (densely connected regions of short edges) and centrality (critical edges with large weights).
Problem Formulation
Problem definitionWe define the rigid graph alignment problem by first reviewing existing graph and structure alignment formulations, and use these to motivate our new problem.