Community Detection

The primary optimization objective used to evaluate the quality of a network partition. Modularity measures the density of edges inside communities compared to the density expected in a random null model. In legal citation graphs, high modularity indicates a partition where cases within a cluster cite each other far more frequently than they cite cases outside the cluster, often revealing distinct doctrinal silos or circuit splits. The Louvain algorithm and Leiden algorithm are the most common greedy optimization methods for maximizing this score on large-scale authority graphs.

A method that builds a dendrogram—a tree-like structure—representing nested community partitions at multiple levels of granularity. In citation networks, this reveals how broad legal domains like 'constitutional law' recursively subdivide into finer topics such as 'Fourth Amendment' and then 'vehicle searches'. Agglomerative approaches start with each case as its own cluster and iteratively merge the most similar, while divisive approaches recursively split the network. This multi-resolution view is critical for building navigable legal knowledge graphs.

A near-linear time community detection algorithm where each node iteratively adopts the majority label of its neighbors. In a citation graph, a case initially assigned a unique identifier will quickly converge to share the label of the precedential cluster that cites it most heavily. Its speed makes it suitable for dynamic, temporal citation analysis on massive corpuses, though it can produce unstable partitions. It is often used as a fast baseline before applying more computationally intensive modularity-based methods.

An information-theoretic approach that detects communities by modeling random walks on the network. The algorithm compresses the description of a random walker's path by using a two-level code: one level for community labels and another for node identifiers within each community. In a citation network, a random walker following citations will become trapped for extended periods within dense doctrinal clusters. Infomap finds the partition that minimizes the description length of these walks, often identifying structures that modularity-based methods miss.

A generative statistical framework that assumes the network was produced by a probabilistic process where the likelihood of a citation between two cases depends solely on their community memberships. Fitting an SBM to a citation graph involves inferring the latent block structure that most plausibly generated the observed citation patterns. Unlike modularity, SBMs can detect disassortative structures—where citations flow between rather than within groups—which can model hierarchical relationships between lower and appellate courts.

Standard algorithms assign each case to a single cluster, but legal doctrines are inherently overlapping. A Supreme Court decision on administrative law and statutory interpretation belongs to both communities simultaneously. Algorithms like the Clique Percolation Method or Mixed-Membership Stochastic Block Models allow nodes to possess fractional membership in multiple groups. This is essential for accurately modeling seminal cases that serve as bridges connecting distinct doctrinal silos in the authority graph.