Shortest Path

Common graph queries:

Unweighted shortest path: BFS.

O(V + E).

Weighted shortest path: Dijkstra.

O((V + E) log V).

A*: Dijkstra with heuristic. Faster when goal is known.

Bidirectional: search from both ends; meet in the middle. ~2x speedup.

All-pairs shortest path (Floyd-Warshall): O(V³). Used for small graphs.

In Cypher:

MATCH p = shortestPath((a:Person {name: 'Alice'})-[*]-(b:Person {name: 'Bob'}))
RETURN p

PageRank (recursive importance):

PR(v) = (1-d)/N + d * sum(PR(u) / out_degree(u) for u in incoming(v))

Solved by iteration until convergence. d = 0.85 typical.

Community detection: Louvain, Label Propagation. Find clusters of densely-connected nodes.

Centrality: betweenness (how often node lies on shortest paths), closeness (avg shortest path length to all others).

These are the staple algorithms. Graph DBs typically include them as built-in functions or via Graph Data Science (GDS) plugins.