Reference

Structural ranking

Behavior

Algorithm	Signal	Appropriate comparison
Degree centrality	Immediate connectivity	Local prominence
Closeness centrality	Mean shortest-path distance	Reachability from one node
Betweenness centrality	Fraction of shortest paths crossing a node	Bridges and bottlenecks
HITS	Mutually reinforcing hubs and authorities	Directed link structure
Personalized PageRank	Stationary visit probability from seeds	Query-biased graph retrieval

How it works

Centrality functions receive node IDs and neighbor callbacks. Personalized PageRank accepts a weighted adjacency mapping through the retrieval API. They return scores, with labels left to callers. Betweenness uses the unweighted Brandes algorithm. HITS and PageRank expose iteration and convergence settings.

Restrict both the node collection and neighbor callbacks to the same authorized graph. Degree and closeness can follow neighbors outside the supplied node collection. Closeness follows the callback direction, so incoming and outgoing adjacency answer different questions. For large graphs, repeated all-source traversals make closeness and betweenness more expensive than degree ranking.

Compare caller-selected structural signals

from mari_components.graph import (
    betweenness_centrality,
    degree_centrality,
    hits,
)

degree = degree_centrality(node_ids, neighbors=undirected_neighbors)
bridges = betweenness_centrality(node_ids, neighbors=outgoing_neighbors)
hub_authority = hits(node_ids, successors=outgoing_neighbors)

degree_by_node = dict(degree)
bridges_by_node = dict(bridges)
features = {
    node: {
        "degree": degree_by_node.get(node, 0.0),
        "betweenness": bridges_by_node.get(node, 0.0),
    }
    for node in node_ids
}

Measures

Property	Check
Numerical conformance	Compare small fixtures with NetworkX
Directionality	Run asymmetric graphs with explicit successor callbacks
Disconnected graphs	Verify closeness normalization and unreachable nodes
Convergence	Record iterations, tolerance, and residual
Retrieval value	Compare evidence recall across baseline and structural-score runs

Papers and implementations

Brandes betweennessHITSPageRankNetworkX

NetworkX is the BSD-3-Clause differential oracle. Mari accepts graph access through callbacks. The caller sets any global ranking policy.