Calculates Alpha Centrality for all nodes.

Alpha centrality is a generalization of Katz centrality for directed graphs. It measures the total number of paths from a node, weighted by path length with attenuation factor alpha.

Unlike Katz, alpha centrality does not include a constant beta term and is particularly useful for analyzing influence in directed networks.

Formula: C(v) = α * Σ C(u) for all predecessors u (or neighbors for undirected)

Time Complexity: O(max_iterations * (V + E))

Parameters

• alpha: Attenuation factor (typically 0.1-0.5)
• initial: Initial centrality value for all nodes
• max_iterations: Maximum number of iterations
• tolerance: Convergence threshold

Example

centrality.alpha_centrality(graph, alpha: 0.3, initial: 1.0, max_iterations: 100, tolerance: 0.0001)
// => dict.from_list([#(1, 2.0), #(2, 3.0), #(3, 2.0)])

Interpreting Alpha Centrality

Unlike Katz, alpha centrality has no baseline beta term, so isolated nodes converge to 0.0 rather than retaining a minimum score.

Calculates Betweenness Centrality for all nodes.

Betweenness centrality of a node v is the sum of the fraction of all-pairs shortest paths that pass through v.

Time Complexity: O(VE) for unweighted, O(VE + V²logV) for weighted.

Interpreting Betweenness Centrality

A high betweenness node is critical for network connectivity: removing it can fragment the graph or severely increase path lengths.

Betweenness centrality with Float weights.

Betweenness centrality with Int weights.

Calculates Closeness Centrality for all nodes.

Closeness centrality measures how close a node is to all other nodes in the graph. It is calculated as the reciprocal of the sum of the shortest path distances from the node to all other nodes.

Formula: C(v) = (n - 1) / Σ d(v, u) for all u ≠ v

Note: In disconnected graphs, nodes that cannot reach all other nodes will have a centrality of 0.0. Consider harmonic_centrality for disconnected graphs.

Time Complexity: O(V * (V + E) log V) using Dijkstra from each node

Parameters

• zero: The identity element for distances (e.g., 0 for integers)
• add: Function to add two distances
• compare: Function to compare two distances
• to_float: Function to convert distance type to Float for final score

Example

centrality.closeness(
  graph,
  with_zero: 0,
  with_add: int.add,
  with_compare: int.compare,
  with_to_float: int.to_float,
)
// => dict.from_list([#(1, 0.666), #(2, 1.0), #(3, 0.666)])

Interpreting Closeness Centrality

Closeness centrality with Float weights. Uses 0.0 as zero, float.add, float.compare, and identity.

Closeness centrality with Int weights (e.g., unweighted graphs). Uses 0 as zero, int.add, int.compare, and int.to_float.

Default PageRank options (damping=0.85, max_iterations=100, tolerance=0.0001).

Calculates the Degree Centrality for all nodes in the graph.

For directed graphs, use mode to specify which edges to count. For undirected graphs, the mode is ignored.

Interpreting Degree Centrality

Degree centrality with default options for undirected graphs. Uses TotalDegree mode.

Calculates Eigenvector Centrality for all nodes.

Eigenvector centrality measures a node’s influence based on the centrality of its neighbors. A node is important if it is connected to other important nodes. Uses power iteration to converge on the principal eigenvector.

Time Complexity: O(max_iterations * (V + E))

Parameters

• max_iterations: Maximum number of power iterations
• tolerance: Convergence threshold for L2 norm

Example

centrality.eigenvector(graph, max_iterations: 100, tolerance: 0.0001)
// => dict.from_list([#(1, 0.707), #(2, 1.0), #(3, 0.707)])

Interpreting Eigenvector Centrality

Eigenvector scores are normalized (L2 norm = 1.0), so they represent relative importance rather than absolute counts.

Calculates Harmonic Centrality for all nodes.

Harmonic centrality is a variation of closeness centrality that handles disconnected graphs gracefully. It sums the reciprocals of the shortest path distances from a node to all other reachable nodes.

Formula: H(v) = Σ (1 / d(v, u)) / (n - 1) for all u ≠ v

Time Complexity: O(V * (V + E) log V)

Interpreting Harmonic Centrality

Unlike closeness, disconnected nodes still receive credit for the neighbors they can reach rather than being penalized with 0.0.

Harmonic centrality with Float weights.

Harmonic centrality with Int weights.

Calculates Katz Centrality for all nodes.

Katz centrality is a variant of eigenvector centrality that adds an attenuation factor (alpha) to prevent the infinite accumulation of centrality in cycles. It also includes a constant term (beta) to give every node some base centrality.

Formula: C(v) = α * Σ C(u) + β for all neighbors u

Time Complexity: O(max_iterations * (V + E))

Parameters

• alpha: Attenuation factor (must be < 1/largest_eigenvalue, typically 0.1-0.3)
• beta: Base centrality (typically 1.0)
• max_iterations: Maximum number of iterations
• tolerance: Convergence threshold

Example

centrality.katz(graph, alpha: 0.1, beta: 1.0, max_iterations: 100, tolerance: 0.0001)
// => dict.from_list([#(1, 2.5), #(2, 3.0), #(3, 2.5)])

Interpreting Katz Centrality

Because of the constant beta term, even isolated nodes receive a non-zero score, making Katz more forgiving than eigenvector centrality.

Calculates PageRank centrality for all nodes.

PageRank measures node importance based on the quality and quantity of incoming links. A node is important if it is linked to by other important nodes. Originally developed for ranking web pages, it’s useful for:

• Ranking nodes in directed networks
• Identifying influential nodes in citation networks
• Finding important entities in knowledge graphs
• Recommendation systems

The algorithm uses a “random surfer” model: with probability damping, the surfer follows a random outgoing link; otherwise, they jump to any random node. This models both link-following behavior and the possibility of starting a new browsing session.

Time Complexity: O(max_iterations × (V + E))

When to Use PageRank

• Directed graphs where link direction matters
• When you care about link quality (links from important nodes count more)
• Citation networks, web graphs, recommendation systems

For undirected graphs, consider eigenvector/3 instead.

Example

// Use default options (recommended for most cases)
let options = centrality.default_pagerank_options()
let scores = centrality.pagerank(graph, options)
// => dict.from_list([#(1, 0.256), #(2, 0.488), #(3, 0.256)])

// Custom options for faster convergence or different damping
let custom = centrality.PageRankOptions(
  damping: 0.9,        // Follow more links before jumping
  max_iterations: 50,  // Faster but less precise
  tolerance: 0.001,    // Less strict convergence
)
let scores = centrality.pagerank(graph, custom)

Interpreting PageRank

PageRank scores always sum to 1.0 across all nodes. A node with rank 0.5 in a 2-node graph means it captures half the total importance in the network.