yog/centrality

Centrality measures for identifying important nodes in graphs.

Provides degree, closeness, harmonic, betweenness, and PageRank centrality. All functions return a Dict(NodeId, Float) mapping nodes to their scores.

Overview

Measure	Function	Best For
Degree	`degree/2`	Local connectivity
Closeness	`closeness/5`	Distance to all others
Harmonic	`harmonic_centrality/5`	Disconnected graphs
Betweenness	`betweenness/5`	Bridge/gatekeeper detection
PageRank	`pagerank/2`	Link-quality importance

Types

Centrality

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A mapping of Node IDs to their calculated centrality scores.

pub type Centrality =
  dict.Dict(Int, Float)

DegreeMode

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Specifies which edges to consider for directed graphs.

pub type DegreeMode {
  InDegree
  OutDegree
  TotalDegree
}

Constructors

```
InDegree
```
Consider only incoming edges (Prestige).
```
OutDegree
```
Consider only outgoing edges (Gregariousness).
```
TotalDegree
```
Consider both incoming and outgoing edges.

PageRankOptions

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Configuration options for the PageRank algorithm.

PageRank models a “random surfer” who follows links with probability damping and jumps to a random page with probability 1 - damping.

Fields

damping: Probability of continuing to follow links (typically 0.85). Higher values mean the surfer follows more links before random jumping.
max_iterations: Maximum iterations before returning current scores.
tolerance: Convergence threshold. Algorithm stops when the L1 norm of score changes falls below this value.

Default Options

Use default_pagerank_options() for standard settings:

damping: 0.85
max_iterations: 100
tolerance: 0.0001

pub type PageRankOptions {
  PageRankOptions(
    damping: Float,
    max_iterations: Int,
    tolerance: Float,
  )
}

Constructors

PageRankOptions(
  damping: Float,
  max_iterations: Int,
  tolerance: Float,
)

Values

alpha_centrality

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pub fn alpha_centrality(
  graph: model.Graph(n, e),
  alpha: Float,
  initial: Float,
  max_iterations: Int,
  tolerance: Float,
) -> dict.Dict(Int, Float)

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(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)])

betweenness

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pub fn betweenness(
  graph: model.Graph(n, e),
  with_zero zero: e,
  with_add add: fn(e, e) -> e,
  with_compare compare: fn(e, e) -> order.Order,
  with_to_float to_float: fn(e) -> Float,
) -> dict.Dict(Int, Float)

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.

betweenness_float

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pub fn betweenness_float(
  graph: model.Graph(n, Float),
) -> dict.Dict(Int, Float)

Betweenness centrality with Float weights.

betweenness_int

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pub fn betweenness_int(
  graph: model.Graph(n, Int),
) -> dict.Dict(Int, Float)

Betweenness centrality with Int weights.

closeness

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pub fn closeness(
  graph: model.Graph(n, e),
  with_zero zero: e,
  with_add add: fn(e, e) -> e,
  with_compare compare: fn(e, e) -> order.Order,
  with_to_float to_float: fn(e) -> Float,
) -> dict.Dict(Int, Float)

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)])

closeness_float

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pub fn closeness_float(
  graph: model.Graph(n, Float),
) -> dict.Dict(Int, Float)

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

closeness_int

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pub fn closeness_int(
  graph: model.Graph(n, Int),
) -> dict.Dict(Int, Float)

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

default_pagerank_options

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pub fn default_pagerank_options() -> PageRankOptions

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

degree

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pub fn degree(
  graph: model.Graph(n, e),
  mode: DegreeMode,
) -> dict.Dict(Int, Float)

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.

degree_total

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pub fn degree_total(
  graph: model.Graph(n, e),
) -> dict.Dict(Int, Float)

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

eigenvector

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pub fn eigenvector(
  graph: model.Graph(n, e),
  max_iterations: Int,
  tolerance: Float,
) -> dict.Dict(Int, Float)

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)])

harmonic_centrality

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pub fn harmonic_centrality(
  graph: model.Graph(n, e),
  with_zero zero: e,
  with_add add: fn(e, e) -> e,
  with_compare compare: fn(e, e) -> order.Order,
  with_to_float to_float: fn(e) -> Float,
) -> dict.Dict(Int, Float)

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)

harmonic_centrality_float

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pub fn harmonic_centrality_float(
  graph: model.Graph(n, Float),
) -> dict.Dict(Int, Float)

Harmonic centrality with Float weights.

harmonic_centrality_int

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pub fn harmonic_centrality_int(
  graph: model.Graph(n, Int),
) -> dict.Dict(Int, Float)

Harmonic centrality with Int weights.

katz

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pub fn katz(
  graph: model.Graph(n, e),
  alpha: Float,
  beta: Float,
  max_iterations: Int,
  tolerance: Float,
) -> dict.Dict(Int, Float)

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)])

pagerank

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pub fn pagerank(
  graph: model.Graph(n, e),
  options: PageRankOptions,
) -> dict.Dict(Int, Float)

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)