# `Yog.Centrality`
[🔗](https://github.com/code-shoily/yog_ex/blob/v0.97.0/lib/yog/centrality.ex#L1)

Centrality measures for identifying important nodes in graphs.

Provides degree, closeness, harmonic, betweenness, and PageRank centrality.
All functions return a map of Node IDs to their scores.

## Overview

| Measure | Function | Best For |
|---------|----------|----------|
| Degree | `degree/2` | Local connectivity |
| Closeness | `closeness/2` | Distance to all others |
| Harmonic | `harmonic/2` | Disconnected graphs |
| Betweenness | `betweenness/2` | Bridge/gatekeeper detection |
| PageRank | `pagerank/2` | Link-quality importance |
| HITS | `hits/2` | Hub and authority scores |
| Eigenvector | `eigenvector/2` | Influence based on neighbor importance |
| Katz | `katz/2` | Attenuated influence with base score |
| Alpha | `alpha/2` | Directed graph influence |

## Centrality Visualization (Betweenness)

Betweenness centrality identifies "bridge" nodes that act as gatekeepers between different parts of a network.

<div class="graphviz">
graph G {
  bgcolor="transparent";
  node [shape=circle, fontname="inherit"];
  edge [fontname="inherit", fontsize=10];

  // High betweenness node (Bridge/Broker)
  Broker [label="Broker", color="#6366f1", penwidth=2.5, style=bold];
  
  // Community A
  Broker -- A1; Broker -- A2; Broker -- A3;
  A1 -- A2; A2 -- A3; A3 -- A1;
  
  // Community B
  Broker -- B1; Broker -- B2; Broker -- B3;
  B1 -- B2; B2 -- B3; B3 -- B1;
}
</div>

    iex> alias Yog.Centrality
    iex> graph = Yog.from_edges(:undirected, [
    ...>   {"Broker", "A1", 1}, {"Broker", "A2", 1}, {"Broker", "A3", 1},
    ...>   {"A1", "A2", 1}, {"A2", "A3", 1}, {"A3", "A1", 1},
    ...>   {"Broker", "B1", 1}, {"Broker", "B2", 1}, {"Broker", "B3", 1},
    ...>   {"B1", "B2", 1}, {"B2", "B3", 1}, {"B3", "B1", 1}
    ...> ])
    iex> scores = Centrality.betweenness(graph)
    iex> # Broker should have the highest score
    ...> scores["Broker"] > scores["A1"]
    true

# `centrality_scores`

```elixir
@type centrality_scores() :: %{required(Yog.node_id()) =&gt; float()}
```

A mapping of Node IDs to their calculated centrality scores.

# `degree_mode`

```elixir
@type degree_mode() :: :in_degree | :out_degree | :total_degree
```

Specifies which edges to consider for directed graphs.
- `:in_degree` - Consider only incoming edges (Prestige)
- `:out_degree` - Consider only outgoing edges (Gregariousness)
- `:total_degree` - Consider both incoming and outgoing edges

# `alpha`

```elixir
@spec alpha(
  Yog.graph(),
  keyword()
) :: centrality_scores()
```

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.

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

## Interpreting Alpha Centrality

| Value | Meaning |
|-------|---------|
| **High** | The node has many paths from other central nodes |
| **Low** | The node is at the edge of the network with few incoming paths |
| `0.0` | Isolated node — no incoming paths to accumulate influence |

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

## Options

- `:alpha` - Attenuation factor (typically 0.1-0.5)
- `:initial` - Initial centrality value for all nodes
- `:max_iterations` - Maximum number of iterations (default: 100)
- `:tolerance` - Convergence threshold (default: 0.0001)

## Example

    iex> graph =
    ...>   Yog.directed()
    ...>   |> Yog.add_node(1, "A")
    ...>   |> Yog.add_node(2, "B")
    ...>   |> Yog.add_node(3, "C")
    ...>   |> Yog.add_edges!([{1, 2, 1}, {1, 3, 1}, {2, 3, 1}])
    iex> scores = Yog.Centrality.alpha(graph, alpha: 0.3, initial: 1.0)
    iex> map_size(scores)
    3

# `betweenness`

```elixir
@spec betweenness(
  Yog.graph(),
  keyword()
) :: centrality_scores()
```

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

| Value | Meaning |
|-------|---------|
| **High** | The node is a bridge or gatekeeper — many shortest paths go through it |
| **Low** | The node is peripheral — most paths bypass it |
| `0.0` | The node lies on no shortest paths between any other pair |

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

## Options

- `:zero` - The identity element for distances
- `:add` - Function to add two distances
- `:compare` - Function to compare two distances (returns `:lt`, `:eq`, or `:gt`)
- `:to_float` - Function to convert distance type to Float

## Example

    iex> graph =
    ...>   Yog.directed()
    ...>   |> Yog.add_node(1, "A")
    ...>   |> Yog.add_node(2, "B")
    ...>   |> Yog.add_node(3, "C")
    ...>   |> Yog.add_edges!([{1, 2, 1}, {2, 3, 1}])
    iex> scores = Yog.Centrality.betweenness(graph,
    ...>   zero: 0,
    ...>   add: &Kernel.+/2,
    ...>   compare: fn a, b ->
    ...>     cond do a < b -> :lt; a > b -> :gt; true -> :eq end
    ...>   end,
    ...>   to_float: fn x -> x * 1.0 end
    ...> )
    iex> # In a path 1->2->3, node 2 lies on the shortest path from 1 to 3
    iex> scores[2] |> Float.round(3)
    1.0

# `closeness`

```elixir
@spec closeness(
  Yog.graph(),
  keyword()
) :: centrality_scores()
```

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/2` for disconnected graphs.

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

## Interpreting Closeness Centrality

| Value | Meaning |
|-------|---------|
| `1.0` | The node is one hop away from all others (e.g. center of a star) |
| `0.5` | The node is typically 2 hops away from others |
| `0.0` | The node cannot reach everyone (disconnected or isolated) |

## Options

- `:zero` - The identity element for distances (e.g., 0 for integers)
- `:add` - Function to add two distances
- `:compare` - Function to compare two distances (returns `:lt`, `:eq`, or `:gt`)
- `:to_float` - Function to convert distance type to Float for final score

## Example

    iex> graph =
    ...>   Yog.undirected()
    ...>   |> Yog.add_node(1, "A")
    ...>   |> Yog.add_node(2, "B")
    ...>   |> Yog.add_node(3, "C")
    ...>   |> Yog.add_edges!([{1, 2, 1}, {2, 3, 1}, {3, 1, 1}])
    iex> scores = Yog.Centrality.closeness(graph,
    ...>   zero: 0,
    ...>   add: &Kernel.+/2,
    ...>   compare: fn a, b ->
    ...>     cond do a < b -> :lt; a > b -> :gt; true -> :eq end
    ...>   end,
    ...>   to_float: fn x -> x * 1.0 end
    ...> )
    iex> # In a triangle, all nodes have closeness 1.0
    iex> scores[1] |> Float.round(3)
    1.0

# `degree`

```elixir
@spec degree(Yog.graph(), degree_mode()) :: centrality_scores()
```

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

| Value | Meaning |
|-------|---------|
| `1.0` | The node is connected to every other node (hub) |
| `0.5` | The node is connected to half the other nodes |
| `0.0` | Isolated node — no connections |

## Example

    iex> graph =
    ...>   Yog.undirected()
    ...>   |> Yog.add_node(1, "A")
    ...>   |> Yog.add_node(2, "B")
    ...>   |> Yog.add_node(3, "C")
    ...>   |> Yog.add_edges!([{1, 2, 1}, {2, 3, 1}, {3, 1, 1}])
    iex> scores = Yog.Centrality.degree(graph)
    iex> # In a triangle, all nodes have degree 2, normalized is 2/2 = 1.0
    iex> scores[1] |> Float.round(3)
    1.0

    iex> # Directed graph with different modes
    iex> graph =
    ...>   Yog.directed()
    ...>   |> Yog.add_node(1, "A")
    ...>   |> Yog.add_node(2, "B")
    ...>   |> Yog.add_node(3, "C")
    ...>   |> Yog.add_edges!([{1, 2, 1}, {2, 3, 1}])
    iex> # Out-degree: Node 1 has 1 outgoing edge
    iex> scores = Yog.Centrality.degree(graph, :out_degree)
    iex> scores[1] |> Float.round(3)
    0.5

# `degree_total`

```elixir
@spec degree_total(Yog.graph()) :: centrality_scores()
```

Degree centrality with default options for undirected graphs.
Uses `:total_degree` mode.

Same as `degree(graph, :total_degree)`.

# `eigenvector`

```elixir
@spec eigenvector(
  Yog.graph(),
  keyword()
) :: centrality_scores()
```

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))

## Interpreting Eigenvector Centrality

| Value | Meaning |
|-------|---------|
| **High** | The node is connected to other highly central nodes |
| **Low** | The node is connected to peripheral or unimportant nodes |
| `0.0` | Isolated node with no connections |

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

## Options

- `:max_iterations` - Maximum number of power iterations (default: 100)
- `:tolerance` - Convergence threshold for L2 norm (default: 0.0001)

## Example

    iex> graph =
    ...>   Yog.undirected()
    ...>   |> Yog.add_node(1, "center")
    ...>   |> Yog.add_node(2, "A")
    ...>   |> Yog.add_node(3, "B")
    ...>   |> Yog.add_edges!([{1, 2, 1}, {1, 3, 1}])
    iex> scores = Yog.Centrality.eigenvector(graph)
    iex> scores[1] > scores[2]
    true

# `harmonic`

```elixir
@spec harmonic(
  Yog.graph(),
  keyword()
) :: centrality_scores()
```

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 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

| Value | Meaning |
|-------|---------|
| `1.0` | The node is directly connected to all others |
| `0.5` | The node is directly connected to half the others |
| `0.0` | Isolated node — cannot reach anyone else |

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

## Options

- `:zero` - The identity element for distances
- `:add` - Function to add two distances
- `:compare` - Function to compare two distances (returns `:lt`, `:eq`, or `:gt`)
- `:to_float` - Function to convert distance type to Float

## Example

    iex> graph =
    ...>   Yog.undirected()
    ...>   |> Yog.add_node(1, "Center")
    ...>   |> Yog.add_node(2, "A")
    ...>   |> Yog.add_node(3, "B")
    ...>   |> Yog.add_edges!([{1, 2, 1}, {1, 3, 1}])
    iex> scores = Yog.Centrality.harmonic(graph,
    ...>   zero: 0,
    ...>   add: &Kernel.+/2,
    ...>   compare: fn a, b ->
    ...>     cond do a < b -> :lt; a > b -> :gt; true -> :eq end
    ...>   end,
    ...>   to_float: fn x -> x * 1.0 end
    ...> )
    iex> # Center node: (1/1 + 1/1) / 2 = 1.0
    iex> scores[1] |> Float.round(3)
    1.0

# `hits`

```elixir
@spec hits(
  Yog.graph(),
  keyword()
) :: %{hubs: centrality_scores(), authorities: centrality_scores()}
```

Calculates HITS hub and authority scores for all nodes.

HITS identifies two types of important nodes in directed graphs:
- **Authorities** — nodes with good content (pointed to by good hubs)
- **Hubs** — nodes that point to good authorities

Unlike PageRank's single score, HITS provides dual scores using iterative
mutual reinforcement. Authority scores are the sum of hub scores of
predecessors; hub scores are the sum of authority scores of successors.

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

## Interpreting HITS

| Score | Meaning |
|-------|---------|
| **High authority** | Many high-quality incoming links |
| **High hub** | Points to many high-authority nodes |
| **Both low** | Peripheral node, neither cited nor curating |

## Options

- `:max_iterations` - Maximum power iterations (default: 100)
- `:tolerance` - Convergence threshold for L2 norm (default: 1.0e-6)

## Example

    iex> graph =
    ...>   Yog.directed()
    ...>   |> Yog.add_node(:a, nil)
    ...>   |> Yog.add_node(:b, nil)
    ...>   |> Yog.add_node(:c, nil)
    ...>   |> Yog.add_node(:d, nil)
    ...>   |> Yog.add_edges!([
    ...>     {:a, :b, 1}, {:a, :c, 1},
    ...>     {:b, :c, 1},
    ...>     {:d, :a, 1}, {:d, :b, 1}
    ...>   ])
    iex> %{hubs: hubs, authorities: auths} = Yog.Centrality.hits(graph)
    iex> # a is a strong hub (points to top authorities b and c)
    ...> hubs[:a] > hubs[:d]
    true
    iex> # b is the best authority (pointed to by multiple hubs)
    ...> auths[:b] > auths[:a]
    true

# `katz`

```elixir
@spec katz(
  Yog.graph(),
  keyword()
) :: centrality_scores()
```

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))

## Interpreting Katz Centrality

| Value | Meaning |
|-------|---------|
| **High** | The node has many short paths to other important nodes |
| **Low** | The node is distant from the network core |
| `≈ beta` | Isolated node — only receives the baseline score |

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

## Options

- `: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 (default: 100)
- `:tolerance` - Convergence threshold (default: 0.0001)

## Example

    iex> graph =
    ...>   Yog.undirected()
    ...>   |> Yog.add_node(1, "A")
    ...>   |> Yog.add_node(2, "B")
    ...>   |> Yog.add_node(3, "C")
    ...>   |> Yog.add_edges!([{1, 2, 1}, {2, 3, 1}, {3, 1, 1}])
    iex> scores = Yog.Centrality.katz(graph, alpha: 0.1, beta: 1.0)
    iex> # All scores should be >= beta
    iex> scores[1] >= 1.0
    true

# `pagerank`

```elixir
@spec pagerank(
  Yog.graph(),
  keyword()
) :: centrality_scores()
```

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.

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

## Interpreting PageRank

| Value | Meaning |
|-------|---------|
| **High** | The node is linked to by many other important nodes |
| **Low** | The node has few or low-quality incoming links |
| `1.0` | Single-node graph (trivial case) |

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.

## 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/2` instead.

## Options

- `:damping` - Probability of continuing to follow links (default: 0.85)
- `:max_iterations` - Maximum iterations before returning (default: 100)
- `:tolerance` - Convergence threshold (default: 0.0001)

## Example

    iex> graph =
    ...>   Yog.directed()
    ...>   |> Yog.add_node(1, "A")
    ...>   |> Yog.add_node(2, "B")
    ...>   |> Yog.add_node(3, "C")
    ...>   |> Yog.add_edges!([{1, 2, 1}, {2, 3, 1}])
    iex> scores = Yog.Centrality.pagerank(graph)
    iex> # Scores should sum to approximately 1.0
    iex> Enum.sum(Map.values(scores)) |> Float.round(2)
    1.0
    iex> # With custom options
    iex> scores = Yog.Centrality.pagerank(graph, damping: 0.9, max_iterations: 50, tolerance: 0.001)
    iex> map_size(scores)
    3

---

*Consult [api-reference.md](api-reference.md) for complete listing*