Closeness Centrality

Closeness centrality measures how close a node is to all other nodes on average.

Function Signature

pg.centrality.closeness(graph: Union[PyGraph, PyDiGraph]) -> Dict[int, float]

Parameters

• graph: The graph to analyze

Returns

Dictionary mapping node IDs to closeness centrality scores (0 to 1).

Description

Closeness is computed as:

C(u) = (n-1) / Σ d(u, v)

Where d(u, v) is the shortest distance between u and v, and n is the number of nodes.

Higher scores indicate nodes closer to all others.

Time Complexity

O(V·E) - shortest path computation from each node

Space Complexity

O(V)

Example

import pygraphina as pg

# Create a graph
g = pg.PyGraph()
nodes = [g.add_node(i) for i in range(5)]

# Create a path: 0-1-2-3-4
for i in range(4):
    g.add_edge(nodes[i], nodes[i+1], 1.0)

# Middle node has highest closeness
closeness = pg.centrality.closeness(g)
for node, score in sorted(closeness.items()):
    print(f"Node {node}: {score:.4f}")

Use Cases

• Finding central hubs in networks
• Optimal placement of facilities
• Communication efficiency analysis
• Identifying nodes with shortest average distance to others

Advantages

• Intuitive interpretation (inverse of average distance)
• Captures global structure
• Works for directed and undirected graphs

Disadvantages

• Undefined for disconnected components (if unreachable)
• Computationally expensive for large graphs
• Sensitive to network connectivity

Variants

• Harmonic Centrality: Handles disconnected graphs better
• Normalized Closeness: Scales by network size