Basic Graph Analysis

Project Overview

The clustering coefficient is a measure used in network analysis to describe how tightly knit a node's neighborhood is. It essentially tells you the degree to which nodes in a graph tend to cluster together.

Visualization of a graph with 10 nodes and 14 edges, showing clustering patterns

Graph Algorithm Functions and Uses

From interconnected highways to social media networks—graphs can represent many real-world systems. The issue is that observing such systems is no simple matter. In this project, you will find simple methods to calculate:

• Degree distribution
• Number of edges
• Number of nodes
• A sophisticated heuristic for getting the diameter of the graph
• The O(d^2 * n) Clustering Coefficient algorithm which counts the number of triangles

Clustering Coefficient Algorithm: Counting Triangles

Naive Approach: O(n^3)

Loop through every node three times and count each triangle that exists.

Optimized Approach: O("degeneracy"^2 * n)

A graph is said to be 𝑘-degenerate if every subgraph of that graph has at least one vertex with degree ≤k.

for u in all nodes:     O(n)
    for v in nodes adjacent to u:   O(# nodes adjacent to u), O(max degree) O(n)
        for w in nodes adjacent to u: (or adjacent to v) O(# nodes adjacent to u)
            if v adjacent to w: (or u adjacent to w)
                add to count

Notes:

• To avoid triple counting, only check u, v, w where u < v < w.
• Degeneracy pop nodes in sorted order biggest to smallest (bucket).

Graph Metrics

The example graph shown above has the following metrics:

• Nodes: 10
• Edges: 14
• Diameter: 5
• Clustering Coefficient: 0.40
• Degree Distribution: {2: 5, 5: 1, 4: 1, 3: 3}

These metrics provide valuable insights into the structure and connectivity patterns of the graph, which can be used for various analytical purposes.

Applications

Whether you are tackling traffic congestion, managing complex relations, or fitting any problem that can be represented by a graph—these graph algorithms can be used for analytics and visualization. Some specific applications include:

• Social network analysis
• Transportation network optimization
• Biological network analysis (protein interactions, neural networks)
• Recommendation systems
• Web page ranking and link analysis