Dynamic Graphs: Triangle Counting (TC)

Chapter 8: Triangle Counting (TC) in Dynamic Graphs

Introduction

Triangle Counting is a common graph algorithm used to identify the number of triangles in a graph. Triangles, or cycles of length 3, are considered a sign of a well-connected, or "cliquish," network. Maintaining an accurate triangle count in dynamic graphs, where edges and nodes can be added or removed, poses unique challenges. This chapter will discuss techniques to address these challenges.

Standard Triangle Counting Algorithms

Several methods exist for counting triangles in a static graph, including node-iterator and edge-iterator algorithms. Below is a simplified outline of a node-iterator algorithm.

  NodeIteratorTriangleCount(Graph G):
    1. Initialize count = 0
    2. For each node u in G:
        3. For each pair of neighbors (v, w) of u:
            4. If (v, w) is an edge in G:
                5. Increment count
    6. Return count / 3

Dynamic Graphs and Triangle Counting

The primary challenges in dynamic graphs are:

• When a new edge is added, new triangles may form.
• When an edge is removed, existing triangles may be broken.

Triangle Counting in Dynamic Graphs

Strategies for maintaining triangle counts in dynamic graphs include:

1. Recomputing from Scratch: This is computationally expensive.
2. Incremental Updates: Update the triangle count incrementally based on the changes in the graph.

Incremental Triangle Counting Algorithm

  IncrementalTriangleCount(Graph G_t, Triangle_Count):
    1. Detect changes (added/removed edges)
    2. For each change:
        3. Identify the affected triangles
        4. Update Triangle_Count accordingly

Conclusion

Triangle Counting is a key metric for understanding the cohesiveness of a network. Incremental algorithms allow for efficient updates in dynamic graphs, providing an up-to-date understanding of the network structure as it evolves.