### Dynamic Graphs: Triangle Counting (TC)

Chapter 8: Triangle Counting in Dynamic Graphs

# 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):
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.