Motivation

In this lecture, we will

• Describe the use and purpose of Steiner trees .
• Learn how to construct an approximate Steiner tree solution.
• Introduce the notion of clustering.
• Introduce graph Laplacian matrix.
• Conceptualize mathematically an ideal version of clustering as an optimization problem.
• Relax the ideal clustering problem to obtain a computationally tractable problem.
• Understand the importance of eigenvalues and eigenvectors of the graph Laplacian and learn spectral clustering .
• Introduce modularity maximization for the purpose of clustering.

Steiner Trees

• Use and Purpose: The Steiner tree problem seeks to connect a given set of nodes (called terminals) in a weighted graph with the minimum total edge weight. This is useful in network design, such as minimizing the cost of laying cables or pipelines connecting specific locations.
• Application: Useful in designing efficient networks in telecommunications, transportation, and logistics, where minimizing infrastructure cost is crucial.

Approximate Steiner Tree Solution

• Construction: Constructing an exact Steiner tree is an NP-hard problem, meaning it's computationally infeasible for large graphs. Approximate solutions are therefore used, employing heuristics and algorithms that provide near-optimal solutions within a reasonable time frame.
• Example Algorithms: Examples include the Minimum Spanning Tree (MST) heuristic, which connects terminals using an MST and then prunes unnecessary edges.

Clustering

• Definition: Clustering involves grouping nodes in a graph such that nodes within the same cluster are more densely connected to each other than to nodes in other clusters. This technique is used to identify community structures within networks.
• Application: Clustering is widely used in social network analysis, image segmentation, pattern recognition, and bioinformatics to identify natural groupings in data.