What is a metric?

A nonnegative function g(x,y) describing the "distance" between neighboring points for a given set

$g(x,y) \ge 0$

$g(x,y) = g(y,x)$

$g(x,y) = 0 \quad iff.\ x = y$

$g(x,y) + g(y,z) \ge g(x,z)$

Similarity metrics can express a broad range of relationships

• Spatial ($\ell_1$ or $\ell_2$ norm)
• Information (Hamming distance)
• Inner product space (Cosine similarity)
• Kernel function (RBF)

H. Chase Stevens

How?

Minimize $\sum_{k=1}^K \sum_{x_n \in C_k} ||x_n - \mu_k||^2$ with respect to $C_k$ and $\mu_k$

Lloyd's algorithm: two steps

Given $k$ and inital cluster centers, $\mu_k$

1. Assign points to closest centroid
2. Recalculate $\mu_k$

Repeat (1) and (2) until $\sum_{k = 1}^K|\mu_{k_{t+1}} - \mu_{k_{t}}| \le \epsilon$

More detail at The Data Science Lab

H. Chase Stevens

Advanced Clustering

What if your data looks like this?

K-means Solution

Spectral Clustering Solution

Spectral Clustering

Some advantages

1. Can be solved efficiently by standard linear algebra software
2. Often outperforms traditional clustering algorithms such as the k-means algorithm

Luxburg (2007). A Tutorial on Spectral Clustering

Spectral Clustering: Steps

1. Compute the similarity matrix, S (e.g. Kernel function)
2. Calculate the affinity matrix, A, from S (e.g. k-nearest neighbors algorithm)
3. Calculate a graph Laplacian, L
4. Perform k-means clustering on matrix, Z, consisting of eigenvectors for k smallest eigenvalues, ignoring the smallest (constant) eigenvector of L

Data ⇨ Transform and Embed

1. Compute the similarity matrix, S (e.g. Kernel function like RBF)

2. Calculate the affinity matrix, A, from S (e.g. k-nearest neighbors algorithm)

3. Calculate a graph Laplacian, L

$L = D − W$

Wikipedia: Laplacian matrix

4. Perform k-means clustering on matrix, Z, consisting of eigenvectors for k smallest eigenvalues of L

Aside: Can also be formulated as Mincut

Graph Partition

Mincut

Eigen value decomposition ⇨ Fiedler vector

Graph Partition

Overview for $k>2$

astropy

Link-community clustering

Yong-Yeol et al (2010). Link communities reveal multiscale complexity in networks

Cluster edges rather than nodes

Yong-Yeol et al (2010). Link communities reveal multiscale complexity in networks

Link-community clustering: Steps

1. Compute similarity score between every pair of edges
2. Perform agglomerative clustering on edge similarities
3. Cut dendrogram at height that maximizes cluster density

1. Compute similarity score between every pair of edges

Yong-Yeol et al (2010). Link communities reveal multiscale complexity in networks

2. Perform agglomerative clustering on edge similarities

Yong-Yeol et al (2010). Link communities reveal multiscale complexity in networks

3. Cut dendrogram at height that maximizes cluster density

Yong-Yeol et al (2010). Link communities reveal multiscale complexity in networks

Resulting link communities are multiscale...

Yong-Yeol et al (2010). Link communities reveal multiscale complexity in networks

Resulting link communities are multiscale...

Yong-Yeol et al (2010). Link communities reveal multiscale complexity in networks

...and overlapping

Yong-Yeol et al (2010). Link communities reveal multiscale complexity in networks

Examples

Flavor Network

Yong-Yeol et al (2011). Flavor network and the principles of food pairing

EGRIN2.0

Gene co-regulation network

Brooks and Reiss et al (2014). A system‐level model for the microbial regulatory genome

Conditionally Co-regulated Modules (Corems)

Brooks and Reiss et al (2014). A system‐level model for the microbial regulatory genome

Similarities between 4567 human genes

Selected from their relationship to 120 genes involved in mitosis, DNA mismatch repair, and BMP signaling

Combined Kernel Network

For more info: Practical 1

How to detect community structure?

1. Node-based spectral clustering
2. Edge-based link-community clustering