\(\)
Hierarchical Clustering is a method of clustering which build a hierarchy of clusters. It could be Agglomerative or Divisive.
 Agglomerative: At the first step, every item is a cluster, then clusters based on their distances are merged and form bigger clusters till all data is in one cluster (Bottom Up). The complexity is \( O (n^2log(n) ) \).
 Divisive: At the beginning, all items are in one big cluster. Then iteratively we break this cluster into smaller clusters (Top Down). The complexity is \( O (2^n) \).
To merge or divide the clusters we need to know the shortest distance between clusters. The common metrics for the distance between clusters are:
 Single Link: smallest distance between points.
 Complete Link: largest distance between points.
 Average Link: average distance between points
 Centroid: distance between centroids.
Depending on the definition of 'shortest distance' (single/ complete/ average/ centroid link ) we have different hierarchical clustering method.
Hierarchical Algorithms:
 Single Link: at each iteration, two clusters that have the closest pair of elements will be merged into a bigger cluster.
 Average Link: distance between clusters is the average distance between all points in between clusters. Clusters with the minimum of these distances merge into a bigger cluster.
 Complete Link: distance between clusters is the distance between those two points that are farthest away from each other. Two clusters with the minimum of these distances merge into a bigger cluster.
 Minimum spanning tree (MST): In a connected graph without any cycle, a spanning tree is a subset tree in which all vertex are still connected. If edges have weight, MST is a span tree in which the edges have the minimum weight. MST may not be unique.
to visualize the outcome of the hierarchical clustering we often use "Dendrogram".
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

import scipy.cluster.hierarchy as hierarchal_clustering import matplotlib.pyplot as plt data=[ [0],[1],[2],[3],[4] ] data_dict = {'A':0,'B':1,'C':2,'D':3,'E':4} data_labels = [data_dict.keys()[data_dict.values().index(i)] for i in range(0, len(data_dict))] method='complete' plt.title(method) hierarchal_clusteringlink = hierarchal_clustering.linkage(data,method,metric) dendro = hierarchal_clustering.dendrogram(hierarchal_clusteringlink,labels=data_labels) plt.show() method='centroid' plt.title(method) hierarchal_clusteringlink = hierarchal_clustering.linkage(data,method,metric) dendro = hierarchal_clustering.dendrogram(hierarchal_clusteringlink,labels=data_labels) plt.show() 
The following graph represents the following matrix :

A,B,C,D,E A,0,1,2,2,3 B,1,0,2,4,3 C,2,2,0,1,5 D,2,4,1,0,3 E,3,3,5,3,0 
Minimum spanning tree of the graph.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

import pandas as pd import networkx as nx import matplotlib.pyplot as plt import csv import numpy as np def make_label_dict(labels): l = {} for i, label in enumerate(labels): l[i] = label return l input_data = pd.read_csv('data/adjacency_matrix.csv', index_col=0) #print input_data.head print input_data.values G = nx.DiGraph(input_data.values) with open('data/adjacency_matrix.csv', 'r') as f: d_reader = csv.DictReader(f) headers = d_reader.fieldnames #print headers labels=make_label_dict(headers) #print labels edge_labels = dict( ((u, v), d["weight"]) for u, v, d in G.edges(data=True) ) pos = nx.spring_layout(G) nx.draw(G, pos) nx.draw_networkx_edge_labels(G, pos, edge_labels=edge_labels) nx.draw(G,pos,node_size=500, labels=labels, with_labels=True) plt.show() from scipy.sparse import csr_matrix from scipy.sparse.csgraph import minimum_spanning_tree X = csr_matrix(np.array(input_data.values) ) Tcsr = minimum_spanning_tree(X) MST= Tcsr.toarray().astype(int) print MST G_MST = nx.DiGraph(MST) edge_labels = dict( ((u, v), d["weight"]) for u, v, d in G_MST.edges(data=True) ) pos = nx.spring_layout(G_MST) nx.draw(G_MST, pos) nx.draw_networkx_edge_labels(G_MST, pos, edge_labels=edge_labels) nx.draw(G_MST,pos,node_size=500, labels=labels, with_labels=True) plt.show() 