Hierarchical clustering in minimum spanning trees

Abstract

The identification of clusters or communities in complex networks is a reappearing problem. The minimum spanning tree (MST), the tree connecting all nodes with minimum total weight, is regarded as an important transport backbone of the original weighted graph. We hypothesize that the clustering of the MST reveals insight in the hierarchical structure of weighted graphs. However, existing theories and algorithms have difficulties to define and identify clusters in trees. Here, we first define clustering in trees and then propose a tree agglomerative hierarchical clustering (TAHC) method for the detection of clusters in MSTs. We then demonstrate that the TAHC method can detect clusters in artificial trees, and also in MSTs of weighted social networks, for which the clusters are in agreement with the previously reported clusters of the original weighted networks. Our results therefore not only indicate that clusters can be found in MSTs, but also that the MSTs contain information about the underlying clusters of the original weighted network. Clustering or community structure has been regarded as one of the most significant features of complex networks. The minimum spanning tree (MST) is the spanning tree in a weighted graph for which the sum of the weights of its constituting links is minimal. There are many different kinds of algorithms for identifying clusters in general networks, but a general definition and algorithm for the detection of clusters in trees has not been established. In our current study, we first define clusters in trees and propose a tree agglomerative hierarchical clustering (TAHC) method for the detection of clusters in MSTs. By detecting clusters in both artificial trees and the MSTs of two weighted social networks, we demonstrate that the clustering of the MSTs reveals the underlying clusters of the original weighted graphs.

EEG Resting State Functional Connectivity in Adult Dyslexics Using Phase Lag Index and Graph Analysis

“…The dependence of weighted network measures on connectivity strength may be exacerbated by current group differences in PLI. Future studies may take advantage of recent developments in the analysis of MST clustering structure (Yu et al, 2015) and tree dissimilarities across the groups (Yu et al, 2016). Such analyses would contribute information about the community structure of the network while avoiding the limitations of weighted networks.…”

A Resting-State Brain Functional Network Study in MDD Based on Minimum Spanning Tree Analysis and the Hierarchical Clustering

“…Among the multitudinous hierarchical clustering algorithms, a tree agglomerative hierarchical clustering (TAHC) method can successfully detect clusters in both artificial trees and the MSTs of weighted social networks. This method was raised in [43]. Moreover, the hierarchical clustering has regarded the MST as a foundation of the complex networks [30].…”

“…The clustering and community structures have been regarded as one of the most significant features of complex networks [43]. In the brain networks, clustering or community structure was defined as a subset of highly interconnected nodes which had similar characteristics [44].…”

“…In this study, TAHC method was used for the detection of the clusters in the aforementioned MSTs. The summary of TAHC algorithm [43] is as follows: first we use geodesic distances between all possible pairs of nodes of the given graph as an input to the agglomerative hierarchical clustering algorithm. Next, compute the similarity between every node pairs in an MST based on geodesic distances.…”

Visualization and cellular hierarchy inference of single-cell data using SPADE

“…An alternative approach that does not use t-SNE would be to perform clustering for the SPADE MST. One approach based on hierarchical clustering of MST has been proposed by Yu et al 28 . The clustering output from such an approach can be used in a supervised setting to determine important lineage markers.…”