Centroid based Tree-Structured Data Clustering Using Vertex/Edge Overlap and Graph Edit Distance

• Published in: Annals of operations research 2020-06, Vol.289 (1), p.85-122
• Main Authors: Dinler, Derya, Tural, Mustafa Kemal, Ozdemirel, Nur Evin
• Format: Article
• Language: English
• Subjects: Algorithms; Brain; Business and Management; Centroids; Clustering; Combinatorics; Operations research; Operations Research/Decision Theory; Outliers (statistics); S.I.: OR in Neuroscience II; Structured data; Theory of Computation; Trees
• ISSN: 0254-5330; 1572-9338
• DOI: 10.1007/s10479-019-03505-7

We consider a clustering problem in which the data objects are rooted m -ary trees with known node correspondence. We assume that the nodes of the trees are unweighted, but the edges can be unweighted or weighted. We measure the similarity and distance between two trees using vertex/edge overlap (VEO) and graph edit distance (GED), respectively. For both measures, we first study the problem of finding a centroid tree of a given cluster of trees in both the unweighted and weighted edge cases. We compute the optimal centroid tree of a given cluster for all measures except the weighted VEO for which a heuristic is developed. We then propose k-means based algorithms that repeat cluster assignment and centroid update steps until convergence. The initial centroid trees are constructed based on the properties of the data. The assignment steps utilize unweighted or weighted versions of VEO or GED to assign each tree to the most similar centroid tree. In the update steps, each centroid tree is updated by considering the trees assigned to it. The proposed algorithms are compared with the traditional k-modes and k-means on randomly generated datasets and shown to be more effective and robust (to outliers) in separating trees into clusters. We also apply our algorithms on a real world brain artery data and show that the previously observed age and sex effects on brain artery structures can be revealed better by means of clustering with our algorithms than the traditional k-modes and k-means.