We present a novel hierarchical graph clustering algorithm inspired by modularity-based clustering techniques. The algorithm is agglomerative and based on a simple distance between clusters induced by the probability of sampling node pairs. We prove that this distance is reducible, which enables the use of the nearest-neighbor chain to speed up the agglomeration. The output of the algorithm is a regular dendrogram, which reveals the multi-scale structure of the graph. The results are illustrated on both synthetic and real datasets.
Usual hierarchical clustering techniques apply to vector data @cite_13 @cite_12 . They do not directly apply to graphs, unless the graph is embedded in some metric space, through spectral techniques for instance @cite_26 @cite_3 . A number of hierarchical clustering algorithms have been developped specifically for graphs. The most popular algorithms are agglomerative and characterized by some distance between clusters, see @cite_25 @cite_9 @cite_23 @cite_0 . None of these distances has been proved to be reducible, a key property of our algorithm. Among non-agglomerative algorithms, the divisive approach of @cite_20 is based on the notion of edge betweenness while the iterative approach of @cite_21 and @cite_14 look for local maxima of modularity or some fitness function; other approaches rely on statistical interence @cite_17 , replica correlations @cite_7 and graph wavelets @cite_6 . To our knowledge, non of these algorithms has been proved to lead to regular dendrograms (that is, without inversion).
@cite_26 @cite_14 @cite_7 @cite_9 @cite_21 @cite_3 @cite_6 @cite_0 @cite_23 @cite_20 @cite_13 @cite_25 @cite_12 @cite_17
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We present a new method for clustering based on compression. The method does not use subject-specific features or background knowledge, and works as follows: First, we determine a parameter-free, universal, similarity distance, the normalized compression distance or NCD, computed from the lengths of compressed data files (singly and in pairwise concatenation). Second, we apply a hierarchical clustering method. The NCD is not restricted to a specific application area, and works across application area boundaries. A theoretical precursor, the normalized information distance, co-developed by one of the authors, is provably optimal. However, the optimality comes at the price of using the noncomputable notion of Kolmogorov complexity. We propose axioms to capture the real-world setting, and show that the NCD approximates optimality. To extract a hierarchy of clusters from the distance matrix, we determine a dendrogram (ternary tree) by a new quartet method and a fast heuristic to implement it. The method is implemented and available as public software, and is robust under choice of different compressors. To substantiate our claims of universality and robustness, we report evidence of successful application in areas as diverse as genomics, virology, languages, literature, music, handwritten digits, astronomy, and combinations of objects from completely different domains, using statistical, dictionary, and block sorting compressors. In genomics, we presented new evidence for major questions in Mammalian evolution, based on whole-mitochondrial genomic analysis: the Eutherian orders and the Marsupionta hypothesis against the Theria hypothesis. A novel graph theoretic approach for data clustering is presented and its application to the image segmentation problem is demonstrated. The data to be clustered are represented by an undirected adjacency graph G with arc capacities assigned to reflect the similarity between the linked vertices. Clustering is achieved by removing arcs of G to form mutually exclusive subgraphs such that the largest inter-subgraph maximum flow is minimized. For graphs of moderate size ( approximately 2000 vertices), the optimal solution is obtained through partitioning a flow and cut equivalent tree of G, which can be efficiently constructed using the Gomory-Hu algorithm (1961). However for larger graphs this approach is impractical. New theorems for subgraph condensation are derived and are then used to develop a fast algorithm which hierarchically constructs and partitions a partially equivalent tree of much reduced size. This algorithm results in an optimal solution equivalent to that obtained by partitioning the complete equivalent tree and is able to handle very large graphs with several hundred thousand vertices. The new clustering algorithm is applied to the image segmentation problem. The segmentation is achieved by effectively searching for closed contours of edge elements (equivalent to minimum cuts in G), which consist mostly of strong edges, while rejecting contours containing isolated strong edges. This method is able to accurately locate region boundaries and at the same time guarantees the formation of closed edge contours. > We consider the following general correlation-clustering problem [N. Bansal, A. Blum, S. Chawla, Correlation clustering, in: Proc. 43rd Annu. IEEE Symp. on Foundations of Computer Science, Vancouver, Canada, November 2002, pp. 238-250]: given a graph with real nonnegative edge weights and a 〈+〉 〈-〉 edge labelling, partition the vertices into clusters to minimize the total weight of cut 〈+〉 edges and uncut 〈-〉 edges. Thus, 〈+〉 edges with large weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster while 〈-〉 edges with large weights encourage the endpoints to belong to different clusters. In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are effectively chosen to be best possible by the problem definition.Correlation clustering was introduced by [Correlation clustering, in: Proc. 43rd Annu. IEEE Syrup. on Foundations of Computer Science, Vancouver, Canada, November 2002, pp. 238-250], motivated by both document clustering and agnostic learning. They proved NP-hardness and gave constant-factor approximation algorithms for the special case in which the graph is complete (full information) and every edge has the same weight. We give an O(log n)-approximation algorithm for the general case based on a linear-programming rounding and the "region-growing" technique. We also prove that this linear program has a gap of Ω(log n), and therefore our approximation is tight under this approach. We also give an O(r3)-approximation algorithm for Kr, r-minor-free graphs. On the other hand, we show that the problem is equivalent to minimum multicut, and therefore APX-hard and difficult to approximate better than Θ(log n). Hierarchical clustering is common method used to determine clusters of similar data points in multi-dimensional spaces. @math algorithms, where @math is the number of points to cluster, have long been known for this problem. This paper discusses parallel algorithms to perform hierarchical clustering using various distance metrics. I describe @math time algorithms for clustering using the single link, average link, complete link, centroid, median, and minimum variance metrics on an @math node CRCW PRAM and @math algorithms for these metrics (except average link and complete link) on @math node butterfly networks or trees. Thus, optimal efficiency is achieved for a significant number of processors using these distance metrics. A general algorithm is given that can be used to perform clustering with the complete link and average link metrics on a butterfly. While this algorithm achieves optimal efficiency for the general class of metrics, it is not optimal for the specific cases of complete link and average link clustering.