For the last few years, the amount of data has significantly increased in the companies. It is the reason why data analysis methods have to evolve to meet new demands. In this article, we introduce a practical analysis of a large database from a telecommunication operator. The problem is to segment a territory and characterize the retrieved areas owing to their inhabitant behavior in terms of mobile telephony. We have call detail records collected during five months in France. We propose a two stages analysis. The first one aims at grouping source antennas which originating calls are similarly distributed on target antennas and conversely for target antenna w.r.t. source antenna. A geographic projection of the data is used to display the results on a map of France. The second stage discretizes the time into periods between which we note changes in distributions of calls emerging from the clusters of source antennas. This enables an analysis of temporal changes of inhabitants behavior in every area of the country.
Numerous methods have been proposed to extract satisfactory clusters of vertices. Some of them @cite_20 are based on the optimization of criteria that favor partitions with homogeneous blocks, especially with pure zero-blocks as recommended in @cite_11 . More recent deterministic approaches have focused on optimizing criteria that quantify how well the co-clustering summarizes the input data @cite_19 (see e.g. @cite_17 for details on such criteria). Other approaches include blockmodeling. In those generative models, a latent cluster indicator variable is associated to each vertex. Conditionally to the latent variables, the probability of observing an edge between two actors follows some standard distribution (a Bernoulli distribution in the simplest case) whose parameters only depend on the pair of clusters designated by the latent variables. In early approaches, the number of clusters is chosen by the user @cite_15 . More recent techniques automatically determine the number of clusters using a Dirichlet Process @cite_13 . Finally, some recent approaches consider non-boolean latent variables: cluster assignments are not strong and a vertex has an affiliation degree to each cluster @cite_5 .
@cite_11 @cite_19 @cite_5 @cite_15 @cite_13 @cite_20 @cite_17
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This paper considers the inverse problem with observed variables Y = BGX ⊕Z, where BG is the incidence matrix of a graph G, X is the vector of unknown vertex variables with a uniform prior, and Z is a noise vector with Bernoulli(e) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery of X is possible if and only the graph G is connected, with a sharp threshold at the edge probability log(n) n for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability p needs to scale to allow exact recovery in the presence of noise. Defining the degree (oversampling) rate of the graph by α = np log(n), it is shown that exact recovery is possible if and only if α > 2 (1−2e)+o(1 (1−2e)). In other words, 2 (1−2e) is the information theoretic threshold for exact recovery at lowSNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. Full version available in [1]. 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 problem of clustering a collection of elements based on pairwise judgments of similarity and dissimilarity. N. (2002) cast the problem thus: given a graph G whose edges are labeled "+" (similar) or "-" (dissimilar), partition the vertices into clusters so that the number of pairs correctly (resp. incorrectly) classified with respect to the input labeling is maximized (resp. minimized). Complete graphs, where the classifier labels every edge, and general graphs, where some edges are not labeled, are both worth studying. We answer several questions left open by N. (2002) and provide a sound overview of clustering with qualitative information. We give a factor 4 approximation for minimization on complete graphs, and a factor O(log n) approximation for general graphs. For the maximization version, a PTAS for complete graphs is shown by N. (2002); we give a factor 0.7664 approximation for general graphs, noting that a PTAS is unlikely by proving APX-hardness. We also prove the APX-hardness of minimization on complete graphs. We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2 Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g.