An energy-based model to optimize cluster visualization

National audienceGraphs are mathematical structures that provide natural means for complex-data representation. Graphs capture the structure and thus help modeling a wide range of complex real-life data in various domains. Moreover graphs are especially suitable for information visualization. Indeed the intuitive visualabstraction (dots and lines) they provide is intimately associated with graphs. Visualization paves the way to interactive exploratory data-analysis and to important goals such as identifying groups and subgroups among data and helping to understand how these groups interact with each other. In this paper, we present a graph drawing approach that helps to better appreciate the cluster structure in data and the interactions that may exist between clusters. In this work, we assume that the clusters are already extracted and focus rather on the visualization aspects. We propose an energy-based model for graph drawing that produces an esthetic drawing that ensures each cluster will occupy a separate zone within thevisualization layout. This method emphasizes the inter-groups interactions and still shows the inter-nodes interactions. The drawing areas assigned to the clusters can be user-specified (prefixed areas) or automatically crafted (free areas). The approach we suggest also enables handling geographically-based clustering. In the case of free areas, we illustrate the use of our drawing method through an example. In the case of prefixed areas, we firstuse an example from citation networks and then use another exampleto compare the results of our method to those of the divide and conquer approach. In the latter case, we show that while the two methods successfully point out the cluster structure our method better visualize the global structure

Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

Functionals of the Brownian motion, localization and metric graphs

We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of the Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization). Two aspects of the physics of the one-dimensional strong localization are reviewed : some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues). Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schr\"odinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated. Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of the planar Brownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and conclusion added. Several references adde

Modeling interdisciplinary interactions among Physics, Mathematics & Computer Science

Interdisciplinarity has over the recent years have gained tremendous importance and has become one of the key ways of doing cutting edge research. In this paper we attempt to model the citation flow across three different fields -- Physics (PHY), Mathematics (MA) and Computer Science (CS). For instance, is there a specific pattern in which these fields cite one another? We carry out experiments on a dataset comprising more than 1.2 million articles taken from these three fields. We quantify the citation interactions among these three fields through temporal bucket signatures. We present numerical models based on variants of the recently proposed relay-linking framework to explain the citation dynamics across the three disciplines. These models make a modest attempt to unfold the underlying principles of how citation links could have been formed across the three fields over time.Comment: Accepted at Journal of Physics: Complexit