Section: New Results

Graph statistics and graph clustering

Participants : Daniel Archambault, Romain Bourqui, Frédéric Gilbert, Guy Melançon, Arnaud Sallaberry, Paolo Simonetto, Faraz Zaidi.

Community detection in social networks

Figure 4. Community detection in social networks – Framework overview

Community detection in social networks still remains a topic of interest, since the decomposition of networks as a set of smaller subgraphs allow for alternative and more readable visual representations of these networks. Dealing with time variations has now become the focus of recent research and remains a challenging problem whereby efficient visualization of evolving relationships and implicit hierarchical structure are important tasks. Our efforts have been devoted to the design and development of a framework to analyze such social networks [13] . The proposed framework is based on dynamic graph discretization and graph decomposition (see Fig. 4 ). The framework allows detection of major structural changes over time, identifies events analyzing temporal dimension and reveals command hierarchies in social networks. To do so, we presented a decomposition algorithm running in O(|E|·deg_max² + |V|·log(|V|)) time that is stable enough to guarantee major changes detections. We also introduced a method inspired by the work of Memon [74] , [72] , [73] to extract a command hierarchy from the network. We used the Catalano/Vidro dataset (URL to VAST Contest 2008) for empirical evaluation and observed that our framework provides a satisfactory assessment of the social and hierarchical structure present in the dataset.

Graph comparison

Comparing graphs remains a first order challenge because of its obvious and wide applicability. Because graphs may be large, or because one might not want to perform a rough comparison of graphs, visually comparing graphs may appear as a good trade-off solution. The exact comparison of graphs anyhow is intractable as it relies rely on the subgraph isomorphism problem. It turns out that many application domain only require the identification of quasi-similar graph patterns, allowing the design of efficient and lower complexity approaches.

Previous methods we developed are based on a heuristic exploiting structural properties of graphs. Quasi-similar patterns are paired and colored accordingly. A method was first developed and used to compare trees to help locate differences in hierarchies [28] . The method was further adapted to compare and draw RNA secondary structures [4] , for instance. We also had developed a full framework to compare graphs in the context of video indexation and search [39] , [41] , [40] .

Figure 5. Result of our decomposition algorithm on a subgraph of the "Hollywood graph" (actors graph) containing 421 movies. Our algorithm detected 404 of these movies.

This type of visual comparison was extended and applied to the integrated visualization of gene expression data from time series experiments in gene regulation networks and metabolic networks [14] .

Such an integration is necessary, since it provides the link between the measurements at the transcriptional level and the observable characteristics of an organism at the functional level. Our application can (i) visualize the data from time series experiments in the context of a regulatory network and a metabolic network; (ii) identify and visualize active regulatory subnetworks from the gene expression data; (iii) perform a statistical test to identify and subsequently visualize affected metabolic subnetworks. Initial results show that our integrated approach speeds up data analysis, and that it can reproduce results of a traditional approach that involves many manual and time-consuming steps.

Figure 6. Screenshot of our tool to visualize gene expression data in both gene regulatory and metabolic networks.

We also investigated what we call difference maps: a single graph G _diff = (N _diff , E _diff ) constructed from the union of two input graphs G₁ = (N₁, E₁) and G₂ = (N₂, E₂) encoding all of the differences between the node and edge sets of G₁ and G₂ . These diagrams can be used to visualize the differences and the similarities between two graphs.

These difference maps can be very large, possibly on the order of twice the number of nodes and edges in the two input graphs. As these sets can be on the order of tens of thousands of edges, a direct drawing of the difference map suffers from high computational cost and extensive visual clutter. It may be advantageous to consider abstracting away areas of the difference map with similar meaning. In the field of information visualization and graph drawing, graph hierarchies offer one way of abstracting the data.

In this work, we designed a method to divide the difference map in a hierarchical way so that each metanode contains nodes and edges that appear in one graph, the other, or both [10] .