Wojciech Czech

Invariants of distance k-graphs for graph embedding

Graph comparison algorithms based on metric space embedding have been proven robust in graph clustering and classification. In this paper we propose graph embedding method exploiting ordered invariants of distance k-graphs, which encode structure of shortest-paths. We study degree histograms of those graphs and use them to construct permutation invariant graph representation called vertex B-matrix. In order to extract more information from structural patterns we also define edge distance k-graphs and associated edge B-matrix. Next, several new graph characteristics obtained by condensing information stored in B-matrices are introduced. We demonstrate that our approach provides stable embedding, which captures relevant graph features. Experiments on classification with satellite photo and mutagenicity benchmark datasets revealed, that new descriptors allow for distinguishing graphs with non trivial structural differences. Moreover, they appear to outperform descriptors based on heat kernel matrix, being at the same time more effective computationally. In the end we test feature selection on B-matrices showing that selecting right B-submatrix can improve classification rate on testing datasets.

Graph descriptors from B-matrix representation

In this paper we propose graph descriptors derived from B-matrices, which are built on the basis of distances between graph vertices. The B-matrices, being invariant under graph isomorphism, are a rich source of information about graph structure. We explore this representation and propose several new graph characteristics that can be used for efficient graph comparison. Experiments on clusterization and classification with synthetic and real-world data revealed, that new descriptors allow for distinguishing graphs with non trivial structural differences. Moreover, they appear to outperform descriptors based on heat kernel matrix, being at the same time more effective computationally.

Interactive Visualization Tool for Planning Cancer Treatment

We discuss the components and main requirements of the interactive visualization and simulation system intended for better understanding the dynamics of solid tumor proliferation. The heterogeneous Complex Automata, discrete-continuum model is used as the simulation engine. It combines Cellular Automata paradigm, particle dynamics and continuum approaches to model mechanical interactions of tumor with the rest of tissue. We show that to provide interactivity, the system has to be efficiently implemented on workstations with multiple cores CPUs controlled by OpenMP interface and/or empowered by GPGPU accelerators. Currently, the computational power of modern CPU and GPU processors enable to simulate the tumors of a few millimeters in diameter in its both avascular and angiogenic phases. To validate the results of simulation with real tumors, we plan to integrate the tumor modeling simulator with the Graph Investigator tool. Then one can validate the simulation results on the base of topological similarity between the tumor vascular networks obtained from its direct observation and simulation. The interactive visualization system can have both educational and research aspects. It can be used as a tool for clinicians and oncologists for educational purposes and, in the nearest future, in medical in silico labs doing research in anticancer drug design and/or in planning cancer treatment.

Efficient Graph Comparison and Visualization Using GPU

This paper presents application of several graph algorithms for comparison and visualization of real-world networks. In order to obtain interactive and robust framework for analysis of large graphs we use CUDA implementations of all-shortest-paths (APSP) and breadth-first-search (BFS) algorithms along with CULA matrix decomposition routines. Such an approach allows for efficient computation of graph feature vectors, visualization with graph B-matrices and accelerating dimensionality reduction methods used to embed graphs into low-dimensional metric spaces. Graph analysis algorithms implemented in CUDA were integrated with Graph Investigator Java application via Java Native Interface (JNI) what makes them more convenient to use. We further present two real-world usage scenarios i.e. analysis and visualization of vascular networks in presence of tumor and clusterization based on graph representations of satelite photos.

Ubiquitous Interactive Visualization of 3-D Mantle Convection through Web Applications Using Java

We have designed a new system for real-time interactive visualization of results taken directly from large-scale simulations of 3-D mantle convection and other large-scale simulations. This approach allows for intense visualization sessions for a couple of hours as opposed to storing massive amounts of data in a storage system. Our data sets consist of 3-D data for volume rendering with sets ranging as high as over 10 million unknowns at each timestep. Large scale visualization on a display wall holding around 13 million pixels has already been accomplished with extension to hand-held devices, such as the OQO and Nokia N800. We are developing web-based software in Java to extend the use of this system across long distances. The software is aimed at creating an interactive and functional application capable of running on multiple browsers by taking advantage of two AJAX-enabled web frameworks: Echo2 and Google Web Toolkit.

Distributed computing of distance‐based graph invariants for analysis and visualization of complex networks

We present a new framework for analysis and visualization of complex networks based on structural information retrieved from their distance k‐graphs and B‐matrices. The construction of B‐matrices for graphs with more than 1 million edges requires massive Breadth‐First Search (BFS) computations and is facilitated using new software prepared for distributed environments. Our framework benefits from data parallelism inherent to all‐pair shortest‐path problem and extends Cassovary, an open‐source in‐memory graph processing engine, to enable multinode computation of distance k‐graphs and related graph descriptors. We also introduce a new type of B‐matrix, constructed using clustering coefficient vertex invariant, which can be generated with a computational effort comparable with the one required for a previously known degree B‐matrix, while delivering an additional set of information about graph structure. Our approach enables efficient generation of expressive, multidimensional descriptors useful in graph embedding and graph mining tasks. The experiments showed that the new framework is scalable and for specific all‐pair shortest‐path task provides better performance than existing generic graph processing frameworks. We further present how the developed tools helped in the analysis and visualization of real‐world graphs from Stanford Large Network Dataset Collection. Copyright

Application of algebraic graph descriptors for clustering of real-world structures

We propose several vector graph descriptors created on the basis of vertex rank measures such as PageRank, Hubs and Authorities or Betweenness Centrality. The descriptors are used for clustering artificial and real-world data. We present the comparison of descriptors with the use of criteria such as computational complexity, size and quality of clustering. The experiments were performed mainly on the sets of aerial photos transformed to graphs with the use of Harris corner detection and Delaunay triangulation. The results show that the introduced pattern vectors can be a lower dimensional, less computationally expensive and graph size independent alternative for spectral descriptors, such as defined by Wilson, Hancock and Luo in [1].

A Method of Analysis and Visualization of Structured Datasets Based on Centrality Information

We present a new method of quantitative graph analysis and visualization based on vertex centrality measures and distance matrices. After generating distance k-graphs and collecting frequency information about their vertex descriptors, we obtain generic, multidimensional representation of a graph, invariant to graph isomorphism. The histograms of vertex centrality measures, organized in a form of B-matrices, allow to capture subtle changes in network structure during its evolution and provide robust tool for graph comparison and classification. We show that different types of B-matrices and their extensions are useful in graph analysis tasks performed on benchmark complex networks from Koblenz and IAM datasets. We compare the results obtained for proposed B-matrix extensions with performance of other state-of-art graph descriptors showing that our method is superior to others.

Comparison of Large Graphs Using Distance Information

We present a new framework for analysis and visualization of large complex networks based on structural information retrieved from their distance k-graphs and B-matrices. The construction of B-matrices for graphs with more than 1 million edges requires massive BFS computations and is facilitated using Cassovary - an open-source in-memory graph processing engine. The approach described in this paper enables efficient generation of expressive, multi-dimensional descriptors useful in graph embedding and graph mining tasks. In experimental section, we present how the developed tools helped in the analysis of real-world graphs from Stanford Large Network Dataset Collection.

Review of Graph Invariants for Quantitative Analysis of Structure Dynamics

In this work we review graph invariants used for quantitative analysis of evolving graphs. Focusing on graph datasets derived from structural pattern recognition and complex networks fields, we demonstrate how to capture relevant topological features of networks. In an experimental setup, we study structural properties of graphs representing rotating 3D objects and show how they are related to characteritics of undelying images. We present how evolving strucure of Autonomous Systems (ASs) network is reflected by non-trivial changes in scalar graph descriptors. We also inspect characteristics of growing tumor vascular networks, obtained from a simulation. Additionally, the overview of currently used graph invariants with several possible groupings is provided.