Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs

Drawings of highly connected (dense) graphs can be very difficult to read. Power Graph Analysis offers an alternate way to draw a graph in which sets of nodes with common neighbours are shown grouped into modules. An edge connected to the module then implies a connection to each member of the module. Thus, the entire graph may be represented with much less clutter and without loss of detail. A recent experimental study has shown that such lossless compression of dense graphs makes it easier to follow paths. However, computing optimal power graphs is difficult. In this paper, we show that computing the optimal power-graph with only one module is NP-hard and therefore likely NP-hard in the general case. We give an ILP model for power graph computation and discuss why ILP and CP techniques are poorly suited to the problem. Instead, we are able to find optimal solutions much more quickly using a custom search method. We also show how to restrict this type of search to allow only limited back-tracking to provide a heuristic that has better speed and better results than previously known heuristics.Comment: Extended technical report accompanying the PacificVis 2013 paper of the same nam

Scalability considerations for multivariate graph visualization

Real-world, multivariate datasets are frequently too large to show in their entirety on a visual display. Still, there are many techniques we can employ to show useful partial views-sufficient to support incremental exploration of large graph datasets. In this chapter, we first explore the cognitive and architectural limitations which restrict the amount of visual bandwidth available to multivariate graph visualization approaches. These limitations afford several design approaches, which we systematically explore. Finally, we survey systems and studies that exhibit these design strategies to mitigate these perceptual and architectural limitations

Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology

Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we propose a novel method using persistent homology to quantify structural changes in time-varying graphs. Specifically, we transform each instance of the time-varying graph into metric spaces, extract topological features using persistent homology, and compare those features over time. We provide a visualization that assists in time-varying graph exploration and helps to identify patterns of behavior within the data. To validate our approach, we conduct several case studies on real world data sets and show how our method can find cyclic patterns, deviations from those patterns, and one-time events in time-varying graphs. We also examine whether persistence-based similarity measure as a graph metric satisfies a set of well-established, desirable properties for graph metrics

Persistent Homology Guided Force-Directed Graph Layouts

Graphs are commonly used to encode relationships among entities, yet their abstractness makes them difficult to analyze. Node-link diagrams are popular for drawing graphs, and force-directed layouts provide a flexible method for node arrangements that use local relationships in an attempt to reveal the global shape of the graph. However, clutter and overlap of unrelated structures can lead to confusing graph visualizations. This paper leverages the persistent homology features of an undirected graph as derived information for interactive manipulation of force-directed layouts. We first discuss how to efficiently extract 0-dimensional persistent homology features from both weighted and unweighted undirected graphs. We then introduce the interactive persistence barcode used to manipulate the force-directed graph layout. In particular, the user adds and removes contracting and repulsing forces generated by the persistent homology features, eventually selecting the set of persistent homology features that most improve the layout. Finally, we demonstrate the utility of our approach across a variety of synthetic and real datasets

A bitwise clique detection approach for accelerating power graph computation and clustering dense graphs

Graphs are at the essence of many data representations. The visual analytics over graphs is usually difficult due to their size, which makes their visual display challenging, and their fundamental algorithms, which are often classified as NP-hard problems. The Power Graph Analysis (PGA) is a method that simplifies networks using reduced representations for complete subgraphs (cliques) and complete bipartite subgraphs (bicliques), in both cases with edge reductions. The benefits of a power graph are the preservation of information and its capacity to show essential information about the original network. However, finding an optimal representation (maximum edges reduction) is also an NPhard problem. In this work, we propose BCD, a greedy algorithm that uses a Bitwise Clique Detection approach to finding power graphs. BCD is faster than competing strategies and allows the analysis of bigger graphs. For the display of larger power graphs, we propose an orthogonal layout to prevent overlapping of edges and vertices. Finally, we describe how the structure induced by the power graph is used for clustering analysis of dense graphs. We demonstrate with several datasets the results obtained by our proposal and compare against competing strategies.Os grafos são essenciais para muitas representações de dados. A análise visual de grafos é usualmente difícil devido ao tamanho, o que representa um desafio para sua visualização. Além de isso, seus algoritmos fundamentais são frequentemente classificados como NP-difícil. Análises dos grafos de potência (PGA em inglês) é um método que simplifica redes usando representações reduzidas para subgrafos completos chamados cliques e subgrafos bipartidos chamados bicliques, em ambos casos com una redução de arestas. Os benefícios da representação de grafo de potência são a preservação de informação e a capacidade de mostrar a informação essencial sobre a rede original. Entretanto, encontrar uma representação ótima (a máxima redução de arestas possível) é também um problema NP-difícil. Neste trabalho, propomos BCD, um algoritmo guloso que usa um abordagem de detecção de bicliques baseado em operações binarias para encontrar representações de grafos de potencia. O BCD é mas rápido que as estratégias atuais da literatura. Finalmente, descrevemos como a estrutura induzida pelo grafo de potência é utilizado para as análises dos grafos densos na detecção de agrupamentos de nodos

VoG: Summarizing and Understanding Large Graphs

How can we succinctly describe a million-node graph with a few simple sentences? How can we measure the "importance" of a set of discovered subgraphs in a large graph? These are exactly the problems we focus on. Our main ideas are to construct a "vocabulary" of subgraph-types that often occur in real graphs (e.g., stars, cliques, chains), and from a set of subgraphs, find the most succinct description of a graph in terms of this vocabulary. We measure success in a well-founded way by means of the Minimum Description Length (MDL) principle: a subgraph is included in the summary if it decreases the total description length of the graph. Our contributions are three-fold: (a) formulation: we provide a principled encoding scheme to choose vocabulary subgraphs; (b) algorithm: we develop \method, an efficient method to minimize the description cost, and (c) applicability: we report experimental results on multi-million-edge real graphs, including Flickr and the Notre Dame web graph.Comment: SIAM International Conference on Data Mining (SDM) 201

JGraphT -- A Java library for graph data structures and algorithms

Mathematical software and graph-theoretical algorithmic packages to efficiently model, analyze and query graphs are crucial in an era where large-scale spatial, societal and economic network data are abundantly available. One such package is JGraphT, a programming library which contains very efficient and generic graph data-structures along with a large collection of state-of-the-art algorithms. The library is written in Java with stability, interoperability and performance in mind. A distinctive feature of this library is the ability to model vertices and edges as arbitrary objects, thereby permitting natural representations of many common networks including transportation, social and biological networks. Besides classic graph algorithms such as shortest-paths and spanning-tree algorithms, the library contains numerous advanced algorithms: graph and subgraph isomorphism; matching and flow problems; approximation algorithms for NP-hard problems such as independent set and TSP; and several more exotic algorithms such as Berge graph detection. Due to its versatility and generic design, JGraphT is currently used in large-scale commercial, non-commercial and academic research projects. In this work we describe in detail the design and underlying structure of the library, and discuss its most important features and algorithms. A computational study is conducted to evaluate the performance of JGraphT versus a number of similar libraries. Experiments on a large number of graphs over a variety of popular algorithms show that JGraphT is highly competitive with other established libraries such as NetworkX or the BGL.Comment: Major Revisio