Planar Drawings of Fixed-Mobile Bigraphs

A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of "layered" 1-bend drawings

StructMatrix: large-scale visualization of graphs by means of structure detection and dense matrices

Given a large-scale graph with millions of nodes and edges, how to reveal macro patterns of interest, like cliques, bi-partite cores, stars, and chains? Furthermore, how to visualize such patterns altogether getting insights from the graph to support wise decision-making? Although there are many algorithmic and visual techniques to analyze graphs, none of the existing approaches is able to present the structural information of graphs at large-scale. Hence, this paper describes StructMatrix, a methodology aimed at high-scalable visual inspection of graph structures with the goal of revealing macro patterns of interest. StructMatrix combines algorithmic structure detection and adjacency matrix visualization to present cardinality, distribution, and relationship features of the structures found in a given graph. We performed experiments in real, large-scale graphs with up to one million nodes and millions of edges. StructMatrix revealed that graphs of high relevance (e.g., Web, Wikipedia and DBLP) have characterizations that reflect the nature of their corresponding domains; our findings have not been seen in the literature so far. We expect that our technique will bring deeper insights into large graph mining, leveraging their use for decision making.Comment: To appear: 8 pages, paper to be published at the Fifth IEEE ICDM Workshop on Data Mining in Networks, 2015 as Hugo Gualdron, Robson Cordeiro, Jose Rodrigues (2015) StructMatrix: Large-scale visualization of graphs by means of structure detection and dense matrices In: The Fifth IEEE ICDM Workshop on Data Mining in Networks 1--8, IEE

Maximizing the Total Resolution of Graphs

A major factor affecting the readability of a graph drawing is its resolution. In the graph drawing literature, the resolution of a drawing is either measured based on the angles formed by consecutive edges incident to a common node (angular resolution) or by the angles formed at edge crossings (crossing resolution). In this paper, we evaluate both by introducing the notion of "total resolution", that is, the minimum of the angular and crossing resolution. To the best of our knowledge, this is the first time where the problem of maximizing the total resolution of a drawing is studied. The main contribution of the paper consists of drawings of asymptotically optimal total resolution for complete graphs (circular drawings) and for complete bipartite graphs (2-layered drawings). In addition, we present and experimentally evaluate a force-directed based algorithm that constructs drawings of large total resolution

Entropy and Graph Based Modelling of Document Coherence using Discourse Entities: An Application

We present two novel models of document coherence and their application to information retrieval (IR). Both models approximate document coherence using discourse entities, e.g. the subject or object of a sentence. Our first model views text as a Markov process generating sequences of discourse entities (entity n-grams); we use the entropy of these entity n-grams to approximate the rate at which new information appears in text, reasoning that as more new words appear, the topic increasingly drifts and text coherence decreases. Our second model extends the work of Guinaudeau & Strube [28] that represents text as a graph of discourse entities, linked by different relations, such as their distance or adjacency in text. We use several graph topology metrics to approximate different aspects of the discourse flow that can indicate coherence, such as the average clustering or betweenness of discourse entities in text. Experiments with several instantiations of these models show that: (i) our models perform on a par with two other well-known models of text coherence even without any parameter tuning, and (ii) reranking retrieval results according to their coherence scores gives notable performance gains, confirming a relation between document coherence and relevance. This work contributes two novel models of document coherence, the application of which to IR complements recent work in the integration of document cohesiveness or comprehensibility to ranking [5, 56]

Discovering Patterns of Interest in IP Traffic Using Cliques in Bipartite Link Streams

Studying IP traffic is crucial for many applications. We focus here on the detection of (structurally and temporally) dense sequences of interactions, that may indicate botnets or coordinated network scans. More precisely, we model a MAWI capture of IP traffic as a link streams, i.e. a sequence of interactions $(t_1 , t_2 , u, v)$ meaning that devices $u$ and $v$ exchanged packets from time $t_1$ to time $t_2$ . This traffic is captured on a single router and so has a bipartite structure: links occur only between nodes in two disjoint sets. We design a method for finding interesting bipartite cliques in such link streams, i.e. two sets of nodes and a time interval such that all nodes in the first set are linked to all nodes in the second set throughout the time interval. We then explore the bipartite cliques present in the considered trace. Comparison with the MAWILab classification of anomalous IP addresses shows that the found cliques succeed in detecting anomalous network activity

The Swiss Board Directors Network in 2009

We study the networks formed by the directors of the most important Swiss boards and the boards themselves for the year 2009. The networks are obtained by projection from the original bipartite graph. We highlight a number of important statistical features of those networks such as degree distribution, weight distribution, and several centrality measures as well as their interrelationships. While similar statistics were already known for other board systems, and are comparable here, we have extended the study with a careful investigation of director and board centrality, a k-core analysis, and a simulation of the speed of information propagation and its relationships with the topological aspects of the network such as clustering and link weight and betweenness. The overall picture that emerges is one in which the topological structure of the Swiss board and director networks has evolved in such a way that special actors and links between actors play a fundamental role in the flow of information among distant parts of the network. This is shown in particular by the centrality measures and by the simulation of a simple epidemic process on the directors network.Comment: Submitted to The European Physical Journal

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

Graphs with many valencies and few eigenvalues

Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs. As a side result, we characterize the disconnected graphs and the graphs with three distinct eigenvalues in the switching class of a regular two-graph

On the Threshold of Intractability

We study the computational complexity of the graph modification problems Threshold Editing and Chain Editing, adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are NP-complete, resolving a conjecture by Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128, 2001). On the positive side, we show the problem admits a quadratic vertex kernel. Furthermore, we give a subexponential time parameterized algorithm solving Threshold Editing in $2^{O(\surd k \log k)} + \text{poly}(n)$ time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (ISAAC, 2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to the related problem of Chain Editing, as well as the completion and deletion variants of both problems