Modularity and community detection in bipartite networks

The modularity of a network quantifies the extent, relative to a null model network, to which vertices cluster into community groups. We define a null model appropriate for bipartite networks, and use it to define a bipartite modularity. The bipartite modularity is presented in terms of a modularity matrix B; some key properties of the eigenspectrum of B are identified and used to describe an algorithm for identifying modules in bipartite networks. The algorithm is based on the idea that the modules in the two parts of the network are dependent, with each part mutually being used to induce the vertices for the other part into the modules. We apply the algorithm to real-world network data, showing that the algorithm successfully identifies the modular structure of bipartite networks.Comment: RevTex 4, 11 pages, 3 figures, 1 table; modest extensions to conten

Grid-Obstacle Representations with Connections to Staircase Guarding

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an $xy$-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is \textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Continuum-plasma solution surrounding nonemitting spherical bodies

The classical problem of the interaction of a nonemitting spherical body with a zero mean-free-path continuum plasma is solved numerically in the full range of physically allowed free parameters (electron Debye length to body radius ratio, ion to electron temperature ratio, and body bias), and analytically in rigorously defined asymptotic regimes (weak and strong bias, weak and strong shielding, thin and thick sheath). Results include current-voltage characteristics as well as floating potential and capacitance, for both continuum and collisionless electrons. Our numerical computations show that for most combinations of physical parameters, there exists a closest asymptotic regime whose analytic solutions are accurate to 15% or better

Individual Differences in Cerebral Blood Flow in Area 17 Predict the Time to Evaluate Visualized Letters

Sixteen subjects closed their eyes and visualized uppercase letters of the alphabet at two sizes, as small as possible or as large as possible while remaining “visible.” Subjects evaluated a shape characteristic of each letter (e.g., whether it has any curved lines), and responded as quickly as possible. Cerebral blood flow was normalized to the same value for each subject, and relative blood flow was computed for a set of regions of interest. The mean response time for each subject in the task was regressed onto the blood flow values. Blood flow in area 17 was negatively correlated with response time (r = -0.65), as was blood flow in area 19 (r = -0.66), whereas blood flow in the inferior parietal lobe was positively correlated with response time (r = 0.54). The first two effects persisted even when variance due to the other correlations was removed. These findings suggest that individual differences in the activation of specific brain loci are directly related to performance of tasks that rely on processing in those loci.Psycholog

Universality in solar flare and earthquake occurrence

Earthquakes and solar flares are phenomena involving huge and rapid releases of energy characterized by complex temporal occurrence. By analysing available experimental catalogs, we show that the stochastic processes underlying these apparently different phenomena have universal properties. Namely both problems exhibit the same distributions of sizes, inter-occurrence times and the same temporal clustering: we find afterflare sequences with power law temporal correlations as the Omori law for seismic sequences. The observed universality suggests a common approach to the interpretation of both phenomena in terms of the same driving physical mechanism

On the Maximum Crossing Number

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 figure

Finding community structure in networks using the eigenvectors of matrices

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as "modularity" over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a new centrality measure that identifies those vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.Comment: 22 pages, 8 figures, minor corrections in this versio

Researching the use of force: The background to the international project

This article provides the background to an international project on use of force by the police that was carried out in eight countries. Force is often considered to be the defining characteristic of policing and much research has been conducted on the determinants, prevalence and control of the use of force, particularly in the United States. However, little work has looked at police officers’ own views on the use of force, in particular the way in which they justify it. Using a hypothetical encounter developed for this project, researchers in each country conducted focus groups with police officers in which they were encouraged to talk about the use of force. The results show interesting similarities and differences across countries and demonstrate the value of using this kind of research focus and methodology