The Triangle Groups (2, 4, 5) and (2, 5, 5) are not Systolic

In this paper we provide new examples of hyperbolic but nonsystolic groups by showing that the triangle groups (2, 4, 5) and (2, 5, 5) are not systolic. Along the way we prove some results about subsets of systolic complexes stable under involutions

A Spectral Algorithm with Additive Clustering for the Recovery of Overlapping Communities in Networks

This paper presents a novel spectral algorithm with additive clustering designed to identify overlapping communities in networks. The algorithm is based on geometric properties of the spectrum of the expected adjacency matrix in a random graph model that we call stochastic blockmodel with overlap (SBMO). An adaptive version of the algorithm, that does not require the knowledge of the number of hidden communities, is proved to be consistent under the SBMO when the degrees in the graph are (slightly more than) logarithmic. The algorithm is shown to perform well on simulated data and on real-world graphs with known overlapping communities.Comment: Journal of Theoretical Computer Science (TCS), Elsevier, A Para\^itr

From Relational Data to Graphs: Inferring Significant Links using Generalized Hypergeometric Ensembles

The inference of network topologies from relational data is an important problem in data analysis. Exemplary applications include the reconstruction of social ties from data on human interactions, the inference of gene co-expression networks from DNA microarray data, or the learning of semantic relationships based on co-occurrences of words in documents. Solving these problems requires techniques to infer significant links in noisy relational data. In this short paper, we propose a new statistical modeling framework to address this challenge. It builds on generalized hypergeometric ensembles, a class of generative stochastic models that give rise to analytically tractable probability spaces of directed, multi-edge graphs. We show how this framework can be used to assess the significance of links in noisy relational data. We illustrate our method in two data sets capturing spatio-temporal proximity relations between actors in a social system. The results show that our analytical framework provides a new approach to infer significant links from relational data, with interesting perspectives for the mining of data on social systems.Comment: 10 pages, 8 figures, accepted at SocInfo201

Percolation in the classical blockmodel

Classical blockmodel is known as the simplest among models of networks with community structure. The model can be also seen as an extremely simply example of interconnected networks. For this reason, it is surprising that the percolation transition in the classical blockmodel has not been examined so far, although the phenomenon has been studied in a variety of much more complicated models of interconnected and multiplex networks. In this paper we derive the self-consistent equation for the size the global percolation cluster in the classical blockmodel. We also find the condition for percolation threshold which characterizes the emergence of the giant component. We show that the discussed percolation phenomenon may cause unexpected problems in a simple optimization process of the multilevel network construction. Numerical simulations confirm the correctness of our theoretical derivations.Comment: 7 pages, 6 figure

Effects of Contact Network Models on Stochastic Epidemic Simulations

The importance of modeling the spread of epidemics through a population has led to the development of mathematical models for infectious disease propagation. A number of empirical studies have collected and analyzed data on contacts between individuals using a variety of sensors. Typically one uses such data to fit a probabilistic model of network contacts over which a disease may propagate. In this paper, we investigate the effects of different contact network models with varying levels of complexity on the outcomes of simulated epidemics using a stochastic Susceptible-Infectious-Recovered (SIR) model. We evaluate these network models on six datasets of contacts between people in a variety of settings. Our results demonstrate that the choice of network model can have a significant effect on how closely the outcomes of an epidemic simulation on a simulated network match the outcomes on the actual network constructed from the sensor data. In particular, preserving degrees of nodes appears to be much more important than preserving cluster structure for accurate epidemic simulations.Comment: To appear at International Conference on Social Informatics (SocInfo) 201

Spatial correlations in attribute communities

Community detection is an important tool for exploring and classifying the properties of large complex networks and should be of great help for spatial networks. Indeed, in addition to their location, nodes in spatial networks can have attributes such as the language for individuals, or any other socio-economical feature that we would like to identify in communities. We discuss in this paper a crucial aspect which was not considered in previous studies which is the possible existence of correlations between space and attributes. Introducing a simple toy model in which both space and node attributes are considered, we discuss the effect of space-attribute correlations on the results of various community detection methods proposed for spatial networks in this paper and in previous studies. When space is irrelevant, our model is equivalent to the stochastic block model which has been shown to display a detectability-non detectability transition. In the regime where space dominates the link formation process, most methods can fail to recover the communities, an effect which is particularly marked when space-attributes correlations are strong. In this latter case, community detection methods which remove the spatial component of the network can miss a large part of the community structure and can lead to incorrect results.Comment: 10 pages and 7 figure

Random Walks on Stochastic Temporal Networks

In the study of dynamical processes on networks, there has been intense focus on network structure -- i.e., the arrangement of edges and their associated weights -- but the effects of the temporal patterns of edges remains poorly understood. In this chapter, we develop a mathematical framework for random walks on temporal networks using an approach that provides a compromise between abstract but unrealistic models and data-driven but non-mathematical approaches. To do this, we introduce a stochastic model for temporal networks in which we summarize the temporal and structural organization of a system using a matrix of waiting-time distributions. We show that random walks on stochastic temporal networks can be described exactly by an integro-differential master equation and derive an analytical expression for its asymptotic steady state. We also discuss how our work might be useful to help build centrality measures for temporal networks.Comment: Chapter in Temporal Networks (Petter Holme and Jari Saramaki editors). Springer. Berlin, Heidelberg 2013. The book chapter contains minor corrections and modifications. This chapter is based on arXiv:1112.3324, which contains additional calculations and numerical simulation

Temporal Evolution of the Solar-Wind Electron Core Density at Solar Minimum by Correlating SWEA Measurements from STEREO A and B

The twin STEREO spacecraft provide a unique tool to study the temporal evolution of the solar-wind properties in the ecliptic since their longitudinal separation increases with time. We derive the characteristic temporal variations at ∼ 1 AU between two different plasma parcels ejected from the same solar source by excluding the spatial variations from our datasets. As part of the onboard IMPACT instrument suite, the SWEA electron experiment provides the solar-wind electron core density at two different heliospheric vantage points. We analyze these density datasets between March and August 2007 and find typical solar minimum conditions. After adjusting for the theoretical time lag between the two spacecraft, we compare the two density datasets. We find that their correlation decreases as the time difference increases between two ejections. The correlation coefficient is about 0.80 for a time lag of a half day and 0.65 for two days. These correlation coefficients from the electron core density are somewhat lower than the ones from the proton bulk velocity obtained in an earlier study, though they are still high enough to consider the solar wind as persistent after two days. These quantitative results reflect the variability of the solar-wind properties in space and time, and they might serve as input for solar-wind models

Solar-Wind Bulk Velocity Throughout the Inner Heliosphere from Multi-Spacecraft Measurements

We extrapolate solar-wind bulk velocity measurements for different in-ecliptic heliospheric positions by calculating the theoretical time lag between the locations. The solar-wind bulk velocity dataset is obtained from in-situ plasma measurements by STEREO A and B, SOHO, Venus Express, and Mars Express. During their simultaneous measurements between 2007 and 2009 we find typical solar activity minimum conditions. In order to validate our extrapolations of the STEREO A and B data, we compare them with simultaneous in-situ observations from the other spacecraft. This way of cross-calibration we obtain a measure for the goodness of our extrapolations over different heliospheric distances. We find that a reliable solar-wind dataset can be provided in case of a longitudinal separation less than 65 degrees. Moreover, we find that the time lag method assuming constant velocity is a good basis to extrapolate from measurements in Earth orbit to Venus or to Mars. These extrapolations might serve as a good solar-wind input information for planetary studies of magnetospheric and ionospheric processes. We additionally show how the stream-stream interactions in the ecliptic alter the bulk velocity during radial propagation