Cliques in high-dimensional random geometric graphs

International audienceRandom geometric graphs have become now a popular object of research. Defined rather simply, these graphs describe real networks much better than classical Erdős–Rényi graphs due to their ability to produce tightly connected communities. The $n$ vertices of a random geometric graph are points in $d$-dimensional Euclidean space, and two vertices are adjacent if they are close to each other. Many properties of these graphs have been revealed in the case when $d$ is fixed. However, the case of growing dimension $d$ is practically unexplored. This regime corresponds to a real-life situation when one has a data set of n observations with a significant number of features, a quite common case in data science today. In this paper, we study the clique structure of random geometric graphs when $n \to \infty$, and $d \to \infty$, and average vertex degree grows significantly slower than $n$. We show that under these conditions, random geometric graphs do not contain cliques of size 4 a.s. if only $d >> \log^{1+\epsilon}(n)$. As for the cliques of size 3, we present new bounds on the expected number of triangles in the case $\log^2(n) << d << \log^3(n)$ that improve previously known results. In addition, we provide new numerical results showing that the underlying geometry can be detected using the number of triangles even for small $n$

Climate Dynamics: A Network-Based Approach for the Analysis of Global Precipitation

Precipitation is one of the most important meteorological variables for defining the climate dynamics, but the spatial patterns of precipitation have not been fully investigated yet. The complex network theory, which provides a robust tool to investigate the statistical interdependence of many interacting elements, is used here to analyze the spatial dynamics of annual precipitation over seventy years (1941-2010). The precipitation network is built associating a node to a geographical region, which has a temporal distribution of precipitation, and identifying possible links among nodes through the correlation function. The precipitation network reveals significant spatial variability with barely connected regions, as Eastern China and Japan, and highly connected regions, such as the African Sahel, Eastern Australia and, to a lesser extent, Northern Europe. Sahel and Eastern Australia are remarkably dry regions, where low amounts of rainfall are uniformly distributed on continental scales and small-scale extreme events are rare. As a consequence, the precipitation gradient is low, making these regions well connected on a large spatial scale. On the contrary, the Asiatic South-East is often reached by extreme events such as monsoons, tropical cyclones and heat waves, which can all contribute to reduce the correlation to the short-range scale only. Some patterns emerging between mid-latitude and tropical regions suggest a possible impact of the propagation of planetary waves on precipitation at a global scale. Other links can be qualitatively associated to the atmospheric and oceanic circulation. To analyze the sensitivity of the network to the physical closeness of the nodes, short-term connections are broken. The African Sahel, Eastern Australia and Northern Europe regions again appear as the supernodes of the network, confirming furthermore their long-range connection structure. Almost all North-American and Asian nodes vanish, revealing that extreme events can enhance high precipitation gradients, leading to a systematic absence of long-range patterns

SPoT: Representing the Social, Spatial, and Temporal Dimensions of Human Mobility with a Unifying Framework

Modeling human mobility is crucial in the analysis and simulation of opportunistic networks, where contacts are exploited as opportunities for peer-topeer message forwarding. The current approach with human mobility modeling has been based on continuously modifying models, trying to embed in them the mobility properties (e.g., visiting patterns to locations or specific distributions of inter-contact times) as they came up from trace analysis. As a consequence, with these models it is difficult, if not impossible, to modify the features of mobility or to control the exact shape of mobility metrics (e.g., modifying the distribution of inter-contact times). For these reasons, in this paper we propose a mobility framework rather than a mobility model, with the explicit goal of providing a exible and controllable tool for modeling mathematically and generating simulatively different possible features of human mobility. Our framework, named SPoT, is able to incorporate the three dimensions - spatial, social, and temporal - of human mobility. The way SPoT does it is by mapping the different social communities of the network into different locations, whose members visit with a configurable temporal pattern. In order to characterize the temporal patterns of user visits to locations and the relative positioning of locations based on their shared users, we analyze the traces of real user movements extracted from three location-based online social networks (Gowalla, Foursquare, and Altergeo). We observe that a Bernoulli process effectively approximates user visits to locations in the majority of cases and that locations that share many common users visiting them frequently tend to be located close to each other. In addition, we use these traces to test the exibility of the framework, and we show that SPoT is able to accurately reproduce the mobility behavior observed in traces. Finally, relying on the Bernoulli assumption for arrival processes, we provide a throughout mathematical analysis of the controllability of the framework, deriving the conditions under which heavy-tailed and exponentially-tailed aggregate inter-contact times (often observed in real traces) emerge