Ricci Curvature of the Internet Topology

Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's "thin triangle condition", which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin, etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201

A survey on Human Mobility and its applications

Human Mobility has attracted attentions from different fields of studies such as epidemic modeling, traffic engineering, traffic prediction and urban planning. In this survey we review major characteristics of human mobility studies including from trajectory-based studies to studies using graph and network theory. In trajectory-based studies statistical measures such as jump length distribution and radius of gyration are analyzed in order to investigate how people move in their daily life, and if it is possible to model this individual movements and make prediction based on them. Using graph in mobility studies, helps to investigate the dynamic behavior of the system, such as diffusion and flow in the network and makes it easier to estimate how much one part of the network influences another by using metrics like centrality measures. We aim to study population flow in transportation networks using mobility data to derive models and patterns, and to develop new applications in predicting phenomena such as congestion. Human Mobility studies with the new generation of mobility data provided by cellular phone networks, arise new challenges such as data storing, data representation, data analysis and computation complexity. A comparative review of different data types used in current tools and applications of Human Mobility studies leads us to new approaches for dealing with mentioned challenges

Scaling and Universality in City Space Syntax: between Zipf and Matthew

We report about universality of rank-integration distributions of open spaces in city space syntax similar to the famous rank-size distributions of cities (Zipf's law). We also demonstrate that the degree of choice an open space represents for other spaces directly linked to it in a city follows a power law statistic. Universal statistical behavior of space syntax measures uncovers the universality of the city creation mechanism. We suggest that the observed universality may help to establish the international definition of a city as a specific land use pattern.Comment: 24 pages, 5 *.eps figure