Node-weighted measures for complex networks with spatially embedded, sampled, or differently sized nodes

When network and graph theory are used in the study of complex systems, a typically finite set of nodes of the network under consideration is frequently either explicitly or implicitly considered representative of a much larger finite or infinite region or set of objects of interest. The selection procedure, e.g., formation of a subset or some kind of discretization or aggregation, typically results in individual nodes of the studied network representing quite differently sized parts of the domain of interest. This heterogeneity may induce substantial bias and artifacts in derived network statistics. To avoid this bias, we propose an axiomatic scheme based on the idea of node splitting invariance to derive consistently weighted variants of various commonly used statistical network measures. The practical relevance and applicability of our approach is demonstrated for a number of example networks from different fields of research, and is shown to be of fundamental importance in particular in the study of spatially embedded functional networks derived from time series as studied in, e.g., neuroscience and climatology.Comment: 21 pages, 13 figure

Strategies for Generating and Evaluating Large-Scale Powerlaw-Distributed P2P Overlays

Part 1: Full Research PapersInternational audienceA very wide variety of physical, demographic, biological and man-made phenomena have been observed to exhibit powerlaw behavior, including the population of cities and villages, sizes of lakes, etc. The Internet is no exception to this. The connectivity of routers, the popularity of web sites, and the degrees of World Wide Web pages are only a few examples of measurements governed by powerlaw. The study of powerlaw networks has strong implications on the design and function of the Internet.Nevertheless, it is still uncertain how to explicitly generate such topologies at a very large scale. In this paper, we investigate the generation of P2P overlays following a powerlaw degree distribution. We revisit and identify weaknesses of existing strategies. We propose a new methodology for generating powerlaw topologies with predictable characteristics, in a completely decentralized, emerging way. We provide analytical support of our methodology and we validate it by large-scale (simulated) experiments