Understanding the structure and the evolution of spatial networks is crucial for many different fields ranging from urbanism to epidemiology. An important consequence of space on networks is that there is a cost associated to the length of edges which in turn has dramatic effects on their topological structure. Another consequence is that most standard measures for complex networks are irrelevant for this class of graphs and I will review here some of the most interesting and promising measures for characterizing these graphs and their time evolution. In particular, I will illustrate on various real-world examples the simplicity profile, the spatial distribution of the betweenness centrality and, if time allows, the shape distribution of faces.
The SINR coverage process was introduced in stochastic geometry in 2001 to model coverage phenomena in wireless communications. Since that time it received a lot of attention, both in engineering and applied probability community. It is called the shot-noise coverage process in the recent edition of the famous book on Stochastic Geometry and Its Applications. In our lecture we shall focus on some more recent results regarding this coverage model, related to Poisson-Dirichlet processes, which appear in several different contexts, in biology, economics and physics.
We propose a broad geometric view of real-world complex networks, inspired by seminal work on Blau space for social networks. We describe how the examination of mathematical models for on-line social networks led to a hypothesis regarding their dimension. A thesis is presented on how all complex networks possess an underlying geometry related to but distinct from their underlying graph distance metric space.
In the last few years, stochastic geometry has been widely used for system-level modeling, performance evaluation and optimization of several candidate system architectures, network topologies, and transmission technologies for next-generation cellular networks. At present, many tractable mathematical methodologies for analyzing and optimizing (heterogeneous) cellular networks in terms of spectral efficiency, coverage and error probability exist. In this talk, the speaker will first review the most recent approaches that have been proposed for modeling emerging cellular network concepts, by focusing his attention on the many assumptions that are usually made “for mathematical tractability”. Subsequently, the speaker will comment the implications that these assumptions entail for system design and optimization. Finally, the speaker will discuss the merits and demerits of stochastic geometry for modeling cellular networks, as well as its claimed mathematical tractability, which is considered to be its main strength by many but an unjustified hype by many others.
If you are handed a network, can you tell if it is embedded in a low-dimensional space? If you know, or believe, that points closer in space are more likely to be connected than those further apart then the answer is yes, and there are simple techniques, such as spectral methods, that will give you an answer. But what if this assumption is incorrect? For instance, airline networks have a decreased chance of edges at short distances because airlines rarely fly between very closely spaced airports. In this talk I will present statistical data on the types of connection patterns that actually occur in spatial networks and describe a generalized method for inferring "latent space" structure from network topology, no matter what the connection patterns. I will also present work on visualization of spatial networks, which can be a tricky task when network structure is affected by external factors such as population density.