My proposed research concerns mathematical aspects of interaction networks with a special focus into peer-to-peer networks. In the past few years, many features have been investigated: small-world phenomenon, scale-free degree distribution and clustering. Motivated by the above properties that deviate form classical random graphs ones, new models of graphs have emerged. They constitute simple reductions of the description of complex systems. To date there has been very little rigorous mathematical work. This is the first and central objective of the project I propose to undertake. The study of most complex networks has been initiated by a desire to understand various real systems ranging from communication networks to ecological webs. Thus we need to focus on robust measures of the network topology. Moreover, the main hope in understanding the particular structure of real-world graphs seems to lie in formulating completely different models based around some alternative ensembles of structures. In addition, t he ultimate goal of the study of networks is to understand and explain the working of systems built upon those networks. Thus the next logical step after developing models of network structure is to look at the behaviour of physical processes occurring on these networks. As our society and economy grow more network-centric, Peer-to-Peer (P2P) architectures are emerging as a viable technical approach to the construction of distributed information processing and file sharing systems. P2P systems are formed from dynamic connections among autonomous computer nodes, producing large and complex application-layer overlay networks. These overlays in turn are built upon physical network infrastructures like the Internet. A challenging research problem is to develop overlay architectures, taking advantage of the above analysis that can support such applications without overloading systems.
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