Network science, the study of large, interconnected complex systems, has been the focus of much effort on the part of many scientists in recent years. The modeling and analysis of natural, engineered, and social networks, ranging from molecular and biological networks to the Internet, the World Wide Web, and also to online social media like Facebook and Twitter, has emerged as an important multidisciplinary field of study. Progress in this area is rapidly affecting many traditional fields of science and engineering as well as areas like public health, business, marketing, finance, and national security. The project concerns the development and analysis of computational methods for studying the structure of networks and of certain dynamical processes taking place on them. In particular, the project will develop mathematically sound approaches for maximally increasing the connectivity and robustness of networks when only a limited number of links can be added or modified. Network robustness is of paramount importance when designing infrastructure networks, since real world networks must be resilient to damage resulting from natural disasters or by man-made, malicious attacks. In addition, the project will develop efficient techniques for simulating the spread of information, viruses, rumors, etc. on networks, as well as the propagation of shocks. The techniques will be implemented on computers and tested on existing available data sets, and the software will be made available to other researchers.
In the first part of the project we will design efficient algorithms for increasing network robustness and connectivity using as an objective function the total communicability of the network, which is a scalar quantity associated with the adjacency matrix of the graph. In particular, we will investigate different criteria for the addition of links in such a way as to (approximately) maximize the increase in total communicability. Edge rewiring and deletion will also be studied. The second part of the project deals with the numerical analysis of so-called Quantum Graphs. We will study numerical methods, including finite element discretizations and fast linear algebra, for solving PDEs posed on metric graphs (1D simplicial complexes), such as the diffusion and Schroedinger equations. In particular, we will study domain decomposition and iterative substructuring methods for solving the large linear systems arising when solving elliptic and parabolic problems on graphs. Exponential integrators based on Krylov subspace methods will also be investigated.