Operator Graph Theory - A new way to look at dynamical systems?

Classically, a (relational) graph is defined as a tuple {N,E} of a set of nodes N connected by a set of edges E. The resulting adjacency relations uniquely describe a given graph, and form the base on which all graph-theoretical quantities, such as the geodesic distance between nodes or their clustering, can be defined. Although for some specific regular graphs exact analytical expressions for many of these quantities do exist, the class of graphs most widely used in physical sciences, random graphs, unfortunately allow for only asymptotic approaches. In many cases, the latter are, at best, only poor approximations of finite and discrete real-world phenomena.

Borrowing from ideas in quantum field theory, a novel approach to graph theory, called Operator Graph Theory (OGT), will be presented, which views, in contrast to the classical definition, graphs as dynamic objects. The mathematical framework underpinning this approach is constructivism, which in conjunction with discrete, ultrafinite mathematics allows for a rigorous algebraic description of finite graphs. The importance of this new approach cannot be overstated, as to-date most results in applied graph theory are of asymptotic or approximative nature. Some applications of the operator graph-theoretical framework are highlighted, including the rigorous analysis of brain networks and the percolation phenomenon, as well as the extension of the OGT framework to the analytical treatment of complex networks.