Title:

Mathematical analysis of kpath Laplacian operators on simple graphs

A set of links and nodes are the fundamental units or components used to represent complex networks. Over the last few decades, network studies have expanded and matured, increasingly making use of complex mathematical tools. Complex networks play a significant role in the propagation of processes, which include for example the case of epidemic spreading, the diffusion process, synchronisation or the consensus process. Such dynamic processes are critically important in achieving understanding of the behaviour of complex systems at different levels of complexity  examples might be the brain and modern manmade infrastructures. Although part of the study of the diffusion of information in the dynamic processes, it is generally supposed that interactions in networks originate only from a node, spreading to its nearest neighbours, there also exist longrange interactions (LRI), which can be transmitted from a node to others not directly connected. The focus of this study is on dynamic processes on networks where nodes interact with not only their nearest neighbours but also through certain LRIs. The generalised kpath Laplacian operators (LOs) Lk, which account for the hop of a diffusive particle to its nonnearest neighbours in a graph, control this diffusive process, describing hops of nodes vi at distance k; here the distance is measured as the length of the shortest path between two nodes. In this way the introduction of the kpath LOs can facilitate conducting more precise studies of network dynamics in different applications. This thesis aims to study a generalised diffusion equation employing the transformed generalised kpath LOs for a locally finite infinite graph. This generalised diffusion equation promotes both normal and superdiffusive processes on infinite graphs. Furthermore, this thesis develops a new theoretical mathematical framework for describing superdiffusion processes that use a transform of the kpath LOs defined on infinite graphs. The choice of the transform appeared to be vitally important as the probability of a long jump should be great enough. As described by other researchers the fractional diffusion equation (FDE) formed the mathematical framework employed to describe this anomalous diffusion. In this regard, it is taken that the diffusive particle is not just hopping to its nearest node but also to any other node of the network with a probability that scales according to the distance between the two places. Initially, we extend the kpath LOs above to consider a connected and locally finite infinite network with a bounded degree and investigate a number of the properties of these operators, such as their selfadjointness and boundedness. Then, three different transformations of the kpath LOs, i.e. the Laplace, Factorial and Mellin transformations as well as their properties, are studied. In addition, in order to show a number of applications of these operators and the transformed ones, the transformed kpath LOs are used to obtain a generalised diffusion process for onedimensional and twodimensional infinite graphs. First, the infinite path graph is studied, where it is possible to prove that when the Laplacian and factorialtransformed operators are used in the generalised diffusion equation, the diffusive processes observed are always normal, independent of the transform parameters. It is then proven analytically that when the kpath LOs are transformed via a Mellin transform and plugged into the diffusion equation, the result is a superdiffusive process for certain values of the exponent in the transform. Secondly, we generalise the results on the superdiffusive behaviour generated by transforming kpath LOs from onedimensional graphs to 2dimensional ones. Our attention focuses on the Abstract Cauchy problem in an infinite square lattice. A generalised diffusion equation on a square lattice corresponding to Mellin transforms of the kpath Laplacian is investigated. Similar to the onedimensional case also for the graph embedded in twodimensional space, we could observe superdiffusive behaviour for the Mellin transformed kpath Laplacian. In comparison to the onedimensional case, the conclusion reached is that the asymptotic behaviour of the solution of the Cauchy problem is much subtler.
