Stochastic processes on complex networks: techniques and explorations

Networks are everywhere. From social networks to ecological networks, transportation networks to the World Wide Web, the existence of groups of units that are interconnected in some manner is ubiquitous in our modern-day world. In many cases, the units that comprise a network can be in one of a finite number of states. The state of a unit can dynamically change over time, and the network connections forming links between the units can facilitate this change in behaviour. Members of a social network, for example, can have a preference for one of a certain number of objects such as a sports team, musical genre or political ideology, and this preference may change as a result of influence from other individuals to which they are linked. The state of the individual units and the state of the group as a whole are intrinsically coupled, and the structure of the network and its complex connectivity patterns can affect group-wide behaviour such as conformity, factions, sustainability and extinction. In this thesis, we delve into the area of dynamical processes on networks where the units of the network can be in one of a multitude of states. The basis of our studies is the probabilistic framework of stochastic processes, reflecting the random nature ever present in natural and man-made phenomena. We develop novel theoretical methods for both Markovian and non-Markovian processes, expressed in terms of arbitrary rate functions that allow for wide applicability. We examine specific cases of stochastic processes from a diverse range of fields including epidemiology, sociology and the physical sciences, providing important contributions to matters of current scrutinous attention. Our work both expands and gives greater understanding to the study of stochastic processes on networks, and provides a unifying framework for the overarching field of complex systems science.