Dynamics on clustered networks: mathematical modelling and empirical analysis

The world is full of networks. Wherever there is a set of entities interacting in a well-defined way, it can be represented as a network. Some examples are social networks, telecommunication networks and the world wide web. In this thesis we study how things spread on networks, and in particular how the network structure impacts the network dynamics. We model social spreading processes which can have different dynamics to other settings and interact differently with the network structure.

It has been empirically shown in some settings that the adoption of behaviour is more likely after multiple exposures to the behaviour, given that the individual has not already adopted [15]. This type of adoption mechanism is known as a complex contagion. While cluster?ing in networks has been shown to inhibit simple contagions — where there is no social-reinforcement mechanism — for complex contagions, clustering can lead to faster spread [64]. Mathematical models for dy?namics on clustered networks are lacking in comparison to those for locally tree-like networks.

In this thesis, we introduce a framework for modelling complex contagion on networks with different clustering levels. We show how, using this framework we can derive analytical results for some very simple toy networks. These results include the cascade condition and the expected cascade size and find that depending on the level of social reinforcement, we get the largest outbreaks for networks with different levels of clustering. We then go on to produce results for a more general family of clustered networks, deriving full distributions of cascade properties.

While much of this thesis is concerned with developing model theory, we feel that it is important to keep in mind the real-world processes that we model. In the final part of this thesis, we explore the spread of a hashtag relating to an Irish cultural event on Twitter. The dynamics of this are interesting because there are two languages at play; Irish and English. In this piece of research we begin to explore how the Irish language is used online.