For many years networks have been a fruitful source of study for mathematicians, one of the first notable examples of network analysis being Leonard Euler's study of paths on the Königsberg bridges. Since that time the field of graph theory and network science has developed greatly and the problems we want to model have also changed.
Perhaps the most evident modern-day networks are those of online social networks such as Facebook, Twitter, and Instagram. Unlike the networks of the Königsberg bridges which are literally set in stone, the networks of interactions and friendships between users of these networks can grow and disappear within minutes or hours. This means we need to develop new tools and algorithms to be able to analyse them fully.
Working with the data-focused Bloom Agency, Oxford Mathematician Andrew Mellor's goal is to analyse, in real time, the conversations occurring on social media surrounding brands, ideas, and reaction to real world events and television shows. To do this he developed the temporal event graph, a static representation of a temporal network. In the temporal event graph our interactions are now nodes, and interactions are linked if they share participants and occur close together in time. Looking at temporal networks in this way captures both the topology of the interactions between agents and the timings between these connections. This complex interplay between topology and temporal connectivity has a profound effect on the spread of epidemics (or viral content) on the network. We can use methods derived for static networks to efficiently analyse the network. These methods are not restricted to the digital domain and can in fact be applied to any sequence of interactions such as flight and transport networks, brain networks, and proximity networks.
Returning to the original application, Andrew and colleagues are currently using the temporal event graph to understand how conversations evolve online, how a user's behaviour changes depending on the conversational topic, and to classify the zoo of conversation types that have been observed. This gives a unique insight into how we behave online, and gives new methods of characterising behaviour as a function of time.