# The structure of how things relate

Creating mathematical tools for characterizing the structure of ideal graphs and irregular networks, and the behaviour of processes on them.

Creating mathematical tools for characterizing the structure of ideal graphs and irregular networks, and the behaviour of processes on them.

**Background** Networks represent interactions between a number of similar objects. While the increasing availability of data makes it easy to build networks, extracting meaning from network structure remains a challenge. Part of the problem is that mathematical methods for characterizing networks are primitive and sundry. While the theory of graphs—idealized networks containing internal symmetry—is more sophisticated, the mathematical foundations for both networks and graphs, and their relation, are underdeveloped.

**Project** We investigate new mathematical methods to describe the geometry and topology of networks and graphs, without regard to the kind of data from which they are derived. The techniques under development are broad and include applications and extensions of spectral graph theory, tailored graph ensembles, replica methods, community detection and combinatorics.

**Consequences** A more complete understanding of the structure of networks and graphs underpins many other research endeavours. Examples range from biomedical data for personalized medicine to predicting sentiment and behaviour from social interaction networks to the stability of interconnected financial institutions.

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