Our papers are the official record of our discoveries. They allow others to build on and apply our work. Each paper is the result of many months of research, so we make a special effort to make them clear, beautiful and inspirational, and publish them in leading journals.

### QFT and kids’ drawings

Groethendieck's “children’s drawings”, a type of bipartite graph, link number theory, geometry, and the physics of conformal field theory.

### Transitions in loopy graphs

The generation of large graphs with a controllable number of short loops paves the way for building more realistic random networks.

### Memristive networks

A simple solvable model of memristive networks suggests a correspondence between the asymptotic states of memristors and the Ising model.

### Hypercube eigenvalues

Hamming balls, subgraphs of the hypercube, maximise the graph’s largest eigenvalue exactly when the dimension of the cube is large enough.

### Exactly solvable random graphs

An explicit analytical solution reproduces the main features of random graph ensembles with many short cycles under strict degree constraints.

### Spectral partitioning

The spectral density of graph ensembles provides an exact solution to the graph partitioning problem and helps detect community structure.

### Dynamics of memristors

Exact equations of motion provide an analytical description of the evolution and relaxation properties of complex memristive circuits.

### Democracy in networks

Analysis of the hyperbolicity of real-world networks distinguishes between those which are aristocratic and those which are democratic.

### Erdős-Ko-Rado theorem analogue

A random analogue of the Erdős-Ko-Rado theorem sheds light on its stability in an area of parameter space which has not yet been explored.

### Maximum percolation time

A simple formula gives the maximum time for an n x n grid to become entirely infected having undergone a bootstrap percolation process.

### Easily repairable networks

When networks come under attack, a repairable architecture is superior to, and globally distinct from, an architecture that is robust.

### Percolation on Galton-Watson trees

The critical probability for bootstrap percolation, a process which mimics the spread of an infection in a graph, is bounded for Galton-Watson trees.

### Scales in weighted networks

Information theory fixes weighted networks’ degeneracy issues with a generalisation of binary graphs and an optimal scale of link intensities.

### Unbiased randomization

Unbiased randomisation processes generate sophisticated synthetic networks for modelling and testing the properties of real-world networks.

### Tailored random graph ensembles

New mathematical tools quantify the topological structure of large directed networks which describe how genes interact within a cell.