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.

### The rate of innovation

The distribution of product complexity helps explain why some technology sectors tend to exhibit faster innovation rates than others.

### Grain shape inference

The distributions of size and shape of a material’s grains can be constructed from a 2D slice of the material and electron diffraction data.

### Exactly solvable random graph

Controlling analytically second or higher-order properties of networks is a great mathematical challenge.

### Serendipity and strategy

In systems of innovation, the relative usefulness of different components changes as the number of components we possess increases.

### Optimal growth rates

An extension of the Kelly criterion maximises the growth rate of multiplicative stochastic processes when limited resources are available.

### Clusters of neurons

Percolation theory shows that the formation of giant clusters of neurons relies on a few parameters that could be measured experimentally.

### Spin systems on hypercubic Bethe lattices: a Bethe–Peierls approach

Hypercubic Bethe lattices retain many of the loops of the topology of realistic spin systems.

### Immune networks: multitasking capabilities near saturation

The immune system must simultaneously recall multiple defense strategies because many antigens can attack the host at the same time.

### Reconstructing a credit network

New mathematical tools can help infer financial networks from partial data to understand the propagation of distress through the network.

### Weighted network evolution

A statistical procedure identifies dominant edges within weighted networks to determine whether a network has reached its steady state.

### Unbiased randomization

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

### Transfer operator analysis of the parallel dynamics of disordered Ising chains

The dynamics of one-dimensional Ising chains is of interest in the context of ageing phenomena.

### Tailored graph ensembles as proxies or null models for real networks II: results on directed graphs

Our approach gives a rigorous quantitative method for prioritising network properties.

### Tie knots and topology

The topological structure of tie knots categorises them by shape, size and aesthetic appeal and defines the sequence of knots to produce them.