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.

### True scale-free networks

Naturally occurring networks have an underlying scale-free structure that is often clouded by finite-size effects in the sample data.

### Recursively divisible numbers

Recursively divisible numbers are a new kind of number that are highly divisible, whose quotients are highly divisible, and so on.

### The physics of financial networks

Complex network theory unlocks systematic understanding of financial stability and climate finance in pursuit of a more sustainable society.

### Transitions in loopy random graphs

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

### Microstructural coarsening

Rapid temperature cycling from one extreme to another affects the rate at which the mean particle size in solid or liquid solutions changes.

### Exact linear regression

Exact methods supersede approximations used in high-dimensional linear regression to find correlations in statistical physics problems.

### Deep layered machines

The ability of deep neural networks to generalize can be unraveled using path integral methods to compute their typical Boolean functions.

### Ample & pristine numbers

Parallels between the perfect and abundant numbers and their recursive analogs point to deeper structure in the recursive divisor function.

### Replica analysis of overfitting

Statistical methods that normally fail for very high-dimensional data can be rescued via mathematical tools from statistical physics.

### Network valuation in financial systems

Consistent valuation of interbank claims within an interconnected financial system can be found with a recursive update of banks' equities.

### Taming complexity

Insights from biology, physics and business shed light on the nature and costs of complexity and how to manage it in business organizations.

### Replica clustering

We optimize Bayesian data clustering by mapping the problem to the statistical physics of a gas and calculating the lowest entropy state.

### Recursive structure of innovation

A theoretical model of recursive innovation suggests that new technologies are recursively built up from new combinations of existing ones.

### Bursting dynamic networks

A mathematical model captures the temporal and steady state behaviour of networks whose two sets of nodes either generate or destroy links.

### Geometry of discrete space

A phase transition creates the geometry of the continuum from discrete space, but it needs disorder if it is to have the right metric.

### Energy harvesting with AI

Machine learning techniques enhance the efficiency of energy harvesters by implementing reversible energy-conserving operations.

### Scale of non-locality

The number of particles in a higher derivative theory of gravity relates to its effective mass scale, which signals the theory’s viability.

### Renewable resource management

Modern portfolio theory inspires a strategy for allocating renewable energy sources which minimises the impact of production fluctuations.

### Memristive networks

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

### Physics of networks

Statistical physics harnesses links between maximum entropy and information theory to capture null model and real-world network features.

### The rate of innovation

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

### A Hamiltonian recipe

An explicit recipe for defining the Hamiltonian in general probabilistic theories, which have the potential to generalise quantum theory.

### 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.

### Solvable memristive circuits

Exact solutions for the dynamics of interacting memristors predict whether they relax to higher or lower resistance states given random initialisations.

### Information asymmetry

Network users who have access to the network’s most informative node, as quantified by a novel index, the InfoRank, have a competitive edge.

### One-shot statistic

One-shot analogs of fluctuation-theorem results help unify these two approaches for small-scale, nonequilibrium statistical physics.

### Hypercube eigenvalues

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

### Volunteer clouds

A novel approach to volunteer clouds outperforms traditional distributed task scheduling algorithms in the presence of intensive workloads.

### From ecology to finance

Bipartite networks model the structures of ecological and economic real-world systems, enabling hypothesis testing and crisis forecasting.

### Forecasting technology deployment

Forecast errors for simple experience curve models facilitate more reliable estimates for the costs of technology deployment.

### Hierarchies in directed networks

An iterative version of a method to identify hierarchies and rankings of nodes in directed networks can partly overcome its resolution limit.

### The interbank network

The large-scale structure of the interbank network changes drastically in times of crisis due to the effect of measures from central banks.

### Exactly solvable random graphs

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

### The science of strategy

The usefulness of components and the complexity of products inform the best strategy for innovation at different stages of the process.

### Serendipity and strategy

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

### Modelling financial systemic risk

Complex networks model the links between financial institutions and how these channels can transition from diversifying to propagating risk.

### Bayesian analysis of medical data

Bayesian networks describe the evolution of orthodontic features on patients receiving treatment versus no treatment for malocclusion.

### Dirac cones in 2D borane

Theoretical searches propose 2D borane as a new graphene-like material which is stable and semi-metallic with Dirac cone structure.

### Quantum neural networks

We generalise neural networks into a quantum framework, demonstrating the possibility of quantum auto-encoders and teleportation.

### Financial network reconstruction

Statistical mechanics concepts reconstruct connections between financial institutions and the stock market, despite limited data disclosure.

### Bipartite trade network

A new algorithm unveils complicated structures in the bipartite mapping between countries and products of the international trade network.

### Quantum thermodynamics

Spectroscopy experiments show that energy shifts due to photon emission from individual molecules satisfy a fundamental quantum relation.

### Debunking in a world of tribes

When people operate in echo chambers, they focus on information adhering to their system of beliefs. Debunking them is harder than it seems

### 3d grains from 2d slices

Moment-based methods provide a simple way to describe a population of spherical particles and extract 3d information from 2d measurements.

### Spectral partitioning

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

### Memristive networks and learning

Memristive networks preserve memory and have the ability to learn according to analysis of the network’s internal memory dynamics.

### Worst-case work entropic equality

A new equality which depends on the maximum entropy describes the worst-case amount of work done by finite-dimensional quantum systems.

### The secret structure of innovation

Firms can harness the shifting importance of component building blocks to build better products and services and hence increase their chances of sustained success.

### Dynamics of memristors

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

### Pathways towards instability

Processes believed to stabilize financial markets can drive them towards instability by creating cyclical structures that amplify distress.

### Disentangling links in networks

Inference from single snapshots of temporal networks can misleadingly group communities if the links between snapshots are correlated.

### Optimal heat exchange networks

Compact heat exchangers can be designed to run at low power if the exchange is concentrated in a crumpled surface fed by a fractal network.

### Non-linear distress propagation

Non-linear models of distress propagation in financial networks characterise key regimes where shocks are either amplified or suppressed.

### Immunisation of systemic risk

Targeted immunisation policies limit distress propagation and prevent system-wide crises in financial networks according to sandpile models.

### Optimal growth rates

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

### Tunnelling interpreted

In quantum tunnelling, a particle tunnels through a barrier that it classically could not surmount.

### The price of complexity

Increasing the complexity of the network of contracts between financial institutions decreases the accuracy of estimating systemic risk.

### Form and function in gene networks

The structural properties of a network motif predict its functional versatility and relate to gene regulatory networks.

### Cascades in flow networks

Coupled distribution grids are more vulnerable to a cascading systemic failure but they have larger safe regions within their networks.

### Self-organising adaptive networks

An adaptive network of oscillators in fragmented and incoherent states can re-organise itself into connected and synchronized states.

### Instability in complex ecosystems

The community matrix of a complex ecosystem captures the population dynamics of interacting species and transitions to unstable abundances.

### Clusters of neurons

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

### Predicting technological progress

A formulation of Moore’s law estimates the probability that a given technology will outperform another at a certain point in the future.

### Cyclic isotropic cosmologies

In an infinitely bouncing Universe, the scalar field driving the cosmological expansion and contraction carries information between phases.

### Eigenvalues of neutral networks

The principal eigenvalue of small neutral networks determines their robustness, and is bounded by the logarithm of the number of vertices.

### Photonic Maxwell's demon

With inspiration from Maxwell’s classic thought experiment, it is possible to extract macroscopic work from microscopic measurements of photons.

### News sentiment and price dynamics

News sentiment analysis and web browsing data are unilluminating alone, but inspected together, predict fluctuations in stock prices.

### Bootstrap percolation models

A subset of bootstrap percolation models, which stabilise systems of cells on infinite lattices, exhibit non-trivial phase transitions.

### Communities in networks

A new tool derived from information theory quantitatively identifies trees, hierarchies and community structures within complex networks.

### Effect of Twitter on stock prices

When the number of tweets about an event peaks, the sentiment of those tweets correlates strongly with abnormal stock market returns.

### Democracy in networks

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

### Protein interaction experiments

Properties of protein interaction networks test the reliability of data and hint at the underlying mechanism with which proteins recruit each other.

### 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.

### Collective attention to politics

Tweet volume is a good indicator of political parties' success in elections when considered over an optimal time window so as to minimise noise.

### A measure of majorization

Single-shot information theory inspires a new formulation of statistical mechanics which measures the optimal guaranteed work of a system.

### Spin systems on Bethe lattices

Exact equations for the thermodynamic quantities of lattices made of d-dimensional hypercubes are obtainable with the Bethe-Peierls approach.

### DebtRank and shock propagation

A dynamical microscopic theory of instability for financial networks reformulates the DebtRank algorithm in terms of basic accounting principles.

### Organized knowledge economies

The Yule-Simon distribution describes the diffusion of knowledge and ideas in a social network which in turn influences economic growth.

### Structure and stability of salts

The stable structures of calcium and magnesium carbonate at high pressures are crucial for understanding the Earth's deep carbon cycle.

### Fragility of the interbank network

The speed of a financial crisis outbreak sets the maximum delay before intervention by central authorities is no longer effective.

### From memory to scale-free

A local model of preferential attachment with short-term memory generates scale-free networks, which can be readily computed by memristors.

### Dynamics of economic complexity

Dynamical systems theory predicts the growth potential of countries with heterogeneous patterns of evolution where regression methods fail.

### 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.

### Taxonomy and economic growth

Less developed countries have to learn simple capabilities in order to start a stable industrialization and development process.

### Random graphs with short loops

The analysis of real networks which contain many short loops requires novel methods, because they break the assumptions of tree-like models.

### Viscosity of polydisperse spheres

A quick and simple way to evaluate the packing fraction of polydisperse spheres, which is a measure of how they crowd around each other.

### Networks of credit default swaps

Time series data from networks of credit default swaps display no early warnings of financial crises without additional macroeconomic indicators.

### Entropies of graph ensembles

Explicit formulae for the Shannon entropies of random graph ensembles provide measures to compare and reproduce their topological features.

### Easily repairable networks

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

### Entanglement typicality

A review of the achievements concerning typical bipartite entanglement for random quantum states involving a large number of particles.

### Predicting interface structures

Generating random structures in the vicinity of a material’s defect predicts the low and high energy atomic structure at the grain boundary.

### 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.

### Memory effects in stock dynamics

The likelihood of stock prices bouncing on specific values increases due to memory effects in the time series data of the price dynamics.

### Self-healing complex networks

The interplay between redundancies and smart reconfiguration protocols can improve the resilience of networked infrastructures to failures.

### Structural imperfections

Fractal structures need very little mass to support a load; but for current designs, this makes them vulnerable to manufacturing errors.

### Default cascades in networks

The optimal architecture of a financial system is only dependent on its topology when the market is illiquid, and no topology is always superior.

### Multitasking immune networks

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

### Random close packing fractions

Lognormal distributions (and mixtures of same) are a useful model for the size distribution in emulsions and sediments.

### Economic complexity: conceptual grounding of a new metrics for global competitiveness

We introduce a novel method to define a self-consistent and non-monetary metrics for the competitiveness of countries.

### Measuring the intangibles: a metric for the economic complexity of countries and products

We present a framework to define a data-driven metrics to assess the level of competitiveness of a country.

### Low-temperature behaviour of social and economic networks

We define a generalized ensemble of graphs by introducing the concept of graph temperature.

### Scales in weighted networks

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

### Immune networks: multi-tasking capabilities at medium load

An intriguing analogy exists between neural networks and immune networks.

### Towers of strength

The Eiffel Tower was never intended to be a permanent feature of the Parisian landscape.

### Hierarchical structures

The most efficient load-bearing fractals are designed as big structures under gentle loads ... a situation common in aerospace applications.

### Interbank controllability

Complex networks detect the driver institutions of an interbank market and ascertain that intervention policies should be time-scale dependent.

### Reconstructing credit

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

### Complex derivatives

Network-based metrics to assess systemic risk and the importance of financial institutions can help tame the financial derivatives market.

### Bootstrapping topology and systemic risk of complex network using the fitness model

Vulnerability and systemicity do not depend only on GDP but also on the complex network of financial relations.

### Weighted network evolution

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

### Ultralight fractal structures from hollow tubes

A material’s architecture can be controlled over an ever increasing set of length scales.

### Network analysis of export flows

Network theory finds unexpected interactions between the number of products a country produces and the number of countries producing each product.

### A new metric for countries’ fitness and products’ complexity

A snapshot of the bipartite network for the most important countries; size of vertices is the fitness and the complexity

### Networks for medical data

Network analysis of diagnostic data identifies combinations of the key factors which cause Class III malocclusion and how they evolve over time.

### Web search queries can predict stock market volumes

Graphical illustration of the analysis presented in this paper.

### Unbiased randomization

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

### Robust and assortative

Spectral analysis shows that disassortative networks exhibit a higher epidemiological threshold and are therefore easier to immunize.

### Clustering inverted

Edge multiplicity—the number of triangles attached to edges—is a powerful analytic tool to understand and generalize network properties.

### What you see is not what you get: how sampling affects macroscopic features of biological networks

It is vital that we understand in detail how the topological characteristics of a real network relate to those of a finite random network.

### Shear elastic deformation in cells

Analysis of the linear elastic behaviour of plant cell dispersions improves our understanding of how to stabilise and texturise food products.

### Dynamics of disordered Ising chains

A transfer operator formalism solves the macroscopic dynamics of disordered Ising chain systems which are relevant for ageing phenomena.

### Diffusional Monte Carlo model of liquid-phase sintering

Our Monte Carlo model sheds light on the Ostwald ripening, namely the growth of big particles at the expense of the small ones.

### Tailored random graph ensembles

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

### Assessing self-assembly

The information needed to self-assemble a structure quantifies its modularity and explains the prevalence of certain structures over others.

### Ever-shrinking spheres

Techniques from random sphere packing predict the dimension of the Apollonian gasket, a fractal made up of non-overlapping hyperspheres.

### 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.

### Random cellular automata

Of the 256 elementary cellular automata, 28 of them exhibit random behavior over time, but spatio-temporal currents still lurk underneath.

### Single elimination competition

In single elimination competition the best indicator of success is a player's wealth: the accumulated wealth of all defeated players.