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.

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- Author

Machine learning

### The limits of LLMs

LLMs generate texts that can fool us to credit them with human capabilities. Understanding their limitations is a key to using them wisely.

Number theory

### Multiplicativity of sets

Expanding the known multiplicative properties of large difference sets yields a new, quantitative proof on the structure of product sets.

Representation theory

### Infinitely high parallelotopes

The height of an infinite parallelotope is infinite, an essential ingredient to prove the irreducibility of unitary representations of some infinite-dimensional groups.

Combinatorics

### The popularity gap

A finite nonempty subset A of a cyclic group, with small enough |A–A|, contains a nonzero element with at least (2+o(1))|A|²/|A–A| representations as a difference of two elements.

Condensed matter theory

### Mobile impurity

Explicit computation of injection and ejection impurity’s Green’s function reveals a generalisation of the Kubo-Martin-Schwinger relation.

AI-assisted maths

### AI for cluster algebras

Investigating cluster algebras through the lens of modern data science reveals an elegant symmetry in the quiver exchange graph embedding.

Number theory

### Counting recursive divisors

Three new closed-form expressions give the number of recursive divisors and ordered factorisations, which were until now hard to compute.

Algebraic geometry

### Bundled Laplacians

By approximating the basis of eigenfunctions, we computationally determine the harmonic modes of bundle-valued Laplacians on Calabi-Yau manifolds.

Representation theory

### Infinite dimensional irreducibility

An analog of quasi-regular representations can be constructed for an infinite-dimensional group, despite the absence of the Haar measure.

Number theory

### Recursive divisor properties

The recursive divisor function has a simple Dirichlet series that relates it to the divisor function and other standard arithmetic functions.

General relativity

### Absorption with amplitudes

How gravitational waves are absorbed by a black hole is understood, for the first time, through effective on-shell scattering amplitudes.

quantum field theory

### Peculiar betas

The beta function for a class of sigma models is not found to be geometric, but rather has an elegant form in the context of algebraic data.

Machine learning

### DeepPavlov dream

A new open-source platform is specifically tailored for developing complex dialogue systems, like generative conversational AI assistants.

Computational linguistics

### Cross-lingual knowledge

Models trained on a Russian topical dataset, of knowledge-grounded human-human conversation, are capable of real-world tasks across languages.

Combinatorics

### Representation for sum-product

A new way to estimate indices via representation theory reveals links to the sum-product phenomena and Zaremba’s conjecture in number theory.

Machine learning

### Speaking DNA

A family of transformer-based DNA language models can interpret genomic sequences, opening new possibilities for complex biological research.

Algebraic geometry

### Genetic polytopes

Genetic algorithms, which solve optimisation problems in a natural selection-inspired way, reveal previously unconstructed Calabi-Yau manifolds.

Condensed matter theory

### Spin diffusion

The spin-spin correlation function of the Hubbard model reveals that finite temperature spin transport in one spatial dimension is diffusive.

Statistical physics

### Exponential Kauffman scaling

Surprisingly, the number of attractors in the critical Kauffman model with connectivity one grows exponentially with the size of the network.

Algebraic geometry

### Mahler measuring amoebae

Genetic symbolic regression methods reveal the relationship between amoebae from tropical geometry and the Mahler measure from number theory.

Combinatorics

### Ungrouped machines

A new connection between continued fractions and the Bourgain–Gamburd machine reveals a girth-free variant of this widely-celebrated theorem.

Complex systems

### Complex digital cities

A complexity-science approach to digital twins of cities views them as self-organising phenomena, instead of machines or logistic systems.

Number theory

### Sum-product with few primes

For a finite set of integers with few prime factors, improving the lower bound on its sum and product sets affirms the Erdös-Szemerédi conjecture.

Group theory

### On John McKay

This obituary celebrates the life and work of John Keith Stuart McKay, highlighting the mathematical miracles for which he will be remembered.

Machine learning

### BERT enhanced with recurrence

The quadratic complexity of attention in transformers is tackled by combining token-based memory and segment-level recurrence, using RMT.

Number theory

### Higher energies

Generalising the recent Kelley–Meka result on sets avoiding arithmetic progressions of length three leads to developments in the theory of the higher energies.

Combinatorics

### In life, there are few rules

The bipartite nature of regulatory networks means gene-gene logics are composed, which severely restricts which ones can show up in life.

Number theory

### Random Chowla conjecture

The distribution of partial sums of a Steinhaus random multiplicative function, of polynomials in a given form, converges to the standard complex Gaussian.

Algebraic geometry

### Symmetric spatial curves

The geometry of symmetric spatial curves reveals characterisations of general one-parameter families of complex univariate polynomials with fully-symmetric Galois groups.

Statistical physics

### Landau meets Kauffman

A new, simple approach to the critical Kauffman model with connectivity one sharpens the bounds on the number and length of attractors.

AI-assisted maths

### AI for arithmetic curves

AI can predict invariants of low genus arithmetic curves, including those key to the Birch-Swinnerton-Dyer conjecture—a millennium prize problem.

Statistical physics

### Single-input Boolean networks

A new, simpler approach to the critical Kauffman model with connectivity one reveals that it has more attractors than previously believed.

Synthetic biology

### Cell soup in screens

Bursting cells can introduce noise in transcription factor screens, but modelling this process allows us to discern true counts from false.

Gravity

### Kerr black holes symmetry

Effective field theories for Kerr black holes, showing the 3-point Kerr amplitudes are uniquely predicted using higher-spin gauge symmetry.

Statistical physics

### Network renormalization

Applying diffusion-based graph operators to complex networks identifies the proper spatiotemporal scales by overcoming small-world effects.

AI-assisted maths

### Clustered cluster algebras

Cluster variables in Grassmannian cluster algebras can be classified with HPC by applying the tableaux method up to a fixed number of columns.

Number theory

### Bounding Zaremba’s conjecture

Using methods related to the Bourgain–Gamburd machine refines the previous bound on Zaremba’s conjecture in the theory of continued fractions.

Neurocomputing

### Optimal electronic reservoirs

Balancing memory from linear components with nonlinearities from memristors optimises the computational capacity of electronic reservoirs.

String theory

### Gauge theory and integrability

The algebra of a toric quiver gauge theory recovers the Bethe ansatz, revealing the relation between gauge theories and integrable systems.

Evolvability

### Flowers of immortality

The eigenvalues of the mortality equation fall into two classes—the flower and the stem—but only the stem eigenvalues control the dynamics.

Combinatorics

### Structure of genetic computation

The structural and functional building blocks of gene regulatory networks correspond, which tell us how genetic computation is organised.

Gravity

### AI classifies space-time

A neural network learns to classify different types of spacetime in general relativity according to their algebraic Petrov classification.

String theory

### Algebra of melting crystals

Certain states in quantum field theories are described by the geometry and algebra of melting crystals via properties of partition functions.

Combinatorics

### Set additivity and growth

The additive dimension of a set, which is the size of a maximal dissociated subset, is closely connected to the rapid growth of higher sumsets.

AI-assisted maths

### Machine learning Hilbert series

Neural networks find efficient ways to compute the Hilbert series, an important counting function in algebraic geometry and gauge theory.

AI-assisted maths

### Line bundle connections

Neural networks find numerical solutions to Hermitian Yang-Mills equations, a difficult system of PDEs crucial to mathematics and physics.

AI-assisted maths

### Calabi-Yau anomalies

Unsupervised machine-learning of the Hodge numbers of Calabi-Yau hypersurfaces detects new patterns with an unexpected linear dependence.

String theory

### Mahler measure for quivers

Mahler measure from number theory is used for the first time in physics, yielding “Mahler flow” which extrapolates different phases in QFT.

Number theory

### Recursively divisible numbers

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

AI-assisted maths

### Learning the Sato–Tate conjecture

Machine-learning methods can distinguish between Sato-Tate groups, promoting a data-driven approach for problems involving Euler factors.

Machine learning

### Universes as big data

Machine-learning is a powerful tool for sifting through the landscape of possible Universes that could derive from Calabi-Yau manifolds.

Network theory

### True scale-free networks

The underlying scale invariance properties of naturally occurring networks are often clouded by finite-size effects due to the sample data.

Number theory

### Reflexions on Mahler

With physically-motivated Newton polynomials from reflexive polygons, we find the Mahler measure and dessin d’enfants are in 1-to-1 correspondence.

Graph 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.

Statistical physics

### Coexistence in diverse ecosystems

Scale-invariant plant clusters explain the ability for a diverse range of plant species to coexist in ecosystems such as Barra Colorado.

Mathematical medicine

### Cancer screening with MRI

An ongoing study tests the feasibility of using MRI scans to screen men for prostate cancer in place of unreliable antigen blood tests.

Neural networks

### Quick quantum neural nets

The notion of quantum superposition speeds up the training process for binary neural networks and ensures that their parameters are optimal.

Quantum physics

### Going, going, gone

A solution to the information paradox uses standard quantum field theory to show that black holes can evaporate in a predictable way.

Mathematical medicine

### Tumour infiltration

A delicate balance between white blood cell protein expression and the molecules on the surface of tumour cells determines cancer prognoses.

Neurocomputing

### Breaking classical barriers

Circuits of memristors, resistors with memory, can exhibit instabilities which allow classical tunnelling through potential energy barriers.

String theory

### 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.

Machine learning

### Neurons on amoebae

Machine-learning 2-dimensional amoeba in algebraic geometry and string theory is able to recover the complicated conditions from so-called lopsidedness.

Group theory

### New approaches to the Monster

Editorial of the last set of lectures given by the founder, McKay, of Moonshine Conjectures, the proof of which got Borcherds the Fields Medal.

Network theory

### Physics of financial networks

Statistical physics contributes to new models and metrics for the study of financial network structure, dynamics, stability and instability.

Economic complexity

### Channels of contagion

Fire sales of common asset holdings can whip through a channel of contagion between banks, insurance companies and investments funds.

Financial risk

### Risky bank interactions

Networks where risky banks are mostly exposed to other risky banks have higher levels of systemic risk than those with stable bank interactions.

Mathematical medicine

### Cancer and coronavirus

Cancer patients who contract and recover from Coronavirus-2 exhibit long-term immune system weaknesses, depending on the type of cancer.

Particle physics

### 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.

Evolvability

### I want to be forever young

The mortality equation governs the dynamics of an evolving population with a given maximum age, offering a theory for programmed ageing.

Inference

### Exact linear regression

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

Combinatorics

### Biological logics are restricted

The fraction of logics that are biologically permitted can be bounded and shown to be tiny, which makes inferring them from experiments easier.

Number theory

### Energy bounds for roots

Bounds for additive energies of modular roots can be generalised and improved with tools from additive combinatorics and algebraic number theory.

Number theory

### Ample and pristine numbers

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

Financial markets

### Network valuation in finance

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

Neural networks

### Deep layered machines

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

Statistical physics

### Replica analysis of overfitting

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

Theory of innovation

### 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.

Inference, Statistical physics

### Replica clustering

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

Theory of innovation

### Recursive structure of innovation

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

Network theory

### 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.

Economic complexity

### Renewable resource management

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

Thermodynamics

### Energy harvesting with AI

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

Geometry

### 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.

Neurocomputing

### Memristive networks

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

Statistical physics

### Physics of networks

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

Theory of innovation

### The rate of innovation

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

Thermodynamics

### One-shot statistic

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

Complex networks

### 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.

Quantum physics

### A Hamiltonian recipe

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

Neurocomputing

### Solvable memristive circuits

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

Inference

### 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.

Ignoble research

### Volunteer clouds

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

Financial networks

### From ecology to finance

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

Graph theory

### Hypercube eigenvalues

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

Technological progress

### Forecasting technology deployment

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

Financial networks

### 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.

Graph theory

### Exactly solvable random graphs

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

Financial networks

### 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.

Theory of innovation

### 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.

Theory of materials

### Dirac cones in 2D borane

The structure of two-dimensional borane, a new semi-metallic single-layered material, has two Dirac cones that meet right at the Fermi energy.

Financial risk

### Modelling financial systemic risk

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

Mathematical medicine

### Bayesian analysis of medical data

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

Theory of innovation

### 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.

Neural networks

### Quantum neural networks

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

Network theory

### 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

Neurocomputing

### Memristive networks and learning

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

Neurocomputing

### Dynamics of memristors

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

Financial networks

### Bipartite trade network

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

Sphere packing

### 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.

Complex systems

### Disentangling links in networks

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

Thermodynamics

### Quantum jumps in thermodynamics

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

Financial markets

### Financial network reconstruction

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

Financial risk

### Pathways towards instability

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

Thermodynamics

### 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.

Theory of innovation

### Serendipity and strategy

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

Graph theory

### Spectral partitioning

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

Complex networks, Financial risk

### Non-linear distress propagation

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

Financial risk

### Immunisation of systemic risk

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

Complex networks

### Optimal growth rates

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

Quantum computing

### Tunnelling interpreted

Quantum tunnelling only occurs if either the Wigner function is negative, or the tunnelling rate operator has a negative Wigner function.

Financial risk

### The price of complexity

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

Thermodynamics

### Photonic Maxwell's demon

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

Graph theory

### 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.

Network theory

### Cascades in flow networks

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

Percolation theory

### Self-organising adaptive networks

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

Thermodynamics

### 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.

Financial markets

### Instability in complex ecosystems

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

Discrete dynamics

### Form and function in gene networks

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

Percolation theory

### Clusters of neurons

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

Gravity

### Cyclic isotropic cosmologies

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

Technological progress

### 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.

Percolation theory

### Bootstrap percolation models

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

Financial markets

### News sentiment and price dynamics

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

Network theory

### Communities in networks

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

Financial markets

### 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.

Complex networks

### Democracy in networks

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

Biological networks

### 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.

Graph theory

### 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.

Complex networks

### 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.

Thermodynamics

### A measure of majorization

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

Financial risk

### DebtRank and shock propagation

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

Statistical physics

### 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.

Financial risk, Network theory

### 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.

Theory of materials

### 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.

Economic complexity

### Dynamics of economic complexity

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

Neurocomputing

### 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.

Percolation theory

### 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.

Graph theory

### 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.

Economic complexity

### Taxonomy and economic growth

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

Sphere packing

### Viscosity of polydisperse spheres

A fast and simple way to measure how polydisperse spheres crowd around each other, termed the packing fraction, agrees well with rheological data.

Financial risk

### 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.

Graph theory

### Entropies of graph ensembles

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

Network theory

### Easily repairable networks

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

Quantum thermodynamics

### Entanglement typicality

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

Theory of materials

### 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 theory

### 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.

Financial risk

### 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.

Network theory

### Self-healing complex networks

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

Fractals

### Structural imperfections

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

Financial risk

### 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.

Sphere packing

### Random close packing fractions

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

Biological networks

### Multitasking immune networks

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

Economic complexity

### Metrics for global competitiveness

A new non-monetary metric captures diversification, a dominant effect on the globalised market, and the effective complexity of products.

Economic complexity

### Measuring the intangibles

Coupled non-linear maps extract information about the competitiveness of countries to the complexity of their products from trade data.

Network theory

### The temperature of networks

A new concept, graph temperature, enables the prediction of distinct topological properties of real-world networks simultaneously.

Network theory

### Scales in weighted networks

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

Mathematical medicine

### Multi-tasking in immune networks

Associative networks with different loads model the ability of the immune system to respond simultaneously to multiple distinct antigen invasions.

Fractals

### Gentle loads

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

Financial networks

### Interbank controllability

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

Financial networks

### Reconstructing credit

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

Financial networks

### Complex derivatives

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

Technological progress

### Organized knowledge economies

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

Financial risk

### Bootstrapping topology and risk

Information about 10% of the links in a complex network is sufficient to reconstruct its main features and resilience with the fitness model.

Network theory

### Weighted network evolution

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

Fractals

### Ultralight fractal structures

The transition from solid to hollow beams changes the scaling of stability versus loading analogously to increasing the hierarchical order by one.

Fractals

### Hierarchical space frames

A systematic way to vary the power-law scaling relations between loading parameters and volume of material aids the hierarchical design process.

Economic complexity

### 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.

Economic complexity

### Metric for fitness and complexity

A quantitative assessment of the non-monetary advantage of diversification represents a country’s hidden potential for development and growth.

Mathematical medicine

### 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.

Financial markets

### Search queries predict stocks

Analysis of web search queries about a given stock, from the seemingly uncoordinated activity of many users, can anticipate the trading peak.

Graph theory

### Unbiased randomization

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

Network theory

### Robust and assortative

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

Network theory

### Clustering inverted

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

Biological networks

### What you see is not what you get

Methods from tailored random graph theory reveal the relation between true biological networks and the often-biased samples taken from them.

Ignoble research

### 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.

Statistical physics

### Dynamics of Ising chains

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

Theory of materials

### Diffusional liquid-phase sintering

A Monte Carlo model simulates the microstructural evolution of metallic and ceramic powders during the consolidation process liquid-phase sintering.

Graph theory

### Tailored random graph ensembles

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

Information theory

### Assessing self-assembly

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

Sphere packing

### Ever-shrinking spheres

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

Discrete dynamics

### Random cellular automata

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

Statistical physics

### Single elimination competition

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