Standard epidemic models exhibit one continuous, second order phase transition to macroscopic outbreaks. However, interventions to control outbreaks may fundamentally alter epidemic dynamics. Here we reveal how such interventions modify the type of phase transition. In particular, we uncover three distinct types of explosive phase transitions for epidemic dynamics with capacity-limited interventions. Depending on the capacity limit, interventions may (i) leave the standard second order phase transition unchanged but exponentially suppress the probability of large outbreaks, (ii) induce a first-order discontinuous transition to macroscopic outbreaks, or (iii) cause a secondary explosive yet continuous third-order transition. These insights highlight inherent limitations in predicting and containing epidemic outbreaks. More generally our study offers a cornerstone example of a third-order explosive phase transition in complex systems.

Understanding the relationship between the heterogeneous structure of complex networks and cooperative phenomena occurring on them remains a key problem in network science. Mean-field theories of spin models on networks constitute a fundamental tool to tackle this problem and a cornerstone of statistical physics, with an impressive number of applications in condensed matter, biology, and computer science. In this work we derive the mean-field equations for the equilibrium behavior of vector spin models on high-connectivity random networks with an arbitrary degree distribution and with randomly weighted links. We demonstrate that the high-connectivity limit of spin models on networks is not universal in that it depends on the full degree distribution. Such nonuniversal behavior is akin to a remarkable mechanism that leads to the breakdown of the central limit theorem when applied to the distribution of effective local fields. Traditional mean-field theories on fully-connected models, such as the Curie–Weiss, the Kuramoto, and the Sherrington–Kirkpatrick model, are only valid if the network degree distribution is highly concentrated around its mean degree. We obtain a series of results that highlight the importance of degree fluctuations to the phase diagram of mean-field spin models by focusing on the Kuramoto model of synchronization and on the Sherrington–Kirkpatrick model of spin-glasses. Numerical simulations corroborate our theoretical findings and provide compelling evidence that the present mean-field theory describes an intermediate regime of connectivity, in which the average degree c scales as a power c ∝ N b (b < 1) of the total number N ≫ 1 of spins. Our findings put forward a novel class of spin models that incorporate the effects of degree fluctuations and, at the same time, are amenable to exact analytic solutions.

We construct a system of N interacting particles on the unit sphere in d-dimensional space, which has d-body interactions only. The equations have a gradient formulation derived from a rotationally-invariant potential of a determinantal form summed over all nodes, with antisymmetric coefficients. For d = 3, for example, all trajectories lie on the two-sphere and the potential is constructed from the triple scalar product summed over all oriented two-simplices. We investigate the cases d = 3, 4, 5 in detail, and find that the system synchronizes from generic initial values for both positive and negative coupling coefficients, to a static final configuration in which the particles lie equally spaced on . Completely synchronized configurations also exist, but are unstable under the d-body interactions. We compare the relative effect of two-body and d-body forces by adding the well-studied two-body interactions to the potential, and find that higher-order interactions enhance the synchronization of the system, specifically, synchronization to a final configuration consisting of equally spaced particles occurs for all d-body and two-body coupling constants of any sign, unless the attractive two-body forces are sufficiently strong relative to the d-body forces. In this case the system completely synchronizes as the two-body coupling constant increases through a positive critical value, with either a continuous transition for d = 3, or discontinuously for d = 5. Synchronization also occurs if the nodes have distributed natural frequencies of oscillation, provided that the frequencies are not too large in amplitude, even in the presence of repulsive two-body interactions which by themselves would result in asynchronous behaviour.

Cognitive networks have evolved to cope with uncertain environments in order to make reliable decisions. Such decision making circuits need to respond to the external world in efficient and flexible ways, and one potentially general mechanism of achieving this is grounded in critical states. Mounting evidence has shown that brains operate close to such critical boundaries consistent with self-organized criticality (SOC). Is this also taking place in small-scale living systems, such as cells? Here, we explore a recent model of engineered gene networks that have been shown to exploit the feedback between order and control parameters (as defined by expression levels of two coupled genes) to achieve an SOC state. We suggest that such SOC motif could be exploited to generate adaptive behavioral patterns and might help design fast responses in synthetic cellular and multicellular organisms.

In this review, we discuss critical dynamics of simple nonequilibrium models on large connectomes, obtained by diffusion MRI, representing the white matter of the human brain. In the first chapter, we overview graph theoretical and topological analysis of these networks, pointing out that universality allows selecting a representative network, the KKI-18, which has been used for dynamical simulation. The critical and sub-critical behaviour of simple, two- or three-state threshold models is discussed with special emphasis on rare-region effects leading to robust Griffiths phases (GP). Numerical results of synchronization phenomena, studied by the Kuramoto model, are also shown, leading to a continuous analog of the GP, termed frustrated synchronization. The models presented here exhibit dynamical scaling behaviour with exponents in agreement with brain experimental data if local homeostasis is provided.

We present a brief review of power laws and correlation functions as measures of criticality and the relation between them. By comparing phenomenology from rain, brain and the forest fire model we discuss the relevant features of self-organisation to the vicinity about a critical state. We conclude that organisation to a region of extended correlations and approximate power laws may be behaviour of interest shared between the three considered systems.

Discovering and quantifying the drivers of language change is a major challenge. Hypotheses about causal factors proliferate, but are difficult to rigorously test. Here we ask a simple question: can 20th century changes in English be explained as a consequence of spatial diffusion, or have other processes created bias in favour of certain linguistic forms? Using two of the most comprehensive spatial datasets available, which measure the state of English at the beginning and end of the 20th century, we calibrate a simple spatial model so that, initialised with the early state, it evolves into the later. Our calibrations reveal that while some changes can be explained by diffusion alone, others are clearly the result of substantial asymmetries between variants. We discuss the origins of these asymmetries and, as a by-product, we generate a full spatio–temporal prediction for the spatial evolution of English features over the 20th century, and a prediction of the future.

We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of three-cycles (triangles). We find that such ensembles generally exhibit two transitions, a clustering and a shattering transition, separating three distinct regimes. At the clustering transition, the graphs change from typically having only isolated cycles to forming cycle clusters. At the shattering transition the graphs break up into many small cliques to achieve the desired three-cycle density. The locations of both transitions depend nontrivially on the system size. We derive a general formula for the three-cycle density in the regime of isolated cycles, for graphs with degree distributions that have finite first and second moments. For bounded degree distributions we present further analytical results on cycle densities and phase transition locations, which, while non-rigorous, are all validated via MCMC sampling simulations. We show that the shattering transition is of an entropic nature, occurring for all three-cycle density values, provided the system is large enough.

We investigate the ground states of spin models defined on networks that we imprint (e.g., non-complex random networks like Erdos–Renyi, or complex networks like Watts–Strogatz, and Barabasi–Albert), and their response to decohering processes which we model with network attacks. We quantify the complexity of these ground states, and their response to the attacks, by calculating distributions of network measures of an emergent network whose link weights are the pairwise mutual information between spins. We focus on attacks which projectively measure spins. We find that the emergent networks in the ground state do not satisfy the usual criteria for complexity, and their average properties are captured well by a single dimensionless parameter in the Hamiltonian. While the response of classical networks to attacks is well-studied, where classical complex networks are known to be more robust to random attacks than random networks, we find counter-intuitive results for our quantum networks. We find that the ground states for Hamiltonians defined on different classes of imprinted networks respond similarly to all our attacks, and the attacks rescale the average properties of the emergent network by a constant factor. Mean field theory explains these results for relatively dense networks, but we also find the simple rescaling behavior away from the regime of validity of mean field theory. Our calculations indicate that complex spin networks are not more robust to projective measurement attacks, and presumably also other quantum attacks, than non-complex spin networks, in contrast to the classical case. Understanding the response of the spin networks to decoherence and attacks will have applications in understanding the physics of open quantum systems, and in designing robust complex quantum systems—possibly even a robust quantum internet in the long run—that is maximally resistant to decoherence.

We use methods from computational algebraic topology to study functional brain networks in which nodes represent brain regions and weighted edges encode the similarity of functional magnetic resonance imaging (fMRI) time series from each region. With these tools, which allow one to characterize topological invariants such as loops in high-dimensional data, we are able to gain understanding of low-dimensional structures in networks in a way that complements traditional approaches that are based on pairwise interactions. In the present paper, we use persistent homology to analyze networks that we construct from task-based fMRI data from schizophrenia patients, healthy controls, and healthy siblings of schizophrenia patients. We thereby explore the persistence of topological structures such as loops at different scales in these networks. We use persistence landscapes and persistence images to represent the output of our persistent-homology calculations, and we study the persistence landscapes and persistence images using k-means clustering and community detection. Based on our analysis of persistence landscapes, we find that the members of the sibling cohort have topological features (specifically, their one-dimensional loops) that are distinct from the other two cohorts. From the persistence images, we are able to distinguish all three subject groups and to determine the brain regions in the loops (with four or more edges) that allow us to make these distinctions.