How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness on gradual addition of links, was long believed to be continuous, but recent numerical evidence of ‘explosive percolation’ suggests that it might also be discontinuous if links compete for addition. Here we analyse the microscopic mechanisms underlying discontinuous percolation processes and reveal a strong impact of single-link additions. We show that in generic competitive percolation processes, including those showing explosive percolation, single links do not induce a discontinuous gap in the largest cluster size in the thermodynamic limit. Nevertheless, our results highlight that for large finite systems single links may still induce substantial gaps, because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively

The emergence of large-scale connectivity on an underlying network or lattice, the so-called percolation transition, has a profound impact on the system's macroscopic behaviours. There is thus great interest in controlling the location of the percolation transition to either enhance or delay its onset and, more generally, in understanding the consequences of such control interventions. Here we review explosive percolation, the sudden emergence of large-scale connectivity that results from repeated, small interventions designed to delay the percolation transition. These transitions exhibit drastic, unanticipated and exciting consequences that make explosive percolation an emerging paradigm for modelling real-world systems ranging from social networks to nanotubes.T he percolation transition, named for the prototypical mathematical problem of pouring liquid through a porous material, describes the onset of large-scale connectivity on an underlying network or lattice. At times, ensuring large-scale connectivity is essential: a transportation network (such as the world-wide airline network) or a communication system (such the Internet) is useful only if a large fraction of the nodes are connected. Yet, in other contexts, large-scale connectivity is a liability: under certain conditions, a virus spreading on a well-connected social or computer network can reach enough nodes to cause an epidemic. Thus, percolation theory is a theoretical underpinning across a range of fields 1,2 and the desire to enhance or delay the onset of percolation has been of interest for many years. The consequences of delaying the transition have only recently been established and here we review explosive percolation (EP), the phenomenon that usually results from repeated, small interventions designed to delay the percolation transition. The onset can indeed be significantly delayed, but once the percolation transition is inevitably reached, large-scale connectivity emerges suddenly.The traditional approach for constructing a random graph, the Erdős-Rényi model, considers a collection of N isolated nodes, with each possible edge between two distinct nodes added to the graph with probability p (refs 3-5). This is a static formulation with no dependence on the history of how edges have been added to the graph. A mathematically equivalent kinetic formulation is initialized with N originally isolated nodes with a randomly sampled edge added at each discrete time step 6 . Letting T denote the number of steps, the process is parameterized by the relative number of introduced edges t = T /N , and typically analysed in the thermodynamic limit of infinite size N . Below some critical t = t c the resulting graph is disjoint, consisting of small isolated clusters (or components) of connected nodes. (See Fig. 1c for an illustration of distinct components.) Let C denote the largest component and |C| its size. For the Erdős-Rényi model, the order parameter |C| undergoes a second-order transition at t c = 1/2 where, below t c , |C| is logarithmic in N and, a...

Morbidity and mortality attributable to third-generation-cephalosporin-resistant E. coli BSI is significant. If prevailing resistance trends continue, high societal and economic costs can be expected. Better management of infections caused by resistant E. coli is becoming essential.

EPO levels can be inappropriately low in critically ill patients, so that EPO deficiency may contribute to the development of anaemia in these patients. This phenomenon is observed not only in the presence of acute renal failure, but also in the presence of sepsis.

. Their excess LOS was 9.2 days. MSSA patients also had higher 30-day (aOR ‫؍‬ 2.4) and hospital (aHR ‫؍‬ 3.1) mortality and an excess LOS of 8.6 days. When the outcomes from the two cohorts were compared, an effect attributable to methicillin resistance was found for 30-day mortality (OR ‫؍‬ 1.8; P ‫؍‬ 0.04), but not for hospital mortality (HR ‫؍‬ 1.1; P ‫؍‬ 0.63) or LOS (difference ‫؍‬ 0.6 days; P ‫؍‬ 0.96). Irrespective of methicillin susceptibility, S. aureus BSI has a significant impact on morbidity and mortality. In addition, MRSA BSI leads to a fatal outcome more frequently than MSSA BSI. Infection control efforts in hospitals should aim to contain infections caused by both resistant and susceptible S. aureus.The emergence of resistant bacteria is a natural consequence of antibiotic use and complicates the treatment of infected patients. Staphylococcus aureus resistant to isoxazolyl penicillins (methicillin-resistant S. aureus [MRSA]) is one of the most frequent pathogens causing resistant infections in hospitals worldwide (14,21,33). The questions are whether and to what extent resistance affects survival and the duration of hospital admission in patients with bacterial infections. Previous studies compared patients with MRSA infections to those infected by methicillin-susceptible S. aureus (MSSA), using mortality as one of the main endpoints. New insights challenge this approach for a number of reasons.Several studies have shown that patients with MRSA bloodstream infection (BSI) differ in many ways from those with MSSA BSI; they are older, have more comorbidities, and experience longer hospital admissions before the onset of infection (6,22,37). If these two groups of patients are compared directly, bias is introduced, compromising the validity of the results. Of all hospitalized patients at risk of acquiring MRSA BSI, the younger, relatively more healthy MSSA patients are selected as the control group, magnifying the possible impact of resistance (20). Moreover, time-dependent distortions are introduced, as patients staying in the hospital for a shorter period, like MSSA patients, have a smaller chance of acquiring MRSA BSI than patients hospitalized for longer periods, who for many reasons are more likely to die, thus leading to overestimation of the clinical impact of resistance (36).

Although human musical performances represent one of the most valuable achievements of mankind, the best musicians perform imperfectly. Musical rhythms are not entirely accurate and thus inevitably deviate from the ideal beat pattern. Nevertheless, computer generated perfect beat patterns are frequently devalued by listeners due to a perceived lack of human touch. Professional audio editing software therefore offers a humanizing feature which artificially generates rhythmic fluctuations. However, the built-in humanizing units are essentially random number generators producing only simple uncorrelated fluctuations. Here, for the first time, we establish long-range fluctuations as an inevitable natural companion of both simple and complex human rhythmic performances. Moreover, we demonstrate that listeners strongly prefer long-range correlated fluctuations in musical rhythms. Thus, the favorable fluctuation type for humanizing interbeat intervals coincides with the one generically inherent in human musical performances.

The emergence of large-scale connectivity and synchronization are crucial to the structure, function and failure of many complex socio-technical networks. Thus, there is great interest in analyzing phase transitions to large-scale connectivity and to global synchronization, including how to enhance or delay the onset. These phenomena are traditionally studied as second-order phase transitions where, at the critical threshold, the order parameter increases rapidly but continuously. In 2009, an extremely abrupt transition was found for a network growth process where links compete for addition in attempt to delay percolation. This observation of "explosive percolation" was ultimately revealed to be a continuous transition in the thermodynamic limit, yet with very atypical finite-size scaling, and it started a surge of work on explosive phenomena and their consequences. Many related models are now shown to yield discontinuous percolation transitions and even hybrid transitions. Explosive percolation enables many other features such as multiple giant components, modular structures, discrete scale invariance and non-self-averaging, relating to properties found in many real phenomena such as explosive epidemics, electric breakdowns and the emergence of molecular life. Models of explosive synchronization provide an analytic framework for the dynamics of abrupt transitions and reveal the interplay between the distribution in natural frequencies and the network structure, with applications ranging from epileptic seizures to waking from anesthesia. Here we review the vast literature on explosive phenomena in networked systems and synthesize the fundamental connections between models and survey the application areas. We attempt to classify explosive phenomena based on underlying mechanisms and to provide a coherent overview and perspective for future research to address the many vital questions that remained unanswered. 34

The Copenhagen problem is a simple model in celestial mechanics. It serves to investigate the behavior of a small body under the gravitational influence of two equally heavy primary bodies. We present a partition of orbits into classes of various kinds of regular motion, chaotic motion, escape and crash. Collisions of the small body onto one of the primaries turn out to be unexpectedly frequent, and their probability displays a scale-free dependence on the size of the primaries. The analysis reveals a high degree of complexity so that long term prediction may become a formidable task. Moreover, we link the results to chaotic scattering theory and the theory of leaking Hamiltonian systems.