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.