The basic notion of percolation in physics assumes the emergence of a giant connected (percolation) cluster in a large disordered system when the density of connections exceeds some critical value. Until recently, the percolation phase transitions were believed to be continuous, however, in 2009, a remarkably different, discontinuous phase transition was reported in a new so-called "explosive percolation" problem. Each link in this problem is established by a specific optimization process. Here, employing strict analytical arguments and numerical calculations, we find that in fact the "explosive percolation" transition is continuous though with an uniquely small critical exponent of the percolation cluster size. These transitions provide a new class of critical phenomena in irreversible systems and processes. 1The modern understanding of disordered systems in statistical and condensed matter physics is essentially based on the notion of percolation (1). When one increases progressively the number of connections between nodes in a network, above some critical number (percolation threshold) a giant connected (percolation) cluster emerges in addition to finite clusters. This percolation cluster contains a finite fraction of nodes and links in a network. The percolation transition was widely believed to be a typical continuous phase transition for various networks architectures and space dimensionalities (2), so it shows standard scaling features, including a power-law size distribution of finite cluster sizes at the percolation threshold. Recently, however, it was reported that a remarkable percolation problem exists in which the percolation cluster emerges discontinuously and already contains a finite fraction of nodes at the percolation threshold (3). This conclusion was based on numerical simulations of a model in which each new connection is made in the following way: choose at random two links that could be added to the network, but add only one of them, namely the link connecting two clusters with the smallest product of their sizes. To emphasize this surprising discontinuity, this kind of percolation was named "explosive" (3). Further investigations of "explosive percolation" in this and similar systems, also mainly based on numerical simulations, supported this strong result but, in addition, surprisingly for abrupt transitions, revealed power-law critical distributions of cluster sizes (4-10) resembling those found in continuous percolation transitions. This selfcontradicting combination of discontinuity and scaling have made explosive percolation one of the challenging and urgent issues in the physics of disordered systems.Here we resolve this confusion. We show that there is not actually any discontinuity at the "explosive percolation" threshold, contrary to the conclusions of the previous investigators. We consider a simple representative model demonstrating this new kind of percolation and show that the "explosive percolation" transition is a continuous, second-order phase transition but, importantly, ...

Abstract-Motivated by streaming applications with stringent delay constraints, we consider the design of online network coding algorithms with timely delivery guarantees. Assuming that the sender is providing the same data to multiple receivers over independent packet erasure channels, we focus on the case of perfect feedback and heterogeneous erasure probabilities. Based on a general analytical framework for evaluating the decoding delay, we show that existing ARQ schemes fail to ensure that receivers with weak channels are able to recover from packet losses within reasonable time. To overcome this problem, we redefine the encoding rules in order to break the chains of linear combinations that cannot be decoded after one of the packets is lost. Our results show that sending uncoded packets at key times ensures that all the receivers are able to meet specific delay requirements with very high probability.

Percolation refers to the emergence of a giant connected cluster in a disordered system when the number of connections between nodes exceeds a critical value. The percolation phase transitions were believed to be continuous until recently when in a new so-called "explosive percolation" problem for a competition driven process, a discontinuous phase transition was reported. The analysis of evolution equations for this process showed however that this transition is actually continuous though with surprisingly tiny critical exponents. For a wide class of representative models, we develop a strict scaling theory of this exotic transition which provides the full set of scaling functions and critical exponents. This theory indicates the relevant order parameter and susceptibility for the problem, and explains the continuous nature of this transition and its unusual properties.PACS numbers: 64.60.ah, 05.40.-a, 64.60.F-The percolation phase transition is one of the central issues for disordered systems [1][2][3][4]. Phase transitions in classical percolation problems are very well known to be continuous, that is, the relative size of the percolation cluster S, which is the order parameter for these models, emerges continuously, without a jump at the percolation threshold. As a continuous phase transition, the ordinary percolation transition is characterized by a power-law distribution of cluster sizes at the percolation threshold and a set of standard scaling properties and relations.This common understanding of percolation was shaken by work [5] that reported a discontinuous percolation phase transition in models whose evolution was driven by local optimization algorithms. Based on a computer experiment for a 512 , 000 node system [5], it was concluded that the percolation transition for these irreversible processes is discontinuous, and that is why this kind of percolation was termed "explosive percolation". This conclusion was supported by a number of simulations of models of this kind [6][7][8][9][10][11][12][13][14][15]. Surprisingly, these and other studies, in addition, reported power-law cluster size distributions at the critical point and scaling features below and above t c (see ), unexpected for discontinuous transitions.We resolved this contradiction by showing that the explosive percolation transition is actually continuous though with a uniquely small critical exponent β of the percolation cluster size [19]. We obtained this result by analyzing evolution equations for this process in the infinite system size limit. Thanks to the smallness of the exponent β, the continuous transition looks so "sharp" that it is virtually impossible to distinguish it from a discontinuous one in computer experiments even for very large systems [19]. More recently, the fact that this transition is continuous was also supported by mathematicians [23]. Nonetheless, in the physics sense, the quest * sdorogov@ua.pt FIG. 1. Illustration of rules in the model of explosive percolation. At each step, two sets of m nodes are chosen at ...

We develop the theory of sparse multiplex networks with partially overlapping links based on their local treelikeness. This theory enables us to find the giant mutually connected component in a two-layer multiplex network with arbitrary correlations between connections of different types. We find that correlations between the overlapping and nonoverlapping links markedly change the phase diagram of the system, leading to multiple hybrid phase transitions. For assortative correlations we observe recurrent hybrid phase transitions.

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