We consider percolation on interdependent locally treelike networks, recently introduced by Buldyrev et al., Nature 464, 1025, and demonstrate that the problem can be simplified conceptually by deleting all references to cascades of failures. Such cascades do exist, but their explicit treatment just complicates the theory -which is a straightforward extension of the usual epidemic spreading theory on a single network. Our method has the added benefits that it is directly formulated in terms of an order parameter and its modular structure can be easily extended to other problems, e.g. to any number of interdependent networks, or to networks with dependency links.

We study four Achlioptas type processes with "explosive" percolation transitions. All transitions are clearly continuous, but their finite size scaling functions are not entire holomorphic. The distributions of the order parameter, the relative size smax/N of the largest cluster, are doublehumped. But -in contrast to first order phase transitions -the distance between the two peaks decreases with system size N as N −η with η > 0. We find different positive values of β (defined via smax/N ∼ (p − pc) β for infinite systems) for each model, showing that they are all in different universality classes. In contrast, the exponent Θ (defined such that observables are homogeneous functions of (p − pc)N Θ ) is close to -or even equal to -1/2 for all models. Percolation is a pervasive concept in statistical physics and probability theory and has been studied in extenso in the past. It came thus as a surprise to many, when Achlioptas et al. [1] claimed that a seemingly mild modification of standard percolation models leads to a discontinuous phase transition -named "explosive percolation" (EP) by them -in contrast to the continuous phase transition seen in ordinary percolation. Following [1] there appeared a flood of papers [2-20] studying various aspects and generalizations of EP. In all cases, with one exception [20], the authors agreed that the transition is discontinuous: the "order parameter", defined as the fraction of vertices/sites in the largest cluster, makes a discrete jump at the percolation transition. In the present paper we join the dissenting minority and add further convincing evidence that the EP transition is continuous in all models, but with unusual finite size behavior.From the physical point of view, the model seems somewhat unnatural, since it involves non-local control (there is a 'supervisor' who has to compare distant pairs of nodes to chose the actual bonds to be established [21]). Also, notwithstanding [8], no realistic applications have been proposed. It is well known that the usual concept of universality classes in critical phenomena is invalidated by the presence of long range interactions. Thus it is not surprising that a percolation model with global control can show completely different behavior [22].Usually, e.g. in thermal equilibrium systems, discontinuous phase transitions are identified with "first order" transitions, while continuous transitions are called "second order". This notation is also often applied to percolative transitions. But EP lacks most attributes -except possibly for the discontinuous order parameter jump -considered essential for first order transitions. None of these other attributes (cooperativity, phase coexistence, and nucleation) is observed in Achlioptas type processes, although they are observed in other percolationtype transitions [23]. Thus EP should never have been viewed as a first order transition, and it is gratifying that it is also not discontinuous.Apart from the behavior of the average value m of the order parameter m, phase transitions can also b...

To identify communities in directed networks, we propose a generalized form of modularity in directed networks by presenting the quantity LinkRank, which can be considered as the PageRank of links. This generalization is consistent with the original modularity in undirected networks and the modularity optimization methods developed for undirected networks can be directly applied to directed networks by optimizing our modified modularity. Also, a model network, which can be used as a benchmark network in further community studies, is proposed to verify our method. Our method is supposed to find communities effectively in citation- or reference-based directed networks.

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Recent advance of high-power laser processing allows for rapid, continuous, area-selective material fabrication, typically represented by laser crystallization of silicon or oxides for display applications. Two-dimensional materials such as graphene exhibit remarkable physical properties and are under intensive development for the manufacture of flexible devices. Here we demonstrate an area-selective ultrafast nanofabrication method using low intensity infrared or visible laser irradiation to direct the self-assembly of block copolymer films into highly ordered manufacturing-relevant architectures at the scale below 12 nm. The fundamental principles underlying this light-induced nanofabrication mechanism include the self-assembly of block copolymers to proceed across the disorder-order transition under large thermal gradients, and the use of chemically modified graphene films as a flexible and conformal light-absorbing layers for transparent, nonplanar, and mechanically flexible surfaces.