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...

Discontinuous percolation transitions and the associated tricritical points are manifest in a wide range of both equilibrium and nonequilibrium cooperative phenomena. To demonstrate this, we present and relate the continuous and first-order behaviors in two different classes of models: The first are generalized epidemic processes that describe in their spatially embedded version--either on or off a regular lattice--compact or fractal cluster growth in random media at zero temperature. A random graph version of these processes is mapped onto a model previously proposed for complex social contagion. We compute detailed phase diagrams and compare our numerical results at the tricritical point in d = 3 with field theory predictions of Janssen et al. [Phys. Rev. E 70, 026114 (2004)]. The second class consists of exponential ("Hamiltonian," i.e., formally equilibrium) random graph models and includes the Strauss and the two-star model, where "chemical potentials" control the densities of links, triangles, or two-stars. When the chemical potentials in either graph model are O(logN), the percolation transition can coincide with a first-order phase transition in the density of links, making the former also discontinuous. Hysteresis loops can then be of mixed order, with second-order behavior for decreasing link fugacity, and a jump (first order) when it increases.

Irreversible aggregation is revisited in view of recent work on renormalization of complex networks. Its scaling laws and phase transitions are related to percolation transitions seen in the latter. We illustrate our points by giving the complete solution for the probability to find any given state in an aggregation process (k + 1)X → X, given a fixed number of unit mass particles in the initial state. Exactly the same probability distributions and scaling are found in one dimensional systems (a trivial network) and well-mixed solutions. This reveals that scaling laws found in renormalization of complex networks do not prove that they are self-similar. PACS numbers: 89.75.Hc, 02.10.Ox, 05.70.Ln Droplets beget rain, goblets coagulate to make butter or cream, and dust particles stick together to form aggregates that can eventually coalesce into planets. At the microscopic level, irreversible aggregation of atoms and molecules creates many familiar forms of matter such as aerosols, colloids, gels, suspensions, clusters and solids [1]. Almost a century ago, Smoluchowski proposed a theory based on rate equations to describe processes governed by diffusion, collision and irreversible merging of aggregates [2]. The theory predicts how many small and large clusters exist at any given time and yields a mass distribution that depends on certain details such as the initial conditions, reactions present, relative rates, the presence or absence of spatial structure, etc. A key interest to physicists has been to derive scaling laws that characterize different universality classes [3, and references therein].By contrast, wide interest in complex networks [4-7] has emerged recently. Vast applications to physics, computer science, biology, and sociology [8-10, and references therein] continue to be vigorously investigated. An important question is whether or not complex networks exhibit self-similarity at different length scales and if they can be grouped into universality classes on that basis. Renormalization schemes for networks were proposed [11][12][13][14] to address this question. Scaling of the mass or degree distribution of the renormalized nodes was used to argue that many complex networks are selfsimilar. The semi-sequential renormalization group (RG) flow underlying the box covering of [11][12][13][14] was studied carefully in [15,16], where it was found that scaling laws may be related to an "RG fixed point" which was observed for a wide variety of networks. A convenient, fully sequential scheme called random sequential renormalization (RSR) was introduced [17]. At each RSR step, one node is selected at random, and all nodes within a fixed distance ℓ of it are replaced by a single super-node.We point out a simple mapping between RSR and irreversible aggregation on any graph. Hence any conclusion drawn for one process holds also for the other. Indeed, a local coarse-graining step to produce a new super-node represents one aggregation event, where a 'molecule' aggregates with all its neighbors within distance ℓ to p...

We introduce the concept of random sequential renormalization (RSR) for arbitrary networks. RSR is a graph renormalization procedure that locally aggregates nodes to produce a coarse grained network. It is analogous to the (quasi)parallel renormalization schemes introduced by C. Song et al. [C. Song et al., Nature (London) 433, 392 (2005)] and studied by F. Radicchi et al. [F. Radicchi et al., Phys. Rev. Lett. 101, 148701 (2008)], but much simpler and easier to implement. Here we apply RSR to critical trees and derive analytical results consistent with numerical simulations. Critical trees exhibit three regimes in their evolution under RSR.where N is the number of nodes at some step in the renormalization and N 0 is the initial size of the tree, RSR is described by a mean-field theory, and fluctuations from one realization to another are small. The exponent ν = 1/2 is derived using random walk and other arguments. The degree distribution becomes broader under successive steps, reaching a power law p k ∼ 1/k γ with γ = 2 and a variance that diverges as N