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When dynamics in a system proceeds under suppressive external bias, the system can undergo an abrupt phase transition, as can happen when an epidemic spreads. Recently, an explosive percolation (EP) model was introduced to understand such phenomena. The order of the EP transition has not been clarified in a unified framework covering low-dimensional systems and the mean-field limit. We introduce a stochastic model in which a rule for dynamics is designed to avoid the formation of a spanning cluster through competitive selection in Euclidean space. We use heuristic arguments to show that in the thermodynamic limit and depending on a control parameter, the EP transition can be either continuous or discontinuous if d < d(c) and is always continuous if d ≥ d(c), where d(c) is the spatial dimension and d is the upper critical dimension.

The evolution of the Erdos-Rényi (ER) network by adding edges is a basis model for irreversible kinetic aggregation phenomena. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel Kij approximately ij , where ij is the product of the sizes of two merging clusters. Here we study that when the giant cluster is discouraged to develop by a sublinear kernel Kij approximately (ij)omega with 0

Consider growing a network, in which every new connection is made between two disconnected nodes. At least one node is chosen randomly from a subset consisting of g fraction of the entire population in the smallest clusters. Here we show that this simple strategy for improving connection exhibits a phase transition barely studied before, namely a hybrid percolation transition exhibiting the properties of both first-order and second-order phase transitions. The cluster size distribution of finite clusters at a transition point exhibits power-law behavior with a continuously varying exponent τ in the range 2 < τ (g) ≤ 2.5. This pattern reveals a necessary condition for a hybrid transition in cluster aggregation processes, which is comparable to the power-law behavior of the avalanche size distribution arising in models with link-deleting processes in interdependent networks.PACS numbers: 64.60. De,64.60.ah,89.75.Da Transport or communication systems grow by adding new connections. Often certain constraints are imposed by society and if these constraint involve global knowledge about connectivity, the transition to a percolating system can become first order, as happens for instance when suppressing the spanning cluster, when imposing a cluster size [1] or when favoring the most disconnected sites [2]. Typically this effect is accompanied by the loss of critical scaling making these abrupt transitions less predictable and thus more dangerous. We will show here, that for a specific case, namely a variant of the model introduced in Ref.[3], critical fluctuations and power-law distributions can prevail and for the first time identify a hybrid transition in explosive percolation.Hybrid phase transitions have been observed recently in many complex network systems [4,5]; in these transitions, the order parameter m(t) exhibits behaviors of both first-order and second-order transitions simultaneously aswhere m 0 and r are constants and β is the critical exponent of the order parameter, and t is a control parameter. Examples of such behavior include k-core percolation [6,7], the cascading failure model on interdependent complex networks [8,9], and the Kuramoto synchronization model with a correlation between the natural frequencies and degrees of each node on complex networks [10,11], etc. For the models in [6-9], a critical behavior appears as nodes or links are deleted from a percolating cluster above the percolation threshold until reaching a transition point t c . As t is decreased infinitesimally further as 1/N beyond t c in finite systems, the order parameter decreases suddenly to zero and a first-order phase transition occurs. Thus, a hybrid phase transition occurs at t = t c in the thermodynamic limit. Next we recall discontinuous percolation transitions occurring in generalized contagion models [12][13][14]. Recent studies [15] of a generalized epidemic model [13] revealed that the discontinuous percolation transition turns out to be a hybrid percolation transition (HPT) represented by (1). For this case, a HPT ...

The finite-size scaling (FSS) theory for continuous phase transitions has been useful in determining the critical behavior from the size-dependent behaviors of thermodynamic quantities. When the phase transition is discontinuous, however, FSS approach has not been well established yet. Here, we develop a FSS theory for the explosive percolation transition arising in the Erdős and Rényi model under the Achlioptas process. A scaling function is derived based on the observed fact that the derivative of the curve of the order parameter at the critical point t(c) diverges with system size in a power-law manner, which is different from the conventional one based on the divergence of the correlation length at t(c). We show that the susceptibility is also described in the same scaling form. Numerical simulation data for different system sizes are well collapsed on the respective scaling functions.