The existence of explosive phase transitions in random (Erdős Rényi-type) networks has been recently documented by Achlioptas et al. [Science 323, 1453[Science 323, (2009] via simulations. In this Letter we describe the underlying mechanism behind these first-order phase transitions and develop tools that allow us to identify (and predict) when a random network will exhibit an explosive transition. Several interesting new models displaying explosive transitions are also presented. PACS numbers: Valid PACS appear hereThe structure and dynamics of networked models and their application to social networks is an important and active area of research encompassing many fields ranging from physics [1, 2, 3] to sociology [4] to combinations thereof [5,6,7]. Ideas from statistical mechanics have contributed greatly to our understanding of such networks and their practical uses [8,9,10]. Of particular importance is the statistical mechanical notion of the order of a phase ('percolation') transition. Phase transitions in random network models are almost always second order [11,12] or higher [12,13]. Thus, it was surprising to many when Achlioptas et al. [14] reported recently that some models of interest in social networks can display first-order (discontinuous) transitions.In that work, they described several random graph models of the Erdős Rényi (ER) variety that exhibit firstorder or what they call "explosive" phase transitions. They provide convincing numerical evidence and a useful characterization of such transitions, but no details on the mechanisms underlying them. They describe several systems which display such transitions and a general class of systems which don't.In this paper we describe the underlying mechanisms behind explosive transitions in ER-type models. We show that, somewhat surprisingly, the key to explosive transitions is not the details of the edge-addition rules at work during the actual "explosion," but rather lies in the period preceding the explosion when a type of "powder keg" develops. In effect, the importance of the rules is to create an explosive situation, which can be detonated with almost any rule. In addition, our analysis provides an understanding of which random network models will have such transitions. This allows us to construct large classes of interesting models that display this behavior. (It also allows us to rule out many other models which will not display explosive transitions.)The prototypical network percolation example is that of pure (non-preferential) ER random graphs [11]. These begin with a set of n nodes, where n is large. Edges are then added to the graph, uniformly at random. As is well known, this system exhibits a phase transition as the number of edges τ increases. For τ < 0.5n all clusters are small (∼ log(n)) while for τ > 0.5n a large cluster (∼ n) appears. In the large-n limit this transition is a secondorder phase transition, i.e., letting s(τ ) be the size of the largest cluster after τ edges have been added, the graph of s(τ )/n against τ /n is continu...

We show that in networks with a hierarchical architecture, critical dynamical behaviors can emerge even when the underlying dynamical processes are not critical. This finding provides explicit insight into current studies of the brain's neuronal network showing power-law avalanches in neural recordings, and provides a theoretical justification of recent numerical findings. Our analysis shows how the hierarchical organization of a network can itself lead to power-law distributions of avalanche sizes and durations, scaling laws between anomalous exponents, and universal functions-even in the absence of self-organized criticality or critical points. This hierarchy-induced phenomenon is independent of, though can potentially operate in conjunction with, standard dynamical mechanisms for generating power laws.

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The effect of distant end walls on the bifurcation to traveling waves is considered. Previous approaches have treated the problem by assuming that.it is a weak perturbation of the translation invariant problem. When the problem is formulated instead in a finite box of length L and the limit L --+ 00 is taken, one obtains amplitude equations that differ from the usual Ginzburg-Landau description by the presence of an additional nonlinear term. This formulation leads to a description in terms of the amplitudes of the primary box modes, which are odd and even parity standing waves. For large L, the equations that result take the form of a Hopf bifurcation with approximate D4 symmetry. These equations are able to describe, qualitatively, not only traveling and "blinking" states, but also asymmetrical blinking states and "repeated transients," all of which have been observed in binary fluid convection experiments. PACS number(s): 47.20. Bp, 47.20.Ky, 03.40.Kf