We study the dynamic scaling hypothesis in invariant surface growth. We show that the existence of power-law scaling of the correlation functions (scale invariance) does not determine a unique dynamic scaling form of the correlation functions, which leads to the different anomalous forms of scaling recently observed in growth models. We derive all the existing forms of anomalous dynamic scaling from a new generic scaling ansatz. The different scaling forms are subclasses of this generic scaling ansatz associated with bounds on the roughness exponent values. The existence of a new class of anomalous dynamic scaling is predicted and compared with simulations.

The probability density function (PDF) of a global measure in a large class of highly correlated systems has been suggested to be of the same functional form. Here, we identify the analytical form of the PDF of one such measure, the order parameter in the low temperature phase of the 2D $\mathrm{XY}$ model. We demonstrate that this function describes the fluctuations of global quantities in other correlated equilibrium and nonequilibrium systems. These include a coupled rotor model, Ising and percolation models, models of forest fires, sandpiles, avalanches, and granular media in a self-organized critical state. We discuss the relationship with both Gaussian and extremal statistics.

We study kinetically rough surfaces which display anomalous scaling in local properties such as the roughness or the height-difference correlation function. By studying the power spectrum of the surface and its relation to the height-difference correlation, we distinguish two independent causes for anomalous scaling. One is superroughening (global roughness exponent larger than or equal to 1), even if the spectrum behaves nonanomalously. Another cause is what we term an intrinsically anomalous spectrum, in whose scaling an independent exponent exists, which induces different scaling properties for small and large length scales. We show that in this case the surface does not need to be superrough in order to display anomalous scaling. The scaling relations we propose for the structure factor and height-difference correlation for intrinsically anomalous surfaces are shown to hold for a random diffusion equation, independently of the value of the global roughness exponent below or above one.

We study surface growth models exhibiting anomalous scaling of the local surface fluctuations. An analytical approach to determine the local scaling exponents of continuum growth models is proposed. The method allows one to predict when a particular growth model will have anomalous properties ( $\alpha \ne {\alpha }_{\mathrm{loc}}$) and to calculate the local exponents. Several continuum growth equations are examined as examples.

Many social, biological, and economic systems can be approached by complex networks of interacting units. In many of these systems relations are directed in the sense that links act only in one direction (outwards or inwards). We investigate the effect of directed links on the behavior of a simple spin-like model evolving on a small-world network. This model may describe for instance the dynamics of public opinion in social influence networks. We show that directed networks may lead to a highly nontrivial phase diagram including first- and second-order phase transitions out of equilibrium.

We argue that fracture surfaces may exhibit anomalous dynamic scaling properties akin to what occurs in some models of kinetic roughening. We determine the complete scaling behavior of the local fluctuations of a brittle fracture in a granite block from experimental data. We obtain a global roughness exponent $\chi =1.2\mathrm{}$ which differs from the local one, ${\chi }_{\mathrm{loc}}=0.79\mathrm{}$. Implications on fracture physics are discussed.