K-core and bootstrap percolation are widely studied models that have been used to represent and understand diverse deactivation and activation processes in natural and social systems. Since these models are considerably similar, it has been suggested
in recent years that they could be complementary. In this manuscript we provide a rigorous analysis that shows that for any degree and threshold distributions heterogeneous bootstrap percolation can be mapped into heterogeneous k-core percolation and vice versa, if the functionality thresholds in both processes satisfy a complementary relation. Another interesting problem in bootstrap and k-core percolation is the fraction of nodes belonging to their giant connected components $P_{infty b}$ and $P_{infty c}$, respectively. We solve this problem analytically for arbitrary randomly connected graphs and arbitrary threshold distributions, and we show that $P_{infty b}$ and $P_{infty c}$ are not complementary. Our theoretical results coincide with computer simulations in the limit of very large graphs. In bootstrap percolation, we show that when using the branching theory to compute the size of the giant component, we must consider two different types of links, which are related to distinct spanning branches of active nodes.
We report the scaling behavior of the Earth and Venus over a wider range of length scales than reported by previous researchers. All landscapes (not only mountains) together follow a consistent scaling behavior, demonstrating a crossover between high
ly correlated (smooth) behavior at short length scales (with a scaling exponent $alpha$=1) and self-affine behavior at long length scales ($alpha$=0.4). The self-affine behavior at long scales is achieved on Earth above 10 km and on Venus above 50 km.
We present a cascading failure model of two interdependent networks in which functional nodes belong to components of size greater than or equal to $s$. We find theoretically and via simulation that in complex networks with random dependency links th
e transition is first-order for $sgeq 3$ and second-order for $s=2$. We find for two square lattices with a distance constraint $r$ in the dependency links that increasing $r$ moves the system from a regime without a phase transition to one with a second-order transition. As $r$ continues to increase the system collapses in a first-order transition. Each regime is associated with a different structure of domain formation of functional nodes.
Detailed empirical studies of publicly traded business firms have established that the standard deviation of annual sales growth rates decreases with increasing firm sales as a power law, and that the sales growth distribution is non-Gaussian with sl
owly decaying tails. To explain these empirical facts, a theory is developed that incorporates both the fluctuations of a single firms sales and the statistical differences among many firms. The theory reproduces both the scaling in the standard deviation and the non-Gaussian distribution of growth rates. Earlier models reproduce the same empirical features by splitting firms into somewhat ambiguous subunits; by decomposing total sales into individual transactions, this ambiguity is removed. The theory yields verifiable predictions and accommodates any form of business organization within a firm. Furthermore, because transactions are fundamental to economic activity at all scales, the theory can be extended to all levels of the economy, from individual products to multinational corporations.
Using event driven molecular dynamics simulations, we study a three dimensional one-component system of spherical particles interacting via a discontinuous potential combining a repulsive square soft core and an attractive square well. In the case of
a narrow attractive well, it has been shown that this potential has two metastable gas-liquid critical points. Here we systematically investigate how the changes of the parameters of this potential affect the phase diagram of the system. We find a broad range of potential parameters for which the system has both a gas-liquid critical point and a liquid-liquid critical point. For the liquid-gas critical point we find that the derivatives of the critical temperature and pressure, with respect to the parameters of the potential, have the same signs: they are positive for increasing width of the attractive well and negative for increasing width and repulsive energy of the soft core. This result resembles the behavior of the liquid-gas critical point for standard liquids. In contrast, for the liquid-liquid critical point the critical pressure decreases as the critical temperature increases. As a consequence, the liquid-liquid critical point exists at positive pressures only in a finite range of parameters. We present a modified van der Waals equation which qualitatively reproduces the behavior of both critical points within some range of parameters, and give us insight on the mechanisms ruling the dependence of the two critical points on the potentials parameters. The soft core potential studied here resembles model potentials used for colloids, proteins, and potentials that have been related to liquid metals, raising an interesting possibility that a liquid-liquid phase transition may be present in some systems where it has not yet been observed.
We investigate the formation of a two-dimensional quasicrystal in a monodisperse system, using molecular dynamics simulations of hard sphere particles interacting via a two-dimensional square-well potential. We find that more than one stable crystall
ine phase can form for certain values of the square-well parameters. Quenching the liquid phase at a very low temperature, we obtain an amorphous phase. By heating this amorphous phase, we obtain a quasicrystalline structure with five-fold symmetry. From estimations of the Helmholtz potentials of the stable crystalline phases and of the quasicrystal, we conclude that the observed quasicrystal phase can be the stable phase in a specific range of temperatures.
