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We analyze a model of interacting agents (e.g. prebiotic chemical species) which are represented by nodes of a network, whereas their interactions are mapped onto directed links between these nodes. On a fast time scale, each agent follows an eigendynamics based on catalytic support from other nodes, whereas on a much slower time scale the network evolves through selection and mutation of its nodes-agent. In the first part of the paper, we explain the dynamics of the model by means of characteristic snapshots of the network evolution and confirm earlier findings on crashes and recoveries in the network structure. In the second part, we focus on the aggregate behavior of the network dynamics. We show that the disruptions in the network structure are smoothed out, so that the average evolution can be described by a growth regime followed by a saturation regime, without an initial random regime. For the saturation regime, we obtain a logarithmic scaling between the average connectivity per node 〈l〉_s and a parameter m, describing the average incoming connectivity, which is independent of the system size N.

Simple arguments based on the general properties of quantum fluctuations have been recently shown to imply that quantum fluctuations of spacetime obey the same scaling laws of the velocity fluctuations in a homogeneous incompressible turbulent flow, as described by Kolmogorov 1941 (K41) scaling theory. Less noted, however, is the fact that this analogy rules out the possibility of a fractal quantum spacetime, in contradiction with growing evidence in quantum gravity research. In this Note, we show that the notion of a fractal quantum spacetime can be restored by extending the analogy between turbulence and quantum gravity beyond the realm of K41 theory. In particular, it is shown that compatibility of a fractal quantum spacetime with the recent Horava–Lifshitz scenario for quantum gravity, implies singular quantum wavefunctions. Finally, we propose an operational procedure, based on extended self-similarity techniques, to inspect the (multi)-scaling properties of quantum gravitational fluctuations.

Explosive percolation proposed by Achlioptas et al. has attracted much attention recently. To understand how the product rule delays the phase transition, in this paper, we will concentrate our efforts on the influence of product rule in the transition from Erdős–Rényi (ER) percolation to Achlioptas explosive percolation. In our model, only a fraction of the adding edges will obey the product rule. With the probability p, the edge is selected under the Achlioptas's product rule; with 1 - p the edge is added between two randomly chosen nodes, that is, the ER rule. When the product rule plays a dominant role in the evolution, p > 0.5, the percolation transition delays more abruptly and instantaneously. The location of the critical points is independent of the network size and satisfies , where and are the critical points of ER percolation and Achlioptas one. We present the largest gap Δs_max and the critical region Δ as a function of p and N. The giant component s in terms of the edge density t above t_c is also analyzed. We find that all the critical values have a linear relationship against p.

The technique of detrended fluctuation analysis (DFA) has been widely used to unveil scaling properties of many different signals. In this paper, we determine scaling properties in the encrypted images by means of a two-dimensional DFA approach. To carry out the image encryption, we use an enhanced cryptosystem based on a rule-90 cellular automaton and we compare the results obtained with its unmodified version and the encryption system AES. The numerical results show that the encrypted images present a persistent behavior which is close to that of the 1/f-noise. These results point to the possibility that the DFA scaling exponent can be used to measure the quality of the encrypted image content.

During sleep there exists a nonlinear dynamic phenomenon, which is called cyclic alternating pattern. This phenomenon is generated in the brain and is composed of a series of events of short duration known as A-phases. It has been shown that A-phases can be found in other physiological systems such as the cardiovascular. However, there is no evidence that shows the temporal influence of the A-phases with the cardiovascular system. For this purpose, we consider the scaling method known as detrended fluctuation analysis (DFA). The analysis was carried out in well sleepers and insomnia people, and the numerical results show an increment in the scaling parameter for the insomnia subjects compared with the normal ones. In addition, the results of the heart dynamics suggests a persistent behavior toward the $1∕f$-noise.

The time series of the states of several well-known hyperchaotic systems are analyzed numerically using the detrended fluctuation analysis based on the discrete wavelet transform. We report the finding of significant scaling behaviors (power-law like) in some of these time series, which can be used as an additional characteristic distinguishing the dynamical evolution of such systems.

In every country except China, COVID-19 first infection cases were imported by travelers, which are either people coming back to their own country or visiting foreigners (international or external tourists). In a global and regional phenomenological analysis of COVID-19 spread, we assume that tourism inflow is a trigger mechanism of worldwide dissemination at the pandemic onset days. Taking into account all countries, a convenient common-time origin timeline was employed as if the beginning of the epidemic would have occurred simultaneously in every country. We obtained very good statistical Pearson and Spearman correlations between accumulated infected cases by country and a positive power of the product $TP$, where $T$ is the tourism inflow before the pandemic and $P$ is the country population.

In the USA, COVID-19 first infection cases were imported by external travelers. At the epidemic onset days, we assume that the disease partially spreads due to domestic passengers air transportation in its densely connected airport network. Taking into account all USA states, we arranged COVID-19 infected cases data in a convenient common time origin timeline as if the beginning of the epidemic would have occurred simultaneously in every state. Looking for a trend between cases and air passengers, we obtained with this timeline very good statistical Pearson and Spearman correlations between accumulated infected cases by state and a positive power of the product $TP$, where $T$ is the domestic flight passengers (travelers) inflow by state before the epidemic and $P$ is its population. We also found a good correlation between percentages of urban area by state and their COVID-19 daily new cases growth rates at onset days.

A scaling theory for the parallel efficiency of simulations on MIMD computers is presented. It is successfully applied to the performance results recently reported by Kohring and Stauffer for different algorithms on an Intel i860 Hypercube. The discrepancies between the measured scaling exponents and our predictions are resolved within the framework of our general theory.

The phase transition of DNA molecules is studied in an exactly solvable formalism with the Morse and Deng–Fan potentials for the interstrand hydrogen bonds of nucleotide base pairs. It is shown that although the two potentials have different short-range behaviors, the thermodynamic quantities of the DNA system in these potentials enjoy the same scaling laws with the associated critical exponents, which are explicitly calculated. These exactly solvable DNA models are shown to exhibit a phase transition of the second order and the results of the analysis agree with previous studies.

This paper proposes a systemic perspective for some aspects of both phylogenesis and ontogenesis by expressing biological organization in terms of "anti-entropy", a notion to be defined below and which conceptually differs from the common use of "negative entropy". To this purpose, we introduce two principles, in addition to the thermodynamic ones, which are (mathematically) compatible with traditional principles but which have no meaning with regard to inert matter. A traditional balance equation for the metabolism will then be extended to the new notion as specified by these principles. We examine far from equilibrium systems and we focus in particular on the production of global entropy associated to the irreversible character of the processes. A close analysis of anti-entropy will be performed from the perspective of a diffusion equation of biomass over "complexity" and, as a complementary approach and as a tool for specifying a source term, in connection to Schrödinger's method regarding his equation in the field of Quantum Mechanics. We borrow only the operatorial approach from this equation and do so using a classical framework, since we use real coefficients instead of complex ones, thus falling outside of the mathematical framework of quantum theories. The first application of our proposal is a simple mathematical reconstruction of Gould's complexity curve of biomass over complexity as it applies to evolution. We then present, based on the existence of different time scales, a partition of ontogenetic time, in reference to entropy and anti-entropy variation. On the grounds of this approach, we analyze the metabolism and scaling laws. This allows to compare various relevant coefficients appearing in these scaling laws, which fit empirical data. Finally, a tentative and quantitative evaluation of complexity is proposed, also in relation to some empirical data (Caenorhabditis elegans).

The quantification of fragmentation of natural, polidisperse, porous media using fractal models is well documented in the literature. However, in many cases, fractal exponents (fractal fragmentation dimension) and coefficients (fractal lacunarity) arising from a power law behavior do not make clear differences between different media. In the present work, the pore-solid fractal (PSF) model was used as a new fractal approach for deriving four scaling parameters (fractal dimension of the particle-size distribution, fractal fragmentation dimension of the fragment-size distribution, probability of fragmentation and scaling factor) from soil initiators subjected to different energy density input. The fractal fragmentation dimension for all soil samples was D_f=2.42±0.16 without correlating with the energy expended in the fragmentation process. By contrast, probability of fragmentation and scaling factor correlated significantly with the energetic term. The PSF model is useful for estimating a group of scaling parameters more appropriate for the quantification of complex patterns associated to fragment-size distributions.

In this paper, we investigate the scaling law relating the size of the boundary of a solid tumor and the rate at which it is lysed by a cell population of non-infiltrating cytotoxic lymphocytes. We do it in the context of enzyme kinetics through geometrical, analytical and numerical arguments. Following the Koch island fractal model, a scale-dependent function that describes the constant rate of the decay process and the fractal dimension is obtained. Then, in silico experiments are accomplished by means of a stochastic hybrid cellular automaton model. This model is used to grow several tumors with varying morphology and to test the power decay law when the cell-mediated immune response is effective, confirming its validity.

The similitude invariants and scaling laws for the buckling and free vibration of antisymmetric angle-ply laminated circular cylindrical shells were derived by directly applying the similitude transformation to the governing differential equations of the problem considered. The scaling laws relate the buckling load and natural frequency of a model to those of a prototype when the similarity requirements between the two systems are fulfilled. In the absence of experimental data, the validity of the scaling laws was verified by calculating theoretically the buckling loads and fundamental frequencies of vibration of the model and substituting them into the scaling laws to yield values for the prototype, which were then compared with those directly computed for the prototype. The numerical studies show that for the case of complete similitude with various stacking sequence, number of plies, and radius ratio, the solutions predicted by the similitude theory agree exactly with those obtained directly from the theory. Some typical partial similitude cases were also investigated. It was demonstrated that the partial similitude model with distortion in stacking sequence is reliable for predicting the behavior of the prototype. On the other hand, models with distortion in material properties are not recommended because of the existence of large discrepancies in the predicted results.

Developed herein are the scaling laws for physical modeling of anti-symmetric cross-ply laminated circular cylindrical shells for buckling and free vibration experiments. In the absence of experimental data, the validity of the scaling laws is verified by numerical experiments. This is accomplished by calculating theoretically the buckling loads and fundamental frequencies of the model and substituting into the scaling laws to obtain the corresponding values of the prototype. The predicted values of the prototype from the scaling laws are then compared with existing closed-form solutions. Examples for the complete similitude cases with various stacking sequences, number of plies, and length-to-radius ratios show exact agreement. The derived relationships between the model and prototype will greatly facilitate and reduce the need for costly experiments. In reality, either due to the complexity of the scaling laws or to economize experimental cost and time, it may not be feasible to construct the model to fulfil the scaling laws completely. Thus, several possible models of partial similitude are investigated numerically. These include models with distortion in laminated material properties, stacking sequences and number of plies. Model with distortion in material properties yields a high percentage of discrepancy and is not recommended.

Tropical countries, like Nigeria, depend on rainfall for agriculture, power generation, transportation and other economic activities. Drought will hinder the performance of these activities, hence, it poses a significant threat to the economy. Understanding fluctuations and structures in droughts will help in forecasting, planning and mitigating its impact on livelihoods. In this study, the multifractal properties of drought at four temporal scales were investigated over different locations across Nigeria. Drought was computed using the standardized precipitation index from monthly precipitation data from 1980 to 2010. Using multifractal detrended fluctuation analysis, meteorological drought was found to have multifractal properties at 1-, 6-, 12- and 24-month temporal scale. The generalized Hurst exponent of drought at different time-scale showed dependence on scaling exponent. Long-range correlations were found to be main source of multifractality at all temporal scales. The multifractal strength increases with increasing temporal scale except for a few locations. The range of spectrum width were found to be 0.306–0.464 and 0.596–0.993 at 1- and 24-month temporal scale, respectively. No significant trend was found in the degree of multifractality across different climatic zones of Nigeria.

We consider simple stochastic climate models, described by slowly time-dependent Langevin equations. We show that when the noise intensity is not too large, these systems can spend substantial amounts of time in metastable equilibrium, instead of adiabatically following the stationary distribution of the frozen system. This behavior can be characterized by describing the location of typical paths, and bounding the probability of atypical paths. We illustrate this approach by giving a quantitative description of phenomena associated with bistability, for three famous examples of simple climate models: Stochastic resonance in an energy balance model describing the Ice Ages; hysteresis in a box model for the Atlantic thermohaline circulation; and bifurcation delay in the case of the Lorenz model for Rayleigh–Bénard convection.

We present an analysis of inter-regional investment stocks within Europe from a complex networks perspective. We consider two different levels: first, we compute the inward–outward investment stocks at the level of firms, based on ownership shares and number of employees; then we estimate the inward–outward investment stock at the level of regions in Europe, by aggregating the ownership network of firms, based on their headquarter location. To our knowledge, there is no similar approach in the literature so far, and we believe that it may lead to important applications for policy making. In the present paper, we focus on the statistical distributions and the scaling laws, while in further studies we will analyze the structure of the network and its relation to geographical space. We find that while outward investment and activity of firms are power law distributed with a similar exponent, for regions these quantities are better described by a log-normal distribution. At both levels we also find scaling laws relating investment to activity and connectivity. In particular, we find that investment stock scales as a power law of the connectivity, as previously found for stock market data.

The practice of detecting power laws and scaling behaviors in economics and finance has gained momentum in the last few years, due to the increased use of concepts and methods first developed in statistical physics. Some disappointment has emerged in the economic profession, however, as regards the models proposed so far to theoretically explain these phenomena. In this paper we aim to address this criticism, showing that scaling behaviors can naturally emerge in a multiagent system with optimizing interacting units characterized by financial fragility.

We investigate phase transitions and critical behaviors of the Kerr–Sen black hole in four dimensions. Computing the involved thermodynamical quantities including the specific heat and using the Ehrenfest scheme, we show that such a black hole undergoes a second-order phase transition. Adopting a new metric form derived from the Gibss free energy scaled by a conformal factor associated with extremal solutions, we calculate the geothermodynamical scalar curvature recovering similar phase transitions. Then, we obtain the scaling laws and the critical exponents, matching perfectly with mean field theory.