Recently, it has been argued that entropy can be a direct measure of complexity, where the smaller value of entropy indicates lower system complexity, while its larger value indicates higher system complexity. We dispute this view and propose a universal measure of complexity that is based on Gell-Mann’s view of complexity. Our universal measure of complexity is based on a non-linear transformation of time-dependent entropy, where the system state with the highest complexity is the most distant from all the states of the system of lesser or no complexity. We have shown that the most complex is the optimally mixed state consisting of pure states, i.e., of the most regular and most disordered which the space of states of a given system allows. A parsimonious paradigmatic example of the simplest system with a small and a large number of degrees of freedom is shown to support this methodology. Several important features of this universal measure are pointed out, especially its flexibility (i.e., its openness to extensions), suitability to the analysis of system critical behaviour, and suitability to study the dynamic complexity.

In this paper, we are addressing the old problem of long-term nonlinear autocorrelation function versus short-term linear autocorrelation function. As continuous-time random walk (CTRW) can describe almost all possible kinds of diffusion, it seems to be an excellent tool to use. To be more precise, for instance, CTRW can successfully describe the short-term negative autocorrelation of returns in high-frequency financial data (caused by the bid-ask bounce phenomena). We observe long-term autocorrelation of absolute values of returns. Can it also be described by the CTRW model? And maybe more importantly, to what extent can it be explained by the same phenomena? To refer to these questions, we propose a new directed CTRW model with memory. The canonical CTRW trajectory consists of spatial jumps preceded by waiting times. In directed CTRW, we consider the case with positive spatial jumps only. We take into account the memory in the model as each spatial jump depends on the previous one. This model, based on simple assumptions, allowed us to obtain the general formula covering most popular types of nonlinear autocorrelation functions.

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In many physical, social, and economic phenomena, we observe changes in a studied quantity only in discrete, irregularly distributed points in time. The stochastic process usually applied to describe this kind of variable is the continuous-time random walk (CTRW). Despite the popularity of these types of stochastic processes and strong empirical motivation, models with a long-term memory within the sequence of time intervals between observations are rare in the physics literature. Here, we fill this gap by introducing a new family of CTRWs. The memory is introduced to the model by assuming that many consecutive time intervals can be the same. Surprisingly, in this process we can observe a slowly decaying nonlinear autocorrelation function without a fat-tailed distribution of time intervals. Our model, applied to high-frequency stock market data, can successfully describe the slope of decay of the nonlinear autocorrelation function of stock market returns. We achieve this result without imposing any dependence between consecutive price changes. This proves the crucial role of inter-event times in the volatility clustering phenomenon observed in all stock markets.

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