Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics

A two-state master equation based decision making model has been shown to generate phase transitions, to be topologically complex and to manifest temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. These properties are entailed by the fundamental assumption that the network elements in the decision making model imperfectly imitate one another. The process of subordination establishes that a single network element can be described by a fractional master equation whose analytic solution yields the observed inverse power-law probability distribution obtained by numerical integration of the two-state master equation to a high degree of accuracy

Spreading processes in Multilayer Networks

Several systems can be modeled as sets of interconnected networks or networks with multiple types of connections, here generally called multilayer networks. Spreading processes such as information propagation among users of an online social networks, or the diffusion of pathogens among individuals through their contact network, are fundamental phenomena occurring in these networks. However, while information diffusion in single networks has received considerable attention from various disciplines for over a decade, spreading processes in multilayer networks is still a young research area presenting many challenging research issues. In this paper we review the main models, results and applications of multilayer spreading processes and discuss some promising research directions.Comment: 21 pages, 3 figures, 4 table

The cause of universality in growth fluctuations

Phenomena as diverse as breeding bird populations, the size of U.S. firms, money invested in mutual funds, the GDP of individual countries and the scientific output of universities all show unusual but remarkably similar growth fluctuations. The fluctuations display characteristic features, including double exponential scaling in the body of the distribution and power law scaling of the standard deviation as a function of size. To explain this we propose a remarkably simple additive replication model: At each step each individual is replaced by a new number of individuals drawn from the same replication distribution. If the replication distribution is sufficiently heavy tailed then the growth fluctuations are Levy distributed. We analyze the data from bird populations, firms, and mutual funds and show that our predictions match the data well, in several respects: Our theory results in a much better collapse of the individual distributions onto a single curve and also correctly predicts the scaling of the standard deviation with size. To illustrate how this can emerge from a collective microscopic dynamics we propose a model based on stochastic influence dynamics over a scale-free contact network and show that it produces results similar to those observed. We also extend the model to deal with correlations between individual elements. Our main conclusion is that the universality of growth fluctuations is driven by the additivity of growth processes and the action of the generalized central limit theorem.Comment: 18 pages, 4 figures, Supporting information provided with the source files