Grzegorz Siudem

Analytical approach to the model of scientific revolutions.

The model of scientific paradigms spreading throughout a community of agents with memory is analyzed using the master equation. The case of two competing ideas is considered for various networks of interactions, including agents placed at Erdős-Rényi graphs or complete graphs. The pace of adopting a new idea by the community is analyzed, along with the distribution of periods after which a new idea replaces the old one. The approach is extended for the chain topology to the more general case when more than two ideas compete. Our analytical results agree with the numerical simulations.

External bias in the model of isolation of communities

We extend a model of community isolation in the d-dimensional lattice to a case with an imposed imbalance between the birth rates of competing communities. We provide analytical and numerical evidences that in the asymmetric two-species model there exists a well-defined value of the asymmetry parameter when the emergence of the isolated (blocked) subgroups is the fastest, i.e., the characteristic time t(c) is minimal. The critical value of the parameter depends only on the lattice dimensionality and is independent of the system size. A similar phenomenon is observed in the multispecies case with a geometric distribution of the birth rates. We also show that blocked subgroups in the multispecies case are absent or very rare when either there is a strictly dominant species that outnumbers the others or there is a large diversity of species. The number of blocked species of different kinds decreases with the dimension of the multispecies system.

Area-width scaling in generalised Motzkin paths

We consider a generalised version of Motzkin paths, where horizontal steps have length l, with l being a fixed positive integer. We first give the general functional equation for the area-width generating function of this model. Using a heuristic ansatz, we then derive the area-width scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schroder paths, before generalising our approach to arbitrary l. We then rigorously derive the tricritical scaling of Schroder paths by applying the generalised method of steepest descents to the known exact solution for their area-width generating function. Our results show that for Dyck and Schroder paths, the heuristic scaling ansatz reproduces the rigorous results.

Agent-based model for the h-index – exact solution

AbstractnHirsch’s h-index is perhaps the most popular citation-basednmeasure of scientific excellence. In 2013, Ionescu and Chopard proposed an agent-basednmodel describing a process for generating publications and citations in an abstractnscientific community [G. Ionescu, B. Chopard, Eur. Phys. J. B 86, 426n(2013)]. Within such a framework, one may simulate a scientist’s activity, and – bynextension – investigate the whole community of researchers. Even though the Ionescu andnChopard model predicts the h-index quite well, the authors provided a solutionnbased solely on simulations. In this paper, we complete their results with exact, analyticnformulas. What is more, by considering a simplified version of the Ionescu-Chopard model,nwe obtained a compact, easy to compute formula for the h-index. The derivednapproximate and exact solutions are investigated on a simulated and real-world datansets.n

Partition Function of the Model of Perfect Gas of Clusters for Interacting Fluids

The last paper of A. Fronczak presents a new, combinatorial approach to the model of perfect gas of clusters of interacting fluids. In the paper the enumerative properties and combinatorial meaning of Bell polynomials have been used to reveal some properties of systems described by grand canonical ensemble. In this paper, an exact proof of one of the important assertions from the work mentioned above is given. The assertion states that for the system in which one can distinguish k non interacting clusters the canonical partition function is given by Bell polynomial of the derivatives of the grand-thermodynamic potential with respect to the fugacity. Some possible applications of the considered theorem are presented.

Diffusion on hierarchical systems of weakly-coupled networks

Abstract We analyzed diffusion dynamics on weakly-coupled networks (interconnected networks) by means of separation of time scales. Using an adiabatic approximation we reduced the system dynamics to a Markov chain with aggregated variables and derived a transport equation that is analogous to Fick’s First Law and includes a driving force. Entropy production is a sum of microscopic entropy transport, which results from the particle’s migration between networks of different topologies and macroscopic entropy production of the Markov chain. Equilibrium particles partition between different sub-networks depends only on internal sub-network parameters. By changing structure of networks one can not only modify diffusion constants but can also induce or reverse the direction of the particles’ flow between different networks. Our framework, confirmed by numerical simulations, is also useful for considering diffusion in nested systems corresponding to hierarchical networks with several different time scales thus it can serve to uncover hidden hierarchy levels from observations of diffusion processes.