Replica-mean-field limits of fragmentation-interaction-aggregation processes

Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called ‘Poisson hypothesis’. However, in most applications this hypothesis is only conjectured. In this paper we establish the Poisson hypothesis for a general class of discrete-time, point-process-based dynamics that we propose to call fragmentation-interaction-aggregation processes, and which are introduced here. These processes feature a network of nodes, each endowed with a state governing their random activation. Each activation triggers the fragmentation of the activated node state and the transmission of interaction signals to downstream nodes. In turn, the signals received by nodes are aggregated to their state. Our main contribution is a proof of the Poisson hypothesis for the replica-mean-field version of any network in this class. The proof is obtained by establishing the propagation of asymptotic independence for state variables in the limit of an infinite number of replicas. Discrete-time Galves–Löcherbach neural networks are used as a basic instance and illustration of our analysis.

Charge radii of potassium isotopes in the RMF(BCS)* approach

We apply the recently proposed RMF(BCS)* ansatz to study the charge radii of the potassium isotopic chain up to $^{52}$K. It is shown that the experimental data can be reproduced rather well, qualitatively similar to the Fayans nuclear density functional theory, but with a slightly better description of the odd-even staggerings (OES). Nonetheless, both methods fail for $^{50}$K and to a lesser extent for $^{48,52}$K. It is shown that if these nuclei are deformed with a $\beta_{20}\approx-0.2$, then one can obtain results consistent with experiments for both charge radii and spin-parities. We argue that beyond mean field studies are needed to properly describe the charge radii of these three nuclei, particularly for $^{50}$K.

Modeling the spread of an epidemic in presence of vaccination using cellular automata

The results of Kermack–McKendrick SIR model are planned to be reproduced by cellular automata (CA) lattice model. The CA algorithms are proposed to study the model of an epidemic, systematically. The basic goal is to capture the effects of spreading of infection over a scale of length. This CA model can provide the rate of growth of the infection over the space which was lacking in the mean-field like susceptible-infected-removed (SIR) model. The motion of the circular front of an infected cluster shows a linear behavior in time. The correlation of a particular site to be infected with respect to the central site is also studied. The outcomes of the CA model are in good agreement with those obtained from SIR model. The results of vaccination have been also incorporated in the CA algorithm with a satisfactory degree of success. The advantage of the present model is that it can shed a considerable amount of light on the physical properties of the spread of a typical epidemic in a simple, yet robust way.