Topological and random spread models with frozen symbols.

When a symbol or a type has been "frozen" (namely, a type of which an individual only produces one individual of the same type), its spread pattern will be changed and this change will affect the long-term behavior of the whole system. However, in a frozen system, the ξ-matrix and the offspring mean matrix are no longer primitive so that the Perron-Frobenius theorem cannot be applied directly when predicting the spread rates. In this paper, our goal is to characterize these key matrices and analyze the spread rate under more general settings both in the topological and random spread models with frozen symbols. More specifically, we propose an algorithm for explicitly computing the spread rate and relate the rate with the eigenvectors of the ξ-matrix or offspring mean matrix. In addition, we reveal that the growth of the population is exponential and that the composition of the population is asymptotically periodic. Furthermore, numerical experiments are provided as supporting evidence for the theory.

Mathematical analysis of topological and random m-order spread models.

This paper focuses on the analysis of two particular models, from deterministic and random perspective respectively, for spreading processes. With a proper encoding of propagation patterns, the spread rate of each pattern is discussed for both models by virtue of the substitution dynamical systems and branching process. In view of this, we are empowered to draw a comparison between two spreading processes according to their spreading models, based on which explanations are proposed on a higher frequency of a pattern in one model than the other. These results are then supported by the numerical evidence later in the article.

Spread rates of spread models with frozen symbols.

This article aims to compare the long-term behavior of a spread model before and after a type in the model becomes frozen, namely, a type of which an individual only produces individuals of the same type. By means of substitution dynamical systems and matrix analysis, the first result of this work gives the spread rates of a 1-spread model with one frozen symbol. Later, in the work, this is shown to hold under more general settings, which include generalized frozen symbols and frozen symbols in m-spread models. Numerical experiments are provided as supporting evidence for the theory.