2023/11/02 by Prashant Kumar, Puneet Sharma, Kumar, Prashant +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Physics and Astronomy · #37B10 #37B20 #37B51 #Cellular Automata and Applications #Complex Network Analysis Techniques #Dynamical Systems (math.DS) #FOS: Mathematics #Gene Regulatory Network Analysis
paper · pdf · doi:10.48550/arxiv.2311.01123
openalex publication_date 2023/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate the dynamical behavior of a two dimensional shift XG (generated by a two dimensional graph G=(H,V)) using the adjacency matrices of the generating graph G. In particular, we investigate properties such as transitivity, directional transitivity, weak mixing, directional weak mixing and mixing for the shift space XG. We prove that if (HV)ij≠ 0 ⇔ (VH)ij≠ 0 (for all i,j), while doubly transitivity (weak mixing) of XH (or XV) ensures the same for two dimensional shift generated by the graph G, directional transitivity (in the direction (r,s)) can be characterized through the block representation of HrVs. We provide necessary and sufficient criteria to establish horizontal (vertical) transitivity for the shift space XG. We also provide examples to establish the necessity of the conditions imposed. Finally, we investigate the decomposability of a given graph into product of graphs with reduced complexity.