On the total signed domination number of the Cartesian product of paths

Let $G$ be a finite connected simple graph with a vertex set $V(G)$ and an edge set $E(G)$. A total signed dominating function of $G$ is a function $f: V(G)\cup E(G)\rightarrow \{-1, 1\}$, such that $\sum_{y\in N_T[x]}f(y) \geq 1$ for all $x\in V(G) \cup E(G)$. The total signed domination number of $G$ is the　minimum weight of a total signed dominating function on $G$. In this　paper, we prove lower and upper bounds on the total signed　domination number of the Cartesian product of two paths, $P_m\Box P_n$

A Dynamic Microblog Network and Information Dissemination in “@” Mode

Social media, especially the microblogs, emerge as a part of our daily life and become a key way to information spread. Thus, information dissemination in the microblog became a research hotspot. Based on some principles that are summarized from the microblog users’ behaviors, this paper proposes a dynamic microblog network model. Through simulations this network has the features of periodicity of average degree, high clustering coefficient, high degree of modularity, and community. Besides, an information dissemination model through “@” in the microblog has been presented. With the microblog network model and the zombie-city model, this paper has modelled an artificial microblog and has simulated the information dissemination in the artificial microblog with different scenes. Therefore, some interesting findings have been presented. (1) Due to a better connectivity, information could spread widely in a random network; (2) information spreads more quickly in a stable microblog network; (3) the decay rate of the relationships will have an effect on information dissemination; that is, with a lower decay rate, information spreads more quickly and widely; (4) the higher active level of users in microblog could promote information spread widely and quickly; (5) the “@” mode of information dissemination makes a high modularity of the information diffusion network