Narrow Results

In this paper, theoretical solutions for degree distribution of decreasing random birth-and-death networks $(0<p<1/2)$ are provided. First, we prove that the degree distribution has the form of Poisson summation, for which degree distribution equations under steady state and probability generating function approach are employed. Then, based on the form of Poisson summation, we further confirm the tail characteristic of degree distribution is Poisson tail. Finally, simulations are carried out to verify these results by comparing the theoretical solutions with computer simulations.

For $n\in \mathbb{N}$, let ${𝒮}in{g}_n$ be the semigroup of all singular mappings on $X_n=\{1,2,\dots ,n\}$. For each nonempty subset $A$ of $X_n$, let ${{𝒮}}_n(A)=\{\alpha \in {𝒮}in{g}_n:(\forall k\in A)x\le k\Rightarrow x\alpha \le k\}$ be the semigroup of all $A$-decreasing mappings on $X_n$. In this paper we determine the rank and idempotent rank of the semigroup ${{𝒮}}_n(A)$.

Let $h:A\to A$ and $𝜀$ be a partial order on $A$. We deal with properties of oriented graphs which correspond to the algebra $(A,h)$ in the case that $h$ is monotone with respect to $𝜀$. We derive that every mono-unary algebra except connected one with a cycle of odd length has the property that there exists a nontrivial partial order such that $(A,h)$ is monotone with respect to it. All mono-unary algebras such that there exists a linear order such that $h$ is monotone with respect to this order will be described; if the number of components of $(A,h)$ is infinite, then the number of such orders is equal to the cardinality of the power set of $A$.