社会图中的弱动态垄断
Weak dynamic monopolies in social graphs
摘要：
动态垄断已经被定义和研究，用于制定社交网络中影响力传播的现象，如疾病，意见，新产品的适应等。网络中受到影响的元素（例如感染或适应意见）被称为活动顶点。在这些模型中，假设当一个元素被激活时，它一直保持活动状态，直到过程结束。但在某些影响力扩散的现象中，这种性质并不成立。例如，在某些疾病中，感染仅持续有限的时间段，或者考虑疾病的传播或计算机病毒的传播以及一些消毒或去污方法。动态垄断对研究后一种现象没有用处。为此，我们引入了一个新的模型，这种影响力的扩散，并称之为弱动态垄断。社交网络由图$G$表示。假设$G$的任何顶点$v$具有阈值$τ(v)\in \Bbb{N}$。如果$V(G)$可以被划分为$D_0=D, D_1, \ldots, D_k$，使得对于任何$i$，$D_i$的任何顶点$v$在$D_{i-1}$中至少具有$τ(v)$邻居，则子集$D\subseteq V(G)$被称为弱动态垄断。在该定义中，$D$的大小和处理时间分别指的是$|D|$和$k$。本文首先研究了弱动态垄断与其他相关概念的关系，然后得到了弱动态垄断的最小规模的一些界。接下来，我们得到了一些结果的处理时间的弱动态垄断的一些图形参数。最后，得到了关于一般图中弱动态垄断最小尺寸的不可近似性的一个困难结果。
英文摘要：
Dynamic monopolies were already defined and studied for the formulation of the phenomena of the spread of influence in social networks such as disease, opinion, adaptation of new product and etc. The elements of the network which have been influenced (e.g. infected or adapted an opinion) are called active vertices. It is assumed in these models that when an element is activated, it remains active until the end of the process. But in some phenomena of the spread of influence this property does not hold. For example in some diseases the infection lasts only a limited period of time or consider the spread of disease or propagation of computer virus together with some quarantination or decontamination methods. Dynamic monopolies are not useful for the study of these latter phenomena. For this purpose, we introduce a new model for such diffusions of influence and call it weak dynamic monopoly. A social network is represented by a graph $G$. Assume that any vertex $v$ of $G$ has a threshold $τ(v)\in \Bbb{N}$. Then a subset $D\subseteq V(G)$ is said to be a weak dynamic monopoly if $V(G)$ can be partitioned into $D_0=D, D_1, \ldots, D_k$ such that for any $i$, any vertex $v$ of $D_i$ has at least $τ(v)$ neighbors in $D_{i-1}$. In this definition, by the size and the processing time of $D$ we mean $|D|$ and $k$, respectively. We first investigate the relationships between weak dynamic monopolies and other related concepts and then obtain some bounds for the smallest size of weak dynamic monopolies. Next we obtain some results concerning the processing time of weak dynamic monopolies in terms of some graph parameters. Finally, a hardness result concerning inapproximibility of the determining the smallest size of weak dynamic monopolies in general graphs is obtained.
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