均匀随机交图的单调增性质及其相变
Monotone Increasing Properties and Their Phase Transitions in Uniform Random Intersection Graphs
摘要：
均匀随机交图已经引起了广泛的兴趣，并在各种应用中得到了广泛的应用。具有$n$节点的均匀随机交叉图被构造如下：每个节点从相同的$n$不同项目池中均匀随机地选择一组$K_n$不同项目，并且两个节点在它们之间建立无向边当且仅当它们共享至少一个项目。对于用$G(n, K_n, P_n)$表示的图，本文给出了如下结果.首先，我们提供了一个精确的概率$G(n, K_n, P_n)$有一个完美的匹配和有一个Hamilton圈，分别在$P_n = ω\big(n (\ln n)^5\big)$（所有渐近符号理解与$n \to \infty$）。分析表明，就像（$k$ -）的连接性在以前的工作中所示，完美匹配的包容性和汉密尔顿循环包容的属性，$G(n, K_n, P_n)$也表现出相变：对于上述每一个属性，随着$K_n$的增加，概率的极限，$G(n, K_n, P_n)$有$0$到$1$增加。其次，我们分别计算了$G(n, K_n, P_n)$在$k$ -连通性（KC）、完全匹配包容性（PMC）和Hamilton圈包容性（HCC）下的相变宽度。对于图属性$R$和正常数$a < \frac{1}{2}$，其中相变宽度$d_n(R, a)$定义为最小$K_n$之间的差，确保$G(n, K_n, P_n)$具有属性$R$，概率至少为$1-a$或$a$，我们针对任何正常数$a<\frac{1}{2}$和$k$证明：（i）如果$P_n=Ω(n)$和$P_n=o(n\ln n)$，则对于每个$n$足够大，$d_n(KC, a)$要么是$1$，要么是$1$。(ii)如果是$P_n=Θ(n\ln n)$，则为$d_n(KC, a)=Θ(1)$。(iii)如果为$P_n=ω(n\ln n)$，则为$d_n(KC, a)=ω(1)$。(iv)如果$P_n=ω\big(n (\ln n)^5\big)$、$d_n(PMC, a)$和$d_n(HCC, a)$均为$ω(1)$。
英文摘要：
Uniform random intersection graphs have received much interest and been used in diverse applications. A uniform random intersection graph with $n$ nodes is constructed as follows: each node selects a set of $K_n$ different items uniformly at random from the same pool of $P_n$ distinct items, and two nodes establish an undirected edge in between if and only if they share at least one item. For such graph denoted by $G(n, K_n, P_n)$, we present the following results in this paper. First, we provide an exact analysis on the probabilities of $G(n, K_n, P_n)$ having a perfect matching and having a Hamilton cycle respectively, under $P_n = ω\big(n (\ln n)^5\big)$ (all asymptotic notation are understood with $n \to \infty$). The analysis reveals that just like ($k$-)connectivity shown in prior work, for both properties of perfect matching containment and Hamilton cycle containment, $G(n, K_n, P_n)$ also exhibits phase transitions: for each property above, as $K_n$ increases, the limit of the probability that $G(n, K_n, P_n)$ has the property increases from $0$ to $1$. Second, we compute the phase transition widths of $G(n, K_n, P_n)$ for $k$-connectivity (KC), perfect matching containment (PMC), and Hamilton cycle containment (HCC), respectively. For a graph property $R$ and a positive constant $a < \frac{1}{2}$, with the phase transition width $d_n(R, a)$ defined as the difference between the minimal $K_n$ ensuring $G(n, K_n, P_n)$ having property $R$ with probability at least $1-a$ or $a$, we show for any positive constants $a<\frac{1}{2}$ and $k$: (i) If $P_n=Ω(n)$ and $P_n=o(n\ln n)$, then $d_n(KC, a)$ is either $0$ or $1$ for each $n$ sufficiently large. (ii) If $P_n=Θ(n\ln n)$, then $d_n(KC, a)=Θ(1)$. (iii) If $P_n=ω(n\ln n)$, then $d_n(KC, a)=ω(1)$. (iv) If $P_n=ω\big(n (\ln n)^5\big)$, $d_n(PMC, a)$ and $d_n(HCC, a)$ are both $ω(1)$.
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