动态网络中异步谣言传播的严密性分析
Tight Analysis of Asynchronous Rumor Spreading in Dynamic Networks
摘要：
异步谣言传播算法在网络中传播一条信息，即所谓的谣言。从单个被通知的节点开始，每个节点与具有速率$1$的指数时钟相关联，并且调用随机邻居以便可能交换谣言。传播时间是网络中所有节点都以高概率被告知的第一次。我们考虑算法在任何动态演化网络中的传播时间，$\mathcal{G}=\{G^{(t)}\}_{t=0}^{\infty}$，它是一个以离散时间步长$t=0,1\ldots$暴露的图序列。我们观察到，除了动态网络的扩展轮廓，节点的度分布随时间的推移影响传播时间。我们建立了上界的传播时间的图形电导和勤奋。对于给定的连通简单图$G=(V,E)$，割集$E(S, \overline{S})$的勤奋度定义为$ρ(S)=\min_{\{u,v\}\in E(S,\overline{S})}\max\{\bar{d}/d_u, \bar{d}/d_v\}$，其中$d_u$是$u$的度，$\bar{d}$是具有较小体积的割的一侧中的节点的平均度（即，${\mathtt{vol}}{(S)}=\sum_{u\in S}d_u$）。$G$的勤奋也被定义为$ρ(G)=\min_{ \emptyset\neq S\subset V}ρ(S)$。我们证明了该算法在$\mathcal{G}$中的传播时间受$T$的约束，其中$T$是$\sum_{t=0}^TΦ(G^{(t)})\cdotρ(G^{(t)})$第一次超过$C\log n$，其中$Φ(G^{(t)})$表示$G^{(t)}$的电导，$C$是一个指定的常数。定义了绝对勤奋度为$\overlineρ(G)=\min_{\{u,v\}\in E}\max\{1/d_u,1/d_v\}$，并建立了绝对勤奋度下的价差上界$T$，首次将$\sum_{t=0}^T\lceilΦ(G^{(t)})\rceil\cdot \overlineρ(G^{(t)})\ge 2n$ .我们提出了动态网络，其中给定的上界几乎是紧的。
英文摘要：
The asynchronous rumor algorithm spreading propagates a piece of information, the so-called rumor, in a network. Starting with a single informed node, each node is associated with an exponential time clock with rate $1$ and calls a random neighbor in order to possibly exchange the rumor. Spread time is the first time when all nodes of a network are informed with high probability. We consider spread time of the algorithm in any dynamic evolving network, $\mathcal{G}=\{G^{(t)}\}_{t=0}^{\infty}$, which is a sequence of graphs exposed at discrete time step $t=0,1\ldots$. We observe that besides the expansion profile of a dynamic network, the degree distribution of nodes over time effect the spread time. We establish upper bounds for the spread time in terms of graph conductance and diligence. For a given connected simple graph $G=(V,E)$, the diligence of cut set $E(S, \overline{S})$ is defined as $ρ(S)=\min_{\{u,v\}\in E(S,\overline{S})}\max\{\bar{d}/d_u, \bar{d}/d_v\}$ where $d_u$ is the degree of $u$ and $\bar{d}$ is the average degree of nodes in the one side of the cut with smaller volume (i.e., ${\mathtt{vol}}{(S)}=\sum_{u\in S}d_u$). The diligence of $G$ is also defined as $ρ(G)=\min_{ \emptyset\neq S\subset V}ρ(S)$. We show that the spread time of the algorithm in $\mathcal{G}$ is bounded by $T$, where $T$ is the first time that $\sum_{t=0}^TΦ(G^{(t)})\cdotρ(G^{(t)})$ exceeds $C\log n$, where $Φ(G^{(t)})$ denotes the conductance of $G^{(t)}$ and $C$ is a specified constant. We also define the absolute diligence as $\overlineρ(G)=\min_{\{u,v\}\in E}\max\{1/d_u,1/d_v\}$ and establish upper bound $T$ for the spread time in terms of absolute diligence, which is the first time when $\sum_{t=0}^T\lceilΦ(G^{(t)})\rceil\cdot \overlineρ(G^{(t)})\ge 2n$. We present dynamic networks where the given upper bounds are almost tight.
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