网络一致性模型中的最优传播
Optimal Spread in Network Consensus Models
摘要：
在基于无向图中随机游走的网络通信模型中，什么样的节点子集（受集合大小的约束）能够最快地传播信息？传播的动态是由一个过程的双重运动从知情的节点到不知情的节点。在此设置中，最优集合$A$最小化在集合外部的节点处开始的随机游走的预期第一命中时间$F(A)$的总和。在本文中，这个问题被重新表述，使搜索的解决方案被限制在一类最佳和“近”最佳的图的子集。我们引入了一个子模块，非递减秩函数$ρ$，允许一些由经典的贪心算法和我们的方法得到的解决方案之间的比较。利用$F$的超模块性和非增性质，证明了解的秩至少是最优集秩的$(1-\frac{1}{e})$倍.当解具有比贪婪解更高的秩时，该常数可以改进为$(1-\frac{1}{e})(1+χ)$，其中$χ>0$是后验确定的。该方法需要评估$F$的集合的一些固定的基数$m$，其中$m$是远远小于基数的最佳设置。当$F$具有前向单元曲率$κ$时，我们可以根据$κ$提供解质量和计算工作量$m$之间的折衷的粗略描述。
英文摘要：
In a model of network communication based on a random walk in an undirected graph, what subset of nodes (subject to constraints on the set size), enable the fastest spread of information? The dynamics of spread is described by a process dual to the movement from informed to uninformed nodes. In this setting, an optimal set $A$ minimizes the sum of the expected first hitting times $F(A)$, of random walks that start at nodes outside the set. In this paper,the problem is reformulated so that the search for solutions is restricted to a class of optimal and "near" optimal subsets of the graph. We introduce a submodular, non-decreasing rank function $ρ$, that permits some comparison between the solution obtained by the classical greedy algorithm and one obtained by our methods. The supermodularity and non-increasing properties of $F$ are used to show that the rank of our solution is at least $(1-\frac{1}{e})$ times the rank of the optimal set. When the solution has a higher rank than the greedy solution this constant can be improved to $(1-\frac{1}{e})(1+χ)$ where $χ>0$ is determined a posteriori. The method requires the evaluation of $F$ for sets of some fixed cardinality $m$, where $m$ is much smaller than the cardinality of the optimal set. When $F$ has forward elemental curvature $κ$, we can provide a rough description of the trade-off between solution quality and computational effort $m$ in terms of $κ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
