We propose a simple and time-optimal algorithm for property testing a graph for its conductance in the CONGEST model. Our algorithm takes only $O(\log n)$ rounds of communication (which is known to be optimal), and consists of simply running multiple random walks of $O(\log n)$ length from a certain number of random sources, at the end of which nodes can decide if the underlying network is a good conductor or far from it. Unlike previous algorithms, no aggregation is required even with a smaller number of walks. Our main technical contribution involves a tight analysis of this process for which we use spectral graph theory. We introduce and leverage the concept of sticky vertices which are vertices in a graph with low conductance such that short random walks originating from these vertices end in a region around them. The present state-of-the-art distributed CONGEST algorithm for the problem by Fichtenberger and Vasudev [MFCS 2018], runs in $O(\log n)$ rounds using three distinct phases : building a rooted spanning tree (\emph{preprocessing}), running $O(n^{100})$ random walks to generate statistics (\emph{Phase~1}), and then convergecasting to the root to make the decision (\emph{Phase~2}). The whole of our algorithm is, however, similar to their Phase~1 running only $O(m^2) = O(n^4)$ walks. Note that aggregation (using spanning trees) is a popular technique but spanning tree(s) are sensitive to node/edge/root failures, hence, we hope our work points to other more distributed, efficient and robust solutions for suitable problems.

翻译：我们提出了一种在CONGEST模型下用于图传导性属性测试的简单且时间最优算法。该算法仅需$O(\log n)$轮通信（已知为最优轮数），其核心机制是从若干随机源独立运行多条长度为$O(\log n)$的随机游走，最终各节点可判定底层网络是否为良导体或远非良导体。与先前算法不同，本算法即便使用更少的游走次数也无需数据聚合。我们的主要技术贡献在于引入谱图理论对该过程进行了严密分析。我们提出并利用了"粘性顶点"概念——这类顶点位于图中低导率区域，使得从其出发的短随机游走终点始终围绕该区域。当前由Fichtenberger与Vasudev [MFCS 2018]提出的分布式CONGEST算法采用三阶段方案：构建有根生成树（预处理阶段）、运行$O(n^{100})$次随机游走生成统计量（阶段1）、通过汇聚播报向根节点传递决策（阶段2）。而本算法整体相当于仅执行其阶段1的$O(m^2)=O(n^4)$次游走。需要指出，基于生成树的聚合技术虽被广泛采用，但生成树对节点/边/根节点故障敏感。因此，我们冀望本工作能为合适问题指明更具分布式特性、高效且鲁棒的解决方案。