平均每当你遇到：社区发现的理论协议
Average whenever you meet: Opportunistic protocols for community detection
摘要：
考虑图$G$上的以下异步机会通信模型：在每一轮中，一个边被随机地均匀且独立地激活，并且（仅）其两个端点可以交换消息并执行本地计算。在此模型下，我们研究了以下随机过程：第一次，一个顶点是一个端点的活动边缘，它选择一个随机数，说$\pm 1$概率为$1/2$ ;然后，在每一轮，当前活动边缘的两个端点更新其值的平均值。我们表明，如果$G$表现出两个社区结构（例如，两个扩展器连接的稀疏切割），由节点持有的值将共同反映在上述过程的一个合适的阶段的底层社区结构，在重要的情况下，允许高效和有效的恢复。更详细地说，我们首先提供了一个第一时刻分析表明，对于一个大类的几乎定期聚类图，包括随机块模型，预期值举行的所有节点，但可以忽略不计的一部分，最终反映了潜在的切割信号。我们证明了这个属性出现后的混合期长度$\mathcal O(n\log n)$。我们进一步提供了一个二阶矩分析的一类更受限制的定期聚类图，其中包括定期随机块模型。对于这种情况下，我们能够表明，大多数节点可以有效地和本地识别他们的社区的参考在一个合适的时间窗口。这导致了第一个机会主义协议，近似恢复社区结构，每个节点只使用多对数工作。即使是上述类别的正则图，我们的二阶矩分析需要新的浓度范围的产品，某些随机矩阵，技术上具有挑战性，可能是独立的利益。
英文摘要：
Consider the following asynchronous, opportunistic communication model over a graph $G$: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say $\pm 1$ with probability $1/2$; then, in each round, the two endpoints of the currently active edge update their values to their average. We show that, if $G$ exhibits a two-community structure (for example, two expanders connected by a sparse cut), the values held by the nodes will collectively reflect the underlying community structure over a suitable phase of the above process, allowing efficient and effective recovery in important cases. In more detail, we first provide a first-moment analysis showing that, for a large class of almost-regular clustered graphs that includes the stochastic block model, the expected values held by all but a negligible fraction of the nodes eventually reflect the underlying cut signal. We prove this property emerges after a mixing period of length $\mathcal O(n\log n)$. We further provide a second-moment analysis for a more restricted class of regular clustered graphs that includes the regular stochastic block model. For this case, we are able to show that most nodes can efficiently and locally identify their community of reference over a suitable time window. This results in the first opportunistic protocols that approximately recover community structure using only polylogarithmic work per node. Even for the above class of regular graphs, our second moment analysis requires new concentration bounds on the product of certain random matrices that are technically challenging and possibly of independent interest.
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