We study the detection and the reconstruction of a large very dense subgraph in a social graph with $n$ nodes and $m$ edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when $m=O(n. \log n)$. A subgraph $S$ is very dense if it has $\Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size $k=O(\sqrt{n}.\log n)$. Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size $\Omega(\sqrt{n})$, then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges.

中文翻译：

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边流中的大的非常密集的子图

我们研究了在具有 $n$ 个节点和 $m$ 条边作为边流的社交图中的大型非常密集子图的检测和重建，当该图遵循幂律度分布时，在 $m 时的状态下=O(n.\log n)$。如果子图 $S$ 具有 $\Omega(|S|^2)$ 边，则它非常密集。我们使用大小为 $k=O(\sqrt{n}.\log n)$ 的 Reservoir 对边缘进行均匀采样。我们的检测算法检查水库是否有一个巨大的组件。我们表明，如果该图包含一个大小为 $\Omega(\sqrt{n})$ 的非常密集的子图，那么检测算法几乎肯定是正确的。另一方面，遵循幂律度分布的随机图几乎肯定没有大的非常密集的子图，并且检测算法几乎肯定是正确的。我们定义了一个新的随机图模型，它遵循幂律度分布并具有很大的非常密集的子图。然后我们证明，在这类随机图上，我们可以以高概率重建非常密集的子图的良好近似。我们将这些结果推广到由边缘流中的滑动窗口定义的动态图。