稀疏随机块模型的互信息
Mutual information for the sparse stochastic block model
AI总结：
该研究针对两社区稀疏随机块模型，提出基于概率测度空间Hamilton-Jacobi方程的互信息极限猜想，证明其为渐近互信息下界，且在跨社区连边更普遍时与已知变分公式吻合。
摘要：
我们研究两社区随机块模型中的社区结构恢复问题，旨在刻画稀疏情形下观测网络与真实社区结构之间的互信息——该情形下节点总数趋于无穷，而单个节点的平均度保持有界。我们的主要贡献包括：提出关于该互信息极限的猜想，将其表示为定义在概率测度空间上的Hamilton-Jacobi方程的解；并证明该猜想的极限是渐近互信息的一个下界。该Hamilton-Jacobi方程的适定性已在我们的配套论文中得到证明。当社区间连边概率高于社区内连边概率时，已知渐近互信息可由一个变分公式给出，我们还证明了此情形下猜想的极限与该变分公式结果一致。
英文摘要：
We consider the problem of recovering the community structure in the stochastic block model with two communities. We aim to describe the mutual information between the observed network and the actual community structure in the sparse regime, where the total number of nodes diverges while the average degree of a given node remains bounded. Our main contributions are a conjecture for the limit of this quantity, which we express in terms of a Hamilton-Jacobi equation posed over a space of probability measures, and a proof that this conjectured limit provides a lower bound for the asymptotic mutual information. The well-posedness of the Hamilton-Jacobi equation is obtained in our companion paper. In the case when links across communities are more likely than links within communities, the asymptotic mutual information is known to be given by a variational formula. We also show that our conjectured limit coincides with this formula in this case.
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