Network reconstruction consists in determining the unobserved pairwise couplings between $N$ nodes given only observational data on the resulting behavior that is conditioned on those couplings -- typically a time-series or independent samples from a graphical model. A major obstacle to the scalability of algorithms proposed for this problem is a seemingly unavoidable quadratic complexity of $\Omega(N^2)$, corresponding to the requirement of each possible pairwise coupling being contemplated at least once, despite the fact that most networks of interest are sparse, with a number of non-zero couplings that is only $O(N)$. Here we present a general algorithm applicable to a broad range of reconstruction problems that significantly outperforms this quadratic baseline. Our algorithm relies on a stochastic second neighbor search (Dong et al., 2011) that produces the best edge candidates with high probability, thus bypassing an exhaustive quadratic search. If we rely on the conjecture that the second-neighbor search finishes in log-linear time (Baron & Darling, 2020; 2022), we demonstrate theoretically that our algorithm finishes in subquadratic time, with a data-dependent complexity loosely upper bounded by $O(N^{3/2}\log N)$, but with a more typical log-linear complexity of $O(N\log^2N)$. In practice, we show that our algorithm achieves a performance that is many orders of magnitude faster than the quadratic baseline -- in a manner consistent with our theoretical analysis -- allows for easy parallelization, and thus enables the reconstruction of networks with hundreds of thousands and even millions of nodes and edges.

翻译：网络重构旨在仅基于观测数据（通常为时间序列或图模型中的独立样本）推断$N$个节点间未观测到的成对耦合关系，这些数据所呈现的行为受耦合关系约束。该问题算法可扩展性的主要障碍在于看似不可避免的$\Omega(N^2)$二次复杂度——尽管大多数目标网络具有稀疏性，非零耦合数仅为$O(N)$，但算法仍需至少遍历所有可能的成对耦合。本文提出一种通用算法，适用于广泛的重构问题，其性能显著优于上述二次复杂度基准。该算法基于随机第二近邻搜索（Dong等，2011），能以高概率生成最优候选边，从而避免穷举式二次搜索。若第二近邻搜索可在对数线性时间内完成（Baron & Darling，2020；2022）的猜想成立，我们理论上证明了算法可在亚二次时间内收敛，其数据依赖复杂度上界为$O(N^{3/2}\log N)$，典型情况下更可达对数线性复杂度$O(N\log^2N)$。实验表明，该算法比二次基准方法快数个数量级——与理论分析一致——且易于并行化，从而实现了对拥有数十万乃至数百万节点和边的网络进行重构。