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受耦合条件约束的观测数据（通常是时间序列或图模型中的独立样本），推断 $N$ 个节点间不可直接观测的成对耦合关系。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——这意味着每个可能的成对耦合至少需被考察一次，尽管大多数目标网络具有稀疏性，其非零耦合数量仅为 $O(N)$。本文提出一种通用算法，适用于广泛的重构问题，显著超越二次基线复杂度。该算法基于随机次近邻搜索（Dong 等，2011），以高概率生成最优边候选集，从而避免穷举二次搜索。若采用次近邻搜索可在对数线性时间内完成（Baron & Darling, 2020; 2022）的猜想，我们从理论上证明算法可在亚二次时间内完成，其数据依赖复杂度上限约为 $O(N^{3/2}\log N)$，而典型复杂度为 $O(N\log^2N)$ 的对数线性形式。实验表明，算法性能比二次基线快多个数量级——这与理论分析一致——且易于并行化，从而能够重构包含数十万乃至数百万节点与边的网络。