AI 中文总结
该研究提出带节点组权重与自校准边读出的fbLM重构流程，基于随机游走共现矩阵实现图重构，在多类网络上获高MCC，重构性能受游走覆盖范围限制。
AI 中文摘要
从随机游走轨迹重构未知图的问题，既出现在天体物理学的空间关联网络中，也出现在网络科学的连通性推断任务中。我们提出一种重构流程，其可观测变量为随机游走共现矩阵，模型为成对边权重基，拟合器为帧平衡列文伯格-马夸尔特（fbLM）方案，该方案带有按节点分组的权重和自校准的边读出。与边际占用不同，共现矩阵在按行求和前保留了有序对，而成对基能够表示加性节点势模型无法描述的结构；这两种改进单独使用均不够有效。我们将该流程应用于电子邮件通信子图、基于COSMO天体目录构建的Delaunay（德洛内）与Voronoi（沃罗诺伊）网络，以及两个受控的12顶点测试图（一个单环图和一个树图），分别在解析噪声和有限游走两种场景下进行测试。重构结果与真实邻接矩阵进行评分，真实邻接矩阵未参与拟合过程，评分指标包括真阳性、假阳性及马修斯相关系数（MCC）。所有测试集在全图规模下均实现了高保真重构：在有限游走数据下，我们恢复了COSMO的Delaunay图和Voronoi图，其全范围的MCC均高于0.98，顶点数分别为119和223，为完整图而非局部截取部分；同时恢复了顶点数为240、边数为417的实证email-Eu-core图。在完整Delaunay图上，图形套索（graphical-lasso）参考方法的MCC为0.540，而fbLM的MCC为0.988。每条重构边带有费雪传播的不确定性，残差缺失几乎完全局限于游走从未遍历的边。因此在有限游走场景下，限制因素是游走覆盖范围而非拟合过程：游走访问的几乎每条边都能被成功重构，故可重构性由图的采样情况而非估计器决定。
英文摘要
Reconstructing an unknown graph from the trajectory of a random walk arises both for spatial correlation networks in astrophysics and for connectivity inference in network science. We present a reconstruction pipeline whose observable is the random-walk co-visitation matrix, whose model is a pairwise edge-weight basis, and whose fitter is a frame-balanced Levenberg-Marquardt (fbLM) scheme with per-node group weights and a self-calibrated edge readout. Unlike the marginal occupation, the co-visitation retains the ordered pair before the row sum is taken, and the pairwise basis can represent structure that an additive node-potential model cannot; neither change suffices alone. We apply the pipeline to an email communication subgraph, to Delaunay and Voronoi networks built from a COSMOS sky catalogue, and to two controlled 12-vertex test graphs, one unicyclic and one a tree, under both analytic-noise and finite-walk regimes. Reconstructions are scored against the ground-truth adjacency, which enters nowhere in the fit, by true/false positives and the Matthews correlation coefficient (MCC). All test-beds are reconstructed with high fidelity at full graph size: on finite-walk data we recover the COSMOS Delaunay and Voronoi graphs at MCC above 0.98 up to their full extent, N=119 and N=223, the whole graph rather than a cut-out of it, and the empirical email-Eu-core graph at N=240 (417 edges). On the full Delaunay graph a graphical-lasso reference returns MCC 0.540 against 0.988 for fbLM. Each reconstructed edge carries a Fisher-propagated uncertainty, and the residual misses are almost entirely confined to edges the walk never traverses. In the finite-walk regime the limiting factor is therefore walk coverage rather than the fit: essentially every edge the walk visits is recovered, so reconstructibility is governed by the sampling of the graph rather than by the estimator.
Comments15 pages, 18 figures, 3 tables. Also archived at Zenodo: doi:10.5281/zenodo.21550779
