A single observed network reflects several mechanisms at once: communities, hubs, and clustering coexist in one graph, each a different model. We treat the network as a combination of candidate mechanisms and study, from a single graph, how strongly each mechanism contributes and how they combine. We address two questions. The first is how to measure each mechanism's contribution when the mechanisms must themselves be estimated from the graph: fitting the mechanisms and their strengths from the same data biases the strengths toward zero, and a correction removes this bias and yields valid confidence intervals. The second is whether the rule of combination is itself recoverable: when a graph is generated by two mechanisms acting together, the graph alone determines whether they combine additively or interact, exactly when the graph is dense enough, a sharp threshold below which no test can decide. The estimate calibrates the candidate mechanisms against the observed edges. We establish matching minimax rate, against a known-design benchmark and the estimated-design problem itself, confirm the methods in simulation, and apply them to real networks, where the signed coefficients recover known structure and, in one case, a confidence interval excludes any positive contribution from a candidate mechanism.

翻译：单个观测到的网络同时反映多种机制：社区结构、枢纽节点和聚类共存于同一张图中，每种机制对应不同模型。我们将网络视为候选机制的组合，并从单张图出发研究各机制的贡献强度及其组合方式。本文解决两个问题：其一，当机制本身需要从图中估计时，如何衡量每个机制的贡献——从同一份数据中拟合机制及其强度会导致强度估计偏向零，通过修正可消除该偏差并得到有效置信区间；其二，组合规则本身是否可恢复——当图由两种机制共同生成时，仅凭图结构可确定它们是加法组合还是交互组合，且该可识别性恰好在图足够稠密时成立，存在一个尖锐阈值，低于该阈值任何检验都无法做出判断。该估计通过观测边对候选机制进行校准。我们建立了与已知设计基准和估计设计问题本身相匹配的极小化极大速率，通过模拟验证了方法有效性，并将其应用于真实网络。在真实网络中，带符号的系数恢复了已知结构，且在一个案例中，置信区间排除了某个候选机制的任何正向贡献。