从单一网络级联中估计与恢复植入稠密子图
Estimation and Recovery of a Planted Dense Subgraph from a Single Network Cascade
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中文总结 AI 辅助
本文研究从单一传播过程推断稀疏随机图中植入稠密子图的问题，通过分析传播中的加速增长阶段，仅凭感染时间一致估计成分大小指数、背景边密度和内部边密度，并在观测顶点身份时实现稠密成分的一致恢复。
中文摘要 AI 辅助
我们研究在稀疏随机图中，从单一传播过程中推断植入稠密成分的问题。该图具有 Erdős-Rényi 背景，边概率为 $p_n$，并包含一个大小为 $n^{\alpha}$ 的植入稠密成分，其中 $\alpha > 1/2$，其内部边密度 $\xi>0$ 为常数。边集不可观测；数据仅包含来自连续时间 SI 过程的单次实现的相继感染时间，该过程具有独立且指数分布的传播时间。我们证明植入稠密成分在传播过程中留下可检测的特征：当探索进入稠密成分后，它会经历一个短暂的加速增长阶段。通过分析该阶段，我们定位其起始点和终点。这些定位结果仅从感染时间即可得到成分大小指数 $\alpha$、背景边密度 $p_n$ 和内部边密度 $\xi$ 的一致估计量。当感染顶点的身份也可观测时，我们进一步建立了对植入稠密成分的一致恢复。
英文摘要
We study the inference of a planted dense component in a sparse random graph from a single spreading process. The graph has an Erdős-Rényi background with edge probability $p_n$ and contains a planted dense component of size $n^α$, with $α> 1/2$, whose internal edge density $ξ>0$ is constant. The edge set is unobserved; the data only consist of the successive infection times from a single realization of a continuous-time SI process with independent and exponentially distributed transmission times. We show that the planted dense component leaves a detectable signature in the spreading process: after the exploration enters the dense component, it undergoes a short phase of accelerated growth. By analyzing this phase, we localize its onset and endpoint. These localization results yield consistent estimators of the component-size exponent $α$, the background edge density $p_n$, and the internal edge density $ξ$ from the infection times alone. When the identities of the infected vertices are also observed, we further establish consistent recovery of the planted dense component.
