Inspired by the works of Goldreich and Ron (J. ACM, 2017) and Nakar and Ron (ICALP, 2021), we initiate the study of property testing in dynamic environments with arbitrary topologies. Our focus is on the simplest non-trivial rule that can be tested, which corresponds to the 1-BP rule of bootstrap percolation and models a simple spreading behavior: Every "infected" node stays infected forever, and each "healthy" node becomes infected if and only if it has at least one infected neighbor. We show various results for both the case where we test a single time step of evolution and where the evolution spans several time steps. In the first, we show that the worst-case query complexity is $O(\Delta/\varepsilon)$ or $\tilde{O}(\sqrt{n}/\varepsilon)$ (whichever is smaller), where $\Delta$ and $n$ are the maximum degree of a node and number of vertices, respectively, in the underlying graph, and we also show lower bounds for both one- and two-sided error testers that match our upper bounds up to $\Delta = o(\sqrt{n})$ and $\Delta = O(n^{1/3})$, respectively. In the second setting of testing the environment over $T$ time steps, we show upper bounds of $O(\Delta^{T-1}/\varepsilon T)$ and $\tilde{O}(|E|/\varepsilon T)$, where $E$ is the set of edges of the underlying graph. All of our algorithms are one-sided error, and all of them are also time-conforming and non-adaptive, with the single exception of the more complex $\tilde{O}(\sqrt{n}/\varepsilon)$-query tester for the case $T = 2$.

翻译：受Goldreich和Ron（J. ACM, 2017）以及Nakar和Ron（ICALP, 2021）工作的启发，我们首次研究具有任意拓扑结构的动态环境下的性质测试问题。我们聚焦于可测试的最简单非平凡规则，即对应于自举渗流中1-BP规则并建模简单扩散行为的规则：每个"已感染"节点永久保持感染状态，而每个"健康"节点当且仅当至少有一个已感染邻居时才会被感染。我们针对单步演化测试和跨多步演化测试两种情况给出了多项结果。在第一种情况下，我们证明最坏情况查询复杂度为$O(\Delta/\varepsilon)$或$\tilde{O}(\sqrt{n}/\varepsilon)$（取较小值），其中$\Delta$和$n$分别表示底层图中的最大节点度数和顶点数，同时我们给出了单侧和双侧错误测试器的下界，该下界分别在$\Delta = o(\sqrt{n})$和$\Delta = O(n^{1/3})$范围内与上界匹配。在第二种跨$T$时间步的测试场景中，我们给出了$O(\Delta^{T-1}/\varepsilon T)$和$\tilde{O}(|E|/\varepsilon T)$的上界，其中$E$为底层图的边集。除$T=2$情况下更为复杂的$\tilde{O}(\sqrt{n}/\varepsilon)$查询测试器外，我们所有算法均为单侧错误、时间一致且非自适应性的。