Synchronous dynamic systems are well-established models that have been used to capture a range of phenomena in networks, including opinion diffusion, spread of disease and product adoption. We study the three most notable problems in synchronous dynamic systems: whether the system will transition to a target configuration from a starting configuration, whether the system will reach convergence from a starting configuration, and whether the system is guaranteed to converge from every possible starting configuration. While all three problems were known to be intractable in the classical sense, we initiate the study of their exact boundaries of tractability from the perspective of structural parameters of the network by making use of the more fine-grained parameterized complexity paradigm. As our first result, we consider treewidth - as the most prominent and ubiquitous structural parameter - and show that all three problems remain intractable even on instances of constant treewidth. We complement this negative finding with fixed-parameter algorithms for the former two problems parameterized by treedepth, a well-studied restriction of treewidth. While it is possible to rule out a similar algorithm for convergence guarantee under treedepth, we conclude with a fixed-parameter algorithm for this last problem when parameterized by treedepth and the maximum in-degree.

翻译：同步动态系统是成熟的模型，已用于捕捉网络中的多种现象，包括观点扩散、疾病传播和产品采纳。我们研究同步动态系统中的三个最显著问题：系统是否会从起始配置过渡到目标配置、系统是否会从起始配置达到收敛、以及系统是否保证能从所有可能的起始配置收敛。尽管这三个问题在经典意义下已知难以处理，我们通过利用更精细的参数化复杂性范式，从网络结构参数的角度首次探索其可处理性的精确边界。作为首要结果，我们考虑树宽——作为最突出且普遍的结构参数——并证明即使实例具有恒定树宽，所有三个问题仍然难以处理。我们通过针对前两个问题的固定参数算法来补充这一负面发现，该算法以树深度（树宽的一种受广泛研究的限制形式）作为参数。虽然可以排除在树深度参数化下为收敛保证问题设计类似算法的可能性，但我们最终给出一个关于该问题在树深度和最大入度参数化下的固定参数算法。