We present an inventory of non-synchronous updating schemes and their effect on the sparse percolation (SP) of damage in finite random Boolean networks (RBNs). Synchronous update is the most thoroughly studied updating scheme, but many non-synchronous updating schemes have not been explored to the same depth. To our knowledge, this work is the first to study the sparse percolation of damage specifically in the context of non-synchronous updating schemes.

In a previous study of finite synchronous RBNs (Rohlf et al. 2007), a characteristic point Ks" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ks was discovered, for which the expectation of damage propagation from a single node was observed to be finite and independent of N" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">N. The existence of such a Ks" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ks has implications for task-solving and robustness. We search for a similar characteristic point Ks" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; line-height: normal; font-size: 16.2px; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Ks using nondeterministic and deterministic asynchronous schemes and cascading schemes. We find that asynchronous schemes do not admit such a point, but cascading schemes do admit one. Our results contribute to better understanding the robustness and information processing capabilities of complex systems with more biologically-plausible updating schemes.