For more than three decades, distributed systems have been described and analyzed using topological tools, primarily using two techniques: protocol complexes and directed algebraic topology. In both cases, the considered computational model generally assumes communication via shared objects (typically a shared memory consisting of a collection of read-write registers) or message-passing enabling direct communication between any pair of processes. This paper aims to examine the use of protocol complexes in the study of network computing. In this case, processes are located at the network nodes and communicate by exchanging messages only along the network's edges (i.e., not every pair of processes can directly communicate).

There are several reasons why applying the topological approach to network computing can be challenging, and a prominent one is that node identifiers yield protocol complexes whose sizes grow exponentially with the size of the underlying network. However, many of the problems studied in this context are of local nature, and their definitions do not depend on the identifiers or the network size. We leverage this independence to meet the above challenge and present local protocol complexes, whose sizes do not depend on the network size. As an application of the "compacted" protocol complexes, we reformulate the celebrated lower bound of Ω(log * n) rounds for 3-coloring the n-node ring in the topological framework.