Topological Aspects of Greedy Self-Organization

We consider self-organization problems, where agents try to agree about the value of a configuration space variable. Problems of consensus and synchronization belong to this category. These are the problems which would often be trivial to solve in a centralized setting, and non-trivial aspects are often directly induced by the process of self-organization itself. We discuss topological reasons as to why simple locally greedy algorithms are not able to create long-range order. The reason why greedy synchronization of a real-valued variable works in a straight forward manner, whereas greedy phase synchronization does not, is topological, in the latter non-trivial homotopy classes in mappings from the interaction graph of the agents to the configuration space exist. We identify higher dimensional configuration spaces with such non-trivial homotopy classes. However, we find that greedy self-organization is able to create long-range order for any higher-dimensional configuration space that does not possess circular components.