Observability Properties of Colored Graphs

A colored graph is a directed graph in which either nodes or edges have been assigned colors that are not necessarily unique. Observability problems in such graphs are concerned with whether an agent observing the colors of edges or nodes traversed on a path in the graph can determine which node they are at currently or which nodes they have visited earlier in the path traversal. Previous research efforts have identified several different notions of observability as well as the associated properties of colored graphs for which those types of observability properties hold. This paper unifies the prior work into a common framework with several new analytic results about relationships between those notions and associated graph properties. The new framework provides an intuitive way to reason about the attainable path reconstruction accuracy as a function of lag and time spent observing, and identifies simple modifications that improve the observability properties of a given graph. This intuition is borne out in a series of numerical experiments. This work has implications for problems that can be described in terms of an agent traversing a colored graph, including the reconstruction of hidden states in a hidden Markov model (HMM).