Asymmetry through time dependency

Given a single network of interactions, asymmetry arises when the links are directed. For example, if protein A upregulates protein B and protein B upregulates protein C, then (in the absence of any further relationships between them) A may affect C but not vice versa. This type of imbalance is reflected in the associated adjacency matrix, which will lack symmetry. A different type of imbalance can arise when interactions appear and disappear over time. If A meets B today and B meets C tomorrow, then (in the absence of any further relationships between them) A may pass a message or disease to C, but not vice versa. Hence, even when each interaction is a two-way exchange, the effect of time ordering can introduce asymmetry. This observation is very closely related to the fact that matrix multiplication is not commutative. In this work, we describe a method that has been designed to reveal asymmetry in static networks and show how it may be combined with a measure that summarizes the potential information flow between nodes in the temporal case. This results in a new method that quantifies the asymmetry arising through time ordering. We show by example that the new tool can be used to visualize and quantify the amount of asymmetry caused by the arrow of time.