Asymmetry through time dependency

Given a single network of interactions, asymmetry arises when the links aredirected. For example, if protein A upregulates protein B and protein Bupregulates protein C, then (in the absence of any further relationships between them) Amay affect C but not vice versa. This type of imbalance is reflected in the associatedadjacency matrix, which will lack symmetry. A different type of imbalance can arise wheninteractions 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 ordisease 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 closelyrelated to the fact that matrix multiplication is not commutative. In this work, wedescribe a method that has been designed to reveal asymmetry in static networks and showhow it may be combined with a measure that summarizes the potential information flowbetween nodes in the temporal case. This results in a new method that quantifies theasymmetry arising through time ordering. We show by example that the new tool can be usedto visualize and quantify the amount of asymmetry caused by the arrow of time.