Networks with Memory

Capturing dynamics of the spread of information and disease with random flow on networks is a paradigm. We show that this conventional approach ignores an important feature of the dynamics: where flow moves to depends on where it comes from. That is, memory matters. We analyze multi-step pathways from different systems and show that ignoring memory overestimates the number of pathways by up to 400% per step, with profound consequences for community detection and ranking as well as for epidemic spreading. Specifically, memoryless dynamics on networks understate the effect of communities and exaggerate the effect of highly connected nodes. Including memory reveals actual travel patterns in air traffic, ranking that favors specialized journals in scientific communication, and diseases that spread more slowly and persist longer in hospitals.