  An avalanche or cascade occurs when one event causes one or more subsequent
events, which in turn may cause further events in a chain reaction. Avalanching
dynamics are studied in many disciplines, with a recent focus on average
avalanche shapes, i.e., the temporal profiles that characterize the growth and
decay of avalanches of fixed duration. At the critical point of the dynamics
the average avalanche shapes for different durations can be rescaled so that
they collapse onto a single universal curve. We apply Markov branching process
theory to derive a simple equation governing the average avalanche shape for
cascade dynamics on networks. Analysis of the equation at criticality
demonstrates that nonsymmetric average avalanche shapes (as observed in some
experiments) occur for certain combinations of dynamics and network topology;
specifically, on networks with heavy-tailed degree distributions. We give
examples using numerical simulations of models for information spreading,
neural dynamics, and behaviour adoption and we propose simple experimental
tests to quantify whether cascading systems are in the critical state.
