Relaxational processes in many complex systems often occur in the form of avalanches resulting frominternal cascades from across the system scale. Here, we probe the space, time, and magnitude signa- tures of avalanching behavior using a network of temporally-directed links subject to a spatial distance criterion between events in the entire catalog. We apply this method onto three systems with avalanche- like characteristics: (i) highly controllable scaled experiments, particularly that of a slowly-driven pile of granular material in a quasi-two-dimensional setup with open edges; (ii) the sandpile, a numerical model of nearest-neighbor interactions in a grid; and (iii) substantially complete empirical data on earthquakes from southern California. Apart from the recovery of the fat-tailed statistics of event sizes, we recover similar power-laws in the spatial and temporal aspects of the networks of these representative systems hinting at possible common underlying generative mechanisms governing them. By consolidating the re- sults from experiments, numerical models, and empirical data, we can gain a better understanding of these highly nonlinear processes in nature.