  Different from the direct contact in epidemics spread, overload failures
propagate through hidden functional dependencies. Many studies focused on the
critical conditions and catastrophic consequences of cascading failures.
However, to understand the network vulnerability and mitigate the cascading
overload failures, the knowledge of how the failures propagate in time and
space is essential but still missing. Here we study the spatio-temporal
propagation behavior of cascading overload failures analytically and
numerically. The cascading overload failures are found to spread radially from
the center of the initial failure with an approximately constant velocity. The
propagation velocity decreases with increasing tolerance, and can be well
predicted by our theoretical framework with one single correction for all the
tolerance values. This propagation velocity is found similar in various model
networks and real network structures. Our findings may help to predict and
mitigate the dynamics of cascading overload failures in realistic systems.
