  This paper considers the dynamics of edges in a network. The Dynamic Bond
Percolation (DBP) process models, through stochastic local rules, the
dependence of an edge $(a,b)$ in a network on the states of its neighboring
edges. Unlike previous models, DBP does not assume statistical independence
between different edges. In applications, this means for example that failures
of transmission lines in a power grid are not statistically independent, or
alternatively, relationships between individuals (dyads) can lead to changes in
other dyads in a social network. We consider the time evolution of the
probability distribution of the network state, the collective states of all the
edges (bonds), and show that it converges to a stationary distribution. We use
this distribution to study the emergence of global behaviors like consensus
(i.e., catastrophic failure or full recovery of the entire grid) or coexistence
(i.e., some failed and some operating substructures in the grid). In
particular, we show that, depending on the local dynamical rule, different
network substructures, such as hub or triangle subgraphs, are more prone to
failure.
