Fault tolerance in networks of bounded degree

We define a new paradigm for distributed computing, almost-everywhere agreement, in which we require only that almost all correct processors reach consensus. Unlike the traditional Byzantine agreement problem, almost-everywhere agreement can be solved on networks of bounded degree. Specifically, we can simulate any sufficiently resilient Byzantine agreement algorithm on a network of bounded degree using our communication scheme described above. Although we 'lose' some correct processors, effectively treating them as faulty, the vast majority of correct processors decide on a common value.