Abstract <p>In this paper, the problem of distributed network survivability is investigated. A distributed system is modeled by a finite connected undirected graph the nodes of which are divided into two types: hosts and switches. Hosts perform computational functions, while switches are used for message passing between hosts. The survivability of the system is understood as its ability to perform the main message passing functions after a failure of some graph edges. The solution to the problem of system survivability is provided by the duplication of message passing paths. The main condition for the correct solution is the absence of message looping in the network for any set of selected paths. A four-path theorem is proved for the case of one sender host and two receiver hosts: if, for each receiver host, there are two paths from a sender host and the sets of edges traversed by these paths are disjoint, then these paths can be selected in such a way that there is no message looping for this set of paths.</p>