A "Chicken & Egg" Network Coding Problem

We consider the multi-source network coding problem in cyclic networks. This problem involves several difficulties not found in acyclic networks, due to additional causality requirements. This paper highlights the difficulty of these causality conditions by analyzing two example cyclic networks which are structurally similar. Both networks have an essentially identical network code which appears to transmit all information from the sources to the sinks; however, this network code is invalid since it violates causality. We show that, in one of the networks, the invalid code can be modified to obey causality, whereas in the other network this is impossible. This unachievability result is proven by a new information inequality for causal coding schemes in a simple cyclic network.