Network coding theory studies the transmission of information in networks whose vertices may perform nontrivial encoding and decoding operations on data as it passes through the network. The main approach to deciding the feasibility of network coding problems aims to reduce the problem to optimization over a polytope of entropic vectors subject to constraints imposed by the network structure. In the case of directed acyclic graphs, these constraints are completely understood, but for general graphs the problem of enumerating them remains open: it is not known how to classify the constraints implied by a property that we call serializability, which refers to the absence of paradoxical circular dependencies in a network code. In this work we initiate the first systematic study of the constraints imposed on a network code by serializability. We find that serializability cannot be detected solely by evaluating the Shannon entropy of edge sets in the graph, but nevertheless, we give a polynomial-time algorithm that decides the serializability of a network code. We define a certificate of non-serializability, called an information vortex, that plays a role in the theory of serializability comparable to the role of fractional cuts in multicommodity flow theory, including a type of min-max relation. Finally, we study the serializability deficit of a network code, defined as the minimum number of extra bits that must be sent in order to make it serializable. For linear codes, we show that it is NP-hard to approximate this parameter within a constant factor, and we demonstrate some surprising facts about the behavior of this parameter under parallel composition of codes.