Schematic of possibilities for network evolution. (a) A simple network is conserved in two lineages, but along one or both lineages, highly accelerated sequence evolution results in loss of detectable homology in modern descendants (e.g. A′ and A are direct sequence descendants of A in the ancestor, and have carried out the same network role throughout evolution, but A and A′ are no longer sequence-alignable). In this case, both network and sequences are monophyletic: from a single origin, the network has retained the same topology, and all sequences have kept the same network role. (b) A simple network independently recruits new elements to elaborate the network (note that the ‘sense’ of the network remains the same, with A still activating the downstream C). In this case, the enhanced network is polyphyletic, as are the new sequences B and D. (c) Along one lineage, the network acquires an independent loop redundant with the B loop, allowing subsequent loss of B along this lineage, without losing network function at any step. In such a case, we consider the network monophyletic, even though the sequences are polyphyletic. Note that in the case of recruitment, B and D could be ancient relatives. Provided D did not carry out the indicated network role in precursor organisms, this still constitutes sequence polyphyly for this network.