Synfire waves are propagating spike packets in synfire chains, which are feedforward chains embedded in random networks. Although synfire waves have proved to be effective quantification for network activity with clear relations to network structure, their utilities are largely limited to feedforward networks with low background activity. To overcome these shortcomings, we describe a novel generalisation of synfire waves, and define 'synconset wave' as a cascade of first spikes within a synchronisation event. Synconset waves would occur in 'synconset chains', which are feedforward chains embedded in possibly heavily recurrent networks with heavy background activity. We probed the utility of synconset waves using simulation of single compartment neuron network models with biophysically realistic conductances, and demonstrated that the spread of synconset waves directly follows from the network connectivity matrix and is modulated by top-down inputs and the resultant oscillations. Such synconset profiles lend intuitive insights into network organisation in terms of connection probabilities between various network regions rather than an adjacency matrix. To test this intuition, we develop a Bayesian likelihood function that quantifies the probability that an observed synfire wave was caused by a given network. Further, we demonstrate it's utility in the inverse problem of identifying the network that caused a given synfire wave. This method was effective even in highly subsampled networks where only a small subset of neurons were accessible, thus showing it's utility in experimental estimation of connectomes in real neuronal-networks. Together, we propose synconset chains/waves as an effective framework for understanding the impact of network structure on function, and as a step towards developing physiology-driven network identification methods. Finally, as synconset chains extend the utilities of synfire chains to arbitrary networks, we suggest utilities of our framework to several aspects of network physiology including cell assemblies, population codes, and oscillatory synchrony.