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Biological neural networks offer some of the most striking and complex examples of emergence ever observed in natural or man-made systems. Individually, the behavior of a single neuron is rather simple, yet these basic building blocks are connected through synapses to form neural networks, which are capable of sophisticated capabilities such as pattern recognition and navigation. Lower-level functionality provided by a given network is combined with other networks to produce more sophisticated capabilities. These capabilities manifest emergently at two vastly different, yet interconnected time scales. At the time scale of neural dynamics, neural networks are responsible for turning noisy external stimuli and internal signals into signals capable of supporting complex computations. A key component in this process is the structure of the network, which itself forms emergently over much longer time scales based on the outputs of its constituent neurons, a process called learning. The analysis and interpretation of the behaviors of these interconnected dynamical systems of neurons should account for the network structure and the collective behavior of the network. The field of graph signal processing (GSP) combines signal processing with network science to study signals defined on irregular network structures. Here, we show that GSP can be a valuable tool in the analysis of emergence in biological neural networks. Beyond any purely scientific pursuits, understanding the emergence in biological neural networks directly impacts the design of more effective artificial neural networks for general machine learning and artificial intelligence tasks across domains, and motivates additional design motifs for novel emergent systems of systems. 

Graph signal processing (GSP) is an emerging field developed for analyzing signals defined on irregular spatial structures modeled as graphs. Given the considerable literature regarding the resilience of infrastructure networks using graph theory, it is not surprising that a number of applications of GSP can be found in the resilience domain. GSP techniques assume that the choice of graphical Fourier transform (GFT) imparts a particular spectral structure on the signal of interest. We assess a number of power distribution systems with respect to metrics of signal structure and identify several correlates to system properties and further demonstrate how these metrics relate to performance of some GSP techniques. We also discuss the feasibility of a data-driven approach that improves these metrics and apply it to a water distribution scenario. Overall, we find that many of the candidate systems analyzed are properly structured in the chosen GFT basis and amenable to GSP techniques, but identify considerable variability and nuance that merits future investigation.