# Brain Network Modularity and Resilience Signaled by Betweenness Centrality Percolation Spiking
## 4 Discussion  
The phase transition behavior of the brain networks in response to betweenness centrality percolations can provide insight into brain resilience and optimization. Brain networks were highly vulnerable to BC attacks (Figure 2 and Figure S6) which has been shown in previous research [6; 14]. However, the associated phase transition preceded by a spike in average BC demonstrates the functional trade-off of relying on being near a phase transition for optimal processing [56; 57]. Recent work has demonstrated healthy brains lie at a near-critical threshold, rather than exactly at a critical point [56]. Remaining at a near-critical point is believed to balance computational efficiency and global dynamics. While many biological research efforts have looked at local criticality through neuronal spiking [58], criticality can also be examined at the global level through brain networks. This present work supports the hypothesis that the brain functions at a near-critical state, demonstrated by the percentage of attacks needed before a phase transition (significant drop in cluster size). If brain networks were maintained at a critical point, it would only take a small number of attacks to produce the phase transition. While this occurs at very high thresholds (\(\leq\)2%), lower thresholds require a larger number of attacks to reach this phase transition (Figure 2), demonstrating a near-critical nature rather than strictly lying at a critical point. This distance, or number of attacks, to a critical point may for example serve as a biomarker for neurodegeneration while providing an indication of robustness against neurodegeneration before significant clinical or functional symptomatology is present. This phase transition signified by the BC spike could also be used to support the critical brain hypothesis. The critical brain hypothesis states the brain functions near a phase transition boundary, or on the edge of chaos, for optimal processing [56]. Support for this theory has been found from microscale spike recordings [58] to macroscale neuroimaging such as fMRI [59; 60]. Evidence has also been found in the connectomic analysis of neuromorphic hardware [61]. However, criticism of the theory stems from the complicated definition of \"criticality\" and where the brain lies on the spectrum of criticality. It is important to stress that in this work, criticality is defined as the connectedness of the network as a unit, or largest cluster. While this definition fits within the bounds of traditional percolation theory regarding crystal structure, it does limit the quantifiability of \"chaos.\" For example, typical neural network processing measures such as Poincare and Lyapunov exponents cannot be utilized since the data are not defined with respect to time, but with respect to the number of nodes removed [62]. This phase transition modularity also offers a unique lens to view brain modularity through transfer and terminal hubs. This modularity configuration of transfer modules has been proposed in previous work [63]. Similarly, rich-club modularity, which has been widely used across connectomic studies [49; 50; 51], uses a similar idea whereby feeder nodes connect non-rich-club nodes to the central rich club. However, instead of individual feeder nodes, this work categorizes nodes critical for information flow between modules as a module itself. This modularity perspective aligns with a neural network framework where the terminal modules act as input and output layers and the information transfer module acts as a hidden layer connecting them. Therefore, one potential application of the phase transition modularity is quantifiably translating brain networks into neural networks (Figure 4). While previous efforts have used connectome networks to generate neural networks [64], this method provides a definitive mapping or allocation of nodes to specific neural network layers.