# Brain Network Modularity and Resilience Signaled by Betweenness Centrality Percolation Spiking
## 4 Discussion
The transfer module can define the hidden layer, and the two other modules, or terminal modules, can act as pseudo-input/output layers, with one terminal module functioning as the input layer and the other as the output layer. It is worth noting that while these terminal layers are interchangeable between input and output layers, the most important finding is the necessity of the transfer module to communicate between the two terminal modules. For example, this network configuration can then input into connectome-based computing software, such as conn2 res [65], to simulate reservoir computing based on human fMRIThis neural network configuration thus enables the quantification of the processing power of the connectome network using clearly defined modules through percolation theory. This work does have some inherent limitations. It is known that the field of connectomics has limitations regarding atlas choice. The atlas chosen can influence the community structures identified using percolation modularity; however, as shown using null models and mouse data, this methodology works regardless of network construction methods. Additionally, this methodology only identified three functional hubs in the brain, as opposed to other modularity algorithms identifying smaller clusters. However, the goal of this work, rather than exclusively identifying modules in the brain that correspond to a specific function, also includes using a new perspective to translate fMRI into a neural network. Thus, three modules fit within the schema of neural networks, including three general layers: input, output, and hidden. Specific to this work is one possible limitation to the widespread practical application of this methodology, the run time, since betweenness centrality is a computationally expensive algorithm that increases in run time with increasing network size. However, specifically for brain networks and connectomics, atlases used to generate connectivity matrices often have fewer than 400 nodes [66]. This work, using 300 nodes, had run times of around four minutes (MacBook Pro 2019 (Apple, Cupertino, CA, USA), 2.6 GHz 6-Core Intel Core i7 (Intel, Santa Clara, CA, USA), 16 GM 2667 MHz DDR4 (SK HYNIX, Icheon, Republic of Korea), AMD Radeon Pro 5300M 4 GB (AMD, Santa Clara, CA, USA), Intel UHD Graphics 630 1536 MG) for each subject at our lowest typical threshold (40%), demonstrating this methodology is useful and time-effective for brain modularity analysis and is far less computationally expensive than small-worldness or rich-club analysis, which requires null networks [49; 50; 51]. Additionally, our findings may be a result of general network properties that may not be specific to brain networks [32; 33]; however, the less pronounced phase transition found in null models that control for degree and strength  
Figure 4: (**A**) Visual representation of the modules identified using individual-level subject percolation. The red and orange nodes represent nodes part of their respective terminal module, while the yellow nodes represent nodes in the transfer module. (**B**) Neural network representation as a potential application of phase transition modularity. Transfer modules identified using phase transition modularity can act as hidden layers, while terminal layers function as either input or output layers. distribution indicate this is most likely a feature of brain networks. An important note is that while our null model controls for network aspects that have been previously shown to be important for percolation response [36; 37; 38; 39], it may not control for every property that is important for the percolation modularity and betweenness centrality spiking in this work. More complex and conservative null models are currently being developed that may provide better insight and comparison to our current work [32]. Thus, future work can explore this percolation modularity and betweenness centrality phenomenon with more extensive null models to enhance the comparative analysis.