Neural network function, density or geometry?

We consider an oriented network in which two subgraphs (modules X and Y), with a given intra-modular edge density, are inter-connected by a fixed number of edges, in both directions. We study the adjacency spectrum of this network, focusing in particular on two aspects: the changes in the spectrum in response to varying the intra and inter-modular edge density, and the effects on the spectrum of perturbing the edge configuration, while keeping the densities fixed. Since the general case is quite complex analytically, we adopted a combination of analytical approaches to particular cases, and numerical simulations for more results. After investigating the behavior of the mean and standard deviation of the eigenvalues, we conjectured the robustness of the adjacency spectrum to variations in edge geometry, when operating under a fixed density profile. We remark that, while this robustness increases with the size N of the network, it emerges at small sizes, and should not be thought of as a property that holds only in the large N limit. We argue that this may be helpful when studying applications on small networks of nodes. We interpret our results in the context of existing literature, which has been placing increasing attention to random graph models, with edges connecting two given nodes with certain probabilities. We discuss whether properties such as Wigner's semicircle law, or effects of community structure, still hold in our context. Finally, we suggest possible applications of the model to understanding synaptic restructuring during learning algorithms, and to classifying emotional responses based on the geometry of the emotion-regulatory neural circuit. In this light, we argue that future directions should be directed towards relating hardwiring to the temporal behavior of the network as a dynamical system .