  Recent studies have been using graph theoretical approaches to model complex
networks (such as social, infrastructural or biological networks), and how
their hardwired circuitry relates to their dynamic evolution in time.
Understanding how configuration reflects on the coupled behavior in a system of
dynamic nodes can be of great importance, for example in the context of how the
brain connectome is affecting brain function. However, the connectivity
patterns that appear in brain networks, and their individual effects on network
dynamics, are far from being fully understood.
  We study the connections between edge configuration and dynamics in a simple
oriented network composed of two interconnected cliques (representative of
brain feedback regulatory circuitry). In this paper, our main goal is to study
the spectra of the graph adjacency and Laplacian matrices, with a focus on
three aspects in particular: (1) the sensitivity/robustness the spectrum in
response to varying the intra and inter-modular edge density, (2) the effects
on the spectrum of perturbing the edge configuration, while keeping the
densities fixed and (3) the effects of increasing the network size. We study
some tractable aspects analytically, then simulate more general results
numerically. This paper aims to clarify, from analytical and modeling
perspectives, the underpinnings of our related work, which further addresses
how graph properties affect the network's temporal dynamics and phase
transitions.
  We propose that this type of results may be helpful when studying small
networks such as macroscopic brain circuits. We suggest potential applications
to understanding synaptic restructuring in learning networks, and the effects
of network configuration to function of emotion-regulatory neural circuits.
