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The Small-World Effect for Interferometer Networks
Complex network theory has focused on properties of networks with real-valued
edge weights. However, in signal transfer networks, such as those representing
the transfer of light across an interferometer, complex-valued edge weights are
needed to represent the manipulation of the signal in both magnitude and phase.
These complex-valued edge weights introduce interference into the signal
transfer, but it is unknown how such interference affects network properties
such as small-worldness. To address this gap, we have introduced a small-world
interferometer network model with complex-valued edge weights and generalized
existing network measures to define the interferometric clustering coefficient,
the apparent path length, and the interferometric small-world coefficient.
Using high-performance computing resources, we generated a large set of
small-world interferometers over a wide range of parameters in system size,
nearest-neighbor count, and edge-weight phase and computed their
interferometric network measures. We found that the interferometric small-world
coefficient depends significantly on the amount of phase on complex-valued edge
weights: for small edge-weight phases, constructive interference led to a
higher interferometric small-world coefficient; while larger edge-weight phases
induced destructive interference which led to a lower interferometric
small-world coefficient. Thus, for the small-world interferometer model,
interferometric measures are necessary to capture the effect of interference on
signal transfer. This model is an example of the type of problem that
necessitates interferometric measures, and applies to any wave-based network
including quantum networks