Eleni Katifori – University of Pennsylvania
Wednesday, November 13th , 10:45AM EDT
Geometry Matters: Spatial Considerations in the Design of Physical Networks
Networks optimized for one or several cost functions are commonplace in nature and engineering. Their topology is usually at least partially determined by the functions the networks need to satisfy. However, when there is a notion of distance between the nodes and the graph is embedded in physical space, geometry also influences the optimal architecture.
In this talk, we explore two instances where spatial embedding determines the optimal form, sometimes in counterintuitive ways. First, we consider a spatially embedded graph traversed by random walkers and investigate the structure of the graph that minimizes the average mean first passage time for the walkers. We generalize our findings to energy landscapes, modeling them as networks with nodes representing energy minima and edges denoting transition pathways with Arrhenius-dependent rates. We find that reducing infinite barriers isolating metastable states can paradoxically increase search time. Additionally, we establish a lower bound for the mean first passage time, showing that search efficiency depends primarily on the ground state’s energy and connectivity. Finally, time permitting, we examine how the spatial embedding of nutrient delivery networks, such as the microvasculature, is affected by geometry, particularly the distance of vessels to the nutrient input source.