Spatial Separability in Hub Location Problems with an Application to Brain Connectivity Networks

Motivated by the need to solve large hub location problems efficiently and accurately, we discover an important characteristic of optimal solutions to p-hub median problems that we call spatial separability. It refers to the partitioning of the network into allocation clusters with nonoverlapping convex hulls. We illustrate numerically that the property persists over a wide range of randomly generated instances and propose a data-driven approach based on an insight from the property to tackle very large problem sizes. Computational experiments corroborate the effectiveness of the proposed approach in generating high-quality solutions within reasonable computational times. We then explore a new application area of hub location problems in brain connectivity networks and introduce the largest and the first set of three-dimensional instances in the literature. Computational results demonstrate the capability of hub location models in successfully depicting the hub organization of the human brain, as validated by the medical literature, thus revealing that hub location models can play an important role in investigating the intricate connectivity of the human brain.

Separating memoranda in depth increases visual working memory performance

Visual working memory is the mechanism supporting the continued maintenance of information after sensory inputs are removed. Although the capacity of visual working memory is limited, memoranda that are spaced farther apart on a 2D display are easier to remember, potentially because neural representations are more distinct within retinotopically-organized areas of visual cortex during memory encoding, maintenance, and/or retrieval. The impact of spatial separability in depth on memory is less clear, even though depth information is essential to guide interactions with objects in the environment. On one account, separating memoranda in depth may facilitate performance if interference between items is reduced. However, depth information must be inferred indirectly from the 2D retinal image, and less is known about how visual cortex represents depth. Thus, an alternative possibility is that separation in depth does not attenuate between-item interference; separation in depth may even impair performance, as attention must be distributed across a larger volume of 3D space. We tested these alternatives using a stereo display while participants remembered the colors of stimuli presented either near or far in the 2D plane or in depth. Increasing separation in-plane and in depth both enhanced performance. Furthermore, participants who were better able to utilize stereo depth cues showed larger benefits when memoranda were separated in depth, particularly for large memory arrays. The observation that spatial separation in the inferred 3D structure of the environment improves memory performance, as is the case in 2D environments, suggests that separating memoranda in depth might reduce neural competition by utilizing cortically separable resources.

Transitivity, Spatially Separable Utility Functions, and United States Migration Streams, 1935–1970

The predictions of spatial-interaction models applied to migration systems may be viewed as the outcome of expected-utility maximization in which the average beliefs and preferences at a given origin are spatially separable. This theory indicates that a test of the spatial separability of the utilities may be performed by examining the degree of transitivity in probabilities and gross flows that is predicted by the spatial-interaction models. United States migration data for four periods between 1935 and 1970 were examined for transitivity at three spatial scales of resolution. These flows all exhibited significantly high degrees of transitivity, although for no period or scale of resolution were migration flows completely without some statistically significant intransitivities in either the probabilities or gross flows. The regions involved in intransitivities varied greatly from period to period, and only weak evidence indicated lower degrees of intransitivity for local aggregates of regions. The hypothesis of spatially separable utilities must be rejected for the migration data examined. Theoretical discussion indicates that several causes may lead to intransitivities, which in turn lead to problems in applying spatial-interaction models to migration data.