Highlights d Mouse S1 and S2 encode overlapping information during a tactile working memory task d Recall responses of previous stimuli are more prevalent in S2 and are relayed to S1 d Category information in S1, but not S2, is necessary for task performance d Network properties of S2 allow task information to persist across behavior states

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I would foremost like to thank my advisor Christopher J. Rasmussen. I have benefited greatly as a mathematician from his guidance and patience. His trust in my ability helped me through times of struggle, while his demand for precision challenged me in times of success. My sincerest thanks go out to Professor Rasmussen, without whom this project would not exist. I also want to thank my committee members David Pollack and Cameron Hill, as well as everyone in the Wesleyan Mathematics Department. In particular, I want to thank the other first year students Nick, Ryan, Josh, Freda, and Noelle, whose conversations I enjoyed, mathematical and otherwise. A special thanks to my family who have supported me throughout the completion of my degrees. Their love and support did not go unnoticed. I hope I made you all proud.

In this paper, we propose a new spectral-based approach to hypothesis testing for populations of networks. The primary goal is to develop a test to determine whether two given samples of networks come from the same random model or distribution. Our test statistic is based on the trace of the third order for a centered and scaled adjacency matrix, which we prove converges to the standard normal distribution as the number of nodes tends to infinity. The asymptotic power guarantee of the test is also provided. The proper interplay between the number of networks and the number of nodes for each network is explored in characterizing the theoretical properties of the proposed testing statistics. Our tests are applicable to both binary and weighted networks, operate under a very general framework where the networks are allowed to be large and sparse, and can be extended to multiple-sample testing. We provide an extensive simulation study to demonstrate the superior performance of our test over existing methods and apply our test to three real datasets.