Saxena, AyushiThe increasing prevalence of graph and network-valued data across various disciplines has prompted significant interest and research in recent years. This dissertation explores the impact of vertex shuffling, or vertex misalignment, on the statistical network inference tasks of hypothesis testing, classification, and clustering. Our focus is within the framework of multiple network inference, where existing methodologies often assume known vertex correspondence across networks. This assumption frequently does not hold in practice. Through theoretical analyses, simulations, and experiments, we aim to reveal the effects of vertex shuffling on different types of performance.Our investigation begins with an examination of two-sample network hypothesis testing, focusing on the decrease in statistical power resulting from vertex shuffling. In this work, our analysis focuses on the random dot product and stochastic block model network settings. Subsequent chapters delve into the effects of shuffling on graph classification and clustering, showcasing how misalignment negatively impacts accuracy in categorizing and clustering graphs (and vertices) based on their structural characteristics. Various machine learning algorithms and clustering methodologies are explored, revealing a theme of consistent performance degradation in the presence of vertex shuffling. We also explore how graph matching algorithms can potentially mitigate the effects of vertex misalignment and recover the lost performance. Our findings also highlight the risk of graph matching as a pre-processing tool, as it can induce artificial signal. These findings highlight the difficulties and subtleties of addressing vertex shuffling across multiple network inference tasks and suggest avenues for future research in order to enhance the robustness of statistical inference methodologies in complex network environments.enDissertationStatisticsMathematicsApplied mathematics