Abstract
The comparison of complex networks is a fundamental problem across scientific domains, including neuroscience, genomics, and the social sciences, where networks are used to represent relationships between variables, individuals, or system components. Despite widespread use, existing approaches to network comparison often rely on edge-wise metrics or single optimal community partitions, which fail to capture higher-order organization and ignore the inherent uncertainty present in network estimation. These limitations are particularly pronounced in correlation networks, where edges represent estimated statistical relationships and are subject to noise, dependence, and preprocessing variability. This dissertation develops a nonparametric probabilistic framework for network comparison that reframes community structure as a distribution over partitions rather than a single deterministic outcome. Building on known limitations of modularity-based methods, such as degeneracy, null-model definition, and sensitivity to network density, this work demonstrates both theoretically and empirically that modularity is ill-suited for direct comparison across networks. To address these issues, Chapter 4 extends classical recovery results from the stochastic block model to sample similarity networks, including correlation networks, establishing conditions under which community structure can be reliably recovered despite estimation noise.
Chapter 5 introduces a partition landscape framework in which each network induces a distribution over partitions via a scoring function. Networks are compared by evaluating their induced landscapes over a shared set of candidate partitions and quantifying agreement using rank-based correlation measures. A formal two-part bootstrap hypothesis test is proposed, to assess whether the rank correlation between the landscapes is insignificant. Simulation studies demonstrate that this approach captures meaningful structural similarity while accounting for uncertainty in network construction, and the process is applied to real-world data examining the effect of diet on microbiomes.
Chapter 6 presents an alternative but complementary method based on the Rashomon principle, defining sets of near-optimal partitions and using their geometry to characterize and compare networks. This perspective provides additional insight into the stability, ambiguity, and multiplicity of plausible community structures supported by a network. Simulation studies establish that these Rashomon sets are both effective for covering the true partition and provide a fuller picture of a noisy network. Additionally, a visualization package is introduced for extensively exploring the Rashomon sets of these networks.
Together, these contributions establish a unified framework for comparing networks that explicitly incorporates uncertainty and avoids reliance on single optimal partitions.
