Statistical analysis and inference 2

Understanding how information is distributed in network systems composed by multiple interacting nodes is a crucial problem in many scientific fields. Partial information decomposition (PID) constitutes a comprehensive framework decomposing the mutual information (MI) that a target random variable shares with a set of source variables into components highlighting how such an information is distributed among the sources, i.e., the unique information exclusively available from each source, the redundant information that can be obtained from at least two different sources, and the synergistic information that is revealed only when multiple sources are considered simultaneously. Since it has been specifically defined for random variables, applying the PID to multivariate time series representing realizations of a vector random process composed by several source processes and a target process may be confounding. In this case, the PID would decompose the information shared by the processes at the same time, thus implicitly disregarding the temporal statistical structure of the multivariate process. To overcome this issue, in this work we formalize a partial information rate decomposition (PIRD) for networks of stationary random processes exhibiting nontrivial temporal correlations, leveraging the use of entropy rates as basic elements of information decomposition. We show that the presence of temporal correlations has a profound impact on the multivariate information shared at lag zero by multiple random processes. Moreover, exploiting the formalism that connects information rates with linear Gaussian regression models and spectral quantities, we provide a joint time and frequency domain formulation of the PIRD which highlights distinct high-order behaviors of oscillatory rhythms.

Pairwise ranking models are widely used in domains such as sports analytics, recommendation systems, and the study of social hierarchies to compare entities and derive meaningful insights. Since the pairwise comparisons inherently involve some degree of uncertainty due to randomness, noise, or incomplete information in the observed outcomes, these models are generally formulated as probabilistic models whose parameters can then be inferred given a dataset. However, in situations with limited or incomplete data, it may be difficult to confidently distinguish between players with similar performance. A shortcoming of many pairwise ranking methods is that they do not inherently account for this sparsity in the data in a nonparametric way, i.e., they do not automatically adjust the number of rankings based on the amount of evidence available. We address this issue by introducing a probabilistic generative model that intrinsically incorporates partial rankings ---rankings where multiple nodes can have the same rank--- in its generative process. This allows us to account for the uncertainty introduced by sparsity in the data by grouping players into the same rank when the data does not provide enough evidence to separate them. Our method is fully Bayesian and can, in principle, accommodate any ranking method. We provide a fast nonparametric agglomerative algorithm to infer the partial rankings, and by fitting our model to a wide range of synthetic and real-world datasets, we find that it can often provide a more parsimonious description of the data under sparse regimes. As a case study, we apply our method to a faculty hiring network amongst Computer Science departments in US universities. The picture that emerges is that of a well-separated hierarchy dominated by a small group of elite universities with very little upward mobility across the rankings.

Belief propagation (BP) is a message passing algorithm widely employed to solve inference problems in probabilistic graphical models such as percolation models, Ising spin models, and Bayesian networks. While BP computes marginal probabilities exactly on tree networks, it also provides good approximations in non-tree networks containing cycles, contributing to its versatility. However, the accuracy of BP in non-tree networks varies greatly depending on the network structure, and this dependency has not yet been fully clarified.This research aims to investigate the performance of BP in non-tree network structures, identifying key factors that influence its accuracy. Using bond percolation and Ising model as examples, we measure the errors of BP predictions by comparing them with Monte Carlo simulation results in various real-world networks. We identify two major sources of error in BP.One source of error is the finite size of the system, which becomes significant in small networks. The conventional BP method exhibits sharp changes in the order parameter so that it fails to capture the smooth transitions in small networks. We demonstrate that a simple modification to the BP algorithm can significantly reduce this finite-size error without compromising computational efficiency.The other source of error arises from the discrepancy between the BP equation and the actual non-tree structure. This is the error that remains even after reducing the finite-size error using our modified BP. We show how the quality of BP's approximations relates to different topological properties, such as degree distributions, clustering coefficients, shortest path lengths, and non-backtracking matrix spectra.Our findings provide insights into the relationship between the applicability of BP and the network structures. This understanding may help develop more accurate and efficient BP algorithms for various non-tree networks.

A central question in the study of human mobility concerns how spatial constraints shape individual and collective behavior. To this end, many macroscopic modeling approaches highlight the role of distance, while microscopic models often rely on individual characteristics of locations to predict mobility patterns. A growing body of work also focuses on mesoscopic patterns, such as geographic/social boundaries between regions or regions with disproportionately high visitation---phenomena that often cannot be captured under standard micro/macroscopic modeling applications. Although existing methods for studying mesoscopic structure in non-spatial data have been used to study interregional mobility flows, these methods do not incorporate the distance between locations, which is known to play an important role. Therefore, we introduce an inferential technique for defining mesoscopic regions that directly incorporates spatial patterns, based on the minimum description length (MDL) principle. Our Bayesian generative model is a modification of the stochastic block model (SBM). The standard SBM assumes a random network as a null model, which may be a suboptimal choice for spatial data because the block structure will build regions using purely spatial patterns. Instead, we build a notion of distance directly into the SBM, allowing the model to disentangle regional effects from macroscopic patterns. To do this, we use the distance between nodes in the planar adjacency graph, formed by linking regions that share a border. We implement an MCMC sampling algorithm that allows us to fit the model to data.We evaluate our method on synthetic and empirical data. Then, we compare its output to that of other regionalization methods in a number of synthetic and experimental settings. Our model can be used as a principled way of defining regions based on human behavioral data, which could be used for a number of downstream analysis tasks.

Latent space (LS) modeling provides a framework for analyzing networks in which nodes are positioned in LS, and the distance between nodes determines their connection probability. This approach simplifies statistical modeling, ensuring dyadic independence of edges conditional on node positions and capturing network features such as clustering via the geometry of the LS. Node and edge covariates are often incorporated to inform node positioning; however, assuming all preselected covariates are informative is unrealistic. Including extraneous covariates can induce bias while reducing estimate precision, particularly in sparse network settings, where covariate information contributes a relatively larger portion of the overall likelihood. Robust methods for covariate selection are thus critical.We extend the framework by Zhang et al., which models the network and node covariates jointly as a function of LS, with two contributions: incorporating a group lasso procedure for covariate regression coefficients and integrating adaptive gradient descent for efficient LS position estimation. These additions automate covariate selection while improving computational efficiency.Through simulations, we compare our enhanced method to the framework by Zhang et al. and a baseline model that estimates LS positions using network data alone. Our method is comparable when all covariates are informative and outperforms both other approaches as the number of noninformative covariates increases. Importantly, this improvement in covariate selection does not degrade the accuracy of network estimation.The proposed enhancements provide an efficient and effective framework for incorporating covariate information in network studies. By leveraging adaptive gradient descent and group lasso regularization, our method improves model robustness and accuracy, particularly in the presence of non-informative covariates, while maintaining the interpretability and flexibility of LS models.

The study of Earth’s dynamics poses significant challenges from methodological and experimental viewpoints. Network science provides an ideal framework to gain insights into the complexityof Earth’s climatology. However, usual approaches do not properly address the uncertainty from observational data to infer the underlying functional connectivity. Traditional methods, such asthresholding, often result in significant information loss and the introduction of spurious connections. Here, we employ a Bayesian approach to reconstruct the dynamics of a climatic networkusing historical reanalysis data from 1970 and future climatic projections until 2100. We utilize temperature and precipitation data, analyzing both historical and projected CMIP6 data for lowand high emission scenarios.Our analysis reveals that the network of structural connectivity has significantly changed in recentdecades and will undergo abrupt shifts in the future in a scenario of high emissions. We focus onthe interdependency among tipping points and find that the intensity of their teleconnections isdecreasing over time compared to the 1970s. The role of the Atlantic Meridional OverturningCirculation (AMOC) is crucial in contributing to this effect, aligning with recent findings suggestingthat this region is on a tipping course. These results are robust over past decades, with deviationsfrom the baseline becoming more profound in future years under high-emission scenarios, thus callingfor a global effort to mitigate the impact of human activities on the environment.