Statistical inference links data and theory in network science

High Reconstructability of Degree-Heterogeneous Networks

Reconstructing complex networks from observed, often noisy data is a fundamental task crucial for understanding complex systems across various domains. Despite numerous methods proposed for network reconstruction, little attention has been given to the relationship between reconstructability and the intrinsic properties of hidden networks. Here, we present a mathematical proof that, for scale-free networks, the reconstruction accuracy increases as the exponent of the power-law degree distribution decreases. This suggests that degree heterogeneity contributes to higher reconstructability. We validate this conclusion in empirical networks, where nodal degrees may not strictly adhere to power laws. Our results demonstrate that the reconstruction accuracy of degree-heterogeneous networks is indeed significantly higher than that of their randomized counterparts.

Determining interaction directionality in complex biochemical networks from stationary measurements

Revealing interactions in complex systems from observed collective dynamics constitutes a fundamental inverse problem in science. Some methods may reveal undirected network topology, e.g., using node-node correlation. Yet, the direction of the interaction, thus a causal inference, remains to be determined - especially in steady-state observations. We introduce a method to infer the directionality within this network only from a "snapshot" of the abundances of the relevant molecules. We examine the validity of the approach for different properties of the system and the data recorded, such as the molecule's level variability, the effect of sampling and measurement errors. Simulations suggest that the given approach successfully infer the reaction rates in various cases.

Detectability constraints on meso-scale structure in complex networks

Community, core-periphery, disassortative and other node partitions allow us to understand the organisation and function of large networks. In this work we study common meso-scale structures using the idea of block modularity. We find that the configuration model imposes strong restrictions on core-periphery and related structures in directed and undirected networks. We derive inequalities expressing when such structures can be detected under the configuration model which are closely related to the resolution limit. Nestedness is closely related to core-periphery and is similarly restricted to only be detectable under certain conditions. We then derive a general equivalence between optimising block modularity and maximum likelihood estimation of the parameters of the degree corrected Stochastic Block Model. This allows us to contrast the two approaches, how they formalise the structure detection problem and understand these constraints in inferential versus descriptive approaches to meso-scale structure detection.

Challenges and opportunities for digital twins in precision medicine from a complex systems perspective

Digital twins (DTs) in precision medicine are increasingly viable, propelled by extensive data collection and advancements in artificial intelligence (AI), alongside traditional biomedical methodologies. We argue that including mechanistic simulations that produce behavior based on explicitly defined biological hypotheses and multiscale mechanisms is beneficial. It enables the exploration of diverse therapeutic strategies and supports dynamic clinical decision-making through insights from network science, quantitative biology, and digital medicine.

Flexible inference in heterogeneous and attributed multilayer networks

Networked datasets can be enriched by different types of information about individual nodes or edges. However, most existing methods for analyzing such datasets struggle to handle the complexity of heterogeneous data, often requiring substantial model-specific analysis. In this article, we develop a probabilistic generative model to perform inference in multilayer networks with arbitrary types of information. Our approach employs a Bayesian framework combined with the Laplace matching technique to ease interpretation of inferred parameters. Furthermore, the algorithmic implementation relies on automatic differentiation, avoiding the need for explicit derivations. This makes our model scalable and flexible to adapt to any combination of input data. We demonstrate the effectiveness of our method in detecting overlapping community structures and performing various prediction tasks on heterogeneous multilayer data, where nodes and edges have different types of attributes. Additionally, we showcase its ability to unveil a variety of patterns in a social support network among villagers in rural India by effectively utilizing all input information in a meaningful way.

Emergence of power law distributions in protein-protein interaction networks through study bias

Degree distributions in protein-protein interaction (PPI) networks are believed to follow a power law (PL). However, technical and study biases affect the experimental procedures for detecting PPIs. For instance, cancer-associated proteins have received disproportional attention. Moreover, bait proteins in large-scale experiments tend to have many false-positive interaction partners. Studying the degree distributions of thousands of PPI networks of controlled provenance, we address the question if PL distributions in observed PPI networks could be explained by these biases alone. Our findings are supported by mathematical models and extensive simulations, and indicate that study bias and technical bias suffice to produce the observed PL distribution. It is, hence, problematic to derive hypotheses about the topology of the true biological interactome from the PL distributions in observed PPI networks. Our study casts doubt on the use of the PL property of biological networks as a modeling assumption or quality criterion in network biology.

Reconstructing networks from simple and complex contagions

Network scientists often use complex dynamic processes to describe network contagions, but tools for fitting contagion models typically assume simple dynamics. Here, we address this gap by developing a nonparametric method to reconstruct a network and dynamics from a series of node states, using a model that breaks the dichotomy between simple pairwise and complex neighborhood-based contagions. We then show that a network is more easily reconstructed when observed through the lens of complex contagions if it is dense or the dynamic saturates, and that simple contagions are better otherwise.

Compression-based inference of network motif sets

Physical and functional constraints on biological networks lead to complex topological patterns across multiple scales in their organization. A particular type of higher-order network feature that has received considerable interest is network motifs, defined as statistically regular subgraphs. These may implement fundamental logical and computational circuits and are referred to as "building blocks of complex networks". Their well-defined structures and small sizes also enable the testing of their functions in synthetic and natural biological experiments. Here, we develop a framework for motif mining based on lossless network compression using subgraph contractions. This provides an alternative definition of motif significance which allows us to compare different motifs and select the collectively most significant set of motifs as well as other prominent network features in terms of their combined compression of the network. Our approach inherently accounts for multiple testing and correlations between subgraphs and does not rely on a priori specification of an appropriate null model. It thus overcomes common problems in hypothesis testing-based motif analysis and guarantees robust statistical inference. We validate our methodology on numerical data and then apply it on synaptic-resolution biological neural networks, as a medium for comparative connectomics, by evaluating their respective compressibility and characterize their inferred circuit motifs.

Prediction of inhibitor development in previously untreated and minimally treated children with severe and moderately severe hemophilia A using a machine-learning network

BACKGROUND

Prediction of inhibitor development in patients with hemophilia A (HA) remains a challenge.

OBJECTIVES

To construct a predictive model for inhibitor development in HA using a network of clinical variables and biomarkers based on the individual similarity network.

METHODS

Previously untreated and minimally treated children with severe/moderately severe HA, participants of the HEMFIL Cohort Study, were followed up until reaching 75 exposure days (EDs) without inhibitor (INH-) or upon inhibitor development (INH+). Clinical data and biological samples were collected before the start of factor (F)VIII replacement (T0). A predictive model (HemfilNET) was built to compare the networks and potential global topological differences between INH- and INH+ at T0, considering the network robustness. For validation, the "leave-one-out" cross-validation technique was employed. Accuracy, precision, recall, and F1-score were used as evaluation metrics for the machine-learning model.

RESULTS

We included 95 children with HA (CHA), of whom 31 (33%) developed inhibitors. The algorithm, featuring 37 variables, identified distinct patterns of networks at T0 for INH+ and INH-. The accuracy of the model was 74.2% for CHA INH+ and 98.4% for INH-. By focusing the analysis on CHA with high-risk F8 mutations for inhibitor development, the accuracy in identifying CHA INH+ increased to 82.1%.

CONCLUSION

Our machine-learning algorithm demonstrated an overall accuracy of 90.5% for predicting inhibitor development in CHA, which further improved when restricting the analysis to CHA with a high-risk F8 genotype. However, our model requires validation in other cohorts. Yet, missing data for some variables hindered more precise predictions.

The Ihara zeta function as a partition function for network structure characterisation

Statistical characterizations of complex network structures can be obtained from both the Ihara Zeta function (in terms of prime cycle frequencies) and the partition function from statistical mechanics. However, these two representations are usually regarded as separate tools for network analysis, without exploiting the potential synergies between them. In this paper, we establish a link between the Ihara Zeta function from algebraic graph theory and the partition function from statistical mechanics, and exploit this relationship to obtain a deeper structural characterisation of network structure. Specifically, the relationship allows us to explore the connection between the microscopic structure and the macroscopic characterisation of a network. We derive thermodynamic quantities describing the network, such as entropy, and show how these are related to the frequencies of prime cycles of various lengths. In particular, the n-th order partial derivative of the Ihara Zeta function can be used to compute the number of prime cycles in a network, which in turn is related to the partition function of Bose-Einstein statistics. The corresponding derived entropy allows us to explore a phase transition in the network structure with critical points at high and low-temperature limits. Numerical experiments and empirical data are presented to evaluate the qualitative and quantitative performance of the resulting structural network characterisations.

Reassessing the modularity of gene co-expression networks using the Stochastic Block Model

Finding communities in gene co-expression networks is a common first step toward extracting biological insight from these complex datasets. Most community detection algorithms expect genes to be organized into assortative modules, that is, groups of genes that are more associated with each other than with genes in other groups. While it is reasonable to expect that these modules exist, using methods that assume they exist a priori is risky, as it guarantees that alternative organizations of gene interactions will be ignored. Here, we ask: can we find meaningful communities without imposing a modular organization on gene co-expression networks, and how modular are these communities? For this, we use a recently developed community detection method, the weighted degree corrected stochastic block model (SBM), that does not assume that assortative modules exist. Instead, the SBM attempts to efficiently use all information contained in the co-expression network to separate the genes into hierarchically organized blocks of genes. Using RNAseq gene expression data measured in two tissues derived from an outbred population of Drosophila melanogaster, we show that (a) the SBM is able to find ten times as many groups as competing methods, that (b) several of those gene groups are not modular, and that (c) the functional enrichment for non-modular groups is as strong as for modular communities. These results show that the transcriptome is structured in more complex ways than traditionally thought and that we should revisit the long-standing assumption that modularity is the main driver of the structuring of gene co-expression networks.

AIGO-DTI: Predicting Drug-Target Interactions Based on Improved Drug Properties Combined with Adaptive Iterative Algorithms

Artificial intelligence-based methods for predicting drug-target interactions (DTIs) aim to explore reliable drug candidate targets rapidly and cost-effectively to accelerate the drug development process. However, current methods are often limited by the topological regularities of drug molecules, making them difficult to generalize to a broader chemical space. Additionally, the use of similarity to measure DTI network links often introduces noise, leading to false DTI relationships and affecting the prediction accuracy. To address these issues, this study proposes an Adaptive Iterative Graph Optimization (AIGO)-DTI prediction framework. This framework integrates atomic cluster information and enhances molecular features through the design of functional group prompts and graph encoders, optimizing the construction of DTI association networks. Furthermore, the optimization of graph structure is transformed into a node similarity learning problem, utilizing multihead similarity metric functions to iteratively update the network structure to improve the quality of DTI information. Experimental results demonstrate the outstanding performance of AIGO-DTI on multiple public data sets and label reversal data sets. Case studies, molecular docking, and existing research validate its effectiveness and reliability. Overall, the method proposed in this study can construct comprehensive and reliable DTI association network information, providing new graphing and optimization strategies for DTI prediction, which contribute to efficient drug development and reduce target discovery costs.

Multimorbidity analysis with low condition counts: a robust Bayesian approach for small but important subgroups

BACKGROUND

Robustly examining associations between long-term conditions may be important in identifying opportunities for intervention in multimorbidity but is challenging when evidence is limited. We have developed a Bayesian inference framework that is robust to sparse data and used it to quantify morbidity associations in the oldest old, a population with limited available data.

METHODS

We conducted a retrospective cross-sectional study of a representative dataset of primary care patients in Scotland as of March 2007. We included 40 long-term conditions and studied their associations in 12,009 individuals aged 90 and older, stratified by sex (3039 men, 8970 women). We analysed associations obtained with Relative Risk (RR), a standard measure in the literature, and compared them with our proposed measure, Associations Beyond Chance (ABC). To enable a broad exploration of interactions between long-term conditions, we built networks of association and assessed differences in their analysis when associations are estimated by RR or ABC.

FINDINGS

Our Bayesian framework was appropriately more cautious in attributing association when evidence is lacking, particularly in uncommon conditions. This caution in reporting association was also present in reporting differences in associations between sex and affected the aggregated measures of multimorbidity and network representations.

INTERPRETATION

Incorporating uncertainty into multimorbidity research is crucial to avoid misleading findings when evidence is limited, a problem that particularly affects small but important subgroups. Our proposed framework improves the reliability of estimations of associations and, more in general, of research into disease mechanisms and multimorbidity.

FUNDING

National Institute for Health and Care Research.

Reconstruction of multiplex networks via graph embeddings

Multiplex networks are collections of networks with identical nodes but distinct layers of edges. They are genuine representations of a large variety of real systems whose elements interact in multiple fashions or flavors. However, multiplex networks are not always simple to observe in the real world; often, only partial information on the layer structure of the networks is available, whereas the remaining information is in the form of aggregated, single-layer networks. Recent works have proposed solutions to the problem of reconstructing the hidden multiplexity of single-layer networks using tools proper for network science. Here, we develop a machine-learning framework that takes advantage of graph embeddings, i.e., representations of networks in geometric space. We validate the framework in systematic experiments aimed at the reconstruction of synthetic and real-world multiplex networks, providing evidence that our proposed framework not only accomplishes its intended task, but often outperforms existing reconstruction techniques.

Multimodal measures of spontaneous brain activity reveal both common and divergent patterns of cortical functional organization

Large-scale functional networks have been characterized in both rodent and human brains, typically by analyzing fMRI-BOLD signals. However, the relationship between fMRI-BOLD and underlying neural activity is complex and incompletely understood, which poses challenges to interpreting network organization obtained using this technique. Additionally, most work has assumed a disjoint functional network organization (i.e., brain regions belong to one and only one network). Here, we employ wide-field Ca2+ imaging simultaneously with fMRI-BOLD in mice expressing GCaMP6f in excitatory neurons. We determine cortical networks discovered by each modality using a mixed-membership algorithm to test the hypothesis that functional networks exhibit overlapping organization. We find that there is considerable network overlap (both modalities) in addition to disjoint organization. Our results show that multiple BOLD networks are detected via Ca2+ signals, and networks determined by low-frequency Ca2+ signals are only modestly more similar to BOLD networks. In addition, the principal gradient of functional connectivity is nearly identical for BOLD and Ca2+ signals. Despite similarities, important differences are also detected across modalities, such as in measures of functional connectivity strength and diversity. In conclusion, Ca2+ imaging uncovers overlapping functional cortical organization in the mouse that reflects several, but not all, properties observed with fMRI-BOLD signals.

Hypergraph reconstruction from uncertain pairwise observations

The network reconstruction task aims to estimate a complex system's structure from various data sources such as time series, snapshots, or interaction counts. Recent work has examined this problem in networks whose relationships involve precisely two entities-the pairwise case. Here, using Bayesian inference, we investigate the general problem of reconstructing a network in which higher-order interactions are also present. We study a minimal example of this problem, focusing on the case of hypergraphs with interactions between pairs and triplets of vertices, measured imperfectly and indirectly. We derive a Metropolis-Hastings-within-Gibbs algorithm for this model to highlight the unique challenges that come with estimating higher-order models. We show that this approach tends to reconstruct empirical and synthetic networks more accurately than an equivalent graph model without higher-order interactions.

Speos: an ensemble graph representation learning framework to predict core gene candidates for complex diseases

Understanding phenotype-to-genotype relationships is a grand challenge of 21st century biology with translational implications. The recently proposed "omnigenic" model postulates that effects of genetic variation on traits are mediated by core-genes and -proteins whose activities mechanistically influence the phenotype, whereas peripheral genes encode a regulatory network that indirectly affects phenotypes via core gene products. Here, we develop a positive-unlabeled graph representation-learning ensemble-approach based on a nested cross-validation to predict core-like genes for diverse diseases using Mendelian disorder genes for training. Employing mouse knockout phenotypes for external validations, we demonstrate that core-like genes display several key properties of core genes: Mouse knockouts of genes corresponding to our most confident predictions give rise to relevant mouse phenotypes at rates on par with the Mendelian disorder genes, and all candidates exhibit core gene properties like transcriptional deregulation in disease and loss-of-function intolerance. Moreover, as predicted for core genes, our candidates are enriched for drug targets and druggable proteins. In contrast to Mendelian disorder genes the new core-like genes are enriched for druggable yet untargeted gene products, which are therefore attractive targets for drug development. Interpretation of the underlying deep learning model suggests plausible explanations for our core gene predictions in form of molecular mechanisms and physical interactions. Our results demonstrate the potential of graph representation learning for the interpretation of biological complexity and pave the way for studying core gene properties and future drug development.

Nonassortative relationships between groups of nodes are typical in complex networks

Decomposing a graph into groups of nodes that share similar connectivity properties is essential to understand the organization and function of complex networks. Previous works have focused on groups with specific relationships between group members, such as assortative communities or core-periphery structures, developing computational methods to find these mesoscale structures within a network. Here, we go beyond these two traditional cases and introduce a methodology that is able to identify and systematically classify all possible community types in directed multi graphs, based on the pairwise relationship between groups. We apply our approach to 53 different networks and find that assortative communities are the most common structures, but that previously unexplored types appear in almost every network. A particularly prevalent new type of relationship, which we call a source-basin structure, has information flowing from a sparsely connected group of nodes (source) to a densely connected group (basin). We look in detail at two online social networks-a new network of Twitter users and a well-studied network of political blogs-and find that source-basin structures play an important role in both of them. This confirms not only the widespread appearance of nonassortative structures but also the potential of hitherto unidentified relationships to explain the organization of complex networks.

Bayesian inference of transition matrices from incomplete graph data with a topological prior

Many network analysis and graph learning techniques are based on discrete- or continuous-time models of random walks. To apply these methods, it is necessary to infer transition matrices that formalize the underlying stochastic process in an observed graph. For weighted graphs, where weighted edges capture observations of repeated interactions between nodes, it is common to estimate the entries of such transition matrices based on the (relative) weights of edges. However in real-world settings we are often confronted with incomplete data, which turns the construction of the transition matrix based on a weighted graph into an inference problem. Moreover, we often have access to additional information, which capture topological constraints of the system, i.e. which edges in a weighted graph are (theoretically) possible and which are not. Examples include transportation networks, where we may have access to a small sample of passenger trajectories as well as the physical topology of connections, or a limited set of observed social interactions with additional information on the underlying social structure. Combining these two different sources of information to reliably infer transition matrices from incomplete data on repeated interactions is an important open challenge, with severe implications for the reliability of downstream network analysis tasks. Addressing this issue, we show that including knowledge on such topological constraints can considerably improve the inference of transition matrices, especially in situations where we only have a small number of observed interactions. To this end, we derive an analytically tractable Bayesian method that uses repeated interactions and a topological prior to perform data-efficient inference of transition matrices. We compare our approach against commonly used frequentist and Bayesian approaches both in synthetic data and in five real-world datasets, and we find that our method recovers the transition probabilities with higher accuracy. Furthermore, we demonstrate that the method is robust even in cases when the knowledge of the topological constraint is partial. Lastly, we show that this higher accuracy improves the results for downstream network analysis tasks like cluster detection and node ranking, which highlights the practical relevance of our method for interdisciplinary data-driven analyses of networked systems.

Unifying pairwise interactions in complex dynamics

Scientists have developed hundreds of techniques to measure the interactions between pairs of processes in complex systems, but these computational methods-from contemporaneous correlation coefficients to causal inference methods-define and formulate interactions differently, using distinct quantitative theories that remain largely disconnected. Here we introduce a large assembled library of 237 statistics of pairwise interactions, and assess their behavior on 1,053 multivariate time series from a wide range of real-world and model-generated systems. Our analysis highlights commonalities between disparate mathematical formulations of interactions, providing a unified picture of a rich interdisciplinary literature. Using three real-world case studies, we then show that simultaneously leveraging diverse methods can uncover those most suitable for addressing a given problem, facilitating interpretable understanding of the quantitative formulation of pairwise dependencies that drive successful performance. Our results and accompanying software enable comprehensive analysis of time-series interactions by drawing on decades of diverse methodological contributions.

The anatomy of a population-scale social network

Large-scale human social network structure is typically inferred from digital trace samples of online social media platforms or mobile communication data. Instead, here we investigate the social network structure of a complete population, where people are connected by high-quality links sourced from administrative registers of family, household, work, school, and next-door neighbors. We examine this multilayer social opportunity structure through three common concepts in network analysis: degree, closure, and distance. Findings present how particular network layers contribute to presumably universal scale-free and small-world properties of networks. Furthermore, we suggest a novel measure of excess closure and apply this in a life-course perspective to show how the social opportunity structure of individuals varies along age, socio-economic status, and education level.

Structure and function in artificial, zebrafish and human neural networks

Network science provides a set of tools for the characterization of the structure and functional behavior of complex systems. Yet a major problem is to quantify how the structural domain is related to the dynamical one. In other words, how the diversity of dynamical states of a system can be predicted from the static network structure? Or the reverse problem: starting from a set of signals derived from experimental recordings, how can one discover the network connections or the causal relations behind the observed dynamics? Despite the advances achieved over the last two decades, many challenges remain concerning the study of the structure-dynamics interplay of complex systems. In neuroscience, progress is typically constrained by the low spatio-temporal resolution of experiments and by the lack of a universal inferring framework for empirical systems. To address these issues, applications of network science and artificial intelligence to neural data have been rapidly growing. In this article, we review important recent applications of methods from those fields to the study of the interplay between structure and functional dynamics of human and zebrafish brain. We cover the selection of topological features for the characterization of brain networks, inference of functional connections, dynamical modeling, and close with applications to both the human and zebrafish brain. This review is intended to neuroscientists who want to become acquainted with techniques from network science, as well as to researchers from the latter field who are interested in exploring novel application scenarios in neuroscience.

Network medicine: an approach to complex kidney disease phenotypes

Scientific reductionism has been the basis of disease classification and understanding for more than a century. However, the reductionist approach of characterizing diseases from a limited set of clinical observations and laboratory evaluations has proven insufficient in the face of an exponential growth in data generated from transcriptomics, proteomics, metabolomics and deep phenotyping. A new systematic method is necessary to organize these datasets and build new definitions of what constitutes a disease that incorporates both biological and environmental factors to more precisely describe the ever-growing complexity of phenotypes and their underlying molecular determinants. Network medicine provides such a conceptual framework to bridge these vast quantities of data while providing an individualized understanding of disease. The modern application of network medicine principles is yielding new insights into the pathobiology of chronic kidney diseases and renovascular disorders by expanding the understanding of pathogenic mediators, novel biomarkers and new options for renal therapeutics. These efforts affirm network medicine as a robust paradigm for elucidating new advances in the diagnosis and treatment of kidney disorders.