Nonparametric weighted stochastic block models

Overlapping community detection in weighted networks via hierarchical clustering

In real-world networks, community structures often appear as tightly connected clusters of nodes, with recent studies suggesting a hierarchical organization where larger groups subdivide into smaller ones across different levels. This hierarchical structure is particularly complex in trade networks, where actors typically belong to multiple communities due to diverse business relationships and contracts. To address this complexity, we present a novel algorithm for detecting hierarchical structures of overlapping communities in weighted networks, focusing on the interdependency between internal and external quality metrics for evaluating the detected communities. The proposed Graph Hierarchical Agglomerative Clustering (GHAC) approach utilizes maximal cliques as the basis units for hierarchical clustering. The algorithm measures dissimilarities between clusters using the minimal closed trail distance (CT-distance) and the size of maximal cliques within overlaps, capturing the density and connectivity of nodes. Through extensive experiments on synthetic networks with known ground truth, we demonstrate that the adjusted Silhouette index is the most reliable internal metric for determining the optimal cut in the dendrogram. Experimental results indicate that the GHAC method is competitive with widely used community detection techniques, particularly in networks with highly overlapping communities. The method effectively reveals the hierarchical structure of communities in weighted networks, as demonstrated by its application to the OECD weighted trade network, which describes the balanced trade value of bilateral trade relations.

The minimal computational substrate of fluid intelligence

The quantification of cognitive powers rests on identifying a behavioural task that depends on them. Such dependence cannot be assured, for the powers a task invokes cannot be experimentally controlled or constrained a priori, resulting in unknown vulnerability to failure of specificity and generalisability. Evaluating a compact version of Raven's Advanced Progressive Matrices (RAPM), a widely used clinical test of fluid intelligence, we show that LaMa, a self-supervised artificial neural network trained solely on the completion of partially masked images of natural environmental scenes, achieves representative human-level test scores a prima vista, without any task-specific inductive bias or training. Compared with cohorts of healthy and focally lesioned participants, LaMa exhibits human-like variation with item difficulty, and produces errors characteristic of right frontal lobe damage under degradation of its ability to integrate global spatial patterns. LaMa's narrow training and limited capacity suggest matrix-style tests may be open to computationally simple solutions that need not necessarily invoke the substrates of reasoning.

Neural network architecture of a mammalian brain

Connectomics research is making rapid advances, although models revealing general principles of connectional architecture are far from complete. Our analysis of 106 published connection reports indicates that the adult rat brain interregional connectome has about 76,940 of a possible 623,310 axonal connections between its 790 gray matter regions mapped in a reference atlas, equating to a network density of 12.3%. We examined the sexually dimorphic network using multiresolution consensus clustering that generated a nested hierarchy of interconnected modules/subsystems with three first-order modules and 157 terminal modules in females. Top-down hierarchy analysis suggests a mirror-image primary module pair in the central nervous system's rostral sector (forebrain-midbrain) associated with behavior control, and a single primary module in the intermediate sector (rhombicbrain) associated with behavior execution; the implications of these results are considered in relation to brain development and evolution. Bottom-up hierarchy analysis reveals known and unfamiliar modules suggesting strong experimentally testable hypotheses. Global network analyses indicate that all hubs are in the rostral module pair, a rich club extends through all three primary modules, and the network exhibits small-world attributes. Simulated lesions of all regions individually enabled ranking their impact on global network organization, and the visual path from the retina was used as a specific example, including the effects of cyclic connection weight changes from the endogenous circadian rhythm generator, suprachiasmatic nucleus. This study elucidates principles of interregional neuronal network architecture for a mammalian brain and suggests a strategy for modeling dynamic structural connectivity.

Deep generative models of protein structure uncover distant relationships across a continuous fold space

Our views of fold space implicitly rest upon many assumptions that impact how we analyze, interpret and understand protein structure, function and evolution. For instance, is there an optimal granularity in viewing protein structural similarities (e.g., architecture, topology or some other level)? Similarly, the discrete/continuous dichotomy of fold space is central, but remains unresolved. Discrete views of fold space bin similar folds into distinct, non-overlapping groups; unfortunately, such binning can miss remote relationships. While hierarchical systems like CATH are indispensable resources, less heuristic and more conceptually flexible approaches could enable more nuanced explorations of fold space. Building upon an Urfold model of protein structure, here we present a deep generative modeling framework, termed DeepUrfold, for analyzing protein relationships at scale. DeepUrfold's learned embeddings occupy high-dimensional latent spaces that can be distilled for a given protein in terms of an amalgamated representation uniting sequence, structure and biophysical properties. This approach is structure-guided, versus being purely structure-based, and DeepUrfold learns representations that, in a sense, define superfamilies. Deploying DeepUrfold with CATH reveals evolutionarily-remote relationships that evade existing methodologies, and suggests a mostly-continuous view of fold space-a view that extends beyond simple geometric similarity, towards the realm of integrated sequence ↔ structure ↔ function properties.

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.

Computational limits to the legibility of the imaged human brain

Our knowledge of the organisation of the human brain at the population-level is yet to translate into power to predict functional differences at the individual-level, limiting clinical applications and casting doubt on the generalisability of inferred mechanisms. It remains unknown whether the difficulty arises from the absence of individuating biological patterns within the brain, or from limited power to access them with the models and compute at our disposal. Here we comprehensively investigate the resolvability of such patterns with data and compute at unprecedented scale. Across 23 810 unique participants from UK Biobank, we systematically evaluate the predictability of 25 individual biological characteristics, from all available combinations of structural and functional neuroimaging data. Over 4526 GPU*hours of computation, we train, optimize, and evaluate out-of-sample 700 individual predictive models, including fully-connected feed-forward neural networks of demographic, psychological, serological, chronic disease, and functional connectivity characteristics, and both uni- and multi-modal 3D convolutional neural network models of macro- and micro-structural brain imaging. We find a marked discrepancy between the high predictability of sex (balanced accuracy 99.7%), age (mean absolute error 2.048 years, R2 0.859), and weight (mean absolute error 2.609Kg, R2 0.625), for which we set new state-of-the-art performance, and the surprisingly low predictability of other characteristics. Neither structural nor functional imaging predicted an individual's psychology better than the coincidence of common chronic disease (p < 0.05). Serology predicted chronic disease (p < 0.05) and was best predicted by it (p < 0.001), followed by structural neuroimaging (p < 0.05). Our findings suggest either more informative imaging or more powerful models will be needed to decipher individual level characteristics from the human brain. We make our models and code openly available.

Brain tumour genetic network signatures of survival

Tumour heterogeneity is increasingly recognized as a major obstacle to therapeutic success across neuro-oncology. Gliomas are characterized by distinct combinations of genetic and epigenetic alterations, resulting in complex interactions across multiple molecular pathways. Predicting disease evolution and prescribing individually optimal treatment requires statistical models complex enough to capture the intricate (epi)genetic structure underpinning oncogenesis. Here, we formalize this task as the inference of distinct patterns of connectivity within hierarchical latent representations of genetic networks. Evaluating multi-institutional clinical, genetic and outcome data from 4023 glioma patients over 14 years, across 12 countries, we employ Bayesian generative stochastic block modelling to reveal a hierarchical network structure of tumour genetics spanning molecularly confirmed glioblastoma, IDH-wildtype; oligodendroglioma, IDH-mutant and 1p/19q codeleted; and astrocytoma, IDH-mutant. Our findings illuminate the complex dependence between features across the genetic landscape of brain tumours and show that generative network models reveal distinct signatures of survival with better prognostic fidelity than current gold standard diagnostic categories.

Statistical models of complex brain networks: a maximum entropy approach

The brain is a highly complex system. Most of such complexity stems from the intermingled connections between its parts, which give rise to rich dynamics and to the emergence of high-level cognitive functions. Disentangling the underlying network structure is crucial to understand the brain functioning under both healthy and pathological conditions. Yet, analyzing brain networks is challenging, in part because their structure represents only one possible realization of a generative stochastic process which is in general unknown. Having a formal way to cope with such intrinsic variability is therefore central for the characterization of brain network properties. Addressing this issue entails the development of appropriate tools mostly adapted from network science and statistics. Here, we focus on a particular class of maximum entropy models for networks, i.e. exponential random graph models, as a parsimonious approach to identify the local connection mechanisms behind observed global network structure. Efforts are reviewed on the quest for basic organizational properties of human brain networks, as well as on the identification of predictive biomarkers of neurological diseases such as stroke. We conclude with a discussion on how emerging results and tools from statistical graph modeling, associated with forthcoming improvements in experimental data acquisition, could lead to a finer probabilistic description of complex systems in network neuroscience.

Estimating the Number of Communities in Weighted Networks

Community detection in weighted networks has been a popular topic in recent years. However, while there exist several flexible methods for estimating communities in weighted networks, these methods usually assume that the number of communities is known. It is usually unclear how to determine the exact number of communities one should use. Here, to estimate the number of communities for weighted networks generated from arbitrary distribution under the degree-corrected distribution-free model, we propose one approach that combines weighted modularity with spectral clustering. This approach allows a weighted network to have negative edge weights and it also works for signed networks. We compare the proposed method to several existing methods and show that our method is more accurate for estimating the number of communities both numerically and empirically.

Graph lesion-deficit mapping of fluid intelligence

Fluid intelligence is arguably the defining feature of human cognition. Yet the nature of its relationship with the brain remains a contentious topic. Influential proposals drawing primarily on functional imaging data have implicated 'multiple demand' frontoparietal and more widely distributed cortical networks, but extant lesion-deficit studies with greater causal power are almost all small, methodologically constrained, and inconclusive. The task demands large samples of patients, comprehensive investigation of performance, fine-grained anatomical mapping, and robust lesion-deficit inference, yet to be brought to bear on it. We assessed 165 healthy controls and 227 frontal or non-frontal patients with unilateral brain lesions on the best-established test of fluid intelligence, Raven's Advanced Progressive Matrices, employing an array of lesion-deficit inferential models responsive to the potentially distributed nature of fluid intelligence. Non-parametric Bayesian stochastic block models were used to reveal the community structure of lesion deficit networks, disentangling functional from confounding pathological distributed effects. Impaired performance was confined to patients with frontal lesions [F(2,387) = 18.491; P < 0.001; frontal worse than non-frontal and healthy participants P < 0.01, P <0.001], more marked on the right than left [F(4,385) = 12.237; P < 0.001; right worse than left and healthy participants P < 0.01, P < 0.001]. Patients with non-frontal lesions were indistinguishable from controls and showed no modulation by laterality. Neither the presence nor the extent of multiple demand network involvement affected performance. Both conventional network-based statistics and non-parametric Bayesian stochastic block modelling heavily implicated the right frontal lobe. Crucially, this localization was confirmed on explicitly disentangling functional from pathology-driven effects within a layered stochastic block model, prominently highlighting a right frontal network involving middle and inferior frontal gyrus, pre- and post-central gyri, with a weak contribution from right superior parietal lobule. Similar results were obtained with standard lesion-deficit analyses. Our study represents the first large-scale investigation of the distributed neural substrates of fluid intelligence in the focally injured brain. Combining novel graph-based lesion-deficit mapping with detailed investigation of cognitive performance in a large sample of patients provides crucial information about the neural basis of intelligence. Our findings indicate that a set of predominantly right frontal regions, rather than a more widely distributed network, is critical to the high-level functions involved in fluid intelligence. Further they suggest that Raven's Advanced Progressive Matrices is a useful clinical index of fluid intelligence and a sensitive marker of right frontal lobe dysfunction.

Feature learning and network structure from noisy node activity data

In the studies of network structures, much attention has been devoted to developing approaches to reconstruct networks and predict missing links when edge-related information is given. However, such approaches are not applicable when we are only given noisy node activity data with missing values. This work presents an unsupervised learning framework to learn node vectors and construct networks from such node activity data. First, we design a scheme to generate random node sequences from node context sets, which are generated from node activity data. Then, a three-layer neural network is adopted training the node sequences to obtain node vectors, which allow us to construct networks and capture nodes with synergistic roles. Furthermore, we present an entropy-based approach to select the most meaningful neighbors for each node in the resulting network. Finally, the effectiveness of the method is validated through both synthetic and real data.

Degree-corrected distribution-free model for community detection in weighted networks

A degree-corrected distribution-free model is proposed for weighted social networks with latent structural information. The model extends the previous distribution-free models by considering variation in node degree to fit real-world weighted networks, and it also extends the classical degree-corrected stochastic block model from un-weighted network to weighted network. We design an algorithm based on the idea of spectral clustering to fit the model. Theoretical framework on consistent estimation for the algorithm is developed under the model. Theoretical results when edge weights are generated from different distributions are analyzed. We also propose a general modularity as an extension of Newman’s modularity from un-weighted network to weighted network. Using experiments with simulated and real-world networks, we show that our method significantly outperforms the uncorrected one, and the general modularity is effective.

Quantifying collective identity online from self-defining hashtags

Mass communication over social media can drive rapid changes in our sense of collective identity. Hashtags in particular have acted as powerful social coordinators, playing a key role in organizing social movements like the Gezi park protests, Occupy Wall Street, #metoo, and #blacklivesmatter. Here we quantify collective identity from the use of hashtags as self-labels in over 85,000 actively-maintained Twitter user profiles spanning 2017-2019. Collective identities emerge from a graph model of individuals' overlapping self-labels, producing a hierarchy of graph clusters. Each cluster is bound together and characterized semantically by specific hashtags key to its formation. We define and apply two information-theoretic measures to quantify the strength of identities in the hierarchy. First we measure collective identity coherence to determine how integrated any identity is from local to global scales. Second, we consider the conspicuousness of any identity given its vocabulary versus the global identity map. Our work reveals a rich landscape of online identity emerging from the hierarchical alignment of uncoordinated self-labeling actions.

Evolution of global development cooperation: An analysis of aid flows with hierarchical stochastic block models

Despite considerable scholarly attention on the institutional and normative aspects of development cooperation, its longitudinal dynamics unfolding at the global level have rarely been investigated. Focusing on aid, we examine the evolving global structure of development cooperation induced by aid flows in its entirety. Representing annual aid flows between donors and recipients from 1970 to 2013 as a series of networks, we apply hierarchical stochastic block models to extensive aid-flow data that cover not only the aid behavior of the major OECD donors but also that of other emerging donors, including China. Despite a considerable degree of external expansion and internal diversification of aid relations over the years, the analysis has uncovered a temporally persistent structure of aid networks. The latter comprises, on the one hand, a limited number of major donors with far-reaching resources and, on the other hand, a large number of mostly poor but globally well-connected recipients. The results cast doubt on the efficacy of recurrent efforts for "aid reform" in substantially changing the global aid flow pattern.

Weighted stochastic block model

We propose a weighted stochastic block model (WSBM) which extends the stochastic block model to the important case in which edges are weighted. We address the parameter estimation of the WSBM by use of maximum likelihood and variational approaches, and establish the consistency of these estimators. The problem of choosing the number of classes in a WSBM is addressed. The proposed model is applied to simulated data and an illustrative data set.

Modularity in Biological Networks

Network modeling, from the ecological to the molecular scale has become an essential tool for studying the structure, dynamics and complex behavior of living systems. Graph representations of the relationships between biological components open up a wide variety of methods for discovering the mechanistic and functional properties of biological systems. Many biological networks are organized into a modular structure, so methods to discover such modules are essential if we are to understand the biological system as a whole. However, most of the methods used in biology to this end, have a limited applicability, as they are very specific to the system they were developed for. Conversely, from the statistical physics and network science perspective, graph modularity has been theoretically studied and several methods of a very general nature have been developed. It is our perspective that in particular for the modularity detection problem, biology and theoretical physics/network science are less connected than they should. The central goal of this review is to provide the necessary background and present the most applicable and pertinent methods for community detection in a way that motivates their further usage in biological research.

The autonomic brain: Multi-dimensional generative hierarchical modelling of the autonomic connectome

The autonomic nervous system governs the body's multifaceted internal adaptation to diverse changes in the external environment, a role more complex than is accessible to the methods-and data scales-hitherto used to illuminate its operation. Here we apply generative graphical modelling to large-scale multimodal neuroimaging data encompassing normal and abnormal states to derive a comprehensive hierarchical representation of the autonomic brain. We demonstrate that whereas conventional structural and functional maps identify regions jointly modulated by parasympathetic and sympathetic systems, only graphical analysis discriminates between them, revealing the cardinal roles of the autonomic system to be mediated by high-level distributed interactions. We provide a novel representation of the autonomic system-a multidimensional, generative network-that renders its richness tractable within future models of its function in health and disease.

Layer reconstruction and missing link prediction of a multilayer network with maximum a posteriori estimation

From social networks to biological networks, different types of interactions among the same set of nodes characterize distinct layers, which are termed multilayer networks. Within a multilayer network, some layers, confirmed through different experiments, could be structurally similar and interdependent. In this paper, we propose a maximum a posteriori-based method to study and reconstruct the structure of a target layer in a multilayer network. Nodes within the target layer are characterized by vectors, which are employed to compute edge weights. Further, to detect structurally similar layers, we propose a method for comparing networks based on the eigenvector centrality. Using similar layers, we obtain the parameters of the conjugate prior. With this maximum a posteriori algorithm, we can reconstruct the target layer and predict missing links. We test the method on two real multilayer networks, and the results show that the maximum a posteriori estimation is promising in reconstructing the target layer even when a large number of links is missing.

Spectral estimation for detecting low-dimensional structure in networks using arbitrary null models

Discovering low-dimensional structure in real-world networks requires a suitable null model that defines the absence of meaningful structure. Here we introduce a spectral approach for detecting a network’s low-dimensional structure, and the nodes that participate in it, using any null model. We use generative models to estimate the expected eigenvalue distribution under a specified null model, and then detect where the data network’s eigenspectra exceed the estimated bounds. On synthetic networks, this spectral estimation approach cleanly detects transitions between random and community structure, recovers the number and membership of communities, and removes noise nodes. On real networks spectral estimation finds either a significant fraction of noise nodes or no departure from a null model, in stark contrast to traditional community detection methods. Across all analyses, we find the choice of null model can strongly alter conclusions about the presence of network structure. Our spectral estimation approach is therefore a promising basis for detecting low-dimensional structure in real-world networks, or lack thereof.

Merge-split Markov chain Monte Carlo for community detection

We present a Markov chain Monte Carlo scheme based on merges and splits of groups that is capable of efficiently sampling from the posterior distribution of network partitions, defined according to the stochastic block model (SBM). We demonstrate how schemes based on the move of single nodes between groups systematically fail at correctly sampling from the posterior distribution even on small networks, and how our merge-split approach behaves significantly better, and improves the mixing time of the Markov chain by several orders of magnitude in typical cases. We also show how the scheme can be straightforwardly extended to nested versions of the SBM, yielding asymptotically exact samples of hierarchical network partitions.

Demarcating geographic regions using community detection in commuting networks with significant self-loops

We develop a method to identify statistically significant communities in a weighted network with a high proportion of self-looping weights. We use this method to find overlapping agglomerations of U.S. counties by representing inter-county commuting as a weighted network. We identify three types of communities; non-nodal, nodal and monads, which correspond to different types of regions. The results suggest that traditional regional delineations that rely on ad hoc thresholds do not account for important and pervasive connections that extend far beyond expected metropolitan boundaries or megaregions.

Core-periphery structure in sectoral international trade networks: A new approach to an old theory

The research on core-periphery structure of global trade from a complex-network perspective has shown that the world system is hierarchically organized into blocks and that countries play different roles in the world economy. Yet, little attention has been paid to investigating whether the sectoral international trade networks conform to a core-periphery structure, hence what is the role of different levels of processing in creating and maintaining structural inequality. This issue is of particular importance given the contemporary focus upon global production networks and reshaping of the international division of labor. With this in mind, we propose a model (LARDEG) from network science to reexamine old theories in economics, such as core-periphery structures in sectoral international trade networks and test whether the global value chains have changed structural positions in terms of the level of processing. The economic background of our model permitting a more accurate sorting of countries into structural positions and the general stability of results have provided for a more solid measurements than has hereto been possible. Our algorithm naturally produces networks with hierarchically nested block structure obtained from an iterative decomposition of the network periphery such that each block represents a vertex set of a maximal size sub-graph existing at different levels. The results not only lend support to the previous hierarchical model of the world-system (core, semi-periphery, and periphery) but also find that, depending on particular industry, the number of analytically identifiable blocks could be more than three. We show that ‘size effect’ is the one that prevails for core block membership at the first hierarchical level, while the GNI per capita is a much poorer proxy for the world-system status. Moreover, the patterns of blocks we label as the second- or third-level ‘core’ are strongly dependent on distance and geographical proximity. Overall, the various configurations of asymmetrical trade patterns between our blocks and the remarkably stable position of core countries at the top of structure clearly indicate that the rise of global production networks has actually restored a huge and unequal international division of labor splitting the world into ‘headquarter’ and ‘factory’ economies.

Mapping the community structure of the rat cerebral cortex with weighted stochastic block modeling

The anatomical architecture of the mammalian brain can be modeled as the connectivity between functionally distinct areas of cortex and sub-cortex, which we refer to as the connectome. The community structure of the connectome describes how the network can be parsed into meaningful groups of nodes. This process, called community detection, is commonly carried out to find internally densely connected communities-a modular topology. However, other community structure patterns are possible. Here we employ the weighted stochastic block model (WSBM), which can identify a wide range of topologies, to the rat cerebral cortex connectome, to probe the network for evidence of modular, core, periphery, and disassortative organization. Despite its algorithmic flexibility, the WSBM identifies substantial modular and assortative topology throughout the rat cerebral cortex connectome, significantly aligning to the modular approach in some parts of the network. Significant deviations from modular partitions include the identification of communities that are highly enriched in core (rich club) areas. A comparison of the WSBM and modular models demonstrates that the former, when applied as a generative model, more closely captures several nodal network attributes. An analysis of variation across an ensemble of partitions reveals that certain parts of the network participate in multiple topological regimes. Overall, our findings demonstrate the potential benefits of adopting the WSBM, which can be applied to a single weighted and directed matrix such as the rat cerebral cortex connectome, to identify community structure with a broad definition that transcends the common modular approach.

Loan maturity aggregation in interbank lending networks obscures mesoscale structure and economic functions

Since the 2007-2009 financial crisis, substantial academic effort has been dedicated to improving our understanding of interbank lending networks (ILNs). Because of data limitations or by choice, the literature largely lacks multiple loan maturities. We employ a complete interbank loan contract dataset to investigate whether maturity details are informative of the network structure. Applying the layered stochastic block model of Peixoto (2015) and other tools from network science on a time series of bilateral loans with multiple maturity layers in the Russian ILN, we find that collapsing all such layers consistently obscures mesoscale structure. The optimal maturity granularity lies between completely collapsing and completely separating the maturity layers and depends on the development phase of the interbank market, with a more developed market requiring more layers for optimal description. Closer inspection of the inferred maturity bins associated with the optimal maturity granularity reveals specific economic functions, from liquidity intermediation to financing. Collapsing a network with multiple underlying maturity layers or extracting one such layer, common in economic research, is therefore not only an incomplete representation of the ILN's mesoscale structure, but also conceals existing economic functions. This holds important insights and opportunities for theoretical and empirical studies on interbank market functioning, contagion, stability, and on the desirable level of regulatory data disclosure.

Generalised thresholding of hidden variable network models with scale-free property

The hidden variable formalism (based on the assumption of some intrinsic node parameters) turned out to be a remarkably efficient and powerful approach in describing and analyzing the topology of complex networks. Owing to one of its most advantageous property - namely proven to be able to reproduce a wide range of different degree distribution forms - it has become a standard tool for generating networks having the scale-free property. One of the most intensively studied version of this model is based on a thresholding mechanism of the exponentially distributed hidden variables associated to the nodes (intrinsic vertex weights), which give rise to the emergence of a scale-free network where the degree distribution p(k) ~ k^−γ is decaying with an exponent of γ = 2. Here we propose a generalization and modification of this model by extending the set of connection probabilities and hidden variable distributions that lead to the aforementioned degree distribution, and analyze the conditions leading to the above behavior analytically. In addition, we propose a relaxation of the hard threshold in the connection probabilities, which opens up the possibility for obtaining sparse scale free networks with arbitrary scaling exponent.

Stochastic block models: A comparison of variants and inference methods

Finding communities in complex networks is a challenging task and one promising approach is the Stochastic Block Model (SBM). But the influences from various fields led to a diversity of variants and inference methods. Therefore, a comparison of the existing techniques and an independent analysis of their capabilities and weaknesses is needed. As a first step, we review the development of different SBM variants such as the degree-corrected SBM of Karrer and Newman or Peixoto's hierarchical SBM. Beside stating all these variants in a uniform notation, we show the reasons for their development. Knowing the variants, we discuss a variety of approaches to infer the optimal partition like the Metropolis-Hastings algorithm. We perform our analysis based on our extension of the Girvan-Newman test and the Lancichinetti-Fortunato-Radicchi benchmark as well as a selection of some real world networks. Using these results, we give some guidance to the challenging task of selecting an inference method and SBM variant. In addition, we give a simple heuristic to determine the number of steps for the Metropolis-Hastings algorithms that lack a usual stop criterion. With our comparison, we hope to guide researches in the field of SBM and highlight the problem of existing techniques to focus future research. Finally, by making our code freely available, we want to promote a faster development, integration and exchange of new ideas.

Analysis of correlation-based biomolecular networks from different omics data by fitting stochastic block models

Background: Biological entities such as genes, promoters, mRNA, metabolites or proteins do not act alone, but in concert in their network context. Modules, i.e., groups of nodes with similar topological properties in these networks characterize important biological functions of the underlying biomolecular system. Edges in such molecular networks represent regulatory and physical interactions, and comparing them between conditions provides valuable information on differential molecular mechanisms. However, biological data is inherently noisy and network reduction techniques can propagate errors particularly to the level of edges. We aim to improve the analysis of networks of biological molecules by deriving modules together with edge relevance estimations that are based on global network characteristics. Methods: The key challenge we address here is investigating the capability of stochastic block models (SBMs) for representing and analyzing different types of biomolecular networks. Fitting them to SBMs both delivers modules of the networks and enables the derivation of edge confidence scores, and it has not yet been investigated for analyzing biomolecular networks. We apply SBM-based analysis independently to three correlation-based networks of breast cancer data originating from high-throughput measurements of different molecular layers: either transcriptomics, proteomics, or metabolomics. The networks were reduced by thresholding for correlation significance or by requirements on scale-freeness.  Results and discussion: We find that the networks are best represented by the hierarchical version of the SBM, and many of the predicted blocks have a biologically and phenotypically relevant functional annotation. The edge confidence scores are overall in concordance with the biological evidence given by the measurements. We conclude that biomolecular networks can be appropriately represented and analyzed by fitting SBMs. As the SBM-derived edge confidence scores are based on global network connectivity characteristics and potential hierarchies within the biomolecular networks are considered, they could be used as additional, integrated features in network-based data comparisons.

Analysis of correlation-based biomolecular networks from different omics data by fitting stochastic block models

Background: Biological entities such as genes, promoters, mRNA, metabolites or proteins do not act alone, but in concert in their network context. Modules, i.e., groups of nodes with similar topological properties in these networks characterize important biological functions of the underlying biomolecular system. Edges in such molecular networks represent regulatory and physical interactions, and comparing them between conditions provides valuable information on differential molecular mechanisms. However, biological data is inherently noisy and network reduction techniques can propagate errors particularly to the level of edges. We aim to improve the analysis of networks of biological molecules by deriving modules together with edge relevance estimations that are based on global network characteristics.  Methods: We propose to fit the networks to stochastic block models (SBM), a method that has not yet been investigated for the analysis of biomolecular networks. This procedure both delivers modules of the networks and enables the derivation of edge confidence scores. We apply it to correlation-based networks of breast cancer data originating from high-throughput measurements of diverse molecular layers such as transcriptomics, proteomics, and metabolomics. The networks were reduced by thresholding for correlation significance or by requirements on scale-freeness.  Results and discussion: We find that the networks are best represented by the hierarchical version of the SBM, and many of the predicted blocks have a biological meaning according to functional annotation. The edge confidence scores are overall in concordance with the biological evidence given by the measurements. As they are based on global network connectivity characteristics and potential hierarchies within the biomolecular networks are taken into account, they could be used as additional, integrated features in network-based data comparisons. Their tight relationship to edge existence probabilities can be exploited to predict missing or spurious edges in order to improve the network representation of the underlying biological system.

Weighted Stochastic Block Models of the Human Connectome across the Life Span

The human brain can be described as a complex network of anatomical connections between distinct areas, referred to as the human connectome. Fundamental characteristics of connectome organization can be revealed using the tools of network science and graph theory. Of particular interest is the network's community structure, commonly identified by modularity maximization, where communities are conceptualized as densely intra-connected and sparsely inter-connected. Here we adopt a generative modeling approach called weighted stochastic block models (WSBM) that can describe a wider range of community structure topologies by explicitly considering patterned interactions between communities. We apply this method to the study of changes in the human connectome that occur across the life span (between 6-85 years old). We find that WSBM communities exhibit greater hemispheric symmetry and are spatially less compact than those derived from modularity maximization. We identify several network blocks that exhibit significant linear and non-linear changes across age, with the most significant changes involving subregions of prefrontal cortex. Overall, we show that the WSBM generative modeling approach can be an effective tool for describing types of community structure in brain networks that go beyond modularity.