Inferring the mesoscale structure of layered, edge-valued, and time-varying networks

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.

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.

Hyperedge prediction and the statistical mechanisms of higher-order and lower-order interactions in complex networks

Complex networked systems often exhibit higher-order interactions, beyond dyadic interactions, which can dramatically alter their observed behavior. Consequently, understanding hypergraphs from a structural perspective has become increasingly important. Statistical, group-based inference approaches are well suited for unveiling the underlying community structure and predicting unobserved interactions. However, these approaches often rely on two key assumptions: that the same groups can explain hyperedges of any order and that interactions are assortative, meaning that edges are formed by nodes with the same group memberships. To test these assumptions, we propose a group-based generative model for hypergraphs that does not impose an assortative mechanism to explain observed higher-order interactions, unlike current approaches. Our model allows us to explore the validity of the assumptions. Our results indicate that the first assumption appears to hold true for real networks. However, the second assumption is not necessarily accurate; we find that a combination of general statistical mechanisms can explain observed hyperedges. Finally, with our approach, we are also able to determine the importance of lower and high-order interactions for predicting unobserved interactions. Our research challenges the conventional assumptions of group-based inference methodologies and broadens our understanding of the underlying structure of hypergraphs.

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.

Generative models for two-ground-truth partitions in networks

A myriad of approaches have been proposed to characterize the mesoscale structure of networks most often as a partition based on patterns variously called communities, blocks, or clusters. Clearly, distinct methods designed to detect different types of patterns may provide a variety of answers' to the networks mesoscale structure. Yet even multiple runs of a given method can sometimes yield diverse and conflicting results, producing entire landscapes of partitions which potentially include multiple (locally optimal) mesoscale explanations of the network. Such ambiguity motivates a closer look at the ability of these methods to find multiple qualitatively different "ground truth" partitions in a network. Here we propose the stochastic cross-block model (SCBM), a generative model which allows for two distinct partitions to be built into the mesoscale structure of a single benchmark network. We demonstrate a use case of the benchmark model by appraising the power of stochastic block models (SBMs) to detect implicitly planted coexisting bicommunity and core-periphery structures of different strengths. Given our model design and experimental setup, we find that the ability to detect the two partitions individually varies by SBM variant and that coexistence of both partitions is recovered only in a very limited number of cases. Our findings suggest that in most instances only one-in some way dominating-structure can be detected, even in the presence of other partitions. They underline the need for considering entire landscapes of partitions when different competing explanations exist and motivate future research to advance partition coexistence detection methods. Our model also contributes to the field of benchmark networks more generally by enabling further exploration of the ability of new and existing methods to detect ambiguity in the mesoscale structure of networks.

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.

Systematic assessment of the quality of fit of the stochastic block model for empirical networks

We perform a systematic analysis of the quality of fit of the stochastic block model (SBM) for 275 empirical networks spanning a wide range of domains and orders of size magnitude. We employ posterior predictive model checking as a criterion to assess the quality of fit, which involves comparing networks generated by the inferred model with the empirical network, according to a set of network descriptors. We observe that the SBM is capable of providing an accurate description for the majority of networks considered, but falls short of saturating all modeling requirements. In particular, networks possessing a large diameter and slow-mixing random walks tend to be badly described by the SBM. However, contrary to what is often assumed, networks with a high abundance of triangles can be well described by the SBM in many cases. We demonstrate that simple network descriptors can be used to evaluate whether or not the SBM can provide a sufficiently accurate representation, potentially pointing to possible model extensions that can systematically improve the expressiveness of this class of models.

Brain Network Organization Following Post-Stroke Neurorehabilitation

Brain network analysis can offer useful information to guide the rehabilitation of post-stroke patients. We applied functional network connection models based on multiplex-multilayer network analysis (MMN) to explore functional network connectivity changes induced by robot-aided gait training (RAGT) using the Ekso, a wearable exoskeleton, and compared it to conventional overground gait training (COGT) in chronic stroke patients. We extracted the coreness of individual nodes at multiple locations in the brain from EEG recordings obtained before and after gait training in a resting state. We found that patients provided with RAGT achieved a greater motor function recovery than those receiving COGT. This difference in clinical outcome was paralleled by greater changes in connectivity patterns among different brain areas central to motor programming and execution, as well as a recruitment of other areas beyond the sensorimotor cortices and at multiple frequency ranges, contemporarily. The magnitude of these changes correlated with motor function recovery chances. Our data suggest that the use of RAGT as an add-on treatment to COGT may provide post-stroke patients with a greater modification of the functional brain network impairment following a stroke. This might have potential clinical implications if confirmed in large clinical trials.

Clustering of heterogeneous populations of networks

Statistical methods for reconstructing networks from repeated measurements typically assume that all measurements are generated from the same underlying network structure. This need not be the case, however. People's social networks might be different on weekdays and weekends, for instance. Brain networks may differ between healthy patients and those with dementia or other conditions. Here we describe a Bayesian analysis framework for such data that allows for the fact that network measurements may be reflective of multiple possible structures. We define a finite mixture model of the measurement process and derive a Gibbs sampling procedure that samples exactly from the full posterior distribution of model parameters. The end result is a clustering of the measured networks into groups with similar structure. We demonstrate the method on both real and synthetic network populations.

Domain‐topic models with chained dimensions: Charting an emergent domain of a major oncology conference

This paper presents a contribution to the study of bibliographic corpora through science mapping. From a graph representation of documents and their textual dimension, stochastic block models can provide a simultaneous clustering of documents and words that we call a domain‐topic model. Previous work investigated the resulting topics, or word clusters, while ours focuses on the study of the document clusters we call domains. To enable the description and interactive navigation of domains, we introduce measures and interfaces that consider the structure of the model to relate both types of clusters. We then present a procedure that extends the block model to cluster metadata attributes of documents, which we call a domain‐chained model, noting that our measures and interfaces transpose to metadata clusters. We provide an example application to a corpus relevant to current science, technology and society (STS) research and an interesting case for our approach: the abstracts presented between 1995 and 2017 at the American Society of Clinical Oncology Annual Meeting, the major oncology research conference. Through a sequence of domain‐topic and domain‐chained models, we identify and describe a group of domains that have notably grown through the last decades and which we relate to the establishment of “oncopolicy” as a major concern in oncology.

Atomic subgraphs and the statistical mechanics of networks

We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges, but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows for the generation of graphs with extensive numbers of triangles and other network motifs commonly observed in many real-world networks. More specifically, we focus on maximum entropy ensembles under constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such models. We also present a procedure for combining distributions of multiple atomic subgraphs that enables the construction of models with fewer parameters. Expanding the model to include atoms with edge and vertex labels we obtain a general class of models that can be parametrized in terms of basic building blocks and their distributions that include many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs, random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree-corrected and directed versions. We show that the entropy for all these models can be derived from a single expression that is characterized by the symmetry groups of atomic subgraphs.

Testing for association in multiview network data

In this paper, we consider data consisting of multiple networks, each composed of a different edge set on a common set of nodes. Many models have been proposed for the analysis of such multiview network data under the assumption that the data views are closely related. In this paper, we provide tools for evaluating this assumption. In particular, we ask: given two networks that each follow a stochastic block model, is there an association between the latent community memberships of the nodes in the two networks? To answer this question, we extend the stochastic block model for a single network view to the two-view setting, and develop a new hypothesis test for the null hypothesis that the latent community memberships in the two data views are independent. We apply our test to protein-protein interaction data from the HINT database. We find evidence of a weak association between the latent community memberships of proteins defined with respect to binary interaction data and the latent community memberships of proteins defined with respect to cocomplex association data. We also extend this proposal to the setting of a network with node covariates. The proposed methods extend readily to three or more network/multivariate data views.

Joint Embedding of Graphs

Feature extraction and dimension reduction for networks is critical in a wide variety of domains. Efficiently and accurately learning features for multiple graphs has important applications in statistical inference on graphs. We propose a method to jointly embed multiple undirected graphs. Given a set of graphs, the joint embedding method identifies a linear subspace spanned by rank one symmetric matrices and projects adjacency matrices of graphs into this subspace. The projection coefficients can be treated as features of the graphs, while the embedding components can represent vertex features. We also propose a random graph model for multiple graphs that generalizes other classical models for graphs. We show through theory and numerical experiments that under the model, the joint embedding method produces estimates of parameters with small errors. Via simulation experiments, we demonstrate that the joint embedding method produces features which lead to state of the art performance in classifying graphs. Applying the joint embedding method to human brain graphs, we find it extracts interpretable features with good prediction accuracy in different tasks.

Inference of edge correlations in multilayer networks

Many recent developments in network analysis have focused on multilayer networks, which one can use to encode time-dependent interactions, multiple types of interactions, and other complications that arise in complex systems. Like their monolayer counterparts, multilayer networks in applications often have mesoscale features, such as community structure. A prominent approach for inferring such structures is the employment of multilayer stochastic block models (SBMs). A common (but potentially inadequate) assumption of these models is the sampling of edges in different layers independently, conditioned on the community labels of the nodes. In this paper, we relax this assumption of independence by incorporating edge correlations into an SBM-like model. We derive maximum-likelihood estimates of the key parameters of our model, and we propose a measure of layer correlation that reflects the similarity between the connectivity patterns in different layers. Finally, we explain how to use correlated models for edge "prediction" (i.e., inference) in multilayer networks. By incorporating edge correlations, we find that prediction accuracy improves both in synthetic networks and in a temporal network of shoppers who are connected to previously purchased grocery products.

Community detection with node attributes in multilayer networks

Community detection in networks is commonly performed using information about interactions between nodes. Recent advances have been made to incorporate multiple types of interactions, thus generalizing standard methods to multilayer networks. Often, though, one can access additional information regarding individual nodes, attributes, or covariates. A relevant question is thus how to properly incorporate this extra information in such frameworks. Here we develop a method that incorporates both the topology of interactions and node attributes to extract communities in multilayer networks. We propose a principled probabilistic method that does not assume any a priori correlation structure between attributes and communities but rather infers this from data. This leads to an efficient algorithmic implementation that exploits the sparsity of the dataset and can be used to perform several inference tasks; we provide an open-source implementation of the code online. We demonstrate our method on both synthetic and real-world data and compare performance with methods that do not use any attribute information. We find that including node information helps in predicting missing links or attributes. It also leads to more interpretable community structures and allows the quantification of the impact of the node attributes given in input.

Unspoken Assumptions in Multi-layer Modularity maximization

A principled approach to recover communities in social networks is to find a clustering of the network nodes into modules (i.e groups of nodes) for which the modularity over the network is maximal. This guarantees partitioning the network nodes into sparsely connected groups of densely connected nodes. A popular extension of modularity has been proposed in the literature so it applies to multi-layer networks, that is, networks that model different types/aspects of interactions among a set of actors. In this extension, a new parameter, the coupling strength ω, has been introduced to couple different copies (i.e nodes) of the same actor with specific weights across different layers. This allows two nodes that refer to the same actor to reward the modularity score with an amount proportional to ω when they appear in the same community. While this extension seems to provide an effective tool to detect communities in multi-layer networks, it is not always clear what kind of communities maximising the generalised modularity can identify in multi-layer networks and whether these communities are inclusive to all possible community structures possible to exist in multi-layer networks. In addition, it has not been thoroughly investigated yet how to interpret ω in real-world scenarios, and whether a proper tuning of ω, if exists, is enough to guarantee an accurate recoverability for different types of multi-layer community structures. In this article, we report the different ways used in the literature to tune ω. We analyse different community structures that can be recovered by maximising the generalised modularity in relation to ω. We propose different models for multi-layer communities in multiplex and time-dependent networks and test if they are recoverable by modularity-maximization community detection methods under any assignment of ω. Our main finding is that only few simple models of multi-layer communities in multiplex and time-dependent networks are recoverable by modularity maximisation methods while more complex models are not accurately recoverable under any assignment of ω.

Short-Term Classification Learning Promotes Rapid Global Improvements of Information Processing in Human Brain Functional Connectome

Classification learning is a preeminent human ability within the animal kingdom but the key mechanisms of brain networks regulating learning remain mostly elusive. Recent neuroimaging advancements have depicted human brain as a complex graph machinery where brain regions are nodes and coherent activities among them represent the functional connections. While long-term motor memories have been found to alter functional connectivity in the resting human brain, a graph topological investigation of the short-time effects of learning are still not widely investigated. For instance, classification learning is known to orchestrate rapid modulation of diverse memory systems like short-term and visual working memories but how the brain functional connectome accommodates such modulations is unclear. We used publicly available repositories (openfmri.org) selecting three experiments, two focused on short-term classification learning along two consecutive runs where learning was promoted by trial-by-trial feedback errors, while a further experiment was used as supplementary control. We analyzed the functional connectivity extracted from BOLD fMRI signals, and estimated the graph information processing in the cerebral networks. The information processing capability, characterized by complex network statistics, significantly improved over runs, together with the subject classification accuracy. Instead, null-learning experiments, where feedbacks came with poor consistency, did not provoke any significant change in the functional connectivity over runs. We propose that learning induces fast modifications in the overall brain network dynamics, definitely ameliorating the short-term potential of the brain to process and integrate information, a dynamic consistently orchestrated by modulations of the functional connections among specific brain regions.

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.

The use of multilayer network analysis in animal behaviour

Network analysis has driven key developments in research on animal behaviour by providing quantitative methods to study the social structures of animal groups and populations. A recent formalism, known as multilayer network analysis, has advanced the study of multifaceted networked systems in many disciplines. It offers novel ways to study and quantify animal behaviour through connected 'layers' of interactions. In this article, we review common questions in animal behaviour that can be studied using a multilayer approach, and we link these questions to specific analyses. We outline the types of behavioural data and questions that may be suitable to study using multilayer network analysis. We detail several multilayer methods, which can provide new insights into questions about animal sociality at individual, group, population and evolutionary levels of organization. We give examples for how to implement multilayer methods to demonstrate how taking a multilayer approach can alter inferences about social structure and the positions of individuals within such a structure. Finally, we discuss caveats to undertaking multilayer network analysis in the study of animal social networks, and we call attention to methodological challenges for the application of these approaches. Our aim is to instigate the study of new questions about animal sociality using the new toolbox of multilayer network analysis.

Tensorial and bipartite block models for link prediction in layered networks and temporal networks

Many real-world complex systems are well represented as multilayer networks; predicting interactions in those systems is one of the most pressing problems in predictive network science. To address this challenge, we introduce two stochastic block models for multilayer and temporal networks; one of them uses nodes as its fundamental unit, whereas the other focuses on links. We also develop scalable algorithms for inferring the parameters of these models. Because our models describe all layers simultaneously, our approach takes full advantage of the information contained in the whole network when making predictions about any particular layer. We illustrate the potential of our approach by analyzing two empirical data sets: a temporal network of e-mail communications, and a network of drug interactions for treating different cancer types. We find that multilayer models consistently outperform their single-layer counterparts, but that the most predictive model depends on the data set under consideration; whereas the node-based model is more appropriate for predicting drug interactions, the link-based model is more appropriate for predicting e-mail communication.

Change points, memory and epidemic spreading in temporal networks

Dynamic networks exhibit temporal patterns that vary across different time scales, all of which can potentially affect processes that take place on the network. However, most data-driven approaches used to model time-varying networks attempt to capture only a single characteristic time scale in isolation — typically associated with the short-time memory of a Markov chain or with long-time abrupt changes caused by external or systemic events. Here we propose a unified approach to model both aspects simultaneously, detecting short and long-time behaviors of temporal networks. We do so by developing an arbitrary-order mixed Markov model with change points, and using a nonparametric Bayesian formulation that allows the Markov order and the position of change points to be determined from data without overfitting. In addition, we evaluate the quality of the multiscale model in its capacity to reproduce the spreading of epidemics on the temporal network, and we show that describing multiple time scales simultaneously has a synergistic effect, where statistically significant features are uncovered that otherwise would remain hidden by treating each time scale independently.

Multiplex core-periphery organization of the human connectome

What is the core of the human brain is a fundamental question that has been mainly addressed by studying the anatomical connections between differently specialized areas, thus neglecting the possible contributions from their functional interactions. While many methods are available to identify the core of a network when connections between nodes are all of the same type, a principled approach to define the core when multiple types of connectivity are allowed is still lacking. Here, we introduce a general framework to define and extract the core-periphery structure of multi-layer networks by explicitly taking into account the connectivity patterns at each layer. We first validate our algorithm on synthetic networks of different size and density, and with tunable overlap between the cores at different layers. We then use our method to merge information from structural and functional brain networks, obtaining in this way an integrated description of the core of the human connectome. Results confirm the role of the main known cortical and subcortical hubs, but also suggest the presence of new areas in the sensori-motor cortex that are crucial for intrinsic brain functioning. Taken together these findings provide fresh evidence on a fundamental question in modern neuroscience and offer new opportunities to explore the mesoscale properties of multimodal brain networks.

Constrained information flows in temporal networks reveal intermittent communities

Many real-world networks represent dynamic systems with interactions that change over time, often in uncoordinated ways and at irregular intervals. For example, university students connect in intermittent groups that repeatedly form and dissolve based on multiple factors, including their lectures, interests, and friends. Such dynamic systems can be represented as multilayer networks where each layer represents a snapshot of the temporal network. In this representation, it is crucial that the links between layers accurately capture real dependencies between those layers. Often, however, these dependencies are unknown. Therefore, current methods connect layers based on simplistic assumptions that do not capture node-level layer dependencies. For example, connecting every node to itself in other layers with the same weight can wipe out dependencies between intermittent groups, making it difficult or even impossible to identify them. In this paper, we present a principled approach to estimating node-level layer dependencies based on the network structure within each layer. We implement our node-level coupling method in the community detection framework Infomap and demonstrate its performance compared to current methods on synthetic and real temporal networks. We show that our approach more effectively constrains information inside multilayer communities so that Infomap can better recover planted groups in multilayer benchmark networks that represent multiple modes with different groups and better identify intermittent communities in real temporal contact networks. These results suggest that node-level layer coupling can improve the modeling of information spreading in temporal networks and better capture intermittent community structure.

Nonparametric weighted stochastic block models

We present a Bayesian formulation of weighted stochastic block models that can be used to infer the large-scale modular structure of weighted networks, including their hierarchical organization. Our method is nonparametric, and thus does not require the prior knowledge of the number of groups or other dimensions of the model, which are instead inferred from data. We give a comprehensive treatment of different kinds of edge weights (i.e., continuous or discrete, signed or unsigned, bounded or unbounded), as well as arbitrary weight transformations, and describe an unsupervised model selection approach to choose the best network description. We illustrate the application of our method to a variety of empirical weighted networks, such as global migrations, voting patterns in congress, and neural connections in the human brain.

A network approach to topic models

One of the main computational and scientific challenges in the modern age is to extract useful information from unstructured texts. Topic models are one popular machine-learning approach that infers the latent topical structure of a collection of documents. Despite their success-particularly of the most widely used variant called latent Dirichlet allocation (LDA)-and numerous applications in sociology, history, and linguistics, topic models are known to suffer from severe conceptual and practical problems, for example, a lack of justification for the Bayesian priors, discrepancies with statistical properties of real texts, and the inability to properly choose the number of topics. We obtain a fresh view of the problem of identifying topical structures by relating it to the problem of finding communities in complex networks. We achieve this by representing text corpora as bipartite networks of documents and words. By adapting existing community-detection methods (using a stochastic block model (SBM) with nonparametric priors), we obtain a more versatile and principled framework for topic modeling (for example, it automatically detects the number of topics and hierarchically clusters both the words and documents). The analysis of artificial and real corpora demonstrates that our SBM approach leads to better topic models than LDA in terms of statistical model selection. Our work shows how to formally relate methods from community detection and topic modeling, opening the possibility of cross-fertilization between these two fields.

Multilink communities of multiplex networks

Multiplex networks describe a large number of complex social, biological and transportation networks where a set of nodes is connected by links of different nature and connotation. Here we uncover the rich community structure of multiplex networks by associating a community to each multilink where the multilinks characterize the connections existing between any two nodes of the multiplex network. Our community detection method reveals the rich interplay between the mesoscale structure of the multiplex networks and their multiplexity. For instance some nodes can belong to many layers and few communities while others can belong to few layers but many communities. Moreover the multilink communities can be formed by a different number of relevant layers. These results point out that mesoscopically there can be large differences in the compressibility of multiplex networks.

Algorithmic detectability threshold of the stochastic block model

The assumption that the values of model parameters are known or correctly learned, i.e., the Nishimori condition, is one of the requirements for the detectability analysis of the stochastic block model in statistical inference. In practice, however, there is no example demonstrating that we can know the model parameters beforehand, and there is no guarantee that the model parameters can be learned accurately. In this study, we consider the expectation-maximization (EM) algorithm with belief propagation (BP) and derive its algorithmic detectability threshold. Our analysis is not restricted to the community structure but includes general modular structures. Because the algorithm cannot always learn the planted model parameters correctly, the algorithmic detectability threshold is qualitatively different from the one with the Nishimori condition.

Modelling sequences and temporal networks with dynamic community structures

In evolving complex systems such as air traffic and social organisations, collective effects emerge from their many components' dynamic interactions. While the dynamic interactions can be represented by temporal networks with nodes and links that change over time, they remain highly complex. It is therefore often necessary to use methods that extract the temporal networks' large-scale dynamic community structure. However, such methods are subject to overfitting or suffer from effects of arbitrary, a priori-imposed timescales, which should instead be extracted from data. Here we simultaneously address both problems and develop a principled data-driven method that determines relevant timescales and identifies patterns of dynamics that take place on networks, as well as shape the networks themselves. We base our method on an arbitrary-order Markov chain model with community structure, and develop a nonparametric Bayesian inference framework that identifies the simplest such model that can explain temporal interaction data.The description of temporal networks is usually simplified in terms of their dynamic community structures, whose identification however relies on a priori assumptions. Here the authors present a data-driven method that determines relevant timescales for the dynamics and uses it to identify communities.

Reconstruction of stochastic temporal networks through diffusive arrival times

Temporal networks have opened a new dimension in defining and quantification of complex interacting systems. Our ability to identify and reproduce time-resolved interaction patterns is, however, limited by the restricted access to empirical individual-level data. Here we propose an inverse modelling method based on first-arrival observations of the diffusion process taking place on temporal networks. We describe an efficient coordinate-ascent implementation for inferring stochastic temporal networks that builds in particular but not exclusively on the null model assumption of mutually independent interaction sequences at the dyadic level. The results of benchmark tests applied on both synthesized and empirical network data sets confirm the validity of our algorithm, showing the feasibility of statistically accurate inference of temporal networks only from moderate-sized samples of diffusion cascades. Our approach provides an effective and flexible scheme for the temporally augmented inverse problems of network reconstruction and has potential in a broad variety of applications.

Reconstruction of time-resolved interactions in networks is more challenging than for the time-independent case, as causal relations limit accessibility to empirical data. Here the authors propose a method based on first-arrival observations of a diffusion process to infer stochastic temporal networks.

Multilayer modeling and analysis of human brain networks

Understanding how the human brain is structured, and how its architecture is related to function, is of paramount importance for a variety of applications, including but not limited to new ways to prevent, deal with, and cure brain diseases, such as Alzheimer's or Parkinson's, and psychiatric disorders, such as schizophrenia. The recent advances in structural and functional neuroimaging, together with the increasing attitude toward interdisciplinary approaches involving computer science, mathematics, and physics, are fostering interesting results from computational neuroscience that are quite often based on the analysis of complex network representation of the human brain. In recent years, this representation experienced a theoretical and computational revolution that is breaching neuroscience, allowing us to cope with the increasing complexity of the human brain across multiple scales and in multiple dimensions and to model structural and functional connectivity from new perspectives, often combined with each other. In this work, we will review the main achievements obtained from interdisciplinary research based on magnetic resonance imaging and establish de facto, the birth of multilayer network analysis and modeling of the human brain.

Community detection, link prediction, and layer interdependence in multilayer networks

Complex systems are often characterized by distinct types of interactions between the same entities. These can be described as a multilayer network where each layer represents one type of interaction. These layers may be interdependent in complicated ways, revealing different kinds of structure in the network. In this work we present a generative model, and an efficient expectation-maximization algorithm, which allows us to perform inference tasks such as community detection and link prediction in this setting. Our model assumes overlapping communities that are common between the layers, while allowing these communities to affect each layer in a different way, including arbitrary mixtures of assortative, disassortative, or directed structure. It also gives us a mathematically principled way to define the interdependence between layers, by measuring how much information about one layer helps us predict links in another layer. In particular, this allows us to bundle layers together to compress redundant information and identify small groups of layers which suffice to predict the remaining layers accurately. We illustrate these findings by analyzing synthetic data and two real multilayer networks, one representing social support relationships among villagers in South India and the other representing shared genetic substring material between genes of the malaria parasite.

The many facets of community detection in complex networks

Community detection, the decomposition of a graph into essential building blocks, has been a core research topic in network science over the past years. Since a precise notion of what constitutes a community has remained evasive, community detection algorithms have often been compared on benchmark graphs with a particular form of assortative community structure and classified based on the mathematical techniques they employ. However, this comparison can be misleading because apparent similarities in their mathematical machinery can disguise different goals and reasons for why we want to employ community detection in the first place. Here we provide a focused review of these different motivations that underpin community detection. This problem-driven classification is useful in applied network science, where it is important to select an appropriate algorithm for the given purpose. Moreover, highlighting the different facets of community detection also delineates the many lines of research and points out open directions and avenues for future research.

Detection of timescales in evolving complex systems

Most complex systems are intrinsically dynamic in nature. The evolution of a dynamic complex system is typically represented as a sequence of snapshots, where each snapshot describes the configuration of the system at a particular instant of time. This is often done by using constant intervals but a better approach would be to define dynamic intervals that match the evolution of the system's configuration. To this end, we propose a method that aims at detecting evolutionary changes in the configuration of a complex system, and generates intervals accordingly. We show that evolutionary timescales can be identified by looking for peaks in the similarity between the sets of events on consecutive time intervals of data. Tests on simple toy models reveal that the technique is able to detect evolutionary timescales of time-varying data both when the evolution is smooth as well as when it changes sharply. This is further corroborated by analyses of several real datasets. Our method is scalable to extremely large datasets and is computationally efficient. This allows a quick, parameter-free detection of multiple timescales in the evolution of a complex system.

Enhanced Detectability of Community Structure in Multilayer Networks through Layer Aggregation

Many systems are naturally represented by a multilayer network in which edges exist in multiple layers that encode different, but potentially related, types of interactions, and it is important to understand limitations on the detectability of community structure in these networks. Using random matrix theory, we analyze detectability limitations for multilayer (specifically, multiplex) stochastic block models (SBMs) in which L layers are derived from a common SBM. We study the effect of layer aggregation on detectability for several aggregation methods, including summation of the layers' adjacency matrices for which we show the detectability limit vanishes as O(L^{-1/2}) with increasing number of layers, L. Importantly, we find a similar scaling behavior when the summation is thresholded at an optimal value, providing insight into the common-but not well understood-practice of thresholding pairwise-interaction data to obtain sparse network representations.

The International Postal Network and Other Global Flows as Proxies for National Wellbeing

The digital exhaust left by flows of physical and digital commodities provides a rich measure of the nature, strength and significance of relationships between countries in the global network. With this work, we examine how these traces and the network structure can reveal the socioeconomic profile of different countries. We take into account multiple international networks of physical and digital flows, including the previously unexplored international postal network. By measuring the position of each country in the Trade, Postal, Migration, International Flights, IP and Digital Communications networks, we are able to build proxies for a number of crucial socioeconomic indicators such as GDP per capita and the Human Development Index ranking along with twelve other indicators used as benchmarks of national well-being by the United Nations and other international organisations. In this context, we have also proposed and evaluated a global connectivity degree measure applying multiplex theory across the six networks that accounts for the strength of relationships between countries. We conclude by showing how countries with shared community membership over multiple networks have similar socioeconomic profiles. Combining multiple flow data sources can help understand the forces which drive economic activity on a global level. Such an ability to infer proxy indicators in a context of incomplete information is extremely timely in light of recent discussions on measurement of indicators relevant to the Sustainable Development Goals.

Clustering network layers with the strata multilayer stochastic block model

Multilayer networks are a useful data structure for simultaneously capturing multiple types of relationships between a set of nodes. In such networks, each relational definition gives rise to a layer. While each layer provides its own set of information, community structure across layers can be collectively utilized to discover and quantify underlying relational patterns between nodes. To concisely extract information from a multilayer network, we propose to identify and combine sets of layers with meaningful similarities in community structure. In this paper, we describe the "strata multilayer stochastic block model" (sMLSBM), a probabilistic model for multilayer community structure. The central extension of the model is that there exist groups of layers, called "strata", which are defined such that all layers in a given stratum have community structure described by a common stochastic block model (SBM). That is, layers in a stratum exhibit similar node-to-community assignments and SBM probability parameters. Fitting the sMLSBM to a multilayer network provides a joint clustering that yields node-to-community and layer-to-stratum assignments, which cooperatively aid one another during inference. We describe an algorithm for separating layers into their appropriate strata and an inference technique for estimating the SBM parameters for each stratum. We demonstrate our method using synthetic networks and a multilayer network inferred from data collected in the Human Microbiome Project.

Emergence of Multiplex Communities in Collaboration Networks

Community structures in collaboration networks reflect the natural tendency of individuals to organize their work in groups in order to better achieve common goals. In most of the cases, individuals exploit their connections to introduce themselves to new areas of interests, giving rise to multifaceted collaborations which span different fields. In this paper, we analyse collaborations in science and among movie actors as multiplex networks, where the layers represent respectively research topics and movie genres, and we show that communities indeed coexist and overlap at the different layers of such systems. We then propose a model to grow multiplex networks based on two mechanisms of intra and inter-layer triadic closure which mimic the real processes by which collaborations evolve. We show that our model is able to explain the multiplex community structure observed empirically, and we infer the strength of the two underlying social mechanisms from real-world systems. Being also able to correctly reproduce the values of intra-layer and inter-layer assortativity correlations, the model contributes to a better understanding of the principles driving the evolution of social networks.

Compensating for population sampling in simulations of epidemic spread on temporal contact networks

Data describing human interactions often suffer from incomplete sampling of the underlying population. As a consequence, the study of contagion processes using data-driven models can lead to a severe underestimation of the epidemic risk. Here we present a systematic method to alleviate this issue and obtain a better estimation of the risk in the context of epidemic models informed by high-resolution time-resolved contact data. We consider several such data sets collected in various contexts and perform controlled resampling experiments. We show how the statistical information contained in the resampled data can be used to build a series of surrogate versions of the unknown contacts. We simulate epidemic processes on the resulting reconstructed data sets and show that it is possible to obtain good estimates of the outcome of simulations performed using the complete data set. We discuss limitations and potential improvements of our method.