arXiv subjects

Tiago P. Peixoto

Publications and source records attributed to Tiago P. Peixoto.

At least 37 records · Page 2Linked to original sources

arXiv2020

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.

physics.soc-ph↗

arXiv2020

Statistical inference of assortative community structures

We develop a principled methodology to infer assortative communities in networks based on a nonparametric Bayesian formulation of the planted partition model. We show that this approach succeeds in finding statistically significant assortative modules in networks, unlike alternatives such as modularity maximization, which systematically overfits both in artificial as well as in empirical examples. In addition, we show that our method is not subject to a resolution limit, and can uncover an arbitrarily large number of communities, as long as there is statistical evidence for them. Our formulation is amenable to model selection procedures, which allow us to compare it to more general approaches based on the stochastic block model, and in this way reveal whether assortativity is in fact the dominating large-scale mixing pattern. We perform this comparison with several empirical networks, and identify numerous cases where the network's assortativity is exaggerated by traditional community detection methods, and we show how a more faithful degree of assortativity can be identified.

physics.soc-ph↗

arXiv2020

Null models for multi-optimized large-scale network structures

We study the emerging large-scale structures in networks subject to selective pressures that simultaneously drive towards higher modularity and robustness against random failures. We construct maximum-entropy null models that isolate the effects of the joint optimization on the network structure from any kind of evolutionary dynamics. Our analysis reveals a rich phase diagram of optimized structures, composed of many combinations of modular, core-periphery and bipartite patterns. Furthermore, we observe parameter regions where the simultaneous optimization can be either synergistic or antagonistic, with the improvement of one criterion directly aiding or hindering the other, respectively. Our results show how interactions between different selective pressures can be pivotal in determining the emerging network structure, and that these interactions can be captured by simple network models.

physics.soc-ph↗

arXiv2019

Limits and trade-offs of topological network robustness

We investigate the trade-off between the robustness against random and targeted removal of nodes from a network. To this end we utilize the stochastic block model to study ensembles of infinitely large networks with arbitrary large-scale structures. We present results from numerical two-objective optimization simulations for networks with various fixed mean degree and number of blocks. The results provide strong evidence that three different blocks are sufficient to realize the best trade-off between the two measures of robustness, i.e.\ to obtain the complete front of Pareto-optimal networks. For all values of the mean degree, a characteristic three block structure emerges over large parts of the Pareto-optimal front. This structure can be often characterized as a core-periphery structure, composed of a group of core nodes with high degree connected among themselves and to a periphery of low-degree nodes, in addition to a third group of nodes which is disconnected from the periphery, and weakly connected to the core. Only at both extremes of the Pareto-optimal front, corresponding to maximal robustness against random and targeted node removal, a two-block core-periphery structure or a one-block fully random network are found, respectively.

cs.NI↗

arXiv2019

Network reconstruction and community detection from dynamics

We present a scalable nonparametric Bayesian method to perform network reconstruction from observed functional behavior that at the same time infers the communities present in the network. We show that the joint reconstruction with community detection has a synergistic effect, where the edge correlations used to inform the existence of communities are also inherently used to improve the accuracy of the reconstruction which, in turn, can better inform the uncovering of communities. We illustrate the use of our method with observations arising from epidemic models and the Ising model, both on synthetic and empirical networks, as well as on data containing only functional information.

physics.soc-ph↗

arXiv2018

Reconstructing networks with unknown and heterogeneous errors

The vast majority of network datasets contains errors and omissions, although this is rarely incorporated in traditional network analysis. Recently, an increasing effort has been made to fill this methodological gap by developing network reconstruction approaches based on Bayesian inference. These approaches, however, rely on assumptions of uniform error rates and on direct estimations of the existence of each edge via repeated measurements, something that is currently unavailable for the majority of network data. Here we develop a Bayesian reconstruction approach that lifts these limitations by not only allowing for heterogeneous errors, but also for single edge measurements without direct error estimates. Our approach works by coupling the inference approach with structured generative network models, which enable the correlations between edges to be used as reliable uncertainty estimates. Although our approach is general, we focus on the stochastic block model as the basic generative process, from which efficient nonparametric inference can be performed, and yields a principled method to infer hierarchical community structure from noisy data. We demonstrate the efficacy of our approach with a variety of empirical and artificial networks.

cs.SI↗

arXiv2018

Nonparametric Bayesian inference of the microcanonical stochastic block model

A principled approach to characterize the hidden structure of networks is to formulate generative models, and then infer their parameters from data. When the desired structure is composed of modules or "communities", a suitable choice for this task is the stochastic block model (SBM), where nodes are divided into groups, and the placement of edges is conditioned on the group memberships. Here, we present a nonparametric Bayesian method to infer the modular structure of empirical networks, including the number of modules and their hierarchical organization. We focus on a microcanonical variant of the SBM, where the structure is imposed via hard constraints, i.e. the generated networks are not allowed to violate the patterns imposed by the model. We show how this simple model variation allows simultaneously for two important improvements over more traditional inference approaches: 1. Deeper Bayesian hierarchies, with noninformative priors replaced by sequences of priors and hyperpriors, that not only remove limitations that seriously degrade the inference on large networks, but also reveal structures at multiple scales; 2. A very efficient inference algorithm that scales well not only for networks with a large number of nodes and edges, but also with an unlimited number of modules. We show also how this approach can be used to sample modular hierarchies from the posterior distribution, as well as to perform model selection. We discuss and analyze the differences between sampling from the posterior and simply finding the single parameter estimate that maximizes it. Furthermore, we expose a direct equivalence between our microcanonical approach and alternative derivations based on the canonical SBM.

physics.data-an↗

arXiv2018

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 which infers the latent topical structure of a collection of documents. Despite their success --- in particular of its 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, e.g. 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. Here we obtain a fresh view on the problem of identifying topical structures by relating it to the problem of finding communities in complex networks. This is achieved by representing text corpora as bipartite networks of documents and words. By adapting existing community-detection methods -- using a stochastic block model (SBM) with non-parametric priors -- we obtain a more versatile and principled framework for topic modeling (e.g., 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. More importantly, our work shows how to formally relate methods from community detection and topic modeling, opening the possibility of cross-fertilization between these two fields.

stat.ML↗

arXiv2018

Consistencies and inconsistencies between model selection and link prediction in networks

A principled approach to understand network structures is to formulate generative models. Given a collection of models, however, an outstanding key task is to determine which one provides a more accurate description of the network at hand, discounting statistical fluctuations. This problem can be approached using two principled criteria that at first may seem equivalent: selecting the most plausible model in terms of its posterior probability; or selecting the model with the highest predictive performance in terms of identifying missing links. Here we show that while these two approaches yield consistent results in most of cases, there are also notable instances where they do not, that is, where the most plausible model is not the most predictive. We show that in the latter case the improvement of predictive performance can in fact lead to overfitting both in artificial and empirical settings. Furthermore, we show that, in general, the predictive performance is higher when we average over collections of models that are individually less plausible, than when we consider only the single most plausible model.

stat.ML↗

arXiv2018

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.

stat.ML↗

arXiv2017

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.

physics.soc-ph↗

arXiv2017

Modeling sequences and temporal networks with dynamic community structures

In evolving complex systems such as air traffic and social organizations, 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.

cs.SI↗

arXiv2016

Disease Localization in Multilayer Networks

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the SIS and SIR dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasi-stationary state method we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: if the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we verified the barrier effect, i.e., for three-layer configuration, when the layer with the largest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems opening new possibilities for the study of spreading processes.

physics.soc-ph↗

arXiv2016

Network structure, metadata and the prediction of missing nodes and annotations

The empirical validation of community detection methods is often based on available annotations on the nodes that serve as putative indicators of the large-scale network structure. Most often, the suitability of the annotations as topological descriptors itself is not assessed, and without this it is not possible to ultimately distinguish between actual shortcomings of the community detection algorithms on one hand, and the incompleteness, inaccuracy or structured nature of the data annotations themselves on the other. In this work we present a principled method to access both aspects simultaneously. We construct a joint generative model for the data and metadata, and a nonparametric Bayesian framework to infer its parameters from annotated datasets. We assess the quality of the metadata not according to its direct alignment with the network communities, but rather in its capacity to predict the placement of edges in the network. We also show how this feature can be used to predict the connections to missing nodes when only the metadata is available, as well as missing metadata. By investigating a wide range of datasets, we show that while there are seldom exact agreements between metadata tokens and the inferred data groups, the metadata is often informative of the network structure nevertheless, and can improve the prediction of missing nodes. This shows that the method uncovers meaningful patterns in both the data and metadata, without requiring or expecting a perfect agreement between the two.

physics.soc-ph↗

arXiv2015

Sampling motif-constrained ensembles of networks

The statistical significance of network properties is conditioned on null models which satisfy spec- ified properties but that are otherwise random. Exponential random graph models are a principled theoretical framework to generate such constrained ensembles, but which often fail in practice, either due to model inconsistency, or due to the impossibility to sample networks from them. These problems affect the important case of networks with prescribed clustering coefficient or number of small connected subgraphs (motifs). In this paper we use the Wang-Landau method to obtain a multicanonical sampling that overcomes both these problems. We sample, in polynomial time, net- works with arbitrary degree sequences from ensembles with imposed motifs counts. Applying this method to social networks, we investigate the relation between transitivity and homophily, and we quantify the correlation between different types of motifs, finding that single motifs can explain up to 60% of the variation of motif profiles.

physics.soc-ph↗

arXiv2015

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

Many network systems are composed of interdependent but distinct types of interactions, which cannot be fully understood in isolation. These different types of interactions are often represented as layers, attributes on the edges or as a time-dependence of the network structure. Although they are crucial for a more comprehensive scientific understanding, these representations offer substantial challenges. Namely, it is an open problem how to precisely characterize the large or mesoscale structure of network systems in relation to these additional aspects. Furthermore, the direct incorporation of these features invariably increases the effective dimension of the network description, and hence aggravates the problem of overfitting, i.e. the use of overly-complex characterizations that mistake purely random fluctuations for actual structure. In this work, we propose a robust and principled method to tackle these problems, by constructing generative models of modular network structure, incorporating layered, attributed and time-varying properties, as well as a nonparametric Bayesian methodology to infer the parameters from data and select the most appropriate model according to statistical evidence. We show that the method is capable of revealing hidden structure in layered, edge-valued and time-varying networks, and that the most appropriate level of granularity with respect to the additional dimensions can be reliably identified. We illustrate our approach on a variety of empirical systems, including a social network of physicians, the voting correlations of deputies in the Brazilian national congress, the global airport network, and a proximity network of high-school students.

physics.soc-ph↗

arXiv2015

Generalized communities in networks

A substantial volume of research has been devoted to studies of community structure in networks, but communities are not the only possible form of large-scale network structure. Here we describe a broad extension of community structure that encompasses traditional communities but includes a wide range of generalized structural patterns as well. We describe a principled method for detecting this generalized structure in empirical network data and demonstrate with real-world examples how it can be used to learn new things about the shape and meaning of networks.

cs.SI↗

arXiv2015

Model selection and hypothesis testing for large-scale network models with overlapping groups

The effort to understand network systems in increasing detail has resulted in a diversity of methods designed to extract their large-scale structure from data. Unfortunately, many of these methods yield diverging descriptions of the same network, making both the comparison and understanding of their results a difficult challenge. A possible solution to this outstanding issue is to shift the focus away from ad hoc methods and move towards more principled approaches based on statistical inference of generative models. As a result, we face instead the more well-defined task of selecting between competing generative processes, which can be done under a unified probabilistic framework. Here, we consider the comparison between a variety of generative models including features such as degree correction, where nodes with arbitrary degrees can belong to the same group, and community overlap, where nodes are allowed to belong to more than one group. Because such model variants possess an increasing number of parameters, they become prone to overfitting. In this work, we present a method of model selection based on the minimum description length criterion and posterior odds ratios that is capable of fully accounting for the increased degrees of freedom of the larger models, and selects the best one according to the statistical evidence available in the data. In applying this method to many empirical unweighted networks from different fields, we observe that community overlap is very often not supported by statistical evidence and is selected as a better model only for a minority of them. On the other hand, we find that degree correction tends to be almost universally favored by the available data, implying that intrinsic node proprieties (as opposed to group properties) are often an essential ingredient of network formation.

physics.data-an↗