Aaron Clauset

Upasana Dutta, Alexander Ray, Aaron Clauset

How does the shape of a network change as its size increases? Although random graph models provide some expectations for such "scaling behaviors" in the structure of networks, relatively little is known about how empirical network structure scales with network size or how well random graphs explain those empirical patterns. Using a large, structurally diverse corpus of networks from four scientific domains, we first characterize the empirical scaling laws of real-world networks, considering how mean degree, transitivity, mean geodesic distance, and degree assortativity vary with network size. We show that networks from all four scientific domains exhibit a consistent set of scaling laws on these measures of network structure, but with differing scaling rates. We then assess the extent to which these empirical scaling laws are explained by three random graph models with different structural assumptions, showing that configuration model random graphs are a remarkably good model of network scaling behavior, although null models with modular structure are slightly better. These findings identify a new set of common patterns in the network structure of complex systems, provide new validation targets for models of network structure, and shed new light on the role of randomness in shaping the large-scale structure of networks.

Bisman Singh, Lucy Van Kleunen, Aaron Clauset

Relational data are ubiquitous in real-world data applications, e.g., in social network analysis or biological modeling, but networks are nearly always incompletely observed. The state-of-the-art for predicting missing links in the hard case of a network without node attributes uses model stacking or neural network techniques. It remains unknown which approach is best, and whether or how the best choice of algorithm depends on the input network's characteristics. We answer these questions systematically using a large, structurally diverse benchmark of 550 real-world networks under two standard accuracy measures (AUC and Top-k), comparing four stacking algorithms with 42 topological link predictors, two of which we introduce here, and two graph neural network algorithms. We show that no algorithm is best across all input networks, all algorithms perform well on most social networks, and few perform well on economic and biological networks. Overall, model stacking with a random forest is both highly scalable and surpasses on AUC or is competitive with graph neural networks on Top-k accuracy. But, algorithm performance depends strongly on network characteristics like the degree distribution, triangle density, and degree assortativity. We introduce a meta-learning algorithm that exploits this variability to optimize link predictions for individual networks by selecting the best algorithm to apply, which we show outperforms all state-of-the-art algorithms and scales to large networks.

Homa Hosseinmardi, Amir Ghasemian, Aaron Clauset et al.

Although it is under-studied relative to other social media platforms, YouTube is arguably the largest and most engaging online media consumption platform in the world. Recently, YouTube's scale has fueled concerns that YouTube users are being radicalized via a combination of biased recommendations and ostensibly apolitical anti-woke channels, both of which have been claimed to direct attention to radical political content. Here we test this hypothesis using a representative panel of more than 300,000 Americans and their individual-level browsing behavior, on and off YouTube, from January 2016 through December 2019. Using a labeled set of political news channels, we find that news consumption on YouTube is dominated by mainstream and largely centrist sources. Consumers of far-right content, while more engaged than average, represent a small and stable percentage of news consumers. However, consumption of anti-woke content, defined in terms of its opposition to progressive intellectual and political agendas, grew steadily in popularity and is correlated with consumption of far-right content off-platform. We find no evidence that engagement with far-right content is caused by YouTube recommendations systematically, nor do we find clear evidence that anti-woke channels serve as a gateway to the far right. Rather, consumption of political content on YouTube appears to reflect individual preferences that extend across the web as a whole.

Amir Ghasemian, Homa Hosseinmardi, Aram Galstyan et al.

Most real-world networks are incompletely observed. Algorithms that can accurately predict which links are missing can dramatically speedup the collection of network data and improve the validity of network models. Many algorithms now exist for predicting missing links, given a partially observed network, but it has remained unknown whether a single best predictor exists, how link predictability varies across methods and networks from different domains, and how close to optimality current methods are. We answer these questions by systematically evaluating 203 individual link predictor algorithms, representing three popular families of methods, applied to a large corpus of 548 structurally diverse networks from six scientific domains. We first show that individual algorithms exhibit a broad diversity of prediction errors, such that no one predictor or family is best, or worst, across all realistic inputs. We then exploit this diversity via meta-learning to construct a series of "stacked" models that combine predictors into a single algorithm. Applied to a broad range of synthetic networks, for which we may analytically calculate optimal performance, these stacked models achieve optimal or nearly optimal levels of accuracy. Applied to real-world networks, stacked models are also superior, but their accuracy varies strongly by domain, suggesting that link prediction may be fundamentally easier in social networks than in biological or technological networks. These results indicate that the state-of-the-art for link prediction comes from combining individual algorithms, which achieves nearly optimal predictions. We close with a brief discussion of limitations and opportunities for further improvement of these results.

Amir Ghasemian, Homa Hosseinmardi, Aaron Clauset

A common data mining task on networks is community detection, which seeks an unsupervised decomposition of a network into structural groups based on statistical regularities in the network's connectivity. Although many methods exist, the No Free Lunch theorem for community detection implies that each makes some kind of tradeoff, and no algorithm can be optimal on all inputs. Thus, different algorithms will over or underfit on different inputs, finding more, fewer, or just different communities than is optimal, and evaluation methods that use a metadata partition as a ground truth will produce misleading conclusions about general accuracy. Here, we present a broad evaluation of over and underfitting in community detection, comparing the behavior of 16 state-of-the-art community detection algorithms on a novel and structurally diverse corpus of 406 real-world networks. We find that (i) algorithms vary widely both in the number of communities they find and in their corresponding composition, given the same input, (ii) algorithms can be clustered into distinct high-level groups based on similarities of their outputs on real-world networks, and (iii) these differences induce wide variation in accuracy on link prediction and link description tasks. We introduce a new diagnostic for evaluating overfitting and underfitting in practice, and use it to roughly divide community detection methods into general and specialized learning algorithms. Across methods and inputs, Bayesian techniques based on the stochastic block model and a minimum description length approach to regularization represent the best general learning approach, but can be outperformed under specific circumstances. These results introduce both a theoretically principled approach to evaluate over and underfitting in models of network community structure and a realistic benchmark by which new methods may be evaluated and compared.

Kansuke Ikehara, Aaron Clauset

The structure of complex networks has been of interest in many scientific and engineering disciplines over the decades. A number of studies in the field have been focused on finding the common properties among different kinds of networks such as heavy-tail degree distribution, small-worldness and modular structure and they have tried to establish a theory of structural universality in complex networks. However, there is no comprehensive study of network structure across a diverse set of domains in order to explain the structural diversity we observe in the real-world networks. In this paper, we study 986 real-world networks of diverse domains ranging from ecological food webs to online social networks along with 575 networks generated from four popular network models. Our study utilizes a number of machine learning techniques such as random forest and confusion matrix in order to show the relationships among network domains in terms of network structure. Our results indicate that there are some partitions of network categories in which networks are hard to distinguish based purely on network structure. We have found that these partitions of network categories tend to have similar underlying functions, constraints and/or generative mechanisms of networks even though networks in the same partition have different origins, e.g., biological processes, results of engineering by human being, etc. This suggests that the origin of a network, whether it's biological, technological or social, may not necessarily be a decisive factor of the formation of similar network structure. Our findings shed light on the possible direction along which we could uncover the hidden principles for the structural diversity of complex networks.

Leto Peel, Daniel B. Larremore, Aaron Clauset

Across many scientific domains, there is a common need to automatically extract a simplified view or coarse-graining of how a complex system's components interact. This general task is called community detection in networks and is analogous to searching for clusters in independent vector data. It is common to evaluate the performance of community detection algorithms by their ability to find so-called "ground truth" communities. This works well in synthetic networks with planted communities because such networks' links are formed explicitly based on those known communities. However, there are no planted communities in real world networks. Instead, it is standard practice to treat some observed discrete-valued node attributes, or metadata, as ground truth. Here, we show that metadata are not the same as ground truth, and that treating them as such induces severe theoretical and practical problems. We prove that no algorithm can uniquely solve community detection, and we prove a general No Free Lunch theorem for community detection, which implies that there can be no algorithm that is optimal for all possible community detection tasks. However, community detection remains a powerful tool and node metadata still have value so a careful exploration of their relationship with network structure can yield insights of genuine worth. We illustrate this point by introducing two statistical techniques that can quantify the relationship between metadata and community structure for a broad class of models. We demonstrate these techniques using both synthetic and real-world networks, and for multiple types of metadata and community structure.

M. E. J. Newman, Aaron Clauset

For many networks of scientific interest we know both the connections of the network and information about the network nodes, such as the age or gender of individuals in a social network, geographic location of nodes in the Internet, or cellular function of nodes in a gene regulatory network. Here we demonstrate how this "metadata" can be used to improve our analysis and understanding of network structure. We focus in particular on the problem of community detection in networks and develop a mathematically principled approach that combines a network and its metadata to detect communities more accurately than can be done with either alone. Crucially, the method does not assume that the metadata are correlated with the communities we are trying to find. Instead the method learns whether a correlation exists and correctly uses or ignores the metadata depending on whether they contain useful information. The learned correlations are also of interest in their own right, allowing us to make predictions about the community membership of nodes whose network connections are unknown. We demonstrate our method on synthetic networks with known structure and on real-world networks, large and small, drawn from social, biological, and technological domains.

Amir Ghasemian, Pan Zhang, Aaron Clauset et al.

We study the fundamental limits on learning latent community structure in dynamic networks. Specifically, we study dynamic stochastic block models where nodes change their community membership over time, but where edges are generated independently at each time step. In this setting (which is a special case of several existing models), we are able to derive the detectability threshold exactly, as a function of the rate of change and the strength of the communities. Below this threshold, we claim that no algorithm can identify the communities better than chance. We then give two algorithms that are optimal in the sense that they succeed all the way down to this limit. The first uses belief propagation (BP), which gives asymptotically optimal accuracy, and the second is a fast spectral clustering algorithm, based on linearizing the BP equations. We verify our analytic and algorithmic results via numerical simulation, and close with a brief discussion of extensions and open questions.

Abigail Z. Jacobs, Aaron Clauset

Research on probabilistic models of networks now spans a wide variety of fields, including physics, sociology, biology, statistics, and machine learning. These efforts have produced a diverse ecology of models and methods. Despite this diversity, many of these models share a common underlying structure: pairwise interactions (edges) are generated with probability conditional on latent vertex attributes. Differences between models generally stem from different philosophical choices about how to learn from data or different empirically-motivated goals. The highly interdisciplinary nature of work on these generative models, however, has inhibited the development of a unified view of their similarities and differences. For instance, novel theoretical models and optimization techniques developed in machine learning are largely unknown within the social and biological sciences, which have instead emphasized model interpretability. Here, we describe a unified view of generative models for networks that draws together many of these disparate threads and highlights the fundamental similarities and differences that span these fields. We then describe a number of opportunities and challenges for future work that are revealed by this view.

Christopher Aicher, Abigail Z. Jacobs, Aaron Clauset

Community detection is an important task in network analysis, in which we aim to learn a network partition that groups together vertices with similar community-level connectivity patterns. By finding such groups of vertices with similar structural roles, we extract a compact representation of the network's large-scale structure, which can facilitate its scientific interpretation and the prediction of unknown or future interactions. Popular approaches, including the stochastic block model, assume edges are unweighted, which limits their utility by throwing away potentially useful information. We introduce the `weighted stochastic block model' (WSBM), which generalizes the stochastic block model to networks with edge weights drawn from any exponential family distribution. This model learns from both the presence and weight of edges, allowing it to discover structure that would otherwise be hidden when weights are discarded or thresholded. We describe a Bayesian variational algorithm for efficiently approximating this model's posterior distribution over latent block structures. We then evaluate the WSBM's performance on both edge-existence and edge-weight prediction tasks for a set of real-world weighted networks. In all cases, the WSBM performs as well or better than the best alternatives on these tasks.

Daniel B. Larremore, Aaron Clauset, Abigail Z. Jacobs

Bipartite networks are a common type of network data in which there are two types of vertices, and only vertices of different types can be connected. While bipartite networks exhibit community structure like their unipartite counterparts, existing approaches to bipartite community detection have drawbacks, including implicit parameter choices, loss of information through one-mode projections, and lack of interpretability. Here we solve the community detection problem for bipartite networks by formulating a bipartite stochastic block model, which explicitly includes vertex type information and may be trivially extended to $k$-partite networks. This bipartite stochastic block model yields a projection-free and statistically principled method for community detection that makes clear assumptions and parameter choices and yields interpretable results. We demonstrate this model's ability to efficiently and accurately find community structure in synthetic bipartite networks with known structure and in real-world bipartite networks with unknown structure, and we characterize its performance in practical contexts.

Leto Peel, Aaron Clauset

Interactions among people or objects are often dynamic in nature and can be represented as a sequence of networks, each providing a snapshot of the interactions over a brief period of time. An important task in analyzing such evolving networks is change-point detection, in which we both identify the times at which the large-scale pattern of interactions changes fundamentally and quantify how large and what kind of change occurred. Here, we formalize for the first time the network change-point detection problem within an online probabilistic learning framework and introduce a method that can reliably solve it. This method combines a generalized hierarchical random graph model with a Bayesian hypothesis test to quantitatively determine if, when, and precisely how a change point has occurred. We analyze the detectability of our method using synthetic data with known change points of different types and magnitudes, and show that this method is more accurate than several previously used alternatives. Applied to two high-resolution evolving social networks, this method identifies a sequence of change points that align with known external "shocks" to these networks.

Christopher Aicher, Abigail Z. Jacobs, Aaron Clauset

We generalize the stochastic block model to the important case in which edges are annotated with weights drawn from an exponential family distribution. This generalization introduces several technical difficulties for model estimation, which we solve using a Bayesian approach. We introduce a variational algorithm that efficiently approximates the model's posterior distribution for dense graphs. In specific numerical experiments on edge-weighted networks, this weighted stochastic block model outperforms the common approach of first applying a single threshold to all weights and then applying the classic stochastic block model, which can obscure latent block structure in networks. This model will enable the recovery of latent structure in a broader range of network data than was previously possible.

Sears Merritt, Abigail Z. Jacobs, Winter Mason et al.

In many complex social systems, the timing and frequency of interactions between individuals are observable but friendship ties are hidden. Recovering these hidden ties, particularly for casual users who are relatively less active, would enable a wide variety of friendship-aware applications in domains where labeled data are often unavailable, including online advertising and national security. Here, we investigate the accuracy of multiple statistical features, based either purely on temporal interaction patterns or on the cooperative nature of the interactions, for automatically extracting latent social ties. Using self-reported friendship and non-friendship labels derived from an anonymous online survey, we learn highly accurate predictors for recovering hidden friendships within a massive online data set encompassing 18 billion interactions among 17 million individuals of the popular online game Halo: Reach. We find that the accuracy of many features improves as more data accumulates, and cooperative features are generally reliable. However, periodicities in interaction time series are sufficient to correctly classify 95% of ties, even for casual users. These results clarify the nature of friendship in online social environments and suggest new opportunities and new privacy concerns for friendship-aware applications that do not require the disclosure of private friendship information.

Aaron Clauset, Ryan Woodard

Quantities with right-skewed distributions are ubiquitous in complex social systems, including political conflict, economics and social networks, and these systems sometimes produce extremely large events. For instance, the 9/11 terrorist events produced nearly 3000 fatalities, nearly six times more than the next largest event. But, was this enormous loss of life statistically unlikely given modern terrorism's historical record? Accurately estimating the probability of such an event is complicated by the large fluctuations in the empirical distribution's upper tail. We present a generic statistical algorithm for making such estimates, which combines semi-parametric models of tail behavior and a nonparametric bootstrap. Applied to a global database of terrorist events, we estimate the worldwide historical probability of observing at least one 9/11-sized or larger event since 1968 to be 11-35%. These results are robust to conditioning on global variations in economic development, domestic versus international events, the type of weapon used and a truncated history that stops at 1998. We then use this procedure to make a data-driven statistical forecast of at least one similar event over the next decade.

Winter Mason, Aaron Clauset

How important are friendships in determining success by individuals and teams in complex collaborative environments? By combining a novel data set containing the dynamics of millions of ad hoc teams from the popular multiplayer online first person shooter Halo: Reach with survey data on player demographics, play style, psychometrics and friendships derived from an anonymous online survey, we investigate the impact of friendship on collaborative and competitive performance. In addition to finding significant differences in player behavior across these variables, we find that friendships exert a strong influence, leading to both improved individual and team performance--even after controlling for the overall expertise of the team--and increased pro-social behaviors. Players also structure their in-game activities around social opportunities, and as a result hidden friendship ties can be accurately inferred directly from behavioral time series. Virtual environments that enable such friendship effects will thus likely see improved collaboration and competition.