Normalized mutual information is widely used as a similarity measure for evaluating the performance of clustering and classification algorithms. In this paper, we argue that results returned by the normalized mutual information are biased for two reasons: first, because they ignore the information content of the contingency table and, second, because their symmetric normalization introduces spurious dependence on algorithm output. We introduce a modified version of the mutual information that remedies both of these shortcomings. As a practical demonstration of the importance of using an unbiased measure, we perform extensive numerical tests on a basket of popular algorithms for network community detection and show that one's conclusions about which algorithm is best are significantly affected by the biases in the traditional mutual information.

Patterns of wins and lo sses in pairwise contests, such as occur in sports and games, consumer research and paired comparison studies, and human and animal social hierarchies, are commonly analyzed using probabilistic models that allow one to quantify the strength of competitors or predict the outcome of future contests. Here, we generalize this approach to incorporate two additional features: an element of randomness or luck that leads to upset wins, and a "depth of competition" variable that measures the complexity of a game or hierarchy. Fitting the resulting model, we estimate depth and luck in a range of games, sports, and social situations. In general, we find that social competition tends to be "deep," meaning it has a pronounced hierarchy with many distinct levels, but also that there is often a nonzero chance of an upset victory. Competition in sports and games, by contrast, tends to be shallow, and in most cases, there is little evidence of upset wins.

Mutual information is commonly used as a measure of similarity between competing labelings of a given set of objects, for example to quantify performance in classification and community detection tasks. As argued recently, however, the mutual information as conventionally defined can return biased results because it neglects the information cost of the so-called contingency table, a crucial component of the similarity calculation. In principle the bias can be rectified by subtracting the appropriate information cost, leading to the modified measure known as the reduced mutual information, but in practice one can only ever compute an upper bound on this information cost, and the value of the reduced mutual information depends crucially on how good a bound is established. In this paper we describe an improved method for encoding contingency tables that gives a substantially better bound in typical use cases, and approaches the ideal value in the common case where the labelings are closely similar, as we demonstrate with extensive numerical results.

We study core-periphery structure in networks using inference methods based on a flexible network model that allows for traditional onionlike cores within cores, but also for hierarchical treelike structures and more general non-nested types of structures. We propose an efficient Monte Carlo scheme for fitting the model to observed networks and report results for a selection of real-world data sets. Among other things, we observe an empirical distinction between networks showing traditional core-periphery structure with a dense core weakly connected to a sparse periphery, and an alternative structure in which the core is strongly connected both within itself and to the periphery. Networks vary in whether they are better represented by one type of structure or the other. We also observe structures that are a hybrid between core-periphery structure and community structure, in which networks have a set of nonoverlapping cores that roughly correspond to communities, surrounded by a single undifferentiated periphery. Computer code implementing our methods is available.

We study the ranking of individuals, teams, or objects, based on pairwise comparisons between them, using the Bradley-Terry model. Estimates of rankings within this model are commonly made using a simple iterative algorithm first introduced by Zermelo almost a century ago. Here we describe an alternative and similarly simple iteration that provably returns identical results but does so much faster -- over a hundred times faster in some cases. We demonstrate this algorithm with applications to a range of example data sets and derive a number of results regarding its convergence.

Networks and network computations have become a primary mathematical tool for analyzing the structure of many kinds of complex systems, ranging from the Internet and transportation networks to biochemical interactions and social networks. A common task in network analysis is the calculation of quantities that reside on the nodes of a network, such as centrality measures, probabilities or model states. In this perspective article we discuss message passing methods, a family of techniques for performing such calculations, based on the propagation of information between the nodes of a network. We introduce the message passing approach with a series of examples, give some illustrative applications and results and discuss the deep connections between message passing and phase transitions in networks. We also point out some limitations of the message passing approach and describe some recently introduced methods that address these limitations.

A fundamental technical challenge in the analysis of network data is the automated discovery of communities - groups of nodes that are strongly connected or that share similar features or roles. In this commentary we review progress in the field over the last 20 years.

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.

The Border Gateway Protocol (BGP) is a distributed protocol that manages interdomain routing without requiring a centralized record of which autonomous systems (ASes) connect to which others. Many methods have been devised to infer the AS topology from publicly available BGP data, but none provide a general way to handle the fact that the data are notoriously incomplete and subject to error. This paper describes a method for reliably inferring AS-level connectivity in the presence of measurement error using Bayesian statistical inference acting on BGP routing tables from multiple vantage points. We employ a novel approach for counting AS adjacency observations in the AS-PATH attribute data from public route collectors, along with a Bayesian algorithm to generate a statistical estimate of the AS-level network. Our approach also gives us a way to evaluate the accuracy of existing reconstruction methods and to identify advantageous locations for new route collectors or vantage points.

Methods for detecting community structure in networks typically aim to identify a single best partition of network nodes into communities, often by optimizing some objective function, but in real-world applications there may be many competitive partitions with objective scores close to the global optimum and one can obtain a more informative picture of the community structure by examining a representative set of such high-scoring partitions than by looking at just the single optimum. However, such a set can be difficult to interpret since its size can easily run to hundreds or thousands of partitions. In this paper we present a method for analyzing large partition sets by dividing them into groups of similar partitions and then identifying an archetypal partition as a representative of each group. The resulting set of archetypal partitions provides a succinct, interpretable summary of the form and variety of community structure in any network. We demonstrate the method on a range of example networks.

The task of ranking individuals or teams, based on a set of comparisons between pairs, arises in various contexts, including sporting competitions and the analysis of dominance hierarchies among animals and humans. Given data on which competitors beat which others, the challenge is to rank the competitors from best to worst. Here we study the problem of computing rankings when there are multiple, potentially conflicting modes of comparison, such as multiple types of dominance behaviors among animals. We assume that we do not know a priori what information each behavior conveys about the ranking, or even whether they convey any information at all. Nonetheless we show that it is possible to compute a ranking in this situation and present a fast method for doing so, based on a combination of an expectation-maximization algorithm and a modified Bradley-Terry model. We give a selection of example applications to both animal and human competition.

The friendship paradox is the observation that the degrees of the neighbors of a node in any network will, on average, be greater than the degree of the node itself. In common parlance, your friends have more friends than you do. In this paper we develop the mathematical theory of the friendship paradox, both in general as well as for specific model networks, focusing not only on average behavior but also on variation about the average and using generating function methods to calculate full distributions of quantities of interest. We compare the predictions of our theory with measurements on a large number of real-world network data sets and find remarkably good agreement. We also develop equivalent theory for the generalized friendship paradox, which compares characteristics of nodes other than degree to those of their neighbors.

Empirical measurements of ecological networks such as food webs and mutualistic networks are often rich in structure but also noisy and error-prone, particularly for rare species for which observations are sparse. Focusing on the case of plant–pollinator networks, we here describe a Bayesian statistical technique that allows us to make accurate estimates of network structure and ecological metrics from such noisy observational data. Our method yields not only estimates of these quantities, but also estimates of their statistical errors, paving the way for principled statistical analyses of ecological variables and outcomes. We demonstrate the use of the method with an application to previously published data on plant–pollinator networks in the Seychelles archipelago and Kosciusko National Park, calculating estimates of network structure, network nestedness, and other characteristics.

Belief propagation is a widely used message passing method for the solution of probabilistic models on networks such as epidemic models, spin models, and Bayesian graphical models, but it suffers from the serious shortcoming that it works poorly in the common case of networks that contain short loops. Here, we provide a solution to this long-standing problem, deriving a belief propagation method that allows for fast calculation of probability distributions in systems with short loops, potentially with high density, as well as giving expressions for the entropy and partition function, which are notoriously difficult quantities to compute. Using the Ising model as an example, we show that our approach gives excellent results on both real and synthetic networks, improving substantially on standard message passing methods. We also discuss potential applications of our method to a variety of other problems.

Most empirical studies of complex networks do not return direct, error-free measurements of network structure. Instead, they typically rely on indirect measurements that are often error prone and unreliable. A fundamental problem in empirical network science is how to make the best possible estimates of network structure given such unreliable data. In this article, we describe a fully Bayesian method for reconstructing networks from observational data in any format, even when the data contain substantial measurement error and when the nature and magnitude of that error is unknown. The method is introduced through pedagogical case studies using real-world example networks, and specifically tailored to allow straightforward, computationally efficient implementation with a minimum of technical input. Computer code implementing the method is publicly available.

The information theoretic measure known as mutual information is widely used as a way to quantify the similarity of two different labelings or divisions of the same set of objects, such as arises, for instance, in clustering and classification problems in machine learning or community detection problems in network science. Here we argue that the standard mutual information, as commonly defined, omits a crucial term which can become large under real-world conditions, producing results that can be substantially in error. We derive an expression for this missing term and hence write a corrected mutual information that gives accurate results even in cases where the standard measure fails. We discuss practical implementation of the new measure and give example applications.

In this paper, we extend a recently proposed framework for message passing on loopy networks to the solution of probabilistic models. We derive a self-consistent set of message passing equations that allow for fast computation of probability distributions in systems that contain short loops, potentially with high density, as well as expressions for the entropy and partition function of such systems, which are notoriously difficult quantities to compute. Using the Ising model as an example, we show that our solutions are asymptotically exact on certain classes of networks with short loops and offer a good approximation on more general networks, improving significantly on results derived from standard belief propagation. We also discuss potential applications of our method to a variety of other problems.

The incorporation of microeconomics concepts into studies using self-administration procedures has provided critical insights into the factors that influence consumption of a wide range of food and drug reinforcers. In particular, the fitting of demand curves to consumption data provides a powerful analytic tool for computing objective metrics of behavior that can be compared across a wide range of reward types in both human and animal experiments. The results of these analyses depend crucially on the mathematical form used to fit the data. The most common choice is an exponential form proposed by Hursh and Silberberg, which is widely used and has provided fundamental insights into relationships between cost and consumption, but it also has some disadvantages. In this paper, we first briefly review the use of demand curves to quantify the motivating effects of food and drugs, then we describe the current methodology and highlight some potential issues that arise in its application. To address these issues, we propose a new mathematical framework for the analysis of consumption data, including a new functional form for the demand curve. We show that this proposed form gives good fits to data for a range of different reinforcers and experimental protocols, while allowing for straightforward calculation of key metrics of demand, including preferred consumption level, maximum response, price at maximum response, and price elasticity of demand. We provide software implementing our entire analysis pipeline, including data fits, data visualization, and the calculation of demand metrics.

We study the structure of heterosexual dating markets in the United States through an analysis of the interactions of several million users of a large online dating website, applying recently developed network analysis methods to the pattern of messages exchanged among users. Our analysis shows that the strongest driver of romantic interaction at the national level is simple geographic proximity, but at the local level, other demographic factors come into play. We find that dating markets in each city are partitioned into submarkets along lines of age and ethnicity. Sex ratio varies widely between submarkets, with younger submarkets having more men and fewer women than older ones. There is also a noticeable tendency for minorities, especially women, to be younger than the average in older submarkets, and our analysis reveals how this kind of racial stratification arises through the messaging decisions of both men and women. Our study illustrates how network techniques applied to online interactions can reveal the aggregate effects of individual behavior on social structure.