To comprehend the multipartite organization of large-scale biological and social systems, we introduce an information theoretic approach that reveals community structure in weighted and directed networks. We use the probability flow of random walks on a network as a proxy for information flows in the real system and decompose the network into modules by compressing a description of the probability flow. The result is a map that both simplifies and highlights the regularities in the structure and their relationships. We illustrate the method by making a map of scientific communication as captured in the citation patterns of >6,000 journals. We discover a multicentric organization with fields that vary dramatically in size and degree of integration into the network of science. Along the backbone of the network-including physics, chemistry, molecular biology, and medicine-information flows bidirectionally, but the map reveals a directional pattern of citation from the applied fields to the basic sciences.clustering ͉ compression ͉ information theory ͉ map of science ͉ bibiometrics B iological and social systems are differentiated, multipartite, integrated, and dynamic. Data about these systems, now available on unprecedented scales, often are schematized as networks. Such abstractions are powerful (1, 2), but even as abstractions they remain highly complex. It therefore is helpful to decompose the myriad nodes and links into modules that represent the network (3-5). A cogent representation will retain the important information about the network and reflect the fact that interactions between the elements in complex systems are weighted, directional, interdependent, and conductive. Good representations both simplify and highlight the underlying structures and the relationships that they depict; they are maps (6, 7).To create a good map, the cartographer must attain a fine balance between omitting important structures by oversimplification and obscuring significant relationships in a barrage of superfluous detail. The best maps convey a great deal of information but require minimal bandwidth: the best maps are also good compressions. By adopting an information-theoretic approach, we can measure how efficiently a map represents the underlying geography, and we can measure how much detail is lost in the process of simplification, which allows us to quantify and resolve the cartographer's tradeoff. Network Maps and Coding TheoryIn this article, we use maps to describe the dynamics across the links and nodes in directed, weighted networks that represent the local interactions among the subunits of a system. These local interactions induce a system-wide flow of information that characterizes the behavior of the full system (8-12). Consequently, if we want to understand how network structure relates to system behavior, we need to understand the flow of information on the network. We therefore identify the modules that compose the network by finding an efficiently coarse-grained description of how information flows on the ne...

BACKGROUND The increasing availability of digital data on scholarly inputs and outputs—from research funding, productivity, and collaboration to paper citations and scientist mobility—offers unprecedented opportunities to explore the structure and evolution of science. The science of science (SciSci) offers a quantitative understanding of the interactions among scientific agents across diverse geographic and temporal scales: It provides insights into the conditions underlying creativity and the genesis of scientific discovery, with the ultimate goal of developing tools and policies that have the potential to accelerate science. In the past decade, SciSci has benefited from an influx of natural, computational, and social scientists who together have developed big data–based capabilities for empirical analysis and generative modeling that capture the unfolding of science, its institutions, and its workforce. The value proposition of SciSci is that with a deeper understanding of the factors that drive successful science, we can more effectively address environmental, societal, and technological problems. ADVANCES Science can be described as a complex, self-organizing, and evolving network of scholars, projects, papers, and ideas. This representation has unveiled patterns characterizing the emergence of new scientific fields through the study of collaboration networks and the path of impactful discoveries through the study of citation networks. Microscopic models have traced the dynamics of citation accumulation, allowing us to predict the future impact of individual papers. SciSci has revealed choices and trade-offs that scientists face as they advance both their own careers and the scientific horizon. For example, measurements indicate that scholars are risk-averse, preferring to study topics related to their current expertise, which constrains the potential of future discoveries. Those willing to break this pattern engage in riskier careers but become more likely to make major breakthroughs. Overall, the highest-impact science is grounded in conventional combinations of prior work but features unusual combinations. Last, as the locus of research is shifting into teams, SciSci is increasingly focused on the impact of team research, finding that small teams tend to disrupt science and technology with new ideas drawing on older and less prevalent ones. In contrast, large teams tend to develop recent, popular ideas, obtaining high, but often short-lived, impact. OUTLOOK SciSci offers a deep quantitative understanding of the relational structure between scientists, institutions, and ideas because it facilitates the identification of fundamental mechanisms responsible for scientific discovery. These interdisciplinary data-driven efforts complement contributions from related fields such as sciento-metrics and the economics and sociology of science. Although SciSci seeks long-standing universal laws and mechanisms that apply across various fields of science, a fundamental challenge going forward is accounting for un...

Many real-world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent. As the tools for doing these simplifications proliferate within the network literature, researchers would benefit from some guidelines about which of the so-called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking. Here we show that different methods highlight different aspects of a network's structure and that the the sort of information that we seek to extract about the system must guide us in our decision. For example, many community detection algorithms, including the popular modularity maximization approach, infer module assignments from an underlying model of the network formation process. However, we are not always as interested in how a system's network structure was formed, as we are in how a network's extant structure influences the system's behavior. To see how structure influences current behavior, we will recognize that links in a network induce movement across the network and result in systemwide interdependence. In doing so, we explicitly acknowledge that most networks carry flow. To highlight and simplify the network structure with respect to this flow, we use the map equation. We present an intuitive derivation of this flow-based and information-theoretic method and provide an interactive on-line application that anyone can use to explore the mechanics of the map equation. The differences between the map equation and the modularity maximization approach are not merely conceptual. Because the map equation attends to patterns of flow on the network and the modularity maximization approach does not, the two methods can yield dramatically different results for some network structures. To illustrate this and build our understanding of each method, we partition several sample networks. We also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation.

Gender disparities appear to be decreasing in academia according to a number of metrics, such as grant funding, hiring, acceptance at scholarly journals, and productivity, and it might be tempting to think that gender inequity will soon be a problem of the past. However, a large-scale analysis based on over eight million papers across the natural sciences, social sciences, and humanities reveals a number of understated and persistent ways in which gender inequities remain. For instance, even where raw publication counts seem to be equal between genders, close inspection reveals that, in certain fields, men predominate in the prestigious first and last author positions. Moreover, women are significantly underrepresented as authors of single-authored papers. Academics should be aware of the subtle ways that gender disparities can occur in scholarly authorship.

whose fates have been followed since 1971. Tissue sampling for DNA analyses started in 1988. Blood samples were taken from all captured sheep until 1993 and stored in preservative at 220 8C. Sampling resumed in 1997, when hair samples were taken from all captured sheep by plucking 50-100 hairs including roots from the back or flank. Hairs were kept either in paper envelopes or plastic bags containing about 5 g of silica at room temperature. From 1998 to 2002, a tissue sample from each captured sheep was taken from the ear with an 8-mm punch. Ear tissue was kept at 220 8C in a solution of 20% dimethylsulphoxide saturated with NaCl. We sampled 433 marked individuals over the course of the study.DNA was extracted from blood with a standard phenol-chloroform method, and from either 20-30 hairs including follicles or about 5 mg of ear tissue, using the QIAamp tissue extraction kit (Qiagen Inc., Mississauga, Ontario). Polymerase chain reaction amplification at 20 ungulate-derived microsatellite loci, 15 as described previously 5 plus MCM527, BM4025, MAF64, OarFCB193 and MAF92 (refs 25, 26), and fragment analysis were performed as described elsewhere 5 . After correction for multiple comparisons, we found no evidence for allelic or genotypic disequilibria at or among these 20 loci.Paternity of 241 individuals was assigned by using the likelihood-based approach described in CERVUS 27 at a confidence level of more than 95% with input parameters given in ref. 5. After paternity analysis, we used KINSHIP 28 to identify 31 clusters of 104 paternal half-sibs among the unassigned offspring. A paternal half-sibship consisted of all pairs of individuals of unassigned paternity that were identified in the KINSHIP analysis as having a likelihood ratio of the probability of a paternal half-sib relationship versus unrelated with an associated P , 0.05 (ref. 28). Members of reconstructed paternal halfsibships were assigned a common unknown paternal identity for the animal model analyses. Paternal identity links in the pedigree were therefore defined for 345 individuals. Animal model analysesBreeding values, genetic variance components and heritabilities were estimated by using a multiple trait restricted-estimate maximum-likelihood (REML) model implemented by the programs PEST 29 and VCE 30 . An animal model was fitted in which the phenotype of each animal was broken down into components of additive genetic value and other random and fixed effects: y ¼ Xb þ Za þ Pc þ e, where y was a vector of phenotypic values, b was a vector of fixed effects, a and c were vectors of additive genetic and permanent environmental, e was a vector of residual values, and X, Z and P were the corresponding design matrices relating records to the appropriate fixed or random effects 18 . Fixed effects included age (factor) and the average weight of yearling ewes in the year of measurement (covariate), which is a better index of resource availability than population size because it accounts for time-lagged effects 4 . The permanent environmental effect grou...

To understand the structure of a large-scale biological, social, or technological network, it can be helpful to decompose the network into smaller subunits or modules. In this article, we develop an information-theoretic foundation for the concept of modularity in networks. We identify the modules of which the network is composed by finding an optimal compression of its topology, capitalizing on regularities in its structure. We explain the advantages of this approach and illustrate them by partitioning a number of real-world and model networks.clustering ͉ compression ͉ information theory M any objects in nature, from proteins to humans, interact in groups that compose social (1), technological (2), or biological systems (3). The groups form a distinct intermediate level between the microscopic and macroscopic descriptions of the system, and group structure may often be coupled to aspects of system function including robustness (3) and stability (4). When we map the interactions among components of a complex system to a network with nodes connected by links, these groups of interacting objects form highly connected modules that are only weakly connected to one other. We can therefore comprehend the structure of a dauntingly complex network by identifying the modules or communities of which it is composed (5-10). When we describe a network as a set of interconnected modules, we are highlighting certain regularities of the network's structure while filtering out the relatively unimportant details. Thus, a modular description of a network can be viewed as a lossy compression of that network's topology, and the problem of community identification as a problem of finding an efficient compression of the structure.This view suggests that we can approach the challenge of identifying the community structure of a complex network as a fundamental problem in information theory (11-13). We provide the groundwork for an information-theoretic approach to community detection and explore the advantages of this approach relative to other methods for community detection. Fig. 1 illustrates our basic framework for identifying communities. We envision the process of describing a complex network by a simplified summary of its module structure as a communication process. The link structure of a complex network is a random variable X; a signaler knows the full form of the network X and aims to convey much of this information in a reduced fashion to a signal receiver. To do so, the signaler encodes information about X as some simplified description Y. She sends the encoded message through a noiseless communication channel. The signal receiver observes the message Y and then ''decodes'' this message, using it to make guesses Z about the structure of the original network X.There are many different ways to describe a network X by a simpler description Y. Which of these is best? The answer to this question of course depends on what you want to do with the description. Nonetheless, information theory offers an appealing general answer to th...

Change is a fundamental ingredient of interaction patterns in biology, technology, the economy, and science itself: Interactions within and between organisms change; transportation patterns by air, land, and sea all change; the global financial flow changes; and the frontiers of scientific research change. Networks and clustering methods have become important tools to comprehend instances of these large-scale structures, but without methods to distinguish between real trends and noisy data, these approaches are not useful for studying how networks change. Only if we can assign significance to the partitioning of single networks can we distinguish meaningful structural changes from random fluctuations. Here we show that bootstrap resampling accompanied by significance clustering provides a solution to this problem. To connect changing structures with the changing function of networks, we highlight and summarize the significant structural changes with alluvial diagrams and realize de Solla Price's vision of mapping change in science: studying the citation pattern between about 7000 scientific journals over the past decade, we find that neuroscience has transformed from an interdisciplinary specialty to a mature and stand-alone discipline.

To comprehend the hierarchical organization of large integrated systems, we introduce the hierarchical map equation, which reveals multilevel structures in networks. In this information-theoretic approach, we exploit the duality between compression and pattern detection; by compressing a description of a random walker as a proxy for real flow on a network, we find regularities in the network that induce this system-wide flow. Finding the shortest multilevel description of the random walker therefore gives us the best hierarchical clustering of the network — the optimal number of levels and modular partition at each level — with respect to the dynamics on the network. With a novel search algorithm, we extract and illustrate the rich multilevel organization of several large social and biological networks. For example, from the global air traffic network we uncover countries and continents, and from the pattern of scientific communication we reveal more than 100 scientific fields organized in four major disciplines: life sciences, physical sciences, ecology and earth sciences, and social sciences. In general, we find shallow hierarchical structures in globally interconnected systems, such as neural networks, and rich multilevel organizations in systems with highly separated regions, such as road networks.