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Abstract Background Antimicrobial resistance is under-documented and commensal Escherichia coli can be used as indicator organisms to study the resistance in the community. We sought to determine the prevalence of resistance to broad-spectrum antimicrobials with particular focus on the quinolones, which have recently been introduced in parts of Africa, including Ghana. Results Forty (13.7%) of 293 E. coli isolates evaluated were nalidixic acid-resistant. Thirteen (52%) of 2006 and 2007 isolates and 10 (66.7%) of 2008 isolates were also resistant to ciprofloxacin. All but one of the quinolone-resistant isolates were resistant to three or more other antimicrobial classes. Sequencing the quinolone-resistance determining regions of gyrA and parC, which encode quinolone targets, revealed that 28 quinolone-resistant E. coli harboured a substitution at position 83 of the gyrA gene product and 20 of these isolates had other gyrA and/or parC substitutions. Horizontally-acquired quinolone-resistance genes qnrB1, qnrB2, qnrS1 or qepA were detected in 12 of the isolates. In spite of considerable overall diversity among E. coli from Ghana, as evaluated by multilocus sequence typing, 15 quinolone-resistant E. coli belonged to sequence type complex 10. Five of these isolates carried qnrS1 alleles. Conclusions Quinolone-resistant E. coli are commonly present in the faecal flora of Accra residents. The isolates have evolved resistance through multiple mechanisms and belong to very few lineages, suggesting clonal expansion. Containment strategies to limit the spread of quinolone-resistant E. coli need to be deployed to conserve quinolone effectiveness and promote alternatives to their use.

Abstract Background Genomic research of cultivated peanut has lagged behind other crop species because of the paucity of polymorphic DNA markers found in this crop. It is necessary to identify additional DNA markers for further genetic research in peanut. Results Microsatellite markers in cultivated peanut were developed using the SSR enrichment procedure. The results showed that the GA/CT repeat was the most frequently dispersed microsatellite in peanut. The primer pairs were designed for fifty-six different microsatellites, 19 of which showed a polymorphism among the genotypes studied. The average number of alleles per locus was 4.25, and up to 14 alleles were found at one locus. This suggests that microsatellite DNA markers produce a higher level of DNA polymorphism than other DNA markers in cultivated peanut. Conclusions It is desirable to isolate and characterize more DNA markers in cultivated peanut for more productive genomic studies, such as genetic mapping, marker-assisted selection, and gene discovery. The development of microsatellite markers holds a promise for such studies.

Background: Laparoscopic live donor nephrectomy has replaced open donor nephrectomy in most patients due to numerous benefits. A live animal model is required to equip surgeons with the necessary skills to perform such a procedure with minimal risk of complications. The aim of this study was to establish the technique for laparoscopic live donor nephrectomy in a porcine (Sus scrofa) model., Materials and Methods: This study was approved by the Animal Ethics Committee of the university. Forty-four pigs underwent laparoscopic live donor nephrectomy. The left kidney was removed with a standardized four-port technique, with a small suprapubic incision to facilitate kidney delivery., Results: All 44 procedures were performed successfully, with no intraoperative complications or conversion to open surgery. There was no apparent damage to any of the kidney grafts. The mean surgical time was 118.3 (±20.7) minutes. There was a small, but statistically insignificant, decrease in surgical time throughout the duration of the study. Several subjects had minor variations in the anatomy of the renal vasculature., Conclusions: This series has developed and proven a training model for laparoscopic donor nephrectomy in pigs. This training model will allow surgeons to develop laparoscopic proficiency in a live donor, to be used in conjunction with human cadaveric training., (Copyright © 2017 Elsevier Inc. All rights reserved.)

We demonstrate an equivalence between two widely used methods of community detection in networks, the method of modularity maximization and the method of maximum likelihood applied to the degree-corrected stochastic block model. Specifically, we show an exact equivalence between maximization of the generalized modularity that includes a resolution parameter and the special case of the block model known as the planted partition model, in which all communities in a network are assumed to have statistically similar properties. Among other things, this equivalence provides a mathematically principled derivation of the modularity function, clarifies the conditions and assumptions of its use, and gives an explicit formula for the optimal value of the resolution parameter.

Community detection, the division of a network into dense subnetworks with only sparse connections between them, has been a topic of vigorous study in recent years. However, while there exist a range of effective methods for dividing a network into a specified number of communities, it is an open question how to determine exactly how many communities one should use. Here we describe a mathematically principled approach for finding the number of communities in a network by maximizing the integrated likelihood of the observed network structure under an appropriate generative model. We demonstrate the approach on a range of benchmark networks, both real and computer generated.

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. Here we demonstrate how this 'metadata' can be used to improve our 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. 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.

In the study of networked systems such as biological, technological, and social networks the available data are often uncertain. Rather than knowing the structure of a network exactly, we know the connections between nodes only with a certain probability. In this paper we develop methods for the analysis of such uncertain data, focusing particularly on the problem of community detection. We give a principled maximum-likelihood method for inferring community structure and demonstrate how the results can be used to make improved estimates of the true structure of the network. Using computer-generated benchmark networks we demonstrate that our methods are able to reconstruct known communities more accurately than previous approaches based on data thresholding. We also give an example application to the detection of communities in a protein-protein interaction network.

Recently, a phase transition has been discovered in the network community detection problem below which no algorithm can tell which nodes belong to which communities with success any better than a random guess. This result has, however, so far been limited to the case where the communities have the same size or the same average degree. Here we consider the case where the sizes or average degrees differ. This asymmetry allows us to assign nodes to communities with better-than-random success by examining their local neighborhoods. Using the cavity method, we show that this removes the detectability transition completely for networks with four groups or fewer, while for more than four groups the transition persists up to a critical amount of asymmetry but not beyond. The critical point in the latter case coincides with the point at which local information percolates, causing a global transition from a less-accurate solution to a more-accurate one.

One of the most widely used methods for community detection in networks is the maximization of the quality function known as modularity. Of the many maximization techniques that have been used in this context, some of the most conceptually attractive are the spectral methods, which are based on the eigenvectors of the modularity matrix. Spectral algorithms have, however, been limited, by and large, to the division of networks into only two or three communities, with divisions into more than three being achieved by repeated two-way division. Here we present a spectral algorithm that can directly divide a network into any number of communities. The algorithm makes use of a mapping from modularity maximization to a vector partitioning problem, combined with a fast heuristic for vector partitioning. We compare the performance of this spectral algorithm with previous approaches and find it to give superior results, particularly in cases where community sizes are unbalanced. We also give demonstrative applications of the algorithm to two real-world networks and find that it produces results in good agreement with expectations for the networks studied.

A substantial volume of research is 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.

Many networks can be usefully decomposed into a dense core plus an outlying, loosely connected periphery. Here we propose an algorithm for performing such a decomposition on empirical network data using methods of statistical inference. Our method fits a generative model of core-periphery structure to observed data using a combination of an expectation-maximization algorithm for calculating the parameters of the model and a belief propagation algorithm for calculating the decomposition itself. We find the method to be efficient, scaling easily to networks with a million or more nodes, and we test it on a range of networks, including real-world examples as well as computer-generated benchmarks, for which it successfully identifies known core-periphery structure with low error rate. We also demonstrate that the method is immune to the detectability transition observed in the related community detection problem, which prevents the detection of community structure when that structure is too weak. There is no such transition for core-periphery structure, which is detectable, albeit with some statistical error, no matter how weak it is.

We study percolation on networks, which is used as a model of the resilience of networked systems such as the Internet to attack or failure and as a simple model of the spread of disease over human contact networks. We reformulate percolation as a message passing process and demonstrate how the resulting equations can be used to calculate, among other things, the size of the percolating cluster and the average cluster size. The calculations are exact for sparse networks when the number of short loops in the network is small, but even on networks with many short loops we find them to be highly accurate when compared with direct numerical simulations. By considering the fixed points of the message passing process, we also show that the percolation threshold on a network with few loops is given by the inverse of the leading eigenvalue of the so-called nonbacktracking matrix.

Random graph models have played a dominant role in the theoretical study of networked systems. The Poisson random graph of Erdős and Rényi, in particular, as well as the so-called configuration model, have served as the starting point for numerous calculations. In this paper we describe another large class of random graph models, which we call equitable random graphs and which are flexible enough to represent networks with diverse degree distributions and many nontrivial types of structure, including community structure, bipartite structure, degree correlations, stratification, and others, yet are exactly solvable for a wide range of properties in the limit of large graph size, including percolation properties, complete spectral density, and the behavior of homogeneous dynamical systems, such as coupled oscillators or epidemic models.

Eigenvector centrality is a common measure of the importance of nodes in a network. Here we show that under common conditions the eigenvector centrality displays a localization transition that causes most of the weight of the centrality to concentrate on a small number of nodes in the network. In this regime the measure is no longer useful for distinguishing among the remaining nodes and its efficacy as a network metric is impaired. As a remedy, we propose an alternative centrality measure based on the nonbacktracking matrix, which gives results closely similar to the standard eigenvector centrality in dense networks where the latter is well behaved but avoids localization and gives useful results in regimes where the standard centrality fails.

Using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure. Both reveal interesting spectral features, including deviations from the Wigner semicircle distribution and phase transitions in the spectra of community structured networks. In this paper we generalize both calculations, giving a prescription for calculating the spectrum of a network with both community structure and an arbitrary degree distribution. In general the spectrum has two parts, a continuous spectral band, which can depart strongly from the classic semicircle form, and a set of outlying eigenvalues that indicate the presence of communities.

We consider three distinct and well-studied problems concerning network structure: community detection by modularity maximization, community detection by statistical inference, and normalized-cut graph partitioning. Each of these problems can be tackled using spectral algorithms that make use of the eigenvectors of matrix representations of the network. We show that with certain choices of the free parameters appearing in these spectral algorithms the algorithms for all three problems are, in fact, identical, and hence that, at least within the spectral approximations used here, there is no difference between the modularity- and inference-based community detection methods, or between either and graph partitioning.

The spread of certain diseases can be promoted, in some cases substantially, by prior infection with another disease. One example is that of HIV, whose immunosuppressant effects significantly increase the chances of infection with other pathogens. Such coinfection processes, when combined with nontrivial structure in the contact networks over which diseases spread, can lead to complex patterns of epidemiological behavior. Here we consider a mathematical model of two diseases spreading through a single population, where infection with one disease is dependent on prior infection with the other. We solve exactly for the sizes of the outbreaks of both diseases in the limit of large population size, along with the complete phase diagram of the system. Among other things, we use our model to demonstrate how diseases can be controlled not only by reducing the rate of their spread, but also by reducing the spread of other infections upon which they depend.

A large number of published studies have examined the properties of either networks of citation among scientific papers or networks of coauthorship among scientists. Here we study an extensive data set covering more than a century of physics papers published in the Physical Review, which allows us to construct both citation and coauthorship networks for the same set of papers. We analyze these networks to gain insight into temporal changes in citation and collaboration over the long time period of the data, as well as correlations and interactions between the two. Among other things, we investigate the change over time in the number of publishing authors, the number of papers they publish, and the number of others with whom they collaborate, changes in the typical number of citations made and received, the extent to which individuals tend to cite themselves or their collaborators more than others, the extent to which they cite themselves or their collaborators more quickly after publication, and the extent to which they tend to return the favor of a citation from another scientist.

We study random graphs with arbitrary distributions of expected degree and derive expressions for the spectra of their adjacency and modularity matrices. We give a complete prescription for calculating the spectra that is exact in the limit of large network size and large vertex degrees. We also study the effect on the spectra of hubs in the network, vertices of unusually high degree, and show that these produce isolated eigenvalues outside the main spectral band, akin to impurity states in condensed matter systems, with accompanying eigenvectors that are strongly localized around the hubs. We give numerical results that confirm our analytic expressions.

We study networks that display community structure--groups of nodes within which connections are unusually dense. Using methods from random matrix theory, we calculate the spectra of such networks in the limit of large size, and hence demonstrate the presence of a phase transition in matrix methods for community detection, such as the popular modularity maximization method. The transition separates a regime in which such methods successfully detect the community structure from one in which the structure is present but is not detected. By comparing these results with recent analyses of maximum-likelihood methods, we are able to show that spectral modularity maximization is an optimal detection method in the sense that no other method will succeed in the regime where the modularity method fails.

Human diseases spread over networks of contacts between individuals and a substantial body of recent research has focused on the dynamics of the spreading process. Here we examine a model of two competing diseases spreading over the same network at the same time, where infection with either disease gives an individual subsequent immunity to both. Using a combination of analytic and numerical methods, we derive the phase diagram of the system and estimates of the expected final numbers of individuals infected with each disease. The system shows an unusual dynamical transition between dominance of one disease and dominance of the other as a function of their relative rates of growth. Close to this transition the final outcomes show strong dependence on stochastic fluctuations in the early stages of growth, dependence that decreases with increasing network size, but does so sufficiently slowly as still to be easily visible in systems with millions or billions of individuals. In most regions of the phase diagram we find that one disease eventually dominates while the other reaches only a vanishing fraction of the network, but the system also displays a significant coexistence regime in which both diseases reach epidemic proportions and infect an extensive fraction of the network.

A fundamental problem in the analysis of network data is the detection of network communities, groups of densely interconnected nodes, which may be overlapping or disjoint. Here we describe a method for finding overlapping communities based on a principled statistical approach using generative network models. We show how the method can be implemented using a fast, closed-form expectation-maximization algorithm that allows us to analyze networks of millions of nodes in reasonable running times. We test the method both on real-world networks and on synthetic benchmarks and find that it gives results competitive with previous methods. We also show that the same approach can be used to extract nonoverlapping community divisions via a relaxation method, and demonstrate that the algorithm is competitively fast and accurate for the nonoverlapping problem.

Group B Streptococcus (GBS) remains a major cause of neonatal sepsis and is an emerging cause of invasive bacterial infections. The 9 known serotypes vary in virulence, and there is little cross-immunity. Key parameters for planning an effective vaccination strategy, such as average length of immunity and transmission probabilities by serotype, are unknown. We simulated GBS spread in a population using a computational model with parameters derived from studies of GBS sexual transmission in a college dormitory. Here we provide estimates of the duration of immunity relative to the transmission probabilities for the 3 GBS serotypes most associated with invasive disease: Ia, III, and V. We also place upper limits on the durations of immunity for serotype Ia (570 days), III (1125 days) and V (260 days). Better transmission estimates are required to establish the epidemiological parameters of GBS infection and determine the best vaccination strategies to prevent GBS disease., (Copyright © 2011 Elsevier B.V. All rights reserved.)

Background: A survey was conducted with nearly 1200 growers in US states (Illinois, Indiana, Iowa, Mississippi, Nebraska and North Carolina) in 2005 with the objective in part of determining the awareness of the potential for development of glyphosate resistance, the experience with glyphosate-resistant (GR) weeds and the sources of information that growers had utilized for information on glyphosate resistance. Growers were asked a series of questions to determine the level of glyphosate resistance awareness and to list the sources of information used to learn about glyphosate resistance issues., Results: The majority of the growers (88%) were aware of a weed's potential to evolve resistance to herbicide, while 44% were aware of state-specific documented cases of GR weeds, and 15% reported having had personal experience with GR weeds. Among sources of information concerning glyphosate resistance issues, farm publications, dealers/retailers and university/extension were the most frequent responses (41, 17 and 14% respectively). Based on a 1-10 effectiveness scale, growers ranked tillage the least effective practice (5.5) and using the correct label rates of herbicides at the proper timing for the size and type of weeds present the most effective practice (8.6) with respect to how effectively the practices mitigated the evolution of GR weeds., Conclusion: Results from this survey can be used by researchers, extension specialists and crop advisors further to bridge the information gap between growers and themselves and better to disseminate information concerning glyphosate resistance and glyphosate resistance management practices through more targeted information and information delivery methods., (Copyright © 2011 Society of Chemical Industry.)

Forest fragmentation is one of the most important threats to global biodiversity, particularly in tropical developing countries. Identifying priority areas for conservation within these forests is essential to their effective management. However, this requires current, accurate environmental information that is often lacking in developing countries. The Cockpit Country, Jamaica, contains forests of international importance in terms of levels of endemism and overall diversity. These forests are under severe threat from the prospect of bauxite mining and other anthropogenic disturbances. In the absence of adequate, up-to-date ecological information, we used satellite remote sensing data and fragmentation analysis to identify interior forested areas that have experienced little or no change as priority conservation sites. We classified Landsat images from 1985, 1989, 1995, 2002, and 2008, using an object-oriented method, which allowed for the inclusion of roads. We conducted our fragmentation analysis using metrics to quantify changes in forest patch number, area, shape, and aggregation. Deforestation and fragmentation fluctuated within the 23-year period but were mostly confined to the periphery of the forest, close to roads and access trails. An area of core forest that remained intact over the period of study was identified within the largest forest patch, most of which was located within the boundaries of a forest reserve and included the last remaining patches of closed-broadleaf forest. These areas should be given highest priority for conservation, as they constitute important refuges for endemic or threatened biodiversity. Minimizing and controlling access will be important in maintaining this core.

Stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks. Most blockmodels, however, ignore variation in vertex degree, making them unsuitable for applications to real-world networks, which typically display broad degree distributions that can significantly affect the results. Here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks. We also propose a heuristic algorithm for community detection using this objective function or its non-degree-corrected counterpart and show that the degree-corrected version dramatically outperforms the uncorrected one in both real-world and synthetic networks.

Traditional random graph models of networks generate networks that are locally treelike, meaning that all local neighborhoods take the form of trees. In this respect such models are highly unrealistic, most real networks having strongly nontreelike neighborhoods that contain short loops, cliques, or other biconnected subgraphs. In this paper we propose and analyze a class of random graph models that incorporates general subgraphs, allowing for nontreelike neighborhoods while still remaining solvable for many fundamental network properties. Among other things we give solutions for the size of the giant component, the position of the phase transition at which the giant component appears, and percolation properties for both site and bond percolation on networks generated by the model.

The response of an ecosystem to perturbations is mediated by both antagonistic and facilitative interactions between species. It is thought that a community's resilience depends crucially on the food web--the network of trophic interactions--and on the food web's degree of compartmentalization. Despite its ecological importance, compartmentalization and the mechanisms that give rise to it remain poorly understood. Here we investigate several definitions of compartments, propose ways to understand the ecological meaning of these definitions, and quantify the degree of compartmentalization of empirical food webs. We find that the compartmentalization observed in empirical food webs can be accounted for solely by the niche organization of species and their diets. By uncovering connections between compartmentalization and species' diet contiguity, our findings help us understand which perturbations can result in fragmentation of the food web and which can lead to catastrophic effects. Additionally, we show that the composition of compartments can be used to address the long-standing question of what determines the ecological niche of a species.

In most models of the spread of disease over contact networks it is assumed that the probabilities per unit time of disease transmission and recovery from disease are constant, implying exponential distributions of the time intervals for transmission and recovery. Time intervals for real diseases, however, have distributions that in most cases are far from exponential, which leads to disagreements, both qualitative and quantitative, with the models. In this paper, we study a generalized version of the susceptible-infected-recovered model of epidemic disease that allows for arbitrary distributions of transmission and recovery times. Standard differential equation approaches cannot be used for this generalized model, but we show that the problem can be reformulated as a time-dependent message passing calculation on the appropriate contact network. The calculation is exact on trees (i.e., loopless networks) or locally treelike networks (such as random graphs) in the large system size limit. On non-tree-like networks we show that the calculation gives a rigorous bound on the size of disease outbreaks. We demonstrate the method with applications to two specific models and the results compare favorably with numerical simulations.

We study random graph models for directed acyclic graphs, a class of networks that includes citation networks, food webs, and feed-forward neural networks among others. We propose two specific models roughly analogous to the fixed edge number and fixed edge probability variants of traditional undirected random graphs. We calculate a number of properties of these models, including particularly the probability of connection between a given pair of vertices, and compare the results with real-world acyclic network data finding that theory and measurements agree surprisingly well-far better than the often poor agreement of other random graph models with their corresponding real-world networks.

We offer a solution to a long-standing problem in the theory of networks, the creation of a plausible, solvable model of a network that displays clustering or transitivity--the propensity for two neighbors of a network node also to be neighbors of one another. We show how standard random-graph models can be generalized to incorporate clustering and give exact solutions for various properties of the resulting networks, including sizes of network components, size of the giant component if there is one, position of the phase transition at which the giant component forms, and position of the phase transition for percolation on the network.

In the last few years we have witnessed the emergence, primarily in online communities, of new types of social networks that require for their representation more complex graph structures than have been employed in the past. One example is the folksonomy, a tripartite structure of users, resources, and tags-labels collaboratively applied by the users to the resources in order to impart meaningful structure on an otherwise undifferentiated database. Here we propose a mathematical model of such tripartite structures that represents them as random hypergraphs. We show that it is possible to calculate many properties of this model exactly in the limit of large network size and we compare the results against observations of a real folksonomy, that of the online photography website Flickr. We show that in some cases the model matches the properties of the observed network well, while in others there are significant differences, which we find to be attributable to the practice of multiple tagging, i.e., the application by a single user of many tags to one resource or one tag to many resources.

Directed acyclic graphs make up a fundamental class of networks that includes citation networks, food webs, and family trees, among others. Here we define a random graph model for directed acyclic graphs and give solutions for a number of the model's properties, including connection probabilities and component sizes, as well as a fast algorithm for simulating the model on a computer. We compare the predictions of the model to a real-world network of citations between physics papers and find surprisingly good agreement, suggesting that the structure of the real network may be quite well described by the random graph.

Networks have in recent years emerged as an invaluable tool for describing and quantifying complex systems in many branches of science. Recent studies suggest that networks often exhibit hierarchical organization, in which vertices divide into groups that further subdivide into groups of groups, and so forth over multiple scales. In many cases the groups are found to correspond to known functional units, such as ecological niches in food webs, modules in biochemical networks (protein interaction networks, metabolic networks or genetic regulatory networks) or communities in social networks. Here we present a general technique for inferring hierarchical structure from network data and show that the existence of hierarchy can simultaneously explain and quantitatively reproduce many commonly observed topological properties of networks, such as right-skewed degree distributions, high clustering coefficients and short path lengths. We further show that knowledge of hierarchical structure can be used to predict missing connections in partly known networks with high accuracy, and for more general network structures than competing techniques. Taken together, our results suggest that hierarchy is a central organizing principle of complex networks, capable of offering insight into many network phenomena.

We study bicomponents in networks, sets of nodes such that each pair in the set is connected by at least two independent paths, so that the failure of no single node in the network can cause them to become disconnected. We show that standard network models predict there to be essentially no small bicomponents in most networks, but there may be a giant bicomponent, whose presence coincides with the presence of the ordinary giant component, and we find that real networks seem by and large to follow this pattern, although there are some interesting exceptions. We also study the size of the giant bicomponent as nodes in the network fail and find in some cases that our networks are quite robust to failure, with large bicomponents persisting until almost all vertices have been removed.

The discovery of community structure is a common challenge in the analysis of network data. Many methods have been proposed for finding community structure, but few have been proposed for determining whether the structure found is statistically significant or whether, conversely, it could have arisen purely as a result of chance. In this paper we show that the significance of community structure can be effectively quantified by measuring its robustness to small perturbations in network structure. We propose a suitable method for perturbing networks and a measure of the resulting change in community structure and use them to assess the significance of community structure in a variety of networks, both real and computer generated.

We consider the problem of finding communities or modules in directed networks. In the past, the most common approach to this problem has been to ignore edge direction and apply methods developed for community discovery in undirected networks, but this approach discards potentially useful information contained in the edge directions. Here we show how the widely used community finding technique of modularity maximization can be generalized in a principled fashion to incorporate information contained in edge directions. We describe an explicit algorithm based on spectral optimization of the modularity and show that it gives demonstrably better results than previous methods on a variety of test networks, both real and computer generated.

We give an exact solution for the complete distribution of component sizes in random networks with arbitrary degree distributions. The solution tells us the probability that a randomly chosen node belongs to a component of size s for any s . We apply our results to networks with the three most commonly studied degree distributions-Poisson, exponential, and power-law-as well as to the calculation of cluster sizes for bond percolation on networks, which correspond to the sizes of outbreaks of epidemic processes on the same networks. For the particular case of the power-law degree distribution, we show that the component size distribution itself follows a power law everywhere below the phase transition at which a giant component forms, but takes an exponential form when a giant component is present.

Networks are widely used in the biological, physical, and social sciences as a concise mathematical representation of the topology of systems of interacting components. Understanding the structure of these networks is one of the outstanding challenges in the study of complex systems. Here we describe a general technique for detecting structural features in large-scale network data that works by dividing the nodes of a network into classes such that the members of each class have similar patterns of connection to other nodes. Using the machinery of probabilistic mixture models and the expectation-maximization algorithm, we show that it is possible to detect, without prior knowledge of what we are looking for, a very broad range of types of structure in networks. We give a number of examples demonstrating how the method can be used to shed light on the properties of real-world networks, including social and information networks.

Background: Antidepressant treatments that achieve a higher remission rate than those currently available are urgently needed. The thyroid hormone triiodothyronine may potentiate antidepressant effects., Objective: To determine the antidepressant efficacy and safety of liothyronine sodium (triiodothyronine) when administered concurrently with the selective serotonin reuptake inhibitor sertraline hydrochloride to patients with major depressive disorder., Design: Double-blind, randomized, 8-week, placebo-controlled trial., Setting: Outpatient referral centers., Patients: A total of 124 adult outpatients meeting unmodified DSM-IV criteria for major depressive disorder without psychotic features., Interventions: Patients were randomized to receive sertraline hydrochloride (50 mg/d for 1 week; 100 mg/d thereafter) plus liothyronine sodium (20-25 microg/d for 1 week; 40-50 microg/d thereafter) or sertraline plus placebo for 8 weeks., Main Outcome Measures: The primary outcome measure was categorical response to treatment (> or =50% decrease in scores on the 21-item Hamilton Rating Scale for Depression from baseline to study end point). Remission rate (final Hamilton Rating Scale for Depression score, < or =6) was a secondary outcome measure., Results: Intent-to-treat Hamilton Rating Scale for Depression response rates were 70% and 50% in the sertraline-liothyronine and sertraline-placebo groups, respectively (P = .02; odds ratio, 2.93; 95% confidence interval, 1.23-7.35); remission rates were 58% with sertraline-liothyronine and 38% with sertraline-placebo (P = .02; odds ratio, 2.69; 95% confidence interval, 1.16-6.49). Baseline T(3) values were lower in patients treated with sertraline-liothyronine who had remissions than in those without remissions (t(48) = 3.36; P<.002). Among patients treated with sertraline-liothyronine, remission was associated with a significant decrease in serum thyrotropin values (F(1,73) = 4.00; P<.05). There were no significant effects of liothyronine supplementation on frequency of adverse effects., Conclusions: These results demonstrate enhancement of the antidepressant effect of sertraline by concurrent treatment with liothyronine without a significant increase in adverse effects. The antidepressant effect of liothyronine may be directly linked to thyroid function.

Models of the convergence of opinion in social systems have been the subject of considerable recent attention in the physics literature. These models divide into two classes, those in which individuals form their beliefs based on the opinions of their neighbors in a social network of personal acquaintances, and those in which, conversely, network connections form between individuals of similar beliefs. While both of these processes can give rise to realistic levels of agreement between acquaintances, practical experience suggests that opinion formation in the real world is not a result of one process or the other, but a combination of the two. Here we present a simple model of this combination, with a single parameter controlling the balance of the two processes. We find that the model undergoes a continuous phase transition as this parameter is varied, from a regime in which opinions are arbitrarily diverse to one in which most individuals hold the same opinion.

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as "modularity" over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

There has been considerable recent interest in the properties of networks, such as citation networks and the worldwide web, that grow by the addition of vertices, and a number of simple solvable models of network growth have been studied. In the real world, however, many networks, including the web, not only add vertices but also lose them. Here we formulate models of the time evolution of such networks and give exact solutions for a number of cases of particular interest. For the case of net growth and so-called preferential attachment--in which newly appearing vertices attach to previously existing ones in proportion to vertex degree--we show that the resulting networks have power-law degree distributions, but with an exponent that diverges as the growth rate vanishes. We conjecture that the low exponent values observed in real-world networks are thus the result of vigorous growth in which the rate of addition of vertices far exceeds the rate of removal. Were growth to slow in the future--for instance, in a more mature future version of the web--we would expect to see exponents increase, potentially without bound.