The impact of network clustering and assortativity on epidemic behaviour

Epidemic models have successfully included many aspects of the complex contact structure apparent in real-world populations. However, it is difficult to accommodate variations in the number of contacts, clustering coefficient and assortativity. Investigations of the relationship between these properties and epidemic behaviour have led to inconsistent conclusions and have not accounted for their interrelationship. In this study, simulation is used to estimate the impact of social network structure on the probability of an SIR (susceptible-infective-removed) epidemic occurring and, if it does, the final size. Increases in assortativity and clustering coefficient are associated with smaller epidemics and the impact is cumulative. Derived values of the basic reproduction ratio (R₀) over networks with the highest property values are more than 20% lower than those derived from simulations with zero values of these network properties.     

Modelling the evolution of genetic regulatory networks

An evolutionary model of genetic regulatory networks is developed, based on a model of network encoding and dynamics called the Artificial Genome (AG). This model derives a number of specific genes and their interactions from a string of (initially random) bases in an idealized manner analogous to that employed by natural DNA. The gene expression dynamics are determined by updating the gene network as if it were a simple Boolean network. The generic behaviour of the AG model is investigated in detail. In particular, we explore the characteristic network topologies generated by the model, their dynamical behaviours, and the typical variance of network connectivities and network structures. These properties are demonstrated to agree with a probabilistic analysis of the model, and the typical network structures generated by the model are shown to lie between those of random networks and scale-free networks in terms of their degree distribution. Evolutionary processes are simulated using a genetic algorithm, with selection acting on a range of properties from gene number and degree of connectivity through periodic behaviour to specific patterns of gene expression. The evolvability of increasingly complex patterns of gene expression is examined in detail. When a degree of redundancy is introduced, the average number of generations required to evolve given targets is reduced, but limits on evolution of complex gene expression patterns remain. In addition, cyclic gene expression patterns with periods that are multiples of shorter expression patterns are shown to be inherently easier to evolve than others. Constraints imposed by the template-matching nature of the AG model generate similar biases towards such expression patterns in networks in initial populations, in addition to the somewhat scale-free nature of these networks. The significance of these results on current understanding of biological evolution is discussed.     

Large-scale properties of clustered networks: implications for disease dynamics

We consider previously proposed procedures for generating clustered networks and investigate how these procedures lead to differences in network properties other than clustering. We interpret our findings in terms of the effect of the network structure on the disease outbreak threshold and disease dynamics. To generate null-model networks for comparison, we implement an assortativity-conserving rewiring algorithm that alters the level of clustering while causing minimal impact on other properties. We show that many theoretical network models used to generate networks with a particular property often lead to significant changes in network properties other than that of interest. For high levels of clustering, different procedures lead to networks that differ in degree heterogeneity and assortativity, and in broader scale measures such as ?(0) and the distribution of shortest path lengths. Hence, care must be taken when investigating the implications of network properties for disease transmission or other dynamic process that the network supports.     

Proximity data-loggers increase the quantity and quality of social network data

Social network analysis is an ideal quantitative tool for advancing our understanding of complex social behaviour. However, this approach is often limited by the challenges of accurately characterizing social structure and measuring network heterogeneity. Technological advances have facilitated the study of social networks, but to date, all such work has focused on large vertebrates. Here, we provide proof of concept for using proximity data-logging to quantify the frequency of social interactions, construct weighted networks and characterize variation in the social behaviour of a lek-breeding bird, the wire-tailed manakin, Pipra filicauda. Our results highlight how this approach can ameliorate the challenges of social network data collection and analysis by concurrently improving data quality and quantity.     

SpecNet: A Spatial Network Algorithm that Generates a Wide Range of Specific Structures

Network measures are used to predict the behavior of different systems. To be able to investigate how various structures behave and interact we need a wide range of theoretical networks to explore. Both spatial and non-spatial methods exist for generating networks but they are limited in the ability of producing wide range of network structures. We extend an earlier version of a spatial spectral network algorithm to generate a large variety of networks across almost all the theoretical spectra of the following network measures: average clustering coefficient, degree assortativity, fragmentation index, and mean degree. We compare this extended spatial spectral network-generating algorithm with a non-spatial algorithm regarding their ability to create networks with different structures and network measures. The spatial spectral network-generating algorithm can generate networks over a much broader scale than the non-spatial and other known network algorithms. To exemplify the ability to regenerate real networks, we regenerate networks with structures similar to two real Swedish swine transport networks. Results show that the spatial algorithm is an appropriate model with correlation coefficients at 0.99. This novel algorithm can even create negative assortativity and managed to achieve assortativity values that spans over almost the entire theoretical range.     

Using network properties to predict disease dynamics on human contact networks

Recent studies have increasingly turned to graph theory to model more realistic contact structures that characterize disease spread. Because of the computational demands of these methods, many researchers have sought to use measures of network structure to modify analytically tractable differential equation models. Several of these studies have focused on the degree distribution of the contact network as the basis for their modifications. We show that although degree distribution is sufficient to predict disease behaviour on very sparse or very dense human contact networks, for intermediate density networks we must include information on clustering and path length to accurately predict disease behaviour. Using these three metrics, we were able to explain more than 98 per cent of the variation in endemic disease levels in our stochastic simulations.     

Representing the UK's cattle herd as static and dynamic networks

Network models are increasingly being used to understand the spread of diseases through sparsely connected populations, with particular interest in the impact of animal movements upon the dynamics of infectious diseases. Detailed data collected by the UK government on the movement of cattle may be represented as a network, where animal holdings are nodes, and an edge is drawn between nodes where a movement of animals has occurred. These network representations may vary from a simple static representation, to a more complex, fully dynamic one where daily movements are explicitly captured. Using stochastic disease simulations, a wide range of network representations of the UK cattle herd are compared. We find that the simpler static network representations are often deficient when compared with a fully dynamic representation, and should therefore be used only with caution in epidemiological modelling. In particular, due to temporal structures within the dynamic network, static networks consistently fail to capture the predicted epidemic behaviour associated with dynamic networks even when parameterized to match early growth rates.     

Emergence of complex social networks from spatial structure and rules of thumb: a modelling approach

Individual-based computer models show that simple heuristic governing individuals’ behavior may suffice to generate complex patterns of social behavior at the group level such as those observed in animal societies. ‘GrooFiWorld’ is an example of such kind of computer models. In this model, self-organization and simple behavioral rules generate complex patterns of social behavior like those described in tolerant and intolerant societies of macaques. Social complexity results from the socio-spatial structure of the group, the nature of which is, in turn, a side-effect of intensity of aggression. The model suggests that a similar mechanism may give rise to complex social structures in macaques. It is, however, unknown if the spatial structure of the model and that of macaques are indeed similar. Here we used social networks analysis as a proxy for spatial structure of the group. Our findings show that the social networks of the model share similar qualitative features with those of macaques. As group size increases, the density and the average individual eigenvector centrality decrease and the modularity and centralization of the network increase. In social networks emerging from simulations resembling intolerant societies the density is lower, the modularity and centralization are higher, and the individuals ranking higher in the dominance hierarchy are more central than in the social networks emerging from simulations resembling egalitarian societies. Given the qualitative similarity between the social networks of the model and that of empirical data, our results suggest that the spatial structure of macaques is similar to that of the model. It seems thus plausible that, as in the model, the spatial structure combined with simple behavioral rules plays a role in the emergence of complex social networks and complex social behavior in macaques.     

Network epidemic models with two levels of mixing

The study of epidemics on social networks has attracted considerable attention recently. In this paper, we consider a stochastic SIR (susceptible-->infective-->removed) model for the spread of an epidemic on a finite network, having an arbitrary but specified degree distribution, in which individuals also make casual contacts, i.e. with people chosen uniformly from the population. The behaviour of the model as the network size tends to infinity is investigated. In particular, the basic reproduction number R(0), that governs whether or not an epidemic with few initial infectives can become established is determined, as are the probability that an epidemic becomes established and the proportion of the population who are ultimately infected by such an epidemic. For the case when the infectious period is constant and all individuals in the network have the same degree, the asymptotic variance and a central limit theorem for the size of an epidemic that becomes established are obtained. Letting the rate at which individuals make casual contacts decrease to zero yields, heuristically, corresponding results for the model without casual contacts, i.e. for the standard SIR network epidemic model. A deterministic model that approximates the spread of an epidemic that becomes established in a large population is also derived. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations work well, even for only moderately sized networks, and that the degree distribution and the inclusion of casual contacts can each have a major impact on the outcome of an epidemic.     

What's in a crowd? Analysis of face-to-face behavioral networks

The availability of new data sources on human mobility is opening new avenues for investigating the interplay of social networks, human mobility and dynamical processes such as epidemic spreading. Here we analyze data on the time-resolved face-to-face proximity of individuals in large-scale real-world scenarios. We compare two settings with very different properties, a scientific conference and a long-running museum exhibition. We track the behavioral networks of face-to-face proximity, and characterize them from both a static and a dynamic point of view, exposing differences and similarities. We use our data to investigate the dynamics of a susceptible-infected model for epidemic spreading that unfolds on the dynamical networks of human proximity. The spreading patterns are markedly different for the conference and the museum case, and they are strongly impacted by the causal structure of the network data. A deeper study of the spreading paths shows that the mere knowledge of static aggregated networks would lead to erroneous conclusions about the transmission paths on the dynamical networks.     

Biological solutions to transport network design

Transport networks are vital components of multicellular organisms, distributing nutrients and removing waste products. Animal and plant transport systems are branching trees whose architecture is linked to universal scaling laws in these organisms. In contrast, many fungi form reticulated mycelia via the branching and fusion of thread-like hyphae that continuously adapt to the environment. Fungal networks have evolved to explore and exploit a patchy environment, rather than ramify through a three-dimensional organism. However, there has been no explicit analysis of the network structures formed, their dynamic behaviour nor how either impact on their ecological function. Using the woodland saprotroph Phanerochaete velutina, we show that fungal networks can display both high transport capacity and robustness to damage. These properties are enhanced as the network grows, while the relative cost of building the network decreases. Thus, mycelia achieve the seemingly competing goals of efficient transport and robustness, with decreasing relative investment, by selective reinforcement and recycling of transport pathways. Fungal networks demonstrate that indeterminate, decentralized systems can yield highly adaptive networks. Understanding how these relatively simple organisms have found effective transport networks through a process of natural selection may inform the design of man-made networks.     

SIR dynamics in random networks with heterogeneous connectivity

Random networks with specified degree distributions have been proposed as realistic models of population structure, yet the problem of dynamically modeling SIR-type epidemics in random networks remains complex. I resolve this dilemma by showing how the SIR dynamics can be modeled with a system of three nonlinear ODE’s. The method makes use of the probability generating function (PGF) formalism for representing the degree distribution of a random network and makes use of network-centric quantities such as the number of edges in a well-defined category rather than node-centric quantities such as the number of infecteds or susceptibles. The PGF provides a simple means of translating between network and node-centric variables and determining the epidemic incidence at any time. The theory also provides a simple means of tracking the evolution of the degree distribution among susceptibles or infecteds. The equations are used to demonstrate the dramatic effects that the degree distribution plays on the final size of an epidemic as well as the speed with which it spreads through the population. Power law degree distributions are observed to generate an almost immediate expansion phase yet have a smaller final size compared to homogeneous degree distributions such as the Poisson. The equations are compared to stochastic simulations, which show good agreement with the theory. Finally, the dynamic equations provide an alternative way of determining the epidemic threshold where large-scale epidemics are expected to occur, and below which epidemic behavior is limited to finite-sized outbreaks.      

The effect of contact heterogeneity and multiple routes of transmission on final epidemic size

Heterogeneity in the number of potentially infectious contacts amongst members of a population increases the basic reproduction ratio (R(0)) and markedly alters disease dynamics compared to traditional mean-field models. Most models describing transmission on contact networks only account for one specific route of transmission. However, for many infectious diseases multiple routes of transmission exist. The model presented here captures transmission through a well defined network of contacts, complemented by mean-field type transmission amongst the nodes of the network that accounts for alternative routes of transmission. The impact of these combined transmission mechanisms on the final epidemic size is investigated analytically. The analytic predictions for the purely mean-field case and the transmission through the network-only case are confirmed by individual-based network simulations. There is a critical transmission potential above which an increased contribution of the mean-field type transmission increases the final epidemic size while an increased contribution of the transmission through the network decreases it. Below the critical transmission potential the opposite effect is observed.     

Combining functional and topological properties to identify core modules in protein interaction networks

Advances in large-scale technologies in proteomics, such as yeast two-hybrid screening and mass spectrometry, have made it possible to generate large Protein Interaction Networks (PINs). Recent methods for identifying dense sub-graphs in such networks have been based solely on graph theoretic properties. Therefore, there is a need for an approach that will allow us to combine domain-specific knowledge with topological properties to generate functionally relevant sub-graphs from large networks. This article describes two alternative network measures for analysis of PINs, which combine functional information with topological properties of the networks. These measures, called weighted clustering coefficient and weighted average nearest-neighbors degree, use weights representing the strengths of interactions between the proteins, calculated according to their semantic similarity, which is based on the Gene Ontology terms of the proteins. We perform a global analysis of the yeast PIN by systematically comparing the weighted measures with their topological counterparts. To show the usefulness of the weighted measures, we develop an algorithm for identification of functional modules, called SWEMODE (Semantic WEights for MODule Elucidation), that identifies dense sub-graphs containing functionally similar proteins. The proposed method is based on the ranking of nodes, i.e., proteins, according to their weighted neighborhood cohesiveness. The highest ranked nodes are considered as seeds for candidate modules. The algorithm then iterates through the neighborhood of each seed protein, to identify densely connected proteins with high functional similarity, according to the chosen parameters. Using a yeast two-hybrid data set of experimentally determined protein-protein interactions, we demonstrate that SWEMODE is able to identify dense clusters containing proteins that are functionally similar. Many of the identified modules correspond to known complexes or subunits of these complexes.     

Epidemic Spread on Weighted Networks

The contact structure between hosts shapes disease spread. Most network-based models used in epidemiology tend to ignore heterogeneity in the weighting of contacts between two individuals. However, this assumption is known to be at odds with the data for many networks (e.g. sexual contact networks) and to have a critical influence on epidemics'' behavior. One of the reasons why models usually ignore heterogeneity in transmission is that we currently lack tools to analyze weighted networks, such that most studies rely on numerical simulations. Here, we present a novel framework to estimate key epidemiological variables, such as the rate of early epidemic expansion () and the basic reproductive ratio (), from joint probability distributions of number of partners (contacts) and number of interaction events through which contacts are weighted. These distributions are much easier to infer than the exact shape of the network, which makes the approach widely applicable. The framework also allows for a derivation of the full time course of epidemic prevalence and contact behaviour, which we validate with numerical simulations on networks. Overall, incorporating more realistic contact networks into epidemiological models can improve our understanding of the emergence and spread of infectious diseases.     

Inferring a District-Based Hierarchical Structure of Social Contacts from Census Data

Researchers have recently paid attention to social contact patterns among individuals due to their useful applications in such areas as epidemic evaluation and control, public health decisions, chronic disease research and social network research. Although some studies have estimated social contact patterns from social networks and surveys, few have considered how to infer the hierarchical structure of social contacts directly from census data. In this paper, we focus on inferring an individual’s social contact patterns from detailed census data, and generate various types of social contact patterns such as hierarchical-district-structure-based, cross-district and age-district-based patterns. We evaluate newly generated contact patterns derived from detailed 2011 Hong Kong census data by incorporating them into a model and simulation of the 2009 Hong Kong H1N1 epidemic. We then compare the newly generated social contact patterns with the mixing patterns that are often used in the literature, and draw the following conclusions. First, the generation of social contact patterns based on a hierarchical district structure allows for simulations at different district levels. Second, the newly generated social contact patterns reflect individuals social contacts. Third, the newly generated social contact patterns improve the accuracy of the SEIR-based epidemic model.     

Formidable females redux:male social integration into female networks and the value of dynamic multilayer networks

The development of multilayer network techniques is a boon for researchers who wish to understand how different interaction layers might influence each other,and how these in turn might influence group dynamics.Here,we investigate how integration between male and female grooming and aggression interaction networks influences male power trajectories in vervet monkeys Chlorocebus pygerythrus.Our previous analyses of this phenomenon used a monolayer approach,and our aim here is to extend these analyses using a dynamic multilayer approach.To do so,we constructed a temporal series of male and female interaction layers.We then used a multivariate multilevel autoregression model to compare cross-lagged associations between a male's centrality in the female grooming layer and changes in male Elo ratings.Our results confirmed our original findings:changes in male centrality within the female grooming network were weakly but positively tied to changes in their Elo ratings.However,the multilayer network approach offered additional insights into this social process,identifying how changes in a male's centrality cascade through the other network layers.This dynamic view indicates that the changes in Elo ratings are likely to be short-lived,but that male centrality within the female network had a much stronger impact throughout the multilayer network as a whole,especially on reducing intermale aggression(i.e.,aggression directed by males toward other males).We suggest that multilayer social network approaches can take advantage of increased amounts of social data that are more commonly collected these days,using a variety of methods.Such data are inherently multilevel and multilayered,and thus offer the ability to quantify more precisely the dynamics of animal social behaviors.     

Social networks in the lek-mating wire-tailed manakin (Pipra filicauda)

How social structure interacts with individual behaviour and fitness remains understudied despite its potential importance to the evolution of cooperation. Recent applications of network theory to social behaviour advance our understanding of the role of social interactions in various contexts. Here we applied network theory to the social system of lek-mating wire-tailed manakins (Pipra filicauda, Pipridae, Aves). We analysed the network of interactions among males in order to begin building a comparative framework to understand where coordinated display behaviour lies along the continuum from solitary to obligately cooperative dual-male displays in the family Pipridae. Network degree (the number of links from a male to others) ranged from 1 to 10, with low mean and high variance, consistent with the theory for the evolution of cooperation within social networks. We also assessed factors that could predict social and reproductive success of males. Four network metrics, degree, eigenvector centrality, information centrality and reach, some of which assess circuitous as well as the shortest (geodesic) paths of male connectivity, predicted male social rise. The duration of a male's territorial tenure during the 4 years of the study predicted his probability of siring offspring.     

Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

In this paper, we outline the theory of epidemic percolation networks and their use in the analysis of stochastic susceptible-infectious-removed (SIR) epidemic models on undirected contact networks. We then show how the same theory can be used to analyze stochastic SIR models with random and proportionate mixing. The epidemic percolation networks for these models are purely directed because undirected edges disappear in the limit of a large population. In a series of simulations, we show that epidemic percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show that epidemic percolation networks can be used to re-derive classical results from several different areas of infectious disease epidemiology. In an Appendix, we show that an epidemic percolation network can be defined for any time-homogeneous stochastic SIR model in a closed population and prove that the distribution of outbreak sizes given the infection of any given node in the SIR model is identical to the distribution of its out-component sizes in the corresponding probability space of epidemic percolation networks. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR models in closed populations.