Is My Network Module Preserved and Reproducible?

In many applications, one is interested in determining which of the properties of a network module change across conditions. For example, to validate the existence of a module, it is desirable to show that it is reproducible (or preserved) in an independent test network. Here we study several types of network preservation statistics that do not require a module assignment in the test network. We distinguish network preservation statistics by the type of the underlying network. Some preservation statistics are defined for a general network (defined by an adjacency matrix) while others are only defined for a correlation network (constructed on the basis of pairwise correlations between numeric variables). Our applications show that the correlation structure facilitates the definition of particularly powerful module preservation statistics. We illustrate that evaluating module preservation is in general different from evaluating cluster preservation. We find that it is advantageous to aggregate multiple preservation statistics into summary preservation statistics. We illustrate the use of these methods in six gene co-expression network applications including 1) preservation of cholesterol biosynthesis pathway in mouse tissues, 2) comparison of human and chimpanzee brain networks, 3) preservation of selected KEGG pathways between human and chimpanzee brain networks, 4) sex differences in human cortical networks, 5) sex differences in mouse liver networks. While we find no evidence for sex specific modules in human cortical networks, we find that several human cortical modules are less preserved in chimpanzees. In particular, apoptosis genes are differentially co-expressed between humans and chimpanzees. Our simulation studies and applications show that module preservation statistics are useful for studying differences between the modular structure of networks. Data, R software and accompanying tutorials can be downloaded from the following webpage: http://www.genetics.ucla.edu/labs/horvath/CoexpressionNetwork/ModulePreservation.     

Local Difference Measures between Complex Networks for Dynamical System Model Evaluation

A faithful modeling of real-world dynamical systems necessitates model evaluation. A recent promising methodological approach to this problem has been based on complex networks, which in turn have proven useful for the characterization of dynamical systems. In this context, we introduce three local network difference measures and demonstrate their capabilities in the field of climate modeling, where these measures facilitate a spatially explicit model evaluation. Building on a recent study by Feldhoff et al. [1] we comparatively analyze statistical and dynamical regional climate simulations of the South American monsoon system. Three types of climate networks representing different aspects of rainfall dynamics are constructed from the modeled precipitation space-time series. Specifically, we define simple graphs based on positive as well as negative rank correlations between rainfall anomaly time series at different locations, and such based on spatial synchronizations of extreme rain events. An evaluation against respective networks built from daily satellite data provided by the Tropical Rainfall Measuring Mission 3B42 V7 reveals far greater differences in model performance between network types for a fixed but arbitrary climate model than between climate models for a fixed but arbitrary network type. We identify two sources of uncertainty in this respect. Firstly, climate variability limits fidelity, particularly in the case of the extreme event network; and secondly, larger geographical link lengths render link misplacements more likely, most notably in the case of the anticorrelation network; both contributions are quantified using suitable ensembles of surrogate networks. Our model evaluation approach is applicable to any multidimensional dynamical system and especially our simple graph difference measures are highly versatile as the graphs to be compared may be constructed in whatever way required. Generalizations to directed as well as edge- and node-weighted graphs are discussed.     

The dichotomy in degree correlation of biological networks

Most complex networks from different areas such as biology, sociology or technology, show a correlation on node degree where the possibility of a link between two nodes depends on their connectivity. It is widely believed that complex networks are either disassortative (links between hubs are systematically suppressed) or assortative (links between hubs are enhanced). In this paper, we analyze a variety of biological networks and find that they generally show a dichotomous degree correlation. We find that many properties of biological networks can be explained by this dichotomy in degree correlation, including the neighborhood connectivity, the sickle-shaped clustering coefficient distribution and the modularity structure. This dichotomy distinguishes biological networks from real disassortative networks or assortative networks such as the Internet and social networks. We suggest that the modular structure of networks accounts for the dichotomy in degree correlation and vice versa, shedding light on the source of modularity in biological networks. We further show that a robust and well connected network necessitates the dichotomy of degree correlation, suggestive of an evolutionary motivation for its existence. Finally, we suggest that a dichotomous degree correlation favors a centrally connected modular network, by which the integrity of network and specificity of modules might be reconciled.     

Spatio-temporal variation in the structure of pollination networks

Pollination networks are representations of all interactions between co-existing plants and their flower visiting animals at a given site. Although the study of networks has become a distinct sub-discipline in pollination biology, few studies have attempted to quantify spatio-temporal variation in species composition and structure of networks. We here investigate patterns of year-to-year change in pollination networks from six different sites spanning a large latitudinal gradient. We quantified level of species persistence and interactions among years, and examined year-to-year variation of network structural parameters in relation to latitude and sampling effort. In addition, we tested for correlations between annual variation in network parameters and short and long-term climate change variables. Numbers of plant and animal species and interactions were roughly constant from one year to another at all sites. However, composition of species and interactions changed from one year to another. Turnover was particularly high for flower visitors and interactions. On the other hand, network structural parameters (connectance, nestedness, modularity and centralization) remained remarkably constant between years, regardless of network size and latitude. Inter-annual variation of network parameters was not related to short or long term variation in climate variables (mean annual temperature and annual precipitation). We thus conclude that pollination networks are highly dynamic and variable in composition of species and interactions among years. However, general patterns of network structure remain constant, indicating that species may be replaced by topologically similar species. These results suggest that pollination networks are to some extent robust against factors affecting species occurrences.     

Condition-dependent functional connectivity: syntax networks in bilinguals

This paper introduces a method to study the variation of brain functional connectivity networks with respect to experimental conditions in fMRI data. It is related to the psychophysiological interaction technique introduced by Friston et al. and extends to networks of correlation modulation (CM networks). Extended networks containing several dozens of nodes are determined in which the links correspond to consistent correlation modulation across subjects. In addition, we assess inter-subject variability and determine networks in which the condition-dependent functional interactions can be explained by a subject-dependent variable. We applied the technique to data from a study on syntactical production in bilinguals and analysed functional interactions differentially across tasks (word reading or sentence production) and across languages. We find an extended network of consistent functional interaction modulation across tasks, whereas the network comparing languages shows fewer links. Interestingly, there is evidence for a specific network in which the differences in functional interaction across subjects can be explained by differences in the subjects' syntactical proficiency. Specifically, we find that regions, including ones that have previously been shown to be involved in syntax and in language production, such as the left inferior frontal gyrus, putamen, insula, precentral gyrus, as well as the supplementary motor area, are more functionally linked during sentence production in the second, compared with the first, language in syntactically more proficient bilinguals than in syntactically less proficient ones. Our approach extends conventional activation analyses to the notion of networks, emphasizing functional interactions between regions independently of whether or not they are activated. On the one hand, it gives rise to testable hypotheses and allows an interpretation of the results in terms of the previous literature, and on the other hand, it provides a basis for studying the structure of functional interactions as a whole, and hence represents a further step towards the notion of large-scale networks in functional imaging.     

Does Human Migration Affect International Trade? A Complex-Network Perspective

This paper explores the relationships between international human migration and merchandise trade using a complex-network approach. We firstly compare the topological structure of worldwide networks of human migration and bilateral trade over the period 1960–2000. Next, we ask whether pairs of countries that are more central in the migration network trade more. We show that: (i) the networks of international migration and trade are strongly correlated, and such correlation can be mostly explained by country economic/demographic size and geographical distance; (ii) centrality in the international-migration network boosts bilateral trade; (iii) intensive forms of country centrality are more trade enhancing than their extensive counterparts. Our findings suggest that bilateral trade between any two countries is not only affected by the presence of migrants from either countries, but also by their relative embeddedness in the complex web of corridors making up the network of international human migration.     

Historical climate‐change influences modularity and nestedness of pollination networks

The structure of species interaction networks is important for species coexistence, community stability and exposure of species to extinctions. Two widespread structures in ecological networks are modularity, i.e. weakly connected subgroups of species that are internally highly interlinked, and nestedness, i.e. specialist species that interact with a subset of those species with which generalist species also interact. Modularity and nestedness are often interpreted as evolutionary ecological structures that may have relevance for community persistence and resilience against perturbations, such as climate‐change. Therefore, historical climatic fluctuations could influence modularity and nestedness, but this possibility remains untested. This lack of research is in sharp contrast to the considerable efforts to disentangle the role of historical climate‐change and contemporary climate on species distributions, richness and community composition patterns. Here, we use a global database of pollination networks to show that historical climate‐change is at least as important as contemporary climate in shaping modularity and nestedness of pollination networks. Specifically, on the mainland we found a relatively strong negative association between Quaternary climate‐change and modularity, whereas nestedness was most prominent in areas having experienced high Quaternary climate‐change. On islands, Quaternary climate‐change had weak effects on modularity and no effects on nestedness. Hence, for both modularity and nestedness, historical climate‐change has left imprints on the network structure of mainland communities, but had comparably little effect on island communities. Our findings highlight a need to integrate historical climate fluctuations into eco‐evolutionary hypotheses of network structures, such as modularity and nestedness, and then test these against empirical data. We propose that historical climate‐change may have left imprints in the structural organisation of species interactions in an array of systems important for maintaining biological diversity.     

Finding Influential Spreaders from Human Activity beyond Network Location

Most centralities proposed for identifying influential spreaders on social networks to either spread a message or to stop an epidemic require the full topological information of the network on which spreading occurs. In practice, however, collecting all connections between agents in social networks can be hardly achieved. As a result, such metrics could be difficult to apply to real social networks. Consequently, a new approach for identifying influential people without the explicit network information is demanded in order to provide an efficient immunization or spreading strategy, in a practical sense. In this study, we seek a possible way for finding influential spreaders by using the social mechanisms of how social connections are formed in real networks. We find that a reliable immunization scheme can be achieved by asking people how they interact with each other. From these surveys we find that the probabilistic tendency to connect to a hub has the strongest predictive power for influential spreaders among tested social mechanisms. Our observation also suggests that people who connect different communities is more likely to be an influential spreader when a network has a strong modular structure. Our finding implies that not only the effect of network location but also the behavior of individuals is important to design optimal immunization or spreading schemes.     

生态网络分析方法研究综述

生态网络分析方法是分析生态系统作用关系、辨识系统内在、整体属性的一种有效的系统分析方法。总结了生态网络分析方法的主要研究成果:网络结构特性、网络稳定性、网络上升性、网络效能等;介绍了构建生态网络模型过程和群落构建规则;以德国西部城市诺伊斯河口氮循环为例,介绍David K是如何运用生态网络分析方法来揭示网络中的微动力流循环规律。生态网络分析方法的主要贡献:(1)对人们凭经验感知的生态系统分室间的关联关系,采用了严密的数学模型和推导进行了描述和证明;(2)为生态系统的微动力流循环的研究提供了方法,对生态系统中物质流的间接循环作用进行了科学论证;(3)不仅为分析生态系统提供了一种科学的数学方法,而且,它为探索生态系统提供了不同与牛顿世界观的崭新的认识论。总结与回顾生态网络分析方法,有益于该方法的运用和进一步完善。     

Evaluating the Connectivity of a Protected Areas' Network under the Prism of Global Change: The Efficiency of the European Natura 2000 Network for Four Birds of Prey

Climate and land use changes are major threats to biodiversity. To preserve biodiversity, networks of protected areas have been established worldwide, like the Natura 2000 network across the European Union (EU). Currently, this reserve network consists of more than 26000 sites covering more than 17% of EU terrestrial territory. Its efficiency to mitigate the detrimental effects of land use and climate change remains an open research question. Here, we examined the potential current and future geographical ranges of four birds of prey under scenarios of both land use and climate changes. By using graph theory, we examined how the current Natura 2000 network will perform in regard to the conservation of these species. This approach determines the importance of a site in regard to the total network and its connectivity. We found that sites becoming unsuitable due to climate change are not a random sample of the network, but are less connected and contribute less to the overall connectivity than the average site and thus their loss does not disrupt the full network. Hence, the connectivity of the remaining network changed only slightly from present day conditions. Our findings highlight the need to establish species-specific management plans with flexible conservation strategies ensuring protection under potential future range expansions. Aquila pomarina is predicted to disappear from the southern part of its range and to become restricted to northeastern Europe. Gyps fulvus, Aquila chrysaetos, and Neophron percnopterus are predicted to locally lose some suitable sites; hence, some isolated small populations may become extinct. However, their geographical range and metapopulation structure will remain relatively unaffected throughout Europe. These species would benefit more from an improved habitat quality and management of the existing network of protected areas than from increased connectivity or assisted migration.     

Discovering Communities through Friendship

We introduce a new method for detecting communities of arbitrary size in an undirected weighted network. Our approach is based on tracing the path of closest-friendship between nodes in the network using the recently proposed Generalized Erds Numbers. This method does not require the choice of any arbitrary parameters or null models, and does not suffer from a system-size resolution limit. Our closest-friend community detection is able to accurately reconstruct the true network structure for a large number of real world and artificial benchmarks, and can be adapted to study the multi-level structure of hierarchical communities as well. We also use the closeness between nodes to develop a degree of robustness for each node, which can assess how robustly that node is assigned to its community. To test the efficacy of these methods, we deploy them on a variety of well known benchmarks, a hierarchal structured artificial benchmark with a known community and robustness structure, as well as real-world networks of coauthorships between the faculty at a major university and the network of citations of articles published in Physical Review. In all cases, microcommunities, hierarchy of the communities, and variable node robustness are all observed, providing insights into the structure of the network.     

Sampling bias affects the relationship between structural importance and species body mass in frugivore-plant interaction networks

Within a community, body mass variation among frugivore species is associated with: a) animal's ecological, physiological and functional traits; b) community-level biogeographic/climatic variables; c) anthropogenic factors, and methodological approaches. Furthermore, frugivore-plant relationships are highly context dependent; thus, variation in species attributes at the community level might determine interaction patterns. An interaction network approach is a useful tool to analyze the relationship between species attributes and species role in maintaining the network connectivity patterns (species structural importance). Particularly, the relationship between species body mass and interspecific interaction patterns could be determined by differences in the community properties and the environmental context of the network's geographic location. We tested the hypotheses that: i) the relationship between frugivore species body mass and its structural importance in the network is determined by the frugivore species body mass coefficient of variation (COV) in the community, and ii) frugivore body mass COV depends on the network context in terms of the local climate variables, level of human impact, and taxa considered within the sampling. We evaluated the relationship between species structural importance and its body mass in 28 frugivore-plant interaction networks from different parts of the world. Species structural importance was calculated as a general measure of centrality, which quantifies the generalization level of the species, the proximity of a species to other species in the network, and the importance of a species as a connector between different parts of the network. A meta-analysis approach was applied to evaluate the influence of local climate and community variables associated with each network on the relationship between species structural importance and its body mass. The relationship between centrality and species body mass was highly fluctuating between networks, and frugivore body mass COV was the variable that best explained this heterogeneity. Moreover, networks with both bird and mammal species showed the highest COV values. Our results show that when there is sufficient body mass variability among species in the community, the largest species take important roles in maintaining the network connectivity patterns. This suggests that the bias towards small species in networks studies may impact the magnitude of the frugivore species body mass COV and, therefore, conceal the importance to larger species in network topologies. Future research in frugivore-plant interaction networks should include the highest possible number of interacting species without limiting the samples to species within a particular body size or taxonomic group.     

Emergence of Connectivity Motifs in Networks of Model Neurons with Short- and Long-Term Plastic Synapses

Recent experimental data from the rodent cerebral cortex and olfactory bulb indicate that specific connectivity motifs are correlated with short-term dynamics of excitatory synaptic transmission. It was observed that neurons with short-term facilitating synapses form predominantly reciprocal pairwise connections, while neurons with short-term depressing synapses form predominantly unidirectional pairwise connections. The cause of these structural differences in excitatory synaptic microcircuits is unknown. We show that these connectivity motifs emerge in networks of model neurons, from the interactions between short-term synaptic dynamics (SD) and long-term spike-timing dependent plasticity (STDP). While the impact of STDP on SD was shown in simultaneous neuronal pair recordings in vitro, the mutual interactions between STDP and SD in large networks are still the subject of intense research. Our approach combines an SD phenomenological model with an STDP model that faithfully captures long-term plasticity dependence on both spike times and frequency. As a proof of concept, we first simulate and analyze recurrent networks of spiking neurons with random initial connection efficacies and where synapses are either all short-term facilitating or all depressing. For identical external inputs to the network, and as a direct consequence of internally generated activity, we find that networks with depressing synapses evolve unidirectional connectivity motifs, while networks with facilitating synapses evolve reciprocal connectivity motifs. We then show that the same results hold for heterogeneous networks, including both facilitating and depressing synapses. This does not contradict a recent theory that proposes that motifs are shaped by external inputs, but rather complements it by examining the role of both the external inputs and the internally generated network activity. Our study highlights the conditions under which SD-STDP might explain the correlation between facilitation and reciprocal connectivity motifs, as well as between depression and unidirectional motifs.     

Scalability of Asynchronous Networks Is Limited by One-to-One Mapping between Effective Connectivity and Correlations

Network models are routinely downscaled compared to nature in terms of numbers of nodes or edges because of a lack of computational resources, often without explicit mention of the limitations this entails. While reliable methods have long existed to adjust parameters such that the first-order statistics of network dynamics are conserved, here we show that limitations already arise if also second-order statistics are to be maintained. The temporal structure of pairwise averaged correlations in the activity of recurrent networks is determined by the effective population-level connectivity. We first show that in general the converse is also true and explicitly mention degenerate cases when this one-to-one relationship does not hold. The one-to-one correspondence between effective connectivity and the temporal structure of pairwise averaged correlations implies that network scalings should preserve the effective connectivity if pairwise averaged correlations are to be held constant. Changes in effective connectivity can even push a network from a linearly stable to an unstable, oscillatory regime and vice versa. On this basis, we derive conditions for the preservation of both mean population-averaged activities and pairwise averaged correlations under a change in numbers of neurons or synapses in the asynchronous regime typical of cortical networks. We find that mean activities and correlation structure can be maintained by an appropriate scaling of the synaptic weights, but only over a range of numbers of synapses that is limited by the variance of external inputs to the network. Our results therefore show that the reducibility of asynchronous networks is fundamentally limited.     

Impact of network structure and cellular response on spike time correlations

Novel experimental techniques reveal the simultaneous activity of larger and larger numbers of neurons. As a result there is increasing interest in the structure of cooperative--or correlated--activity in neural populations, and in the possible impact of such correlations on the neural code. A fundamental theoretical challenge is to understand how the architecture of network connectivity along with the dynamical properties of single cells shape the magnitude and timescale of correlations. We provide a general approach to this problem by extending prior techniques based on linear response theory. We consider networks of general integrate-and-fire cells with arbitrary architecture, and provide explicit expressions for the approximate cross-correlation between constituent cells. These correlations depend strongly on the operating point (input mean and variance) of the neurons, even when connectivity is fixed. Moreover, the approximations admit an expansion in powers of the matrices that describe the network architecture. This expansion can be readily interpreted in terms of paths between different cells. We apply our results to large excitatory-inhibitory networks, and demonstrate first how precise balance--or lack thereof--between the strengths and timescales of excitatory and inhibitory synapses is reflected in the overall correlation structure of the network. We then derive explicit expressions for the average correlation structure in randomly connected networks. These expressions help to identify the important factors that shape coordinated neural activity in such networks.     

Making Large-Scale Networks from fMRI Data

Pairwise correlations are currently a popular way to estimate a large-scale network (> 1000 nodes) from functional magnetic resonance imaging data. However, this approach generally results in a poor representation of the true underlying network. The reason is that pairwise correlations cannot distinguish between direct and indirect connectivity. As a result, pairwise correlation networks can lead to fallacious conclusions; for example, one may conclude that a network is a small-world when it is not. In a simulation study and an application to resting-state fMRI data, we compare the performance of pairwise correlations in large-scale networks (2000 nodes) against three other methods that are designed to filter out indirect connections. Recovery methods are evaluated in four simulated network topologies (small world or not, scale-free or not) in scenarios where the number of observations is very small compared to the number of nodes. Simulations clearly show that pairwise correlation networks are fragmented into separate unconnected components with excessive connectedness within components. This often leads to erroneous estimates of network metrics, like small-world structures or low betweenness centrality, and produces too many low-degree nodes. We conclude that using partial correlations, informed by a sparseness penalty, results in more accurate networks and corresponding metrics than pairwise correlation networks. However, even with these methods, the presence of hubs in the generating network can be problematic if the number of observations is too small. Additionally, we show for resting-state fMRI that partial correlations are more robust than correlations to different parcellation sets and to different lengths of time-series.     

Reconstructing reticulate evolution in species-theory and practice.

We present new methods for reconstructing reticulate evolution of species due to events such as horizontal transfer or hybrid speciation; both methods are based upon extensions of Wayne Maddison's approach in his seminal 1997 paper. Our first method is a polynomial time algorithm for constructing phylogenetic networks from two gene trees contained inside the network.We allow the network to have an arbitrary number of reticulations, but we limit the reticulation in the network so that the cycles in the network are node-disjoint ("galled"). Our second method is a polynomial time algorithm for constructing networks with one reticulation, where we allow for errors in the estimated gene trees. Using simulations, we demonstrate improved performance of this method over both NeighborNet and Maddison's method.     

Neural networks for clustered and longitudinal data using mixed effects models

Although most statistical methods for the analysis of longitudinal data have focused on retrospective models of association, new advances in mobile health data have presented opportunities for predicting future health status by leveraging an individual's behavioral history alongside data from similar patients. Methods that incorporate both individual-level and sample-level effects are critical to using these data to its full predictive capacity. Neural networks are powerful tools for prediction, but many assume input observations are independent even when they are clustered or correlated in some way, such as in longitudinal data. Generalized linear mixed models (GLMM) provide a flexible framework for modeling longitudinal data but have poor predictive power particularly when the data are highly nonlinear. We propose a generalized neural network mixed model that replaces the linear fixed effect in a GLMM with the output of a feed-forward neural network. The model simultaneously accounts for the correlation structure and complex nonlinear relationship between input variables and outcomes, and it utilizes the predictive power of neural networks. We apply this approach to predict depression and anxiety levels of schizophrenic patients using longitudinal data collected from passive smartphone sensor data.     

The Role of Human Transportation Networks in Mediating the Genetic Structure of Seasonal Influenza in the United States

Recent studies have demonstrated the importance of accounting for human mobility networks when modeling epidemics in order to accurately predict spatial dynamics. However, little is known about the impact these movement networks have on the genetic structure of pathogen populations and whether these effects are scale-dependent. We investigated how human movement along the aviation and commuter networks contributed to intra-seasonal genetic structure of influenza A epidemics in the continental United States using spatially-referenced hemagglutinin nucleotide sequences collected from 2003–2013 for both the H3N2 and H1N1 subtypes. Comparative analysis of these transportation networks revealed that the commuter network is highly spatially-organized and more heavily traveled than the aviation network, which instead is characterized by high connectivity between all state pairs. We found that genetic distance between sequences often correlated with distance based on interstate commuter network connectivity for the H1N1 subtype, and that this correlation was not as prevalent when geographic distance or aviation network connectivity distance was assessed against genetic distance. However, these patterns were not as apparent for the H3N2 subtype at the scale of the continental United States. Finally, although sequences were spatially referenced at the level of the US state of collection, a community analysis based on county to county commuter connections revealed that commuting communities did not consistently align with state geographic boundaries, emphasizing the need for the greater availability of more specific sequence location data. Our results highlight the importance of utilizing host movement data in characterizing the underlying genetic structure of pathogen populations and demonstrate a need for a greater understanding of the differential effects of host movement networks on pathogen transmission at various spatial scales.     

Application of network methods for understanding evolutionary dynamics in discrete habitats

In populations occupying discrete habitat patches, gene flow between habitat patches may form an intricate population structure. In such structures, the evolutionary dynamics resulting from interaction of gene‐flow patterns with other evolutionary forces may be exceedingly complex. Several models describing gene flow between discrete habitat patches have been presented in the population‐genetics literature; however, these models have usually addressed relatively simple settings of habitable patches and have stopped short of providing general methodologies for addressing nontrivial gene‐flow patterns. In the last decades, network theory – a branch of discrete mathematics concerned with complex interactions between discrete elements – has been applied to address several problems in population genetics by modelling gene flow between habitat patches using networks. Here, we present the idea and concepts of modelling complex gene flows in discrete habitats using networks. Our goal is to raise awareness to existing network theory applications in molecular ecology studies, as well as to outline the current and potential contribution of network methods to the understanding of evolutionary dynamics in discrete habitats. We review the main branches of network theory that have been, or that we believe potentially could be, applied to population genetics and molecular ecology research. We address applications to theoretical modelling and to empirical population‐genetic studies, and we highlight future directions for extending the integration of network science with molecular ecology.