A Shadowing Problem in the Detection of Overlapping Communities: Lifting the Resolution Limit through a Cascading Procedure

Community detection is the process of assigning nodes and links in significant communities (e.g. clusters, function modules) and its development has led to a better understanding of complex networks. When applied to sizable networks, we argue that most detection algorithms correctly identify prominent communities, but fail to do so across multiple scales. As a result, a significant fraction of the network is left uncharted. We show that this problem stems from larger or denser communities overshadowing smaller or sparser ones, and that this effect accounts for most of the undetected communities and unassigned links. We propose a generic cascading approach to community detection that circumvents the problem. Using real and artificial network datasets with three widely used community detection algorithms, we show how a simple cascading procedure allows for the detection of the missing communities. This work highlights a new detection limit of community structure, and we hope that our approach can inspire better community detection algorithms.     

Learning Gene Networks under SNP Perturbations Using eQTL Datasets

The standard approach for identifying gene networks is based on experimental perturbations of gene regulatory systems such as gene knock-out experiments, followed by a genome-wide profiling of differential gene expressions. However, this approach is significantly limited in that it is not possible to perturb more than one or two genes simultaneously to discover complex gene interactions or to distinguish between direct and indirect downstream regulations of the differentially-expressed genes. As an alternative, genetical genomics study has been proposed to treat naturally-occurring genetic variants as potential perturbants of gene regulatory system and to recover gene networks via analysis of population gene-expression and genotype data. Despite many advantages of genetical genomics data analysis, the computational challenge that the effects of multifactorial genetic perturbations should be decoded simultaneously from data has prevented a widespread application of genetical genomics analysis. In this article, we propose a statistical framework for learning gene networks that overcomes the limitations of experimental perturbation methods and addresses the challenges of genetical genomics analysis. We introduce a new statistical model, called a sparse conditional Gaussian graphical model, and describe an efficient learning algorithm that simultaneously decodes the perturbations of gene regulatory system by a large number of SNPs to identify a gene network along with expression quantitative trait loci (eQTLs) that perturb this network. While our statistical model captures direct genetic perturbations of gene network, by performing inference on the probabilistic graphical model, we obtain detailed characterizations of how the direct SNP perturbation effects propagate through the gene network to perturb other genes indirectly. We demonstrate our statistical method using HapMap-simulated and yeast eQTL datasets. In particular, the yeast gene network identified computationally by our method under SNP perturbations is well supported by the results from experimental perturbation studies related to DNA replication stress response.     

Discovering Link Communities in Complex Networks by an Integer Programming Model and a Genetic Algorithm

Identification of communities in complex networks is an important topic and issue in many fields such as sociology, biology, and computer science. Communities are often defined as groups of related nodes or links that correspond to functional subunits in the corresponding complex systems. While most conventional approaches have focused on discovering communities of nodes, some recent studies start partitioning links to find overlapping communities straightforwardly. In this paper, we propose a new quantity function for link community identification in complex networks. Based on this quantity function we formulate the link community partition problem into an integer programming model which allows us to partition a complex network into overlapping communities. We further propose a genetic algorithm for link community detection which can partition a network into overlapping communities without knowing the number of communities. We test our model and algorithm on both artificial networks and real-world networks. The results demonstrate that the model and algorithm are efficient in detecting overlapping community structure in complex networks.     

The ecological and evolutionary implications of merging different types of networks

Interactions among species drive the ecological and evolutionary processes in ecological communities. These interactions are effectively key components of biodiversity. Studies that use a network approach to study the structure and dynamics of communities of interacting species have revealed many patterns and associated processes. Historically these studies were restricted to trophic interactions, although network approaches are now used to study a wide range of interactions, including for example the reproductive mutualisms. However, each interaction type remains studied largely in isolation from others. Merging the various interaction types within a single integrative framework is necessary if we want to further our understanding of the ecological and evolutionary dynamics of communities. Dividing the networks up is a methodological convenience as in the field the networks occur together in space and time and will be linked by shared species. Herein, we outline a conceptual framework for studying networks composed of more than one type of interaction, highlighting key questions and research areas that would benefit from their study.     

Homologous control of protein signaling networks

In a previous paper we introduced a method called augmented sparse reconstruction (ASR) that identifies links among nodes of ordinary differential equation networks, given a small set of observed trajectories with various initial conditions. The main purpose of that technique was to reconstruct intracellular protein signaling networks.In this paper we show that a recursive augmented sparse reconstruction generates artificial networks that are homologous to a large, reference network, in the sense that kinase inhibition of several reactions in the network alters the trajectories of a sizable number of proteins in comparable ways for reference and reconstructed networks. We show this result using a large in-silico model of the epidermal growth factor receptor (EGF-R) driven signaling cascade to generate the data used in the reconstruction algorithm.The most significant consequence of this observed homology is that a nearly optimal combinatorial dosage of kinase inhibitors can be inferred, for many nodes, from the reconstructed network, a result potentially useful for a variety of applications in personalized medicine.     

Top‐down network analysis characterizes hidden termite–termite interactions

The analysis of ecological networks is generally bottom‐up, where networks are established by observing interactions between individuals. Emergent network properties have been indicated to reflect the dominant mode of interactions in communities that might be mutualistic (e.g., pollination) or antagonistic (e.g., host–parasitoid communities). Many ecological communities, however, comprise species interactions that are difficult to observe directly. Here, we propose that a comparison of the emergent properties from detail‐rich reference communities with known modes of interaction can inform our understanding of detail‐sparse focal communities. With this top‐down approach, we consider patterns of coexistence between termite species that live as guests in mounds built by other host termite species as a case in point. Termite societies are extremely sensitive to perturbations, which precludes determining the nature of their interactions through direct observations. We perform a literature review to construct two networks representing termite mound cohabitation in a Brazilian savanna and in the tropical forest of Cameroon. We contrast the properties of these cohabitation networks with a total of 197 geographically diverse mutualistic plant–pollinator and antagonistic host–parasitoid networks. We analyze network properties for the networks, perform a principal components analysis (PCA), and compute the Mahalanobis distance of the termite networks to the cloud of mutualistic and antagonistic networks to assess the extent to which the termite networks overlap with the properties of the reference networks. Both termite networks overlap more closely with the mutualistic plant–pollinator communities than the antagonistic host–parasitoid communities, although the Brazilian community overlap with mutualistic communities is stronger. The analysis raises the hypothesis that termite–termite cohabitation networks may be overall mutualistic. More broadly, this work provides support for the argument that cryptic communities may be analyzed via comparison to well‐characterized communities.     

Augmented sparse reconstruction of protein signaling networks

The problem of reconstructing and identifying intracellular protein signaling and biochemical networks is of critical importance in biology. We propose a mathematical approach called augmented sparse reconstruction for the identification of links among nodes of ordinary differential equation (ODE) networks, given a small set of observed trajectories with various initial conditions. As a test case, the method is applied to the epidermal growth factor receptor (EGFR) driven signaling cascade, a well-studied and clinically important signaling network. Our method builds a system of representation from a collection of trajectory integrals, selectively attenuating blocks of terms in the representation. The system of representation is then augmented with random vectors, and l₁ minimization is used to find sparse representations for the dynamical interactions of each node. After showing the performance of our method on a model of the EGFR protein network, we sketch briefly the potential future therapeutic applications of this approach.     

What Can Interaction Webs Tell Us About Species Roles?

The group model is a useful tool to understand broad-scale patterns of interaction in a network, but it has previously been limited in use to food webs, which contain only predator-prey interactions. Natural populations interact with each other in a variety of ways and, although most published ecological networks only include information about a single interaction type (e.g., feeding, pollination), ecologists are beginning to consider networks which combine multiple interaction types. Here we extend the group model to signed directed networks such as ecological interaction webs. As a specific application of this method, we examine the effects of including or excluding specific interaction types on our understanding of species roles in ecological networks. We consider all three currently available interaction webs, two of which are extended plant-mutualist networks with herbivores and parasitoids added, and one of which is an extended intertidal food web with interactions of all possible sign structures (+/+, -/0, etc.). Species in the extended food web grouped similarly with all interactions, only trophic links, and only nontrophic links. However, removing mutualism or herbivory had a much larger effect in the extended plant-pollinator webs. Species removal even affected groups that were not directly connected to those that were removed, as we found by excluding a small number of parasitoids. These results suggest that including additional species in the network provides far more information than additional interactions for this aspect of network structure. Our methods provide a useful framework for simplifying networks to their essential structure, allowing us to identify generalities in network structure and better understand the roles species play in their communities.     

Motifs in the assembly of food web networks

The assembly of local communities from regional pools is a multifaceted process that involves the confluence of interactions and environmental conditions at the local scale and biogeographic and evolutionary history at the regional scale. Understanding the relative influence of these factors on community structure has remained a challenge and mechanisms driving community assembly are often inferred from patterns of taxonomic, functional, and phylogenetic diversity. Moreover, community assembly is often viewed through the lens of competition and rarely includes trophic interactions or entire food webs. Here, we use motifs – subgraphs of nodes (e.g. species) and links (e.g. predation) whose abundance within a network deviates significantly as compared to a random network topology – to explore the assembly of food web networks found in the leaves of the northern pitcher plant Sarracenia purpurea. We compared counts of three‐node motifs across a hierarchy of scales to a suite of null models to determine if motifs are over‐, under‐, or randomly represented. We then assessed if the pattern of representation of a motif in a given network matched that of the network it was assembled from. We found that motif representation in over 70% of site networks matched the continental network they were assembled from and over 75% of local networks matched the site networks they were assembled from for the majority of null models. This suggests that the same processes are shaping networks across scales. To generalize our results and effectively use a motif perspective to study community assembly, a theoretical framework detailing potential mechanisms for all possible combinations of motif representation is necessary.     

Detecting behavioural changes in human movement to inform the spatial scale of interventions against COVID-19

On March 23 2020, the UK enacted an intensive, nationwide lockdown to mitigate transmission of COVID-19. As restrictions began to ease, more localized interventions were used to target resurgences in transmission. Understanding the spatial scale of networks of human interaction, and how these networks change over time, is critical to targeting interventions at the most at-risk areas without unnecessarily restricting areas at low risk of resurgence. We use detailed human mobility data aggregated from Facebook users to determine how the spatially-explicit network of movements changed before and during the lockdown period, in response to the easing of restrictions, and to the introduction of locally-targeted interventions. We also apply community detection techniques to the weighted, directed network of movements to identify geographically-explicit movement communities and measure the evolution of these community structures through time. We found that the mobility network became more sparse and the number of mobility communities decreased under the national lockdown, a change that disproportionately affected long distance connections central to the mobility network. We also found that the community structure of areas in which locally-targeted interventions were implemented following epidemic resurgence did not show reorganization of community structure but did show small decreases in indicators of travel outside of local areas. We propose that communities detected using Facebook or other mobility data be used to assess the impact of spatially-targeted restrictions and may inform policymakers about the spatial extent of human movement patterns in the UK. These data are available in near real-time, allowing quantification of changes in the distribution of the population across the UK, as well as changes in travel patterns to inform our understanding of the impact of geographically-targeted interventions.     

Link Community Detection Using Generative Model and Nonnegative Matrix Factorization

Discovery of communities in complex networks is a fundamental data analysis problem with applications in various domains. While most of the existing approaches have focused on discovering communities of nodes, recent studies have shown the advantages and uses of link community discovery in networks. Generative models provide a promising class of techniques for the identification of modular structures in networks, but most generative models mainly focus on the detection of node communities rather than link communities. In this work, we propose a generative model, which is based on the importance of each node when forming links in each community, to describe the structure of link communities. We proceed to fit the model parameters by taking it as an optimization problem, and solve it using nonnegative matrix factorization. Thereafter, in order to automatically determine the number of communities, we extend the above method by introducing a strategy of iterative bipartition. This extended method not only finds the number of communities all by itself, but also obtains high efficiency, and thus it is more suitable to deal with large and unexplored real networks. We test this approach on both synthetic benchmarks and real-world networks including an application on a large biological network, and compare it with two highly related methods. Results demonstrate the superior performance of our approach over competing methods for the detection of link communities.     

A thermodynamic view of networks

Networks can be described by the frequency distribution of the number of links associated with each node (the degree of the node). Of particular interest are the power law distributions, which give rise to the so-called scale-free networks, and the distributions of the form of the simplified canonical law (SCL) introduced by Mandelbrot, which give what we shall call the Mandelbrot networks. Many dynamical methods have been obtained for the construction of scale-free networks, but no dynamical construction of Mandelbrot networks has been demonstrated. Here we develop a systematic technique to obtain networks with any given distribution of the degrees of the nodes. This is done using a thermodynamic approach in which we maximise the entropy associated with degree distribution of the nodes of the network subject to certain constraints. These constraints can be chosen systematically to produce the desired network architecture. For large networks we therefore replace a dynamical approach to the stationary state by a thermodynamical viewpoint. We use the method to generate scale-free and Mandelbrot networks with arbitrarily chosen parameters. We emphasise that this approach opens the possibility of insights into a thermodynamics of networks by suggesting thermodynamic relations between macroscopic variables for networks.     

An Adaptive Complex Network Model for Brain Functional Networks

Brain functional networks are graph representations of activity in the brain, where the vertices represent anatomical regions and the edges their functional connectivity. These networks present a robust small world topological structure, characterized by highly integrated modules connected sparsely by long range links. Recent studies showed that other topological properties such as the degree distribution and the presence (or absence) of a hierarchical structure are not robust, and show different intriguing behaviors. In order to understand the basic ingredients necessary for the emergence of these complex network structures we present an adaptive complex network model for human brain functional networks. The microscopic units of the model are dynamical nodes that represent active regions of the brain, whose interaction gives rise to complex network structures. The links between the nodes are chosen following an adaptive algorithm that establishes connections between dynamical elements with similar internal states. We show that the model is able to describe topological characteristics of human brain networks obtained from functional magnetic resonance imaging studies. In particular, when the dynamical rules of the model allow for integrated processing over the entire network scale-free non-hierarchical networks with well defined communities emerge. On the other hand, when the dynamical rules restrict the information to a local neighborhood, communities cluster together into larger ones, giving rise to a hierarchical structure, with a truncated power law degree distribution.     

Rewiring of peatland plant–microbe networks outpaces species turnover

Enviro–climatic changes are thought to be causing alterations in ecosystem processes through shifts in plant and microbial communities; however, how links between plant and microbial communities change with enviro–climatic change is likely to be less straightforward but may be fundamental for many ecological processes. To address this, we assessed the composition of the plant community and the prokaryotic community – using amplicon-based sequencing – of three European peatlands that were distinct in enviro–climatic conditions. Bipartite networks were used to construct site-specific plant–prokaryote co-occurrence networks. Our data show that between sites, plant and prokaryotic communities differ and that turnover in interactions between the communities was complex. Essentially, turnover in plant–microbial interactions is much faster than turnover in the respective communities. Our findings suggest that network rewiring does largely result from novel or different interactions between species common to all realised networks. Hence, turnover in network composition is largely driven by the establishment of new interactions between a core community of plants and microorganisms that are shared among all sites. Taken together our results indicate that plant–microbe associations are context dependent, and that changes in enviro–climatic conditions will likely lead to network rewiring. Integrating turnover in plant–microbe interactions into studies that assess the impact of enviro–climatic change on peatland ecosystems is essential to understand ecosystem dynamics and must be combined with studies on the impact of these changes on ecosystem processes.     

Volatility of network indices due to undersampling of intraspecific variation in plant–insect interactions

Appropriate sampling effort of interaction networks is necessary to extract robust indices describing the structure of species interactions. Here we show that time-invariant variation in the composition and diversity of interaction partners of plant individuals of the same species explains volatility in aggregate network statistics due to undersampling. Within a multi-species pollinator–plant interaction matrix, we replaced the interactions observed on multiple individuals of a single plant species (Sinapis arvensis, pooled interactions) with the plant–insect interactions observed on a single plant individual. In the resampling approach, we considered the interactions of 1 to 84 S. arvensis individuals in different combinations. For each resampled network, several commonly applied aggregated statistics were calculated to test how intraspecific variation affects the properties of a multi-species network. Our results showed that aggregate statistics are sensitive towards qualitative and quantitative intraspecific variation of flower–visitor interactions within a multi-species network, which may affect the ecological interpretation about the properties of a community. These findings challenge the robustness of commonly applied network indices, confirm the urge for a sufficient and representative sampling of interactions, and emphasize the significance of intraspecific variation in the context of communities and networks.     

Interspecific networks in ground beetle (Coleoptera: Carabidae) assemblages

Although changes to interspecific relationships can significantly alter the composition of insect assemblages, they are often ignored when assessing impacts of environmental change. Long-term ground beetle data were used in this study to analyse ecological networks from three habitats at two sites in Scotland. A Bayesian Network inference algorithm was used to reveal interspecific relationships. The significance and strength of relationships between species (nodes) were estimated along with other network properties. Links were identified as positive relationships if co-occurrences of beetles correlated positively, and as negatives relationships if there was a negative correlation between the occurrences of the species. Most of the species had few links and only 10% of the nodes were connected with several links. Calathus fuscipes, a common carabid in the samples, was the most connected, with nine links to other species. More interspecific relationships were found to be positive than negative, with 48 and 23 links, respectively. The modular structure of the network was assessed and eight separate sub-networks were found. Habitat preferences of the species were clearly represented in the structure of the sets of those five sub-networks containing more than one species and were in line with the findings of the indicator species analysis. In our study, we showed that generated Bayesian networks can model interspecific relationships between carabid species. Due to the relative ease of the collection of field data and the high information content of the results, this method could be incorporated into everyday ecological analysis.     

Time Development in the Early History of Social Networks: Link Stabilization,Group Dynamics,and Segregation

Studies of the time development of empirical networks usually investigate late stages where lasting connections have already stabilized. Empirical data on early network history are rare but needed for a better understanding of how social network topology develops in real life. Studying students who are beginning their studies at a university with no or few prior connections to each other offers a unique opportunity to investigate the formation and early development of link patterns and community structure in social networks. During a nine week introductory physics course, first year physics students were asked to identify those with whom they communicated about problem solving in physics during the preceding week. We use these students'' self reports to produce time dependent student interaction networks. We investigate these networks to elucidate possible effects of different student attributes in early network formation. Changes in the weekly number of links show that while roughly half of all links change from week to week, students also reestablish a growing number of links as they progress through their first weeks of study. Using the Infomap community detection algorithm, we show that the networks exhibit community structure, and we use non-network student attributes, such as gender and end-of-course grade to characterize communities during their formation. Specifically, we develop a segregation measure and show that students structure themselves according to gender and pre-organized sections (in which students engage in problem solving and laboratory work), but not according to end-of-coure grade. Alluvial diagrams of consecutive weeks'' communities show that while student movement between groups are erratic in the beginnning of their studies, they stabilize somewhat towards the end of the course. Taken together, the analyses imply that student interaction networks stabilize quickly and that students establish collaborations based on who is immediately available to them and on observable personal characteristics.     

Spatial Fingerprints of Community Structure in Human Interaction Network for an Extensive Set of Large-Scale Regions

Human interaction networks inferred from country-wide telephone activity recordings were recently used to redraw political maps by projecting their topological partitions into geographical space. The results showed remarkable spatial cohesiveness of the network communities and a significant overlap between the redrawn and the administrative borders. Here we present a similar analysis based on one of the most popular online social networks represented by the ties between more than 5.8 million of its geo-located users. The worldwide coverage of their measured activity allowed us to analyze the large-scale regional subgraphs of entire continents and an extensive set of examples for single countries. We present results for North and South America, Europe and Asia. In our analysis we used the well-established method of modularity clustering after an aggregation of the individual links into a weighted graph connecting equal-area geographical pixels. Our results show fingerprints of both of the opposing forces of dividing local conflicts and of uniting cross-cultural trends of globalization.     

Revealing the Hidden Relationship by Sparse Modules in Complex Networks with a Large-Scale Analysis

One of the remarkable features of networks is module that can provide useful insights into not only network organizations but also functional behaviors between their components. Comprehensive efforts have been devoted to investigating cohesive modules in the past decade. However, it is still not clear whether there are important structural characteristics of the nodes that do not belong to any cohesive module. In order to answer this question, we performed a large-scale analysis on 25 complex networks with different types and scales using our recently developed BTS (bintree seeking) algorithm, which is able to detect both cohesive and sparse modules in the network. Our results reveal that the sparse modules composed by the cohesively isolated nodes widely co-exist with the cohesive modules. Detailed analysis shows that both types of modules provide better characterization for the division of a network into functional units than merely cohesive modules, because the sparse modules possibly re-organize the nodes in the so-called cohesive modules, which lack obvious modular significance, into meaningful groups. Compared with cohesive modules, the sizes of sparse ones are generally smaller. Sparse modules are also found to have preferences in social and biological networks than others.     

藏东南典型暗针叶林不同土壤剖面微生物群落特征

深层土壤中的微生物群落对陆地生态系统养分和能量循环转化过程不可或缺,研究青藏高原典型暗针叶林带土壤微生物群落在土壤垂直剖面的变化特征,对深入认识高寒区域森林生态系统土壤微生物群落构建特征及全球变化影响预测具有重要意义。运用Illumina Miseq高通量测序技术和分子生态网络分析,研究藏东南色季拉山暗针叶林带表层（0-20 cm）和底层土壤（40-60 cm）微生物群落组成及分子生态网络结构。研究结果表明随着土壤深度增加,真菌和细菌的丰富度和Shannon多样性指数显著降低。主坐标分析（PCoA）显示土壤深度显著影响真菌和细菌的群落结构（P < 0.01）。不同微生物种群对土壤深度的响应有显著差异,座囊菌纲（Dothideomycetes）、银耳纲（Tremellomycetes）和拟杆菌门（Bacteroidetes）、变形菌门（Proteobacteria）的相对丰度随剖面加深而显著降低,而古菌根菌纲（Archaeorhizomycetes）和绿弯菌门（Chloroflexi）则显著增加。分子生态网络分析发现,真菌网络以负相关连接为主（占总连接数65%-98%）,而细菌网络以正相关连接为主（69%-75%）,真菌和细菌网络中正相关连接的比例均随剖面加深而增加。底层土壤真菌和细菌网络的平均连接度和平均聚类系数均高于表层土壤,说明微生物网络随土壤深度的增加而变得更复杂。真菌网络的平均路径距离和模块性在底层土壤均大于表层土壤,意味着真菌网络应对环境变化的稳定性随剖面加深而增加,而细菌网络则正相反,在表层土壤的稳定性更强。真菌网络中连接节点的个数随剖面加深而增加,锤舌菌纲（Leotiomycetes）是连接网络模块的关键菌种;在细菌网络中模块枢纽和连接节点则随剖面加深而降低,并且放线菌门、变形菌门等关键种群在分子生态网络中的功能在表层和底层土壤有明显差异。综上所述,藏东南色季拉山暗针叶林带深层土壤中微生物群落特征与表层土壤有显著差别,揭示影响深层土壤微生物网络构建和稳定的关键种群,对深入理解和预测青藏高原森林生态系统对全球变化的响应与反馈有重要意义。