Percolation on Networks with Conditional Dependence Group

Recently, the dependence group has been proposed to study the robustness of networks with interdependent nodes. A dependence group means that a failed node in the group can lead to the failures of the whole group. Considering the situation of real networks that one failed node may not always break the functionality of a dependence group, we study a cascading failure model that a dependence group fails only when more than a fraction β of nodes of the group fail. We find that the network becomes more robust with the increasing of the parameter β. However, the type of percolation transition is always first order unless the model reduces to the classical network percolation model, which is independent of the degree distribution of the network. Furthermore, we find that a larger dependence group size does not always make the networks more fragile. We also present exact solutions to the size of the giant component and the critical point, which are in agreement with the simulations well.     

油菜基因流网络的结构特性与小团体分析

为从系统角度掌握转基因油菜(Brassica campestris)基因流规律, 基于常规油菜与十字花科植物杂交的有关文献,构建了油菜基因流网络拓扑图实例并分析其网络结构特性。研究结果表明, 该网络节点度服从幂律分布, 具有无标度特性。从随机攻击和恶性攻击两个方面对标准结构熵和网络效率2个指标的网络稳健性进行分析, 结果显示在随机移除不到10%的顶点时网络表现较好的鲁棒性, 但在受到选择性移除10%的顶点时, 网络具有极弱的抗攻击性。基于软件UCINET对网络进行了小团体和结构同型性分析, 可将网络中298个节点22类十字花科植物划分为2个小团体和5类结构角色, 其中甘蓝型油菜在基因流网络小团体中具有关键性作用。这些研究结果可为揭示转基因油菜基因流规律提供新思路, 同时也可对转基因油菜商业化种植采取合理的农业生产栽培管理措施提供参考。     

Robustness of Oscillatory Behavior in Correlated Networks

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.     

Failure tolerance of motif structure in biological networks

Complex networks serve as generic models for many biological systems that have been shown to share a number of common structural properties such as power-law degree distribution and small-worldness. Real-world networks are composed of building blocks called motifs that are indeed specific subgraphs of (usually) small number of nodes. Network motifs are important in the functionality of complex networks, and the role of some motifs such as feed-forward loop in many biological networks has been heavily studied. On the other hand, many biological networks have shown some degrees of robustness in terms of their efficiency and connectedness against failures in their components. In this paper we investigated how random and systematic failures in the edges of biological networks influenced their motif structure. We considered two biological networks, namely, protein structure network and human brain functional network. Furthermore, we considered random failures as well as systematic failures based on different strategies for choosing candidate edges for removal. Failure in the edges tipping to high degree nodes had the most destructive role in the motif structure of the networks by decreasing their significance level, while removing edges that were connected to nodes with high values of betweenness centrality had the least effect on the significance profiles. In some cases, the latter caused increase in the significance levels of the motifs.     

Resilience of protein-protein interaction networks as determined by their large-scale topological features

The relationship between the structure and function of biological networks constitutes a fundamental issue in systems biology. Particularly, the structure of protein-protein interaction networks is related to important biological functions. In this work, we investigated how such a resilience is determined by the large scale features of the respective networks. Four species are taken into account, namely yeast Saccharomyces cerevisiae, worm Caenorhabditis elegans, fly Drosophila melanogaster and Homo sapiens. We adopted two entropy-related measurements (degree entropy and dynamic entropy) in order to quantify the overall degree of robustness of these networks. We verified that while they exhibit similar structural variations under random node removal, they differ significantly when subjected to intentional attacks (hub removal). As a matter of fact, more complex species tended to exhibit more robust networks. More specifically, we quantified how six important measurements of the networks topology (namely clustering coefficient, average degree of neighbors, average shortest path length, diameter, assortativity coefficient, and slope of the power law degree distribution) correlated with the two entropy measurements. Our results revealed that the fraction of hubs and the average neighbor degree contribute significantly for the resilience of networks. In addition, the topological analysis of the removed hubs indicated that the presence of alternative paths between the proteins connected to hubs tend to reinforce resilience. The performed analysis helps to understand how resilience is underlain in networks and can be applied to the development of protein network models.     

油菜基因流网络的结构特性与小团体分析

为从系统角度掌握转基因油菜（Brassica campestris）基因流规律,基于常规油菜与十字花科植物杂交的有关文献,构建了油菜基因流网络拓扑图实例并分析其网络结构特性。研究结果表明,该网络节点度服从幂律分布,具有无标度特性。从随机攻击和恶性攻击两个方面对标准结构熵和网络效率2个指标的网络稳健性进行分析,结果显示在随机移除不到10%的顶点时网络表现较好的鲁棒性,但在受到选择性移除10%的顶点时,网络具有极弱的抗攻击性。基于软件UCINET对网络进行了小团体和结构同型性分析,可将网络中298个节点22类十字花科植物划分为2个小团体和5类结构角色,其中甘蓝型油菜在基因流网络小团体中具有关键性作用。这些研究结果可为揭示转基因油菜基因流规律提供新思路,同时也可对转基因油菜商业化种植采取合理的农业生产栽培管理措施提供参考。     

Robustness and state-space structure of Boolean gene regulatory models

Robustness to perturbation is an important characteristic of genetic regulatory systems, but the relationship between robustness and model dynamics has not been clearly quantified. We propose a method for quantifying both robustness and dynamics in terms of state-space structures, for Boolean models of genetic regulatory systems. By investigating existing models of the Drosophila melanogaster segment polarity network and the Saccharomyces cerevisiae cell-cycle network, we show that the structure of attractor basins can yield insight into the underlying decision making required of the system, and also the way in which the system maximises its robustness. In particular, gene networks implementing decisions based on a few genes have simple state-space structures, and their attractors are robust by virtue of their simplicity. Gene networks with decisions that involve many interacting genes have correspondingly more complicated state-space structures, and robustness cannot be achieved through the structure of the attractor basins, but is achieved by larger attractor basins that dominate the state space. These different types of robustness are demonstrated by the two models: the D. melanogaster segment polarity network is robust due to simple attractor basins that implement decisions based on spatial signals; the S. cerevisiae cell-cycle network has a complicated state-space structure, and is robust only due to a giant attractor basin that dominates the state space.     

Community Structure and Multi-Modal Oscillations in Complex Networks

In many types of network, the relationship between structure and function is of great significance. We are particularly interested in community structures, which arise in a wide variety of domains. We apply a simple oscillator model to networks with community structures and show that waves of regular oscillation are caused by synchronised clusters of nodes. Moreover, we show that such global oscillations may arise as a direct result of network topology. We also observe that additional modes of oscillation (as detected through frequency analysis) occur in networks with additional levels of topological hierarchy and that such modes may be directly related to network structure. We apply the method in two specific domains (metabolic networks and metropolitan transport) demonstrating the robustness of our results when applied to real world systems. We conclude that (where the distribution of oscillator frequencies and the interactions between them are known to be unimodal) our observations may be applicable to the detection of underlying community structure in networks, shedding further light on the general relationship between structure and function in complex systems.     

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.     

The Role of Community Mixing Styles in Shaping Epidemic Behaviors in Weighted Networks

The dynamics of infectious diseases that are spread through direct contact have been proven to depend on the strength of community structure or modularity within the underlying network. It has been recently shown that weighted networks with similar modularity values may exhibit different mixing styles regarding the number of connections among communities and their respective weights. However, the effect of mixing style on epidemic behavior was still unclear. In this paper, we simulate the spread of disease within networks with different mixing styles: a dense-weak style (i.e., many edges among the communities with small weights) and a sparse-strong style (i.e., a few edges among the communities with large weights). Simulation results show that, with the same modularity: 1) the mixing style significantly influences the epidemic size, speed, pattern and immunization strategy; 2) the increase of the number of communities amplifies the effect of the mixing style; 3) when the mixing style changes from sparse-strong to dense-weak, there is a ‘saturation point’, after which the epidemic size and pattern become stable. We also provide a mean-field solution of the epidemic threshold and size on weighted community networks with arbitrary external and internal degree distribution. The solution explains the effect of the second moment of the degree distribution, and a symmetric effect of internal and external connections (incl. degree distribution and weight). Our study has both potential significance for designing more accurate metrics for the community structure and exploring diffusion dynamics on metapopulation networks.     

Trade-Offs Between Efficiency and Robustness in Bacterial Metabolic Networks Are Associated with Niche Breadth

The relation between structure and function in biologic networks is a central point of systems biology research. Key functional features—notably, efficiency and robustness—are linked to the topologic structure of a network, and there appears to be a degree of trade-off between these features, i.e., simulation studies indicate that more efficient networks tend to be less robust. Here, we investigate this issue in metabolic networks from 105 lineages of bacteria having a wide range of ecologies. We take quantitative measurements on each network and integrate this network data with ecologic data using a phylogenetic comparative model. In this setting, we find that biologic conclusions obtained with classical phylogenetic comparative methods are sensitive to correlations between model covariates and phylogenetic branch length. To avoid this problem, we propose a revised statistical framework—hierarchical mixed-effect regression—to accommodate phylogenetic nonindependence. Using this approach, we show that the cartography of metabolic networks does indeed reflect a trade-off between efficiency and robustness. Furthermore, ecologic characteristics related to niche breadth are strong predictors of network shape. Given the broad variation in niche breadth seen among species, we predict that there is no universally optimal balance between efficiency and robustness in bacterial metabolic networks and, thus, no universally optimal network structure. These results highlight the biologic relevance of variation in network structure and the potential role of niche breadth in shaping metabolic strategies of efficiency and robustness. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users. Ransom A. Myers Died March 27th, 2007. He will be missed.     

Attack Robustness and Centrality of Complex Networks

Many complex systems can be described by networks, in which the constituent components are represented by vertices and the connections between the components are represented by edges between the corresponding vertices. A fundamental issue concerning complex networked systems is the robustness of the overall system to the failure of its constituent parts. Since the degree to which a networked system continues to function, as its component parts are degraded, typically depends on the integrity of the underlying network, the question of system robustness can be addressed by analyzing how the network structure changes as vertices are removed. Previous work has considered how the structure of complex networks change as vertices are removed uniformly at random, in decreasing order of their degree, or in decreasing order of their betweenness centrality. Here we extend these studies by investigating the effect on network structure of targeting vertices for removal based on a wider range of non-local measures of potential importance than simply degree or betweenness. We consider the effect of such targeted vertex removal on model networks with different degree distributions, clustering coefficients and assortativity coefficients, and for a variety of empirical networks.     

Effects of Ploidy and Recombination on Evolution of Robustness in a Model of the Segment Polarity Network

Many genetic networks are astonishingly robust to quantitative variation, allowing these networks to continue functioning in the face of mutation and environmental perturbation. However, the evolution of such robustness remains poorly understood for real genetic networks. Here we explore whether and how ploidy and recombination affect the evolution of robustness in a detailed computational model of the segment polarity network. We introduce a novel computational method that predicts the quantitative values of biochemical parameters from bit sequences representing genotype, allowing our model to bridge genotype to phenotype. Using this, we simulate 2,000 generations of evolution in a population of individuals under stabilizing and truncation selection, selecting for individuals that could sharpen the initial pattern of engrailed and wingless expression. Robustness was measured by simulating a mutation in the network and measuring the effect on the engrailed and wingless patterns; higher robustness corresponded to insensitivity of this pattern to perturbation. We compared robustness in diploid and haploid populations, with either asexual or sexual reproduction. In all cases, robustness increased, and the greatest increase was in diploid sexual populations; diploidy and sex synergized to evolve greater robustness than either acting alone. Diploidy conferred increased robustness by allowing most deleterious mutations to be rescued by a working allele. Sex (recombination) conferred a robustness advantage through “survival of the compatible”: those alleles that can work with a wide variety of genetically diverse partners persist, and this selects for robust alleles.     

A Novel Top-k Strategy for Influence Maximization in Complex Networks with Community Structure

In complex networks, it is of great theoretical and practical significance to identify a set of critical spreaders which help to control the spreading process. Some classic methods are proposed to identify multiple spreaders. However, they sometimes have limitations for the networks with community structure because many chosen spreaders may be clustered in a community. In this paper, we suggest a novel method to identify multiple spreaders from communities in a balanced way. The network is first divided into a great many super nodes and then k spreaders are selected from these super nodes. Experimental results on real and synthetic networks with community structure show that our method outperforms the classic methods for degree centrality, k-core and ClusterRank in most cases.     

Generating Attributed Networks with Communities

In many modern applications data is represented in the form of nodes and their relationships, forming an information network. When nodes are described with a set of attributes we have an attributed network. Nodes and their relationships tend to naturally form into communities or clusters, and discovering these communities is paramount to many applications. Evaluating algorithms or comparing algorithms for automatic discovery of communities requires networks with known structures. Synthetic generators of networks have been proposed for this task but most solely focus on connectivity and their properties and overlook attribute values and the network properties vis-à-vis these attributes. In this paper, we propose a new generator for attributed networks with community structure that dependably follows the properties of real world networks.     

Dynamic properties of network motifs contribute to biological network organization

Biological networks, such as those describing gene regulation, signal transduction, and neural synapses, are representations of large-scale dynamic systems. Discovery of organizing principles of biological networks can be enhanced by embracing the notion that there is a deep interplay between network structure and system dynamics. Recently, many structural characteristics of these non-random networks have been identified, but dynamical implications of the features have not been explored comprehensively. We demonstrate by exhaustive computational analysis that a dynamical property—stability or robustness to small perturbations—is highly correlated with the relative abundance of small subnetworks (network motifs) in several previously determined biological networks. We propose that robust dynamical stability is an influential property that can determine the non-random structure of biological networks.     

Food webs robustness to biodiversity loss: the roles of connectance, expansibility and degree distribution

We analyse the robustness of food webs against species loss by considering the influence of several structural factors of the networks, such as connectance, degree distribution and expansibility. The last concept refers to the absence of structural bottlenecks in the food web, whose removal separate the network into large isolate clusters. In theory networks with identical connectance can display different expansibility characteristics. Using the spectral scaling method we studied 17 food networks and classified them as good expansion (GE) and not-GE networks. The combination of GE properties and degree distribution of species permitted the classification of food webs into six different classes. These classes characterize the differences in robustness of food webs to species loss. While the webs having uniform degree distributions and displaying GE properties are the most robust to species loss, the presence of bottlenecks and skewed distribution of the number of links per species make food webs very vulnerable to primary removal of species.     

Robustness in phenotypic plasticity and heterogeneity patterns enabled by EMT networks

Epithelial-mesenchymal plasticity (EMP) is a key arm of cancer metastasis and is observed across many contexts. Cells undergoing EMP can reversibly switch between three classes of phenotypes: epithelial (E), mesenchymal (M), and hybrid E/M. While a large number of multistable regulatory networks have been identified to be driving EMP in various contexts, the exact mechanisms and design principles that enable robustness in driving EMP across contexts are not yet fully understood. Here, we investigated dynamic and structural robustness in EMP networks with regard to phenotypic heterogeneity and plasticity. We use two different approaches to simulate these networks: a computationally inexpensive, parameter-independent continuous state space Boolean model, and an ODE-based parameter-agnostic framework (RACIPE), both of which yielded similar phenotypic distributions. While the latter approach is useful for measurements of plasticity, the former model enabled us to extensively investigate robustness in phenotypic heterogeneity. Using perturbations to network topology and by varying network parameters, we show that multistable EMP networks are structurally and dynamically more robust compared with their randomized counterparts, thereby highlighting their topological hallmarks. These features of robustness are governed by a balance of positive and negative feedback loops embedded in these networks. Using a combination of the number of negative and positive feedback loops weighted by their lengths, we identified a metric that can explain the structural and dynamical robustness of these networks. This metric enabled us to compare networks across multiple sizes, and the network principles thus obtained can be used to identify fragilities in large networks without simulating their dynamics. Our analysis highlights a network topology-based approach to quantify robustness in the phenotypic heterogeneity and plasticity emergent from EMP networks.