Preservation of network degree distributions from non-uniform failures

There has been a considerable amount of interest in recent years on the robustness of networks to failures. Many previous studies have concentrated on the effects of node and edge removals on the connectivity structure of a static network; the networks are considered to be static in the sense that no compensatory measures are allowed for recovery of the original structure. Real world networks such as the world wide web, however, are not static and experience a considerable amount of turnover, where nodes and edges are both added and deleted. Considering degree-based node removals, we examine the possibility of preserving networks from these types of disruptions. We recover the original degree distribution by allowing the network to react to the attack by introducing new nodes and attaching their edges via specially tailored schemes. We focus particularly on the case of non-uniform failures, a subject that has received little attention in the context of evolving networks. Using a combination of analytical techniques and numerical simulations, we demonstrate how to preserve the exact degree distribution of the studied networks from various forms of attack.     

Weighted Networks Model Based on Traffic Dynamics with Local Perturbation

In the study of weighted complex networks, the interplay between traffic and topology have been paid much attention. However, the variation of topology and weight brought by new added vertices or edges should also be considered. In this paper, an evolution model of weighted networks driven by traffic dynamics with local perturbation is proposed. The model gives power-law distribution of degree, weight and strength, as confirmed by empirical measurements. By choosing appropriate parameters W and δ, the exponents of various power law distributions can be adjusted to meet real world networks. Nontrivial clustering coefficient C, degree assortativity coefficient r, and strength-degree correlation are also considered. What should be emphasized is that, with the consideration of local perturbation, one can adjust the exponent of strength-degree correlation more effectively. It makes our model more general than previous ones and may help reproducing real world networks more appropriately. PACS numbers： 87.23.Kg, 89.75.Da, 89.75.Fb, 89.75.Hc.     

一类权重网络的加速演化模型

采用动态形成权重网络的方法，研究了在演化过程中新增边具有加速连接情况下权重网络的拓扑特性和强度分布，给出了节点强度与度的解析表达式.分析表明，加速演化的权重模型具有明显的无标度特性.再者，只要权重网络的边权重服从某一概率分布，则在演化过程中强度择优连接与度择优连接对于网络的度分布没有影响，且与具体的概率分布无关.     

A local-world evolving hypernetwork model

Complex hypernetworks are ubiquitous in the real system. It is very important to investigate the evolution mechanisms. In this paper, we present a local-world evolving hypernetwork model by taking into account the hyperedge growth and local-world hyperedge preferential attachment mechanisms. At each time step, a newly added hyperedge encircles a new coming node and a number of nodes from a randomly selected local world. The number of the selected nodes from the local world obeys the uniform distribution and its mean value is m. The analytical and simulation results show that the hyperdegree approximately obeys the power-law form and the exponent of hyperdegree distribution is γ = 2 + 1/m. Furthermore, we numerically investigate the node degree, hyperedge degree, clustering coefficient, as well as the average distance, and find that the hypernetwork model shares the scale-free and small-world properties, which shed some light for deeply understanding the evolution mechanism of the real systems.     

一类高聚类系数的加权无标度网络及其同步能力分析

针对真实世界中大规模网络都具有明显聚类效应的特点, 提出一类具有高聚类系数的加权无标度网络演化模型, 该模型同时考虑了优先连接、三角结构、随机连接和社团结构等四种演化机制. 在模型演化规则中, 以概率p增加单个节点, 以概率1–p增加一个社团. 与以往研究的不同在于新边的建立, 以概率φ在旧节点之间进行三角连接, 以概率1–φ进行随机连接. 仿真分析表明, 所提出的网络度、强度和权值分布都是服从幂律分布的形式, 且具有高聚类系数的特性, 聚类系数的提高与社团结构和随机连接机制有直接的关系. 最后通过数值仿真分析了网络演化机制对同步动态特性的影响, 数值仿真结果表明, 网络的平均聚类系数越小, 网络的同步能力越强. 关键词： 无标度网络 加权网络 聚类系数 同步能力     

蛋白质相互作用网络特征的理论再现

无标度性、小世界性、功能模块结构及度负关联性是大量生物网络共同的特征. 为了理解生物网络无标度性、小世界性和度负关联性的形成机制, 研究者已经提出了各种各样基于复制和变异的网络增长模型. 在本文中,我们从生物学的角度通过引入偏爱小复制原则及变异和非均匀的异源二聚作用构建了一个简单的蛋白质相互作用网络演化模型.数值模拟结果表明,该演化模型几乎可以再现现在实测结果所公认的蛋白质相互作用网络的性质：无标度性、小世界性、度负关联性和功能模块结构. 我们的演化模型对理解蛋白质相互作用网络演化过程中的可能机制提供了一定的帮助. 关键词： 蛋白质相互作用网络 偏爱小 非均匀的异源二聚作用 功能模块结构     

Emergence of modularity and disassortativity in protein-protein interaction networks

In this paper, we present a simple evolution model of protein-protein interaction networks by introducing a rule of small-preference duplication of a node, meaning that the probability of a node chosen to duplicate is inversely proportional to its degree, and subsequent divergence plus nonuniform heterodimerization based on some plausible mechanisms in biology. We show that our model cannot only reproduce scale-free connectivity and small-world pattern, but also exhibit hierarchical modularity and disassortativity. After comparing the features of our model with those of real protein-protein interaction networks, we believe that our model can provide relevant insights into the mechanism underlying the evolution of protein-protein interaction networks.     

How divergence mechanisms influence disassortative mixing property in biology

The duplication–divergence mechanism of network growth has been widely investigated, especially in gene and protein networks. Both the duplication and divergence have a key role in biological network evolution. However, the relative roles of these mechanisms in the influence of disassortative property in protein interaction networks remain to be clarified. It has been shown that duplication can indeed make protein networks evolve towards disassortative networks. To make the relationship between the disassortative property and the duplication–divergence mechanism more clear, we further discuss how the divergence mechanism influences the disassortative property. We tested four different divergence mechanisms, i.e., node deletion, edge deletion, edge addition, and edge rewiring to study their effects on disassortative property. Our study highlights the crucial roles of different divergence evolution mechanisms.     

Nonlinear growth in weighted networks with neighborhood preferential attachment

We propose a nonlinear growing model for weighted networks with two significant characteristics: (i) the new weights triggered by new edges at each time step grow nonlinearly with time; and (ii) a neighborhood local-world exists for local preferential attachment, which is defined as one selected node and its neighbors. Global strength-driven and local weight-driven preferential attachment mechanisms are involved in our model. We study the evolution process through both mathematical analysis and numerical simulation, and find that the model exhibits a wide-range power-law distribution for node degree, strength, and weight. In particular, a nonlinear degree–strength relationship is obtained. This nonlinearity implies that accelerating growth of new weights plays a nontrivial role compared with accelerating growth of edges. Because of the specific local-world model, a small-world property emerges, and a significant hierarchical organization, independent of the parameters, is observed.     

Are networks with more edges easier to synchronize,or not？

In this paper, the relationship between network synchronizability and the edge-addition of its associated graph is investigated. First, it is shown that adding one edge to a cycle definitely decreases the network synchronizability. Then, since sometimes the synchronizability can be enhanced by changing the network structure, the question of whether the networks with more edges are easier to synchronize is addressed. Based on a subgraph and complementary graph method, it is shown by examples that the answer is negative even if the network structure is arbitrarily optimized. This reveals that generally there are redundant edges in a network, which not only make no contributions to synchronization but actually may reduce the synchronizability. Moreover, a simple example shows that the node betweenness centrality is not always a good indicator for the network synchronizability. Finally, some more examples are presented to illustrate how the network synchronizability varies following the addition of edges, where all the examples show that the network synchronizability globally increases but locally fluctuates as the number of added edges increases.     

Constraints and entropy in a model of network evolution

Barabási–Albert’s “Scale Free” model is the starting point for much of the accepted theory of the evolution of real world communication networks. Careful comparison of the theory with a wide range of real world networks, however, indicates that the model is in some cases, only a rough approximation to the dynamical evolution of real networks. In particular, the exponent γ of the power law distribution of degree is predicted by the model to be exactly 3, whereas in a number of real world networks it has values between 1.2 and 2.9. In addition, the degree distributions of real networks exhibit cut offs at high node degree, which indicates the existence of maximal node degrees for these networks. In this paper we propose a simple extension to the “Scale Free” model, which offers better agreement with the experimental data. This improvement is satisfying, but the model still does not explain why the attachment probabilities should favor high degree nodes, or indeed how constraints arrive in non-physical networks. Using recent advances in the analysis of the entropy of graphs at the node level we propose a first principles derivation for the “Scale Free” and “constraints” model from thermodynamic principles, and demonstrate that both preferential attachment and constraints could arise as a natural consequence of the second law of thermodynamics.     

Scale-free network growth by ranking

Network growth is currently explained through mechanisms that rely on node prestige measures, such as degree or fitness. In many real networks, those who create and connect nodes do not know the prestige values of existing nodes but only their ranking by prestige. We propose a criterion of network growth that explicitly relies on the ranking of the nodes according to any prestige measure, be it topological or not. The resulting network has a scale-free degree distribution when the probability to link a target node is any power-law function of its rank, even when one has only partial information of node ranks. Our criterion may explain the frequency and robustness of scale-free degree distributions in real networks, as illustrated by the special case of the Web graph.     

Modular and hierarchical structure of social contact networks

Social contact networks exhibit overlapping qualities of communities, hierarchical structure and spatial-correlated nature. We propose a mixing pattern of modular and growing hierarchical structures to reconstruct social contact networks by using an individual’s geospatial distribution information in the real world. The hierarchical structure of social contact networks is defined based on the spatial distance between individuals, and edges among individuals are added in turn from the modular layer to the highest layer. It is a gradual process to construct the hierarchical structure: from the basic modular model up to the global network. The proposed model not only shows hierarchically increasing degree distribution and large clustering coefficients in communities, but also exhibits spatial clustering features of individual distributions. As an evaluation of the method, we reconstruct a hierarchical contact network based on the investigation data of a university. Transmission experiments of influenza H1N1 are carried out on the generated social contact networks, and results show that the constructed network is efficient to reproduce the dynamic process of an outbreak and evaluate interventions. The reproduced spread process exhibits that the spatial clustering of infection is accordant with the clustering of network topology. Moreover, the effect of individual topological character on the spread of influenza is analyzed, and the experiment results indicate that the spread is limited by individual daily contact patterns and local clustering topology rather than individual degree.     

Detecting hierarchical and overlapping network communities using locally optimal modularity changes

Agglomerative clustering is a well established strategy for identifying communities in networks. Communities are successively merged into larger communities, coarsening a network of actors into a more manageable network of communities. The order in which merges should occur is not in general clear, necessitating heuristics for selecting pairs of communities to merge. We describe a hierarchical clustering algorithm based on a local optimality property. For each edge in the network, we associate the modularity change for merging the communities it links. For each community vertex, we call the preferred edge that edge for which the modularity change is maximal. When an edge is preferred by both vertices that it links, it appears to be the optimal choice from the local viewpoint. We use the locally optimal edges to define the algorithm: simultaneously merge all pairs of communities that are connected by locally optimal edges that would increase the modularity, redetermining the locally optimal edges after each step and continuing so long as the modularity can be further increased. We apply the algorithm to model and empirical networks, demonstrating that it can efficiently produce high-quality community solutions. We relate the performance and implementation details to the structure of the resulting community hierarchies. We additionally consider a complementary local clustering algorithm, describing how to identify overlapping communities based on the local optimality condition.     

Edge-based-attack induced cascading failures on scale-free networks

Most previous existing works on cascading failures only focused on attacks on nodes rather than on edges. In this paper, we discuss the response of scale-free networks subject to two different attacks on edges during cascading propagation, i.e., edge removal by either the descending or ascending order of the loads. Adopting a cascading model with a breakdown probability p of an overload edge and the initial load (k_ik_j)^α of an edge ij, where k_i and k_j are the degrees of the nodes connected by the edge ij and α is a tunable parameter, we investigate the effects of two attacks for the robustness of Barabási-Albert (BA) scale-free networks against cascading failures. In the case of α<1, our investigation by the numerical simulations leads to a counterintuitive finding that BA scale-free networks are more sensitive to attacks on the edges with the lowest loads than the ones with the highest loads, not relating to the breakdown probability. In addition, the same effect of two attacks in the case of α=1 may be useful in furthering studies on the control and defense of cascading failures in many real-life networks. We then confirm by the theoretical analysis these results observed in simulations.     

Recursive filtration method for detecting community structure in networks

Community detection is a topic of considerable recent interest within complex networks, but most methods proposed so far are divisive and agglomerative methods which delete only one edge each time to split the network, or agglomerating only one node each time until no individual node remains. Unlike those, we propose a method to split networks in parallel by deleting many edges in each filtration operation, and propose a community recursive coefficient (CRC) denoted by M instead of Q (modularity) to quantify the effect of the splitting results in this paper. We proved that recursive optimizing of the local M is equivalent to acquiring the maximal global Q value corresponding to good divisions. For a network with m edges, c communities and arbitrary topology, the method split the network at most c+1 times and detected the community structure in time O(m²+(c+1)m). We give several example applications, and show that the method can detect local communities according to the densities of external links to them in increasing order especially in large networks.     

A novel scheme based on minimum delay at the edges for optical burst switching networks

This paper proposes a novel scheme based on minimum delay at the edges (MDE) for optical burst switching (OBS) networks. This scheme is designed to overcome the long delay at the edge nodes of OBS networks. The MDE scheme features simultaneous burst assembly, channel scheduling, and pre-transmission of control packet. It also features estimated setup and explicit release (ESXR) signaling protocol. The MDE scheme can minimize the delay at the edge nodes for data packets, and improve the end-to-end latency performance for OBS networks. In addition, comparing with the conventional scheme, the performances of the MDE scheme are analyzed in this paper.     

Kinetics of node splitting in evolving complex networks

We introduce a collection of complex networks generated by a combination of preferential attachment and a previously unexamined process of “splitting” nodes of degree k$k$ into k$k$ nodes of degree 1$1$. Four networks are considered, each evolves at each time step by either preferential attachment, with probability p$p$, or splitting with probability 1−p$1-p$. Two methods of attachment are considered; first, attachment of an edge between a newly created node and an existing node in the network, and secondly by attachment of an edge between two existing nodes. Splitting is also considered in two separate ways; first by selecting each node with equal probability and secondly, selecting the node with probability proportional to its degree. Exact solutions for the degree distributions are found and scale-free structure is exhibited in those networks where the candidates for splitting are chosen with uniform probability, those that are chosen preferentially are distributed with a power law with exponential cut-off.     

Hierarchical Characterization of Complex Networks

While the majority of approaches to the characterization of complex networks has relied on measurements considering only the immediate neighborhood of each network node, valuable information about the network topological properties can be obtained by considering further neighborhoods. The current work considers the concept of virtual hierarchies established around each node and the respectively defined hierarchical node degree and clustering coefficient (introduced in cond-mat/0408076), complemented by new hierarchical measurements, in order to obtain a powerful set of topological features of complex networks. The interpretation of such measurements is discussed, including an analytical study of the hierarchical node degree for random networks, and the potential of the suggested measurements for the characterization of complex networks is illustrated with respect to simulations of random, scale-free and regular network models as well as real data (airports, proteins and word associations). The enhanced characterization of the connectivity provided by the set of hierarchical measurements also allows the use of agglomerative clustering methods in order to obtain taxonomies of relationships between nodes in a network, a possibility which is also illustrated in the current article.