Growing random networks with fitness

Three models of growing random networks with fitness-dependent growth rates are analysed using the rate equations for the distribution of their connectivities. In the first model (A), a network is built by connecting incoming nodes to nodes of connectivity k and random additive fitness η, with rate (k−1)+η. For η>0 we find the connectivity distribution is power law with exponent γ=〈η〉+2. In the second model (B), the network is built by connecting nodes to nodes of connectivity k, random additive fitness η and random multiplicative fitness ζ with rate ζ(k−1)+η. This model also has a power law connectivity distribution, but with an exponent which depends on the multiplicative fitness at each node. In the third model (C), a directed graph is considered and is built by the addition of nodes and the creation of links. A node with fitness (α,β), i incoming links and j outgoing links gains a new incoming link with rate α(i+1), and a new outgoing link with rate β(j+1). The distributions of the number of incoming and outgoing links both scale as power laws, with inverse logarithmic corrections.     

Assortativity of complementary graphs

Newman's measure for (dis)assortativity, the linear degree correlationρ _D , is widely studied although analytic insight into the assortativity of an arbitrary network remains far from well understood. In this paper, we derive the general relation (2), (3) and Theorem 1 between the assortativity ρ _D (G) of a graph G and the assortativityρ _D (G ^c) of its complement G ^c. Both ρ _D (G) and ρ _D (G ^c) are linearly related by the degree distribution in G. When the graph G(N,p) possesses a binomial degree distribution as in the Erd?s-Rényi random graphs G _p (N), its complementary graph G _p ^c (N) = G _{1- p} (N) follows a binomial degree distribution as in the Erd?s-Rényi random graphs G _{1- p} (N). We prove that the maximum and minimum assortativity of a class of graphs with a binomial distribution are asymptotically antisymmetric: ρ _max(N,p) = -ρ _min(N,p) for N → ∞. The general relation (3) nicely leads to (a) the relation (10) and (16) between the assortativity range ρ _max(G)–ρ _min(G) of a graph with a given degree distribution and the range ρ _max(G ^c)–ρ _min(G ^c) of its complementary graph and (b) new bounds (6) and (15) of the assortativity. These results together with our numerical experiments in over 30 real-world complex networks illustrate that the assortativity range ρ _max–ρ _min is generally large in sparse networks, which underlines the importance of assortativity as a network characterizer.     

Local preferential attachment model for hierarchical networks

In real-life networks, incomers may only connect to a few others in a local area for their limited information, and individuals in a local area are likely to have close relations. Accordingly, we propose a local preferential attachment model. Here, a local-area-network stands for a node and all its neighbors, and the new nodes perform nonlinear preferential attachment, , in local areas. The stable degree distribution and clustering-degree correlations are analytically obtained. With the increasing of α, the clustering coefficient increases, while assortativity decreases from positive to negative. In addition, by adjusting the parameter α, the model can generate different kinds of degree distribution, from exponential to power-law. The hierarchical organization, independent of α, is the most significant character of this model.     

确定度分布条件下可变同配系数的算法构造与影响分析

复杂网络是现实中大量节点和边的抽象拓扑, 如何揭示网络内部拓扑对网络连通性、脆弱性等特征的影响是当前研究的热点. 本文在确定度分布的条件下, 根据Newman提出的同配系数的定义分析其影响因素. 首先在可变同配系数下分别提出了基于度分布的确定算法和基于概率分布的不确定算法, 并分别在三种不同类型的网络(Erdös-Rényi网络, Barabási-Albert网络, Email真实网络)中验证. 实验结果表明: 当网络规模达到一定程度时, 确定算法优于贪婪算法. 以此为基础, 分析了同配系数改变时聚类系数的变化, 发现两者之间存在关联性, 并从网络的微观结构变化中揭示了聚类系数变化的原因.     

Two-population dynamics in a growing network model

We introduce a growing network evolution model with nodal attributes. The model describes the interactions between potentially violent V and non-violent N agents who have different affinities in establishing connections within their own population versus between the populations. The model is able to generate all stable triads observed in real social systems. In the framework of rate equations theory, we employ the mean-field approximation to derive analytical expressions of the degree distribution and the local clustering coefficient for each type of nodes. Analytical derivations agree well with numerical simulation results. The assortativity of the potentially violent network qualitatively resembles the connectivity pattern in terrorist networks that was recently reported. The assortativity of the network driven by aggression shows clearly different behavior than the assortativity of the networks with connections of non-aggressive nature in agreement with recent empirical results of an online social system.     

Quantifying structure in networks

We investigate exponential families of random graph distributions as a framework for systematic quantification of structure in networks. In this paper we restrict ourselves to undirected unlabeled graphs. For these graphs, the counts of subgraphs with no more than k links are a sufficient statistics for the exponential families of graphs with interactions between at most k links. In this framework we investigate the dependencies between several observables commonly used to quantify structure in networks, such as the degree distribution, cluster and assortativity coefficients.     

A Collaboration Network Model with Multiple Evolving Factors

To describe the empirical data of collaboration networks,several evolving mechanisms have been proposed,which usually introduce different dynamics factors controlling the network growth.These models can reasonably reproduce the empirical degree distributions for a number of well-studied real-world collaboration networks.On the basis of the previous studies,in this work we propose a collaboration network model in which the network growth is simultaneously controlled by three factors,including partial preferential attachment,partial random attachment and network growth speed.By using a rate equation method,we obtain an analytical formula for the act degree distribution.We discuss the dependence of the act degree distribution on these different dynamics factors.By fitting to the empirical data of two typical collaboration networks,we can extract the respective contributions of these dynamics factors to the evolution of each networks.     

Open Quantum Random Walks with Decoherence on Coins with n Degrees of Freedom

In this paper we define a new type of decoherent quantum random walk with parameter 0≤p≤1, which becomes a unitary quantum random walk (UQRW) when p=0 and an open quantum random walk (OQRW) when p=1, respectively. We call this process a partially open quantum random walk (POQRW). We study the limiting distribution of a POQRW on Z ¹ subject to decoherence on coins with n degrees of freedom. The limiting distribution of the POQRW converges to a convex combination of normal distributions, under an eigenvalue condition. A Perron-Frobenius type of theorem is established to determine whether or not a POQRW satisfies the eigenvalue condition. Moreover, we explicitly compute the limiting distributions of characteristic equations of the position probability functions when n=2 and 3.     

Evolving networks with bimodal degree distribution

Networks with bimodal degree distribution are most robust to targeted and random attacks. We present a model for constructing a network with bimodal degree distribution. The procedure adopted is to add nodes to the network with a probability p and delete the links between nodes with probability (1 − p). We introduce an additional constraint in the process through an immunity score, which controls the dynamics of the growth process based on the feedback value of the last few time steps. This results in bimodal nature for the degree distribution. We study the standard quantities which characterize the networks, like average path length and clustering coefficient in the context of our growth process and show that the resultant network is in the small world family. It is interesting to note that bimodality in degree distribution is an emergent phenomenon.     

A model for cascading failures in scale-free networks with a breakdown probability

Considering that not all overload nodes will be removed from networks due to some effective measures to protect them, we propose a new cascading model with a breakdown probability. Adopting the initial load of a node j to be L_j=[k_j(∑_m∈Γjk_m)]^α with k_j and Γ_j being the degree of the node j and the set of its neighboring nodes, respectively, where α is a tunable parameter, we investigate the relationship between some parameters and universal robustness characteristics against cascading failures on scale-free networks. According to a new measure originated from a phase transition from the normal state to collapse, the numerical simulations show that Barabási-Albert (BA) networks reach the strongest robustness level against cascading failures when the tunable parameter α=0.5, while not relating to the breakdown probability. We furthermore explore the effect of the average degree 〈k〉 for network robustness, thus obtaining a positive correlation between 〈k〉 and network robustness. We then analyze the effect of the breakdown probability on the network robustness and confirm by theoretical predictions this universal robustness characteristic observed in simulations. Our work may have practical implications for controlling various cascading-failure-induced disasters in the real world.     

Evolution of a Modified Binomial Random Graph by Agglomeration

In the classical Erd?s–Rényi random graph G(n, p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n, p) is homogeneous in the sense that all vertices have the same characteristics. On the other hand, numerous real-world networks are inhomogeneous in this respect. Such an inhomogeneity of vertices may influence the connection probability between pairs of vertices. The purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the Erd?s-Rényi random graph G(n, p). Given a configuration of n vertices arranged in N subsets of vertices (we call each subset a super-vertex), we define a random graph with N super-vertices by letting two super-vertices be connected if and only if there is at least one edge between them in G(n, p). Our main result concerns the threshold for connectedness. We also analyze the phase transition for the emergence of the giant component and the degree distribution. Even though our model begins with G(n, p), it assumes the existence of some community structure encoded in the configuration. Furthermore, under certain conditions it exhibits a power law degree distribution. Both properties are important for real-world applications.     

具有双峰特性的双层超网络模型

随着社会经济的快速发展,社会成员及群体之间的关系呈现出了更复杂、更多元化的特点.超网络作为一种描述复杂多元关系的网络,已在不同领域中得到了广泛的应用.服从泊松度分布的随机网络是研究复杂网络的开创性模型之一,而在现有的超网络研究中,基于ER随机图的超网络模型尚属空白.本文首先在基于超图的超网络结构中引入ER随机图理论,提出了一种ER随机超网络模型,对超网络中的节点超度分布进行了理论分析,并通过计算机仿真了在不同超边连接概率条件下的节点超度分布情况,结果表明节点超度分布服从泊松分布,符合随机网络特征并且与理论推导相一致.进一步,为更准确有效地描述现实生活中的多层、异质关系,本文构建了节点超度分布具有双峰特性,层间采用随机方式连接,层内分别为ER-ER,BA-BA和BA-ER三种不同类型的双层超网络模型,理论分析得到了三种双层超网络节点超度分布的解析表达式,三种双层超网络在仿真实验中的节点超度分布均具有双峰特性.     

An evolving model of online bipartite networks

Understanding the structure and evolution of online bipartite networks is a significant task since they play a crucial role in various e-commerce services nowadays. Recently, various attempts have been tried to propose different models, resulting in either power-law or exponential degree distributions. However, many empirical results show that the user degree distribution actually follows a shifted power-law distribution, the so-called Mandelbrot’s law, which cannot be fully described by previous models. In this paper, we propose an evolving model, considering two different user behaviors: random and preferential attachment. Extensive empirical results on two real bipartite networks, Delicious and CiteULike  , show that the theoretical model can well characterize the structure of real networks for both user and object degree distributions. In addition, we introduce a structural parameter p$p$, to demonstrate that the hybrid user behavior leads to the shifted power-law degree distribution, and the region of power-law tail will increase with the increment of p$p$. The proposed model might shed some lights in understanding the underlying laws governing the structure of real online bipartite networks.     

Lobby index in networks

We propose a new node centrality measure in networks, the lobby index, which is inspired by Hirsch’s h-index. It is shown that in scale-free networks with exponent α the distribution of the l-index has power tail with exponent α(α+1). Properties of the l-index and extensions are discussed.     

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.     

Scale-free networks by super-linear preferential attachment rule

A network growth model with geographic limitation of accessible information about the status of existing nodes is investigated. In this model, the probability Π(k) of an existing node of degree k is found to be super-linear with Π(k)∼k^α and α>1 when there are links from new nodes. The numerical results show that the constructed networks have typical power-law degree distributions P(k)∼k^−γ and the exponent γ depends on the constraint level. An analysis of local structural features shows the robust emergence of scale-free network structure in spite of the super-linear preferential attachment rule. This local structural feature is directly associated with the geographical connection constraints which are widely observed in many real networks.     

Evolutionary games on scale-free networks with a preferential selection mechanism

Considering the heterogeneity of individuals’ influence in the real world, we introduce a preferential selection mechanism to evolutionary games (the Prisoner’s Dilemma Game and the Snowdrift Game) on scale-free networks and focus on the cooperative behavior of the system. In every step, each agent chooses an individual from all its neighbors with a probability proportional to k^α indicating the influence of the neighbor, where k is the degree. Simulation results show that the cooperation level has a non-trivial dependence on α. To understand the effect of preferential selection mechanism on the evolution of the system, we investigate the time series of the cooperator frequency in detail. It is found that the cooperator frequency is greatly influenced by the initial strategy of hub nodes when α>0. This observation is confirmed by investigating the system behavior when some hub nodes’ strategies are fixed.     

Degree correlation in scale-free graphs

We obtain closed form expressions for the expected conditional degree distribution and the joint degree distribution of the linear preferential attachment model for network growth in the steady state. We consider the multiple-destination preferential attachment growth model, where incoming nodes at each timestep attach to β existing nodes, selected by degree-proportional probabilities. By the conditional degree distribution p(?|k), we mean the degree distribution of nodes that are connected to a node of degree k. By the joint degree distribution p(k,?), we mean the proportion of links that connect nodes of degrees k and ?. In addition to this growth model, we consider the shifted-linear preferential growth model and solve for the same quantities, as well as a closed form expression for its steady-state degree distribution.     

Random removal of edges from scale free graphs

It has been discovered that many naturally occurring networks (the internet, the power grid of the western US, various biological networks, etc.) satisfy a power-law degree distribution. Such scale-free networks have many interesting properties, one of which is robustness to random damage. This problem has been analyzed from the point of view of node deletion and connectedness. Recently, it has also been considered from the point of view of node deletion and scale preservation. In this paper we consider the problem from the point of view of edge deletion and scale preservation. In agreement with the work on node deletion and scale preservation, we show that a scale-free graph should not be expected to remain scale free when edges are removed at random.     

Distinct Clusterings and Characteristic Path Lengths in Dynamic Small-World Networks with Identical Limit Degree Distribution

Many real-world networks belong to a particular class of structures, known as small-world networks, that display short distance between pair of nodes. In this paper, we introduce a simple family of growing small-world networks where both addition and deletion of edges are possible. By tuning the deletion probability q _t , the model undergoes a transition from large worlds to small worlds. By making use of analytical or numerical means we determine the degree distribution, clustering coefficient and average path length of our networks. Surprisingly, we find that two similar evolving mechanisms, which provide identical degree distribution under a reciprocal scaling as t goes to infinity, can lead to quite different clustering behaviors and characteristic path lengths. It is also worth noting that Farey graphs constitute the extreme case q _t ??0 of our random construction.