Rank-based model for weighted network with hierarchical organization and disassortative mixing

In this paper, we study a rank-based model for weighted network. The evolution rule of the network is based on the ranking of node strength, which couples the topological growth and the weight dynamics. Analytically and by simulations, we demonstrate that the generated networks recover the scale-free distributions of degree and strength in the whole region of the growth dynamics parameter (α>0). Moreover, this network evolution mechanism can also produce scale-free property of weight, which adds deeper comprehension of the networks growth in the presence of incomplete information. We also characterize the clustering and correlation properties of this class of networks. It is showed that at α=1 a structural phase transition occurs, and for α>1 the generated network simultaneously exhibits hierarchical organization and disassortative degree correlation, which is consistent with a wide range of biological networks.     

Modelling Human Cortical Network in Real Brain Space

Highly specific structural organization is of great significance in the topology of cortical networks. We introduce a human cortical network model, taking the specific cortical structure into account, in which nodes are brain sites placed in the actual positions of cerebral cortex and the establishment of edges depends on the spatial path length rather than the linear distance. The resulting network exhibits the essential features of cortical connectivity, properties of small-world networks and multiple clusters structure. Additionally, assortative mixing is also found in this model. All of these findings may be attributed to the specific cortical architecture.     

Duplication： a Mechanism Producing Disassortative Mixing Networks in Biology

Assortative/disassortative mixing is an important topological property of a network. A network is called assortative mixing if the nodes in the network tend to connect to their connectivity peers, or disassortative mixing if nodes with low degrees are more likely to connect with high-degree nodes. We have known that biological networks such as protein-protein interaction networks （PPI）, gene regulatory networks, and metabolic networks tend to be disassortative. On the other hand, in biological evolution, duplication and divergence are two fundamental processes. In order to make the relationship between the property of disassortative mixing and the two basic biological principles clear and to study the cause of the disassortative mixing property in biological networks, we present a random duplication model and an anti-preference duplication model. Our results show that disassortative mixing networks can be obtained by both kinds of models from uncorrelated initial networks. Moreover, with the growth of the network size, the disassortative mixing property becomes more obvious.     

Graph kernels,hierarchical clustering,and network community structure: experiments and comparative analysis

There has been a quickly growing interest in properties of complex networks, such as the small world property, power-law degree distribution, network transitivity, and community structure, which seem to be common to many real world networks. In this study, we consider the community property which is also found in many real networks. Based on the diffusion kernels of networks, a hierarchical clustering approach is proposed to uncover the community structure of different extent of complex networks. We test the method on some networks with known community structures and find that it can detect significant community structure in these networks. Comparison with related methods shows the effectiveness of the method.     

Ordered community structure in networks

Community structure in networks is often a consequence of homophily, or assortative mixing, based on some attribute of the vertices. For example, researchers may be grouped into communities corresponding to their research topic. This is possible if vertex attributes have unordered discrete values, but many networks exhibit assortative mixing by some ordered (discrete or continuous) attribute, such as age or geographical location. In such cases, the identification of discrete communities may be difficult or impossible. We consider how the notion of community structure can be generalized to networks that have assortative mixing by ordered attributes. We propose a method of generating synthetic networks with ordered communities and investigate the effect of ordered community structure on the spread of infectious diseases. We also show that current community detection algorithms fail to recover community structure in ordered networks, and evaluate an alternative method using a layout algorithm to recover the ordering.     

Detecting community structure from coherent oscillation of excitable systems

Many networks are proved to have community structures. On the basis of the fact that the dynamics on networks are intensively affected by the related topology, in this paper the dynamics of excitable systems on networks and a corresponding approach for detecting communities are discussed. Dynamical networks are formed by interacting neurons; each neuron is described using the FHN model. For noisy disturbance and appropriate coupling strength, neurons may oscillate coherently and their behavior is tightly related to the community structure. Synchronization between nodes is measured in terms of a correlation coefficient based on long time series. The correlation coefficient matrix can be used to project network topology onto a vector space. Then by the K-means cluster method, the communities can be detected. Experiments demonstrate that our algorithm is effective at discovering community structure in artificial networks and real networks, especially for directed networks. The results also provide us with a deep understanding of the relationship of function and structure for dynamical networks.     

Concentric circular arrays of Josephson junctions as dressed models of granular superconductors

In this paper networks that optimize a combined measure of local and global synchronizability are evolved. It is shown that for low coupling improvements in the local synchronizability dominate network evolution. This leads to an expressed grouping of elements with similar native frequency into cliques, allowing for an early onset of synchronization, but rendering full synchronization hard to achieve. In contrast, for large coupling the network evolution is governed by improvements towards full synchronization, preventing any expressed community structure. Such networks exhibit strong coupling between dissimilar oscillators. Albeit a rapid transition to full synchronization is achieved, the onset of synchronization is delayed in comparison to the first type of networks. The paper illustrates that an early onset of synchronization (which relates to clustering) and global synchronization are conflicting demands on network topology.     

An empirical study of Chinese language networks

Chinese is spoken by the largest number of people in the world, and it is regarded as one of the most important languages. In this paper, we explore the statistical properties of Chinese language networks (CLNs) within the framework of complex network theory. Based on one of the largest Chinese corpora, i.e. People’s Daily Corpus, we construct two networks (CLN1 and CLN2) from two different respects, with Chinese words as nodes. In CLN1, a link between two nodes exists if they appear next to each other in at least one sentence; in CLN2, a link represents that two nodes appear simultaneously in a sentence. We show that both networks exhibit small-world effect, scale-free structure, hierarchical organization and disassortative mixing. These results indicate that in many topological aspects Chinese language shapes complex networks with organizing principles similar to other previously studied language systems, which shows that different languages may have some common characteristics in their evolution processes. We believe that our research may shed some new light into the Chinese language and find some potentially significant implications.     

Can a few fanatics influence the opinion of a large segment of a society?

Models that provide insight into how extreme positions regarding any social phenomenon may spread in a society or at the global scale are of great current interest. A realistic model must account for the fact that globalization, internet, and other means of mass communications have given rise to scale-free networks of interactions between people. We propose a novel model which takes into account the nature of the interactions network, and provides some key insights into this phenomenon. These include, (1) the existence of a fundamental difference between a hierarchical network whereby people are influenced by those that are higher in the hierarchy but not by those below them, and a symmetrical network where person-on-person influence works mutually, and (2) that a few “fanatics” can influence a large fraction of the population either temporarily (in the hierarchical networks) or permanently (in symmetrical networks). Even if the “fanatics” disappear, the population may still remain susceptible to the positions originally advocated by them. The model is, however, general and applicable to any phenomenon for which there is a degree of enthusiasm or susceptibility to in the population.     

Detect overlapping and hierarchical community structure in networks

Clustering and community structure is crucial for many network systems and the related dynamic processes. It has been shown that communities are usually overlapping and hierarchical. However, previous methods investigate these two properties of community structure separately. This paper proposes an algorithm (EAGLE) to detect both the overlapping and hierarchical properties of complex community structure together. This algorithm deals with the set of maximal cliques and adopts an agglomerative framework. The quality function of modularity is extended to evaluate the goodness of a cover. The examples of application to real world networks give excellent results.     

Inevitable self-similar topology of binary trees and their diverse hierarchical density

Self-similar topology, which can be characterized as power law size distribution, has been found in diverse tree networks ranging from river networks to taxonomic trees. In this study, we find that the statistical self-similar topology is an inevitable consequence of any full binary tree organization. We show this by coding a binary tree as a unique bifurcation string. This coding scheme allows us to investigate trees over the realm from deterministic to entirely random trees. To obtain partial random trees, partial random perturbation is added to the deterministic trees by an operator similar to that used in genetic algorithms. Our analysis shows that the hierarchical density of binary trees is more diverse than has been described in earlier studies. We find that the connectivity structure of river networks is far from strict self-similar trees. On the other hand, organization of some social networks is close to deterministic supercritical trees.     

Network robustness to targeted attacks. The interplay of expansibility and degree distribution

We study the property of certain complex networks of being both sparse and highly connected, which is known as “good expansion” (GE). A network has GE properties if every subset S of nodes (up to 50% of the nodes) has a neighborhood that is larger than some “expansion factor” φ multiplied by the number of nodes in S. Using a graph spectral method we introduce here a new parameter measuring the good expansion character of a network. By means of this parameter we are able to classify 51 real-world complex networks — technological, biological, informational, biological and social — as GENs or non-GENs. Combining GE properties and node degree distribution (DD) we classify these complex networks in four different groups, which have different resilience to intentional attacks against their nodes. The simultaneous existence of GE properties and uniform degree distribution contribute significantly to the robustness in complex networks. These features appear solely in 14% of the 51 real-world networks studied here. At the other extreme we find that ∼40% of all networks are very vulnerable to targeted attacks. They lack GE properties, display skewed DD — exponential or power-law — and their topologies are changed more dramatically by targeted attacks directed at bottlenecks than by the removal of network hubs.     

Assortative mixing in networks

A network is said to show assortative mixing if the nodes in the network that have many connections tend to be connected to other nodes with many connections. Here we measure mixing patterns in a variety of networks and find that social networks are mostly assortatively mixed, but that technological and biological networks tend to be disassortative. We propose a model of an assortatively mixed network, which we study both analytically and numerically. Within this model we find that networks percolate more easily if they are assortative and that they are also more robust to vertex removal.     

A limited resource model of fault-tolerant capability against cascading failure of complex network

We propose a novel capacity model for complex networks against cascading failure. In this model, vertices with both higher loads and larger degrees should be paid more extra capacities, i.e. the allocation of extra capacity on vertex i will be proportional to k_i ^γ , where k_i is the degree of vertex i and γ > 0 is a free parameter. We have applied this model on Barabási-Albert network as well as two real transportation networks, and found that under the same amount of available resource, this model can achieve better network robustness than previous models.     

Epidemic outbreaks in growing scale-free networks with local structure

The class of generative models has already attracted considerable interest from researchers in recent years and much expanded the original ideas described in BA model. Most of these models assume that only one node per time step joins the network. In this paper, we grow the network by adding n interconnected nodes as a local structure into the network at each time step with each new node emanating m new edges linking the node to the preexisting network by preferential attachment. This successfully generates key features observed in social networks. These include power-law degree distribution p_k∼k^−(3+μ), where μ=(n−1)/m is a tuning parameter defined as the modularity strength of the network, nontrivial clustering, assortative mixing, and modular structure. Moreover, all these features are dependent in a similar way on the parameter μ. We then study the susceptible-infected epidemics on this network with identical infectivity, and find that the initial epidemic behavior is governed by both of the infection scheme and the network structure, especially the modularity strength. The modularity of the network makes the spreading velocity much lower than that of the BA model. On the other hand, increasing the modularity strength will accelerate the propagation velocity.     

Is the topology of the Internet network really fit to sustain its function?

The Internet is one of the most interesting realizations of a “complex” network. As a non-supervised growing object, it allows the study of the selective pressure which drives the network to assume its current structure. The DIMES and the ROUTEVIEWS projects are ongoing projects aimed at evaluating the topological structure of the Internet (at the Autonomous System or AS grain-level) on the basis of different types of measurements. The topological analysis of the networks produced by the two projects has allowed us to infer a growth mechanism which has been used to build up synthetic networks with similar properties. These networks have been used as test-beds for the implementation of a model of traffic dynamics, with the aim of assessing the ability of the Internet’s topology to support the basic actions for data traffic handling. Results have been compared with those obtained by using a random network of similar size. The effects of some structural perturbations (arcs and nodes’ removal, traffic localization) have been also evaluated in terms of the induced variations of the network’s efficiency. The resulting scenario is consistent with the hypothesis that the structure of the Internet is only partially fit to host communication processes and that the intelligence of the TCP/IP protocol is partly needed to overcome some “structural” deficiencies.     

How non-uniform tolerance parameter strategy changes the response of scale-free networks to failures

In this paper, we introduce a non-uniform tolerance parameter (TP) strategy (the tolerance parameter is characterized by the proportion between the unused capacity and the capacity of a vertex) and study how the non-uniform TP strategy influences the response of scale-free (SF) networks to cascading failures. Different from constant TP in previous work of Motter and Lai (ML), the TP in the proposed strategy scales as a power-law function of vertex degree with an exponent b. The simulations show that under low construction costs D, when b>0 the tolerance of SF networks can be greatly improved, especially at moderate values of b; When b<0 the tolerance gets worse, compared with the case of constant TP in the ML model. While for high D the tolerance declines slightly with the b, namely b<0 is helpful to the tolerance, and b>0 is harmful. Because for smaller b the cascade of the network is mainly induced by failures of most high-degree vertices; while for larger b, the cascade attributes to damage of most low-degree vertices. Furthermore, we find that the non-uniform TP strategy can cause changes of the structure and the load-degree correlation in the network after the cascade. These results might give insights for the design of both network capacity to improve network robustness under limitation of small cost, and for the design of strategies to defend cascading failures of networks.     

Random field Ising model and community structure in complex networks

We propose a method to determine the community structure of a complex network. In this method the ground state problem of a ferromagnetic random field Ising model is considered on the network with the magnetic field B_s = +∞, B_t = -∞, and B_i≠s,t=0 for a node pair s and t. The ground state problem is equivalent to the so-called maximum flow problem, which can be solved exactly numerically with the help of a combinatorial optimization algorithm. The community structure is then identified from the ground state Ising spin domains for all pairs of s and t. Our method provides a criterion for the existence of the community structure, and is applicable equally well to unweighted and weighted networks. We demonstrate the performance of the method by applying it to the Barabási-Albert network, Zachary karate club network, the scientific collaboration network, and the stock price correlation network. (Ising, Potts, etc.)     

An evolutionary model of social networks

Social networks in communities, markets, and societies self-organise through the interactions of many individuals. In this paper we use a well-known mechanism of social interactions — the balance of sentiment in triadic relations — to describe the development of social networks. Our model contrasts with many existing network models, in that people not only establish but also break up relations whilst the network evolves. The procedure generates several interesting network features such as a variety of degree distributions and degree correlations. The resulting network converges under certain conditions to a steady critical state where temporal disruptions in triangles follow a power-law distribution.     

Local affinity in heterogeneous growing networks

In this paper we present a study of the influence of local affinity in heterogeneous preferential attachment (PA) networks. Heterogeneous PA models are a generalization of the Barabási-Albert model to heterogeneous networks, where the affinity between nodes biases the attachment probability of links. Threshold models are a class of heterogeneous PA models where the affinity between nodes is inversely related to the distance between their states. We propose a generalization of threshold models where network nodes have individual affinity functions, which are then combined to yield the affinity of each potential interaction. We analyze the influence of the affinity functions in the topological properties averaged over a network ensemble. The network topology is evaluated through the distributions of connectivity degrees, clustering coefficients and geodesic distances. We show that the relaxation of the criterion of a single global affinity still leads to a reasonable power-law scaling in the connectivity and clustering distributions under a wide spectrum of assumptions. We also show that the richer behavior of the model often exhibits a better agreement with the empirical observations on real networks.