Symmetry-based structure entropy of complex networks

Precisely quantifying the heterogeneity or disorder of network systems is important and desired in studies of behaviors and functions of network systems. Although various degree-based entropies have been available to measure the heterogeneity of real networks, heterogeneity implicated in the structures of networks can not be precisely quantified yet. Hence, we propose a new structure entropy based on automorphism partition. Analysis of extreme cases shows that entropy based on automorphism partition can quantify the structural heterogeneity of networks more precisely than degree-based entropies. We also summarized symmetry and heterogeneity statistics of many real networks, finding that real networks are more heterogeneous in the view of automorphism partition than what have been depicted under the measurement of degree-based entropies; and that structural heterogeneity is strongly negatively correlated to symmetry of real networks.     

On the construction of complex networks with optimal Tsallis entropy

In this work, we first formulate the Tsallis entropy in the context of complex networks. We then propose a network construction whose topology maximizes the Tsallis entropy. The growing network model has two main ingredients: copy process and random attachment mechanism (C-R model). We show that the resulting degree distribution exactly agrees with the required degree distribution that maximizes the Tsallis entropy. We also provide another example of network model using a combination of preferential and random attachment mechanisms (P-R model) and compare it with the distribution of the Tsallis entropy. In this case, we show that by adequately identifying the exponent factor q, the degree distribution can also be written in the q-exponential form. Taken together, our findings suggest that both mechanisms, copy process and preferential attachment, play a key role for the realization of networks with maximum Tsallis entropy. Finally, we discuss the interpretation of q parameter of the Tsallis entropy in the context of complex networks.     

Quantum tetrahedra and simplicial spin networks

A new link between tetrahedra and the group SU(2) is pointed out: by associating to each face of a tetrahedron an irreducible unitary SU(2) representation and by imposing that the faces close, the concept of the quantum tetrahedron is seen to emerge. The Hilbert space of the quantum tetrahedron is introduced and it is shown that, due to an uncertainty relation, the “geometry of the tetrahedron” exists only in the sense of “mean geometry”.A kinematical model of quantum gauge theory is also proposed, which shares the advantages of the loop representation approach in handling in a simple way gauge- and diff-invariances at a quantum level, but is completely combinatorial. The concept of quantum tetrahedron finds a natural application in this model, giving a possible interpretation of SU(2) spin networks in terms of geometrical objects.     

On the distributions of Laplacian eigenvalues versus node degrees in complex networks

In this paper, the important issue of Laplacian eigenvalue distributions is investigated through theory-guided extensive numerical simulations, for four typical complex network models, namely, the ER random-graph networks, WS and NW small-world networks, and BA scale-free networks. It is found that these four types of complex networks share some common features, particularly similarities between the Laplacian eigenvalue distributions and the node degree distributions.     

On functional determinants of laplacians in polygons and simplicial complexes

The functional determinant of an elliptic operator with positive, discrete spectrum may be defined ase ^–Z' (0), whereZ(s), the zeta function, is the sum analytically continued ins. In this paperZ'(0) is calculated for the Laplace operator with Dirichlet boundary conditions inside polygons with the topology of a disc in the Euclidean plane. Our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries. Our expression can be viewed as the energy for a system of static point particles, corresponding to the corners of the polygon, with self-energy and pair interaction energy. We have completely explicit closed expressions for triangles and regular polygons with an arbitrary number of sides. Among these, there are five special cases (three triangles, the square and the circled), where theZ'(0) are known by other means. One special case fixes an integration constant, and the other provide four independent analytical checks on our calculation.     

Combinatorial basis and non-asymptotic form of the Tsallis entropy function

Using a q-analog of Boltzmann's combinatorial basis of entropy, the non-asymptotic non-degenerate and degenerate combinatorial forms of the Tsallis entropy function are derived. The new measures – supersets of the Tsallis entropy and the non-asymptotic variant of the Shannon entropy – are functions of the probability and degeneracy of each state, the Tsallis parameter q and the number of entities N. The analysis extends the Tsallis entropy concept to systems of small numbers of entities, with implications for the permissible range of q and the role of degeneracy.     

Identifying influential spreaders in complex networks based on entropy weight method and gravity law

In complex networks,identifying influential spreader is of great significance for improving the reliability of networks and ensuring the safe and effective operation of networks.Nowadays,it is widely used in power networks,aviation networks,computer networks,and social networks,and so on.Traditional centrality methods mainly include degree centrality,closeness centrality,betweenness centrality,eigenvector centrality,k-shell,etc.However,single centrality method is onesided and inaccurate,and sometimes many nodes have the same centrality value,namely the same ranking result,which makes it difficult to distinguish between nodes.According to several classical methods of identifying influential nodes,in this paper we propose a novel method that is more full-scaled and universally applicable.Taken into account in this method are several aspects of node’s properties,including local topological characteristics,central location of nodes,propagation characteristics,and properties of neighbor nodes.In view of the idea of the multi-attribute decision-making,we regard the basic centrality method as node’s attribute and use the entropy weight method to weigh different attributes,and obtain node’s combined centrality.Then,the combined centrality is applied to the gravity law to comprehensively identify influential nodes in networks.Finally,the classical susceptible-infected-recovered(SIR)model is used to simulate the epidemic spreading in six real-society networks.Our proposed method not only considers the four topological properties of nodes,but also emphasizes the influence of neighbor nodes from the aspect of gravity.It is proved that the new method can effectively overcome the disadvantages of single centrality method and increase the accuracy of identifying influential nodes,which is of great significance for monitoring and controlling the complex networks.     

Normalized entropy of rank distribution： a novel measure of heterogeneity of complex networks

Many unique properties of complex networks result from heterogeneity. The measure and analysis of heterogeneity are important and desirable to the research of the properties and functions of complex networks. In this paper, the rank distribution is proposed as a new statistic feature of complex networks. Based on the rank distribution, a novel measure of the heterogeneity called a normalized entropy of rank distribution (NERD) is proposed. The NERD accords with the normal meaning of heterogeneity within the context of complex networks compared with conventional measures. The heterogeneity of scale-free networks is studied using the NERD. It is shown that scale-free networks become more heterogeneous as the scaling exponent decreases and the NERD of scale-free networks is independent of the number of vertices, which indicates that the NERD is a suitable and effective measure of heterogeneity for networks with different sizes.     

Approaching human language with complex networks

The interest in modeling and analyzing human language with complex networks is on the rise in recent years and a considerable body of research in this area has already been accumulated. We survey three major lines of linguistic research from the complex network approach: 1) characterization of human language as a multi-level system with complex network analysis; 2) linguistic typological research with the application of linguistic networks and their quantitative measures; and 3) relationships between the system-level complexity of human language (determined by the topology of linguistic networks) and microscopic linguistic (e.g., syntactic) features (as the traditional concern of linguistics). We show that the models and quantitative tools of complex networks, when exploited properly, can constitute an operational methodology for linguistic inquiry, which contributes to the understanding of human language and the development of linguistics. We conclude our review with suggestions for future linguistic research from the complex network approach: 1) relationships between the system-level complexity of human language and microscopic linguistic features; 2) expansion of research scope from the global properties to other levels of granularity of linguistic networks; and 3) combination of linguistic network analysis with other quantitative studies of language (such as quantitative linguistics).     

多重边复杂网络系统的稳定性分析

根据网络中边的不同性质提出了网络拆分的思想,通过引入时滞进行拆分,从而建立了多重边复杂网络的动力学模型. 基于Lyapunov稳定理论研究了多重边复杂网络的稳定性问题,给出了节点动力学无时滞和有时滞两种情况下网络稳定的充分条件. 最后通过数值仿真验证了结论的正确性和有效性.     

Chaotification of complex networks with impulsive control

This paper investigates the chaotification problem of complex dynamical networks (CDN) with impulsive control. Both the discrete and continuous cases are studied. The method is presented to drive all states of every node in CDN to chaos. The proposed impulsive control strategy is effective for both the originally stable and unstable CDN. The upper bound of the impulse intervals for originally stable networks is derived. Finally, the effectiveness of the theoretical results is verified by numerical examples.     

Synchronization in complex networks with age ordering

The propensity for synchronization is studied in a complex network of asymmetrically coupled units, where the asymmetry in a given link is determined by the relative age of the involved nodes. In growing scale-free networks, synchronization is enhanced when couplings from older to younger nodes are dominant. We describe the requirements for such an effect in a more general context and compare with the situations in nongrowing random networks with and without a degree ordering.     

Synchronization in complex networks with switching topology

This Letter investigates synchronization issues of complex dynamical networks with switching topology. By constructing a common Lyapunov function, we show that local and global synchronization for a linearly coupled network with switching topology can be evaluated by the time average of second smallest eigenvalues corresponding to the Laplacians of switching topology. This result is quite powerful and can be further used to explore various switching cases for complex dynamical networks. Numerical simulations illustrate the effectiveness of the obtained results in the end.     

Jamming in complex networks with degree correlation

We study the effects of the degree-degree correlations on the pressure congestion J when we apply a dynamical process on scale free complex networks using the gradient network approach. We find that the pressure congestion for disassortative (assortative) networks is lower (bigger) than the one for uncorrelated networks which allow us to affirm that disassortative networks enhance transport through them. This result agree with the fact that many real world transportation networks naturally evolve to this kind of correlation. We explain our results showing that for the disassortative case the clusters in the gradient network turn out to be as much elongated as possible, reducing the pressure congestion J and observing the opposite behavior for the assortative case. Finally we apply our model to real world networks, and the results agree with our theoretical model.     

Analysis of complex networks associated to seismic clusters near the Itoiz reservoir dam

On September 18, 2004, a 4.6 mbLg earthquake occurred in the western Pyrenees, with an epicenter located approximately 4 km from the Itoiz reservoir dam. We study the aftershock sequence of this earthquake and compare it with others that occurred in the same area in order to evaluate whether this might be a case of reservoir induced seismicity triggered by impounding of water. In order to do so, we first decluster the catalog by means of a link algorithm whose distance is based on the ETAS model. We then analyze the different clusters obtained in terms of their fractal dimension and the properties of the complex network formed by the individual earthquakes belonging to these clusters. We conclude that there are properties of the cluster representing the September 18, 2004 earthquake that are distinctively different from the properties of other clusters found in the same region. This could be an indicator of a different seismic source, potentially produced by the Itoiz dam impoundment.     

Synchronization of complex dynamical networks with nonidentical nodes

This Letter investigates the synchronization problem of a complex network with nonidentical nodes, and proposes two effective control schemes to synchronize the network onto any smooth goal dynamics. By applying open-loop control to all nodes and placing adaptive feedback injections on a small fraction of network nodes, a low-dimensional sufficient condition is derived to guarantee the global synchronization of the complex network with nonidentical nodes. By introducing impulsive effects to the open-loop controlled network, another synchronization scheme is developed for the network composed of nonidentical nodes, and an upper bound of impulsive intervals is estimated to ensure the global stability of the synchronization process. Numerical simulations are given to verify the theoretical results.     

Synchronization of complex dynamical networks with time delays

In the present paper, two kinds of dynamical complex networks are considered. The first is that elements of every node have different time delays but all nodes in such networks have the same time-delay vector. The second is that different nodes have different time-delay vectors, and the elements of each node also have different time delays. Corresponding synchronization theorems are established. Numerical examples show the efficiency of the derived theorems.