Random pseudofractal networks with competition

In this paper, we present a simple rule which assigns fitness to each edge to generate random pseudofractal networks (RPNs). This RPN model is both scale-free and small-world. We obtain the theoretical results that the power-law exponent is γ=2+1/(1+α) for the tunable parameter α>-1, and that the degree distribution is of an exponential form for others. Analytical results also show that an RPN has a large clustering coefficient and can process hierarchical structure as C(k)∼k^-1 that is in accordance with many real networks. And we prove that the mean distance L(N) scales slower logarithmically with network size N. In particular, we explain the effect of nodes with degree 2 on the clustering coefficient. These results agree with numerical simulations very well.     

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.     

Network topology and correlation features affiliated with European airline companies

The physics information of four specific airline flight networks in European Continent, namely the Austrian airline, the British airline, the France-Holland airline and the Lufthhansa airline, was quantitatively analyzed by the concepts of a complex network. It displays some features of small-world networks, namely a large clustering coefficient and small average shortest-path length for these specific airline networks. The degree distributions for the small degree branch reveal power law behavior with an exponent value of 2-3 for the Austrian and the British flight networks, and that of 1-2 for the France-Holland and the Lufthhansa airline flight networks. So the studied four airlines are sorted into two classes according to the topology structure. Similarly, the flight weight distributions show two kinds of different decay behavior with the flight weight: one for the Austrian and the British airlines and another for the France-Holland airline and the Lufthhansa airlines. In addition, the degree-degree correlation analysis shows that the network has disassortative behavior for all the value of degree k, and this phenomenon is different from the international airline network and US airline network. Analysis of the clustering coefficient (C(k)) versus k, indicates that the flight networks of the Austrian Airline and the British Airline reveal a hierarchical organization for all airports, however, the France-Holland Airline and the Lufthhansa Airline show a hierarchical organization mostly for larger airports. The correlation of node strength (S(k)) and degree is also analyzed, and a power-law fit S(k)∼k^1.1 can roughly fit all data of these four airline companies. Furthermore, we mention seasonal changes and holidays may cause the flight network to form a different topology. An example of the Austrian Airline during Christmas was studied and analyzed.     

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.     

An extended clique degree distribution and its heterogeneity in cooperation-competition networks

After Xiao et al. [W.-K. Xiao, J. Ren, F. Qi, Z.W. Song, M.X. Zhu, H.F. Yang, H.Y. Jin, B.-H. Wang, Tao Zhou, Empirical study on clique-degree distribution of networks, Phys. Rev. E 76 (2007) 037102], in this article we present an investigation on so-called k-cliques, which are defined as complete subgraphs of k (k>1) nodes, in the cooperation-competition networks described by bipartite graphs. In the networks, the nodes named actors are taking part in events, organizations or activities, named acts. We mainly examine a property of a k-clique called “k-clique act degree”, q, defined as the number of acts, in which the k-clique takes part. Our analytic treatment on a cooperation-competition network evolution model demonstrates that the distribution of k-clique act degrees obeys Mandelbrot distribution, P(q)∝(q+α)^−γ. To validate the analytical model, we have further studied 13 different empirical cooperation-competition networks with the clique numbers k=2 and k=3. Empirical investigation results show an agreement with the analytic derivations. We propose a new “heterogeneity index”, H, to describe the heterogeneous degree distributions of k-clique and heuristically derive the correlation between H and α and γ. We argue that the cliques, which take part in the largest number of acts, are the most important subgraphs, which can provide a new criterion to distinguish important cliques in the real world networks.     

Influence of network structure on rumor propagation

Rumor propagation in complex networks is studied analytically and numerically by using the SIR model. Analytically, a mean-field theory is worked out by considering the influence of network topological structure and the unequal footings of neighbors of an infected node in propagating the rumor. It is found that the final infected density of population with degree k   is ρ(k)=1−exp(−αk)$\rho (k)=1-\exp (-\alpha k)$, where α is a parameter related to network structure. The number of the total final infected nodes depends on the network topological structure and will decrease when the structure changes from random to scale-free network. Numerical simulations confirm the theoretical predictions.     

Power-law behavior in complex organizational communication networks during crisis

Communication networks can be described as patterns of contacts which are created due to the flow of messages and information shared among participating actors. Contemporary organizations are now commonly viewed as dynamic systems of adaptation and evolution containing several parts, which interact with one another both in internal and in external environment. Although there is limited consensus among researchers on the precise definition of organizational crisis, there is evidence of shared meaning: crisis produces individual crisis, crisis can be associated with positive or negative conditions, crises can be situations having been precipitated quickly or suddenly or situations that have developed over time and are predictable etc. In this research, we study the power-law behavior of an organizational email communication network during crisis from complexity perspective. Power law simply describes that, the probability that a randomly selected node has k links (i.e. degree k) follows P(k)∼k^−γ, where γ is the degree exponent. We used social network analysis tools and techniques to analyze the email communication dataset. We tested two propositions: (1) as organization goes through crisis, a few actors, who are prominent or more active, will become central, and (2) the daily communication network as well as the actors in the communication network exhibit power-law behavior. Our preliminary results support these two propositions. The outcome of this study may provide significant advancement in exploring organizational communication network behavior during crisis.     

Growing networks with preferential deletion and addition of edges

A preferential attachment model for a growing network incorporating the deletion of edges is studied and the expected asymptotic degree distribution is analyzed. At each time step t=1,2,…, with probability π₁>0 a new vertex with one edge attached to it is added to the network and the edge is connected to an existing vertex chosen proportionally to its degree, with probability π₂ a vertex is chosen proportionally to its degree and an edge is added between this vertex and a randomly chosen other vertex, and with probability π₃=1−π₁−π₂<1/2 a vertex is chosen proportionally to its degree and a random edge of this vertex is deleted. The model is intended to capture a situation where high-degree vertices are more dynamic than low-degree vertices in the sense that their connections tend to be changing. A recursion formula is derived for the expected asymptotic fraction p_k of vertices with degree k, and solving this recursion reveals that, for π₃<1/3, we have p_k∼k^{−(3−7π₃)/(1−3π₃)}, while, for π₃>1/3, the fraction p_k decays exponentially at rate (π₁+π₂)/2π₃. There is hence a non-trivial upper bound for how much deletion the network can incorporate without losing the power-law behavior of the degree distribution. The analytical results are supported by simulations.     

Diffusive capture processes for information search

We show how effectively the diffusive capture processes (DCP) on complex networks can be applied to information search in the networks. Numerical simulations show that our method generates only 2% of traffic compared with the most popular flooding-based query-packet-forwarding (FB) algorithm. We find that the average searching time, 〈T〉, of the our model is more scalable than another well known n-random walker model and comparable to the FB algorithm both on real Gnutella network and scale-free networks with γ=2.4. We also discuss the possible relationship between 〈T〉 and 〈k²〉, the second moment of the degree distribution of the networks.     

Isothermal adsorption kinetics on heterogeneous surfaces

Adsorption kinetics on energetically heterogeneous surfaces under isothermal conditions is analyzed using the uniform energy distribution model. Considering the quasi-equilibrium of surface diffusion between the adsorption sites with different energy, the kinetic equations dΘ/dt=(k_ap−A_dK_diff)(1−Θ) for first-order adsorption and dΘ/dt=k_ap(1−Θ)²−A_dK_diffΘ(1−Θ) for dissociative adsorption are obtained, where K_diff is a coefficient describing the surface diffusion equilibrium, which depends on the coverage and the energy distribution. Under isochoric conditions with p decreasing due to adsorption, surface diffusion accelerates the rate towards equilibrium significantly, as observed in static calorimetric adsorption experiments. An approximate solution in Lagergren form is derived for this condition.     

Geographical dispersal of mobile communication networks

In this paper, we analyze statistical properties of a communication network constructed from the records of a mobile phone company. The network consists of 2.5 million customers that have placed 810 million communications (phone calls and text messages) over a period of 6 months and for whom we have geographical home localization information. It is shown that the degree distribution in this network has a power-law degree distribution k⁻⁵ and that the probability that two customers are connected by a link follows a gravity model, i.e. decreases as d⁻², where d is the distance between the customers. We also consider the geographical extension of communication triangles and we show that communication triangles are not only composed of geographically adjacent nodes but that they may extend over large distances. This last property is not captured by the existing models of geographical networks and in a last section we propose a new model that reproduces the observed property. Our model, which is based on the migration and on the local adaptation of agents, is then studied analytically and the resulting predictions are confirmed by computer simulations.     

Evolutionary prisoner’s dilemma game on weighted scale-free networks

This paper investigates the evolutionary prisoner’s dilemma on weighted scale-free networks. The weighted networks are generated by adopting Barabási-Albert scale-free network and assigning link weight with w_ij=(k_i×k_j)^β. Simulation results show that the cooperation frequency has a strong dependence on β. The value of β which is associated with the maximal cooperation frequency has been sought out. Moreover, Gini coefficient and Pareto exponent of the system’s wealth distribution are investigated. The inequality of wealth distribution is minimized at β≈−1.     

A local-world node deleting evolving network model

A new type network growth rule which comprises node addition with the concept of local-world connectivity and node deleting is studied. A series of theoretical analysis and numerical simulation to the LWD network are conducted in this Letter. Firstly, the degree distribution p(k) of this network changes no longer pure scale free but truncates by an exponential tail and the truncation in p(k) increases as pa decreases. Secondly, the connectivity is tighter, as the local-world size M increases. Thirdly, the average path length L increases and the clustering coefficient 〈C〉 decreases as generally node deleting increases. Finally, 〈C〉 trends up when the local-world size M increases, so as to kmax. Hence, the expanding local-world can compensate the infection of the node deleting.     

Triadic Closure in Configuration Models with Unbounded Degree Fluctuations

The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering c(k), i.e., the probability that two neighbors of a degree-k node are neighbors themselves. We show that c(k) progressively falls off with k and the graph size n and eventually for \(k=\varOmega (\sqrt{n})\) settles on a power law \(c(k)\sim n^{5-2\tau }k^{-2(3-\tau )}\) with \(\tau \in (2,3)\) the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting.     

Critical parameters near the ferromagnetic-paramagnetic phase transition in La0.7A0.3(Mn1−xbx)O3 (A=Sr; B=Ti and Al; x=0.0 and 0.05) compounds

The critical parameters provide important information concerning the interaction mechanisms near the paramagnetic-to-ferromagnetic transition. In this paper, we present a thorough study for the critical behavior of La_0.7A_0.3(Mn_1−xB_x)O₃ (A=Sr; B=Ti and Al; x=0.0 and 0.05) polycrystalline samples near ferromagnetic-paramagnetic phase transition temperature by analyzing isothermal magnetization data. We have analyzed our dc-magnetization data near the transition temperature with the help of the modified Arrot plot, Kouvel-Fisher method. We have determined the critical temperature T_C and the critical parameters β, γ and δ. With the values of T_C, β and γ, we plot M×(1−T/T_C)^−β vs. H×(1−T/T_C)^−γ. All the data collapse on one of the two curves. This suggests that the data below and above T_C obey scaling, following a single equation of state. Critical parameters for x=0 and x_Ti=0.05 samples are between those predicted for a 3D-Heisenberg model and mean-field theory and for x_Al=0.05 samples the values obtained for the critical parameters are close to those predicted by the mean-field theory.     

Low-temperature property investigation of the lead indium-niobate-lead nickel-niobate solid solution

The complex perovskite solid solution (1−x) Pb(In_1/2Nb_1/2)O₃-(x) Pb(Ni_1/3Nb_2/3)O₃ has been successfully prepared by the Columbite precursor method. The temperature dependencies of the dielectric constant and pyroelectric coefficient were measured between −261 and 200 °C. Relaxor ferroelectric behavior has been noticed in all compositions across the solid solution. The room-temperature electrostrictive coefficient, Q₃₃, was 1.83×10⁻² C²/m⁴ for x=0.10. No room-temperature piezoelectric activity was detected; however, upon cooling to −261 °C the maximum coupling coefficients k_p=29%, k_t=11%, and k₃₃=31% were observed for the composition x=1.00.     

Scaling relation for earthquake networks

The scaling relation, 2γ−δ=1, for the exponents of the power-law connectivity distribution, γ, and the power-law eigenvalue distribution of the adjacency matrix, δ, is theoretically predicted to be fulfilled by a locally treelike scale-free network in the “effective medium approximation” (i.e., an analog of the mean field approximation). Here, it is shown that such a relation holds well for the reduced simple earthquake networks (i.e., the network without tadpole-loops and multiple edges) constructed from the seismic data taken from California and Japan. This validates the goodness of the effective medium approximation in the earthquake networks and is consistent with the hierarchical organization of the networks. The present result may be useful for modeling seismicity on complex networks.     

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.     

Chemical diffusion in molybdenum disulphide

Ceramic molybdenum disulphide (MoS₂) was equilibrated at an ambient sulphur vapour partial pressure p(S₂), 10 Pa<p(S₂)<1000 Pa. After the step change of p(S₂) to a new value, the equilibration kinetics was monitored by measuring electrical conductivity. The application of the solution of Fick's second law (with the initial condition: no concentration gradient in specimen and the boundary condition: surface concentration constant) to the kinetic data gave the chemical diffusion coefficient. The chemical diffusion coefficient, D_chem, determined at 1273 K, was D_chem=(3.20±0.32)*10⁻⁷ cm² s⁻¹ and was found to be independent of sulphur vapour partial pressure. The usefulness of transient electrical conductivity method for determining real values of diffusion data was discussed in terms of defect structure of the studied material.     

Networking effects on evolutionary snowdrift game in networks with fixed degrees

We study the effects of spatial structures other than the degree distribution on the extent of the emergence of cooperation in an evolutionary snowdrift game. By swapping the links in three different types of regular lattices with a fixed degree k, we study how the frequency of cooperator f_C changes as the clustering coefficient (CC), which signifies how the nearest neighbors of a vertex are connected, and the sharing coefficient (SC), which signifies how the next-nearest neighbors of a vertex are shared by the nearest neighbors, are varied. For small k, a non-vanishing CC tends to suppress f_C. A non-vanishing SC also leads to a suppressed f_C for the networks studied. As the degree increases, the sensitivity of f_C to the network properties is found to become increasingly weak. The result is discussed within the context of the ranking patterns of average payoffs as k changes. An approximation for f_C, which is based on the idea of a finite fully connected network and gives results in good agreement with numerical results, is derived in the limit of large k.