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We consider the preferential attachment model introduced by Deijfen and Lindholm (2009) in which, at every discrete time step: (i) either we add a vertex and connect it to an older vertex; or (ii) we add an edge between two random vertices; or (iii) we delete one edge. We show that, when the deletion probability equals 1 / 3 , the expected degree of any given vertex grows logarithmically, thus correcting a statement made in Lindholm and Vallier (2011). Moreover we show that, when the deletion probability is strictly less than 1 / 3 , then the function which scales the expected degree of a given vertex, identified in Lindholm and Vallier (2011), also guarantees almost sure convergence for the degree process of a given vertex. [ABSTRACT FROM AUTHOR]

Abstract A general population evolution model is considered. Any individual of the population is characterized by its score. Certain general conditions are assumed concerning the number of the individuals and their scores. Asymptotic theorems are obtained for the number of individuals having a fixed score. Then it is proved that the score distribution is scale free. The result is applied to obtain the weight distribution of cliques in a random graph evolution model. [ABSTRACT FROM AUTHOR]

In this article we propose a general methodology for constructing complex networks. Popular selection schemes in Genetic algorithms are used for this construction. Mathematically, it has been shown that, under some weak constraints, the degree distribution of the resulting networks follow power-law, as seen in real world networks. Power-law degree distribution is one of the most significant structural characteristics observed in many real-world complex networks. The main reason behind the emergence of this phenomenon is the mechanism of preferential attachment which states that in a growing network a node with higher degree is more likely to receive new links. However, degree is not the only key factor influencing the network growth leading to power-law degree distribution. Instead, there must be several other factors whose cumulative effect, called fitness of a node, has a significant role in attracting other nodes and thereby producing power law networks. The concept of fitness can be thought of as a generalization of node degree. Heterogeneity in preferential linking also plays an important role in producing power-law networks in this context. The proposed construction methodology, also leading to power law networks, combines the inherent fitness value of a node, drawn from a particular distribution, with various attachment schemes based on the different selection methods commonly used in Genetic algorithms. Six different selection schemes are used in total. Different well known structural measures like average degree of the nearest neighbors, average path length, clustering coefficient, etc. are calculated for each newly generated network to understand their behavior patterns. It has been found that these six schemes can be divided into two distinct groups of three on the basis of their structural properties, where one of these two groups produces proper power-law networks which possess topological properties similar to observed in the real world. Finally, extensive simulations and experiments over scientific collaboration networks validate the effectiveness of the proposed models. [ABSTRACT FROM AUTHOR]

At the n th step of a time-inhomogeneous infinite Polya process, either a new bin is created and a ball is put in that bin, or a ball is put into an existing bin, in which case the bin is chosen according to a preferential attachment type rule. We introduce a new class of such processes and find the asymptotics of the expected number of bins of size k , for k fixed, as n → ∞ . [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to explore the evolving mechanism of urban roadway network. With the consideration of self-organization effect and planning effect during evolution, the authors try to demonstrate the impact of preferential attachment, module scale and module structure on the evolving network model. Design/methodology/approach – The roadway network is built in the form of abstract network by dual approach. By using the evolving model of modular growth, the authors analyze the effects and mechanism of the evolving process. Then through numerical analysis, the impact of evolving effects on urban roadway network topology structure is discussed from the aspects of preferential attachment, module scale and module structure. Findings – The module structure property, small-world property and scale-free property of roadway network can be affected with various degrees by the change of preferential attachment and module scale. However, the impact of module structure on network properties is small, which can be ignored. Therefore, in practice, the self-organization effect and planning effect of evolving network can be reached by changing the preferential attachment and module scale, so as to generate the network structure with specific properties. Research limitations/implications – Some local events, such as road extensions, road demolition and intersection rebuilding, exist during the evolving process under real-world situation. While those cases have not been considered in preferential attachment. Therefore, researchers are encouraged to take these factors into consideration in further research. Practical implications – The paper has implications for practice in urban transportation planning and roadway constructions, which can help to guide the planning of urban roadway and to adjust or restore partial network when broken down according to the evolving law. Originality/value – The impact of preferential attachment, module scale and module structure on the evolving network model is measured. And the relationship between different network properties can be used to build some patterns of network. From this point of view, the development of urban roadway network can be predicted and intervened. [ABSTRACT FROM AUTHOR]

Abstract: Two preferential attachment-type graph models which allow for dynamic addition/deletion of edges/vertices are considered. The focus of this paper is on the limiting expected degree of a fixed vertex. For both models a phase transition is seen to occur, i.e. if the probability with which edges are deleted is below a model-specific threshold value, the limiting expected degree is infinite, but if the probability is higher than the threshold value, the limiting expected degree is finite. In the regime above the critical threshold probability, however, the behaviour of the two models may differ. For one of the models a non-zero (as well as zero) limiting expected degree can be obtained whilst the other only has a zero limit. Furthermore, this phase transition is seen to occur for the same critical threshold probability of removing edges as the one which determines whether the degree sequence is of power-law type or if the tails decays exponentially fast. [Copyright &y& Elsevier]

Abstract: A stochastic model for the spread of a sexually transmitted disease (STD) is presented. To reflect varying degrees of promiscuity among individuals it is assumed that the infectivity of any infected individual is proportional to the number of previous contacts the individual has had with other infected individuals. In both the simple single-sex model and in the more complex two-sex model, the tree graphs of the infection exhibit scale-free network behaviour (i.e. power-law behaviour in the upper tail of the degree distribution). The distributions of the size of the infection and of the ring number (distance from the original source of the infection) are determined. [Copyright &y& Elsevier]

• Spatial and aspatial aspects of racial segregation indicate complexity. • Size distributions of racial enclaves are power-laws. • Spatial footprints of racial enclaves are monofractals. • Complex character of segregation patterns puts constrains on segregation dynamics. • Dynamics described by preferential attachment fulfills these constraints. Cities are complex systems, their complexity manifests itself through fractality of their spatial structures and by power law distributions (scaling) of multiple urban attributes. Here we report on the previously unreported manifestation of urban complexity – scaling in patterns of residential racial segregation. A high-resolution racial grid of a city is segmented into racial enclaves which are patches of stationary racial composition. Empirical PDFs of patch areas and population counts in 41 US cities were analyzed to reveal that these variables have distributions which are either power laws or approximate power laws. Power law holds for a pool of all patches, for patches from individual cities, and patches restricted to specific racial types. The average value of the exponent is 1.64/1.68 for area/population in 1990 and 1.70/1.74 in 2010. The values of exponents for type-specific patches vary, but variations had decreased from 1990 to 2010. We have also performed a multifractal analysis of patterns formed by racial patches and found that these patterns are monofractal with average values of fractal dimensions in the 0.94-1.81 range depending on racial types and the year of analysis. Power law distribution of racial patch sizes and a fractal character of racial patterns present observable and quantifiable constraints on models of racial segregation. We argue that growth by preferential attachment is a plausible mechanism leading to observed patterns of segregation. [ABSTRACT FROM AUTHOR]