Complex networks are characterized by having a scale-free power-law (PL) degree distribution, a small world phenomenon, a high average clustering coefficient, and the emergence of community structure. Most proposed models did not incorporate all of these statistical properties and neglected incorporating the heterogeneous nature of network nodes. Even proposed heterogeneous complex network models were not generalized for different complex networks. We define a novel aspect of node-heterogeneity which is the node connection standard heterogeneity. We introduce our novel model “settling node adaptive model” SNAM which reflects this new nodes' heterogeneous aspect. SNAM was successful in preserving PL degree distribution, small world phenomenon and high clustering coefficient of complex networks. A modified version of SNAM shows the emergence of community structure. We prove using mathematical analysis that networks generated using SNAM have a PL degree distribution.
Top1. Introduction
Complex networks are comprised of sets of numerous interconnected nodes that interact in different ways. Complex networks are large, containing from a thousand to several million or more nodes which are connected by edges. In addition to being large, the structure of complex networks is neither completely regular nor completely random. The structure of a complex network results from the fact that complex systems are self-organizing. As a complex system evolves, interactions, usually represented as edges, among its many constituent units, usually represented as nodes, result in an emergent structure with unforeseen properties. While complex networks do share common characteristics with respect to size, structure, and emergent behavior, there is no single general, precise, and accepted definition of network complexity(Boyd, D. M. and Ellison N. B,2007). Therefore, differentiating complex networks from other types of networks is difficult given this lack of an accepted definition. Network nodes and links can represent different entities and relations depending on the analyzed network. For example, in social networks the users engage with each other for various purposes, including business, entertainment, citation, movies, transport, banking, knowledge sharing, and many other activities. The widespread use of complex networks as models in different fields has made the study of complex networks and their structure an important research topic.
The study and analysis of data extracted from complex networks has revealed a number of distinct features and behavioral patterns that distinguish these networks. Awareness of these features can lead to an improved understanding of the network’s structure and dynamics. Such knowledge can be utilized in different fields to answer questions such as: How can the current World Wide Web (WWW) structure be used to predict future connections between websites? How can one deduce new relationships or reveal potential networks’ vulnerabilities? We might have perfect knowledge of the parts constituting the network, but its large-scale structure and dynamics may not be immediately obvious. Analysis of complex networks analysis can provide insight into the ties and relations linking nodes and an improved understanding of a network’s dynamics. Such analysis can help enhance decision-making dealing with network management and resource allocation in different applications. Analysis indicates that certain statistical properties are common to a large number of these networks (Barabási, A.-L. and R. Albert,1999, Newman, M. E. J., 2003, Barabási, A.-L. and R. Albert, 2002). These are only some of the motivations that make complex networks an important research topic.
Recent enhancements in computational and storage capabilities have made it feasible to pursue these three areas of study. Researchers are now able to gather and analyze large databases resulting from interactions between different nodes in real world networks. These developments allow researchers to identify the properties of complex networks. Real-world network datasets are often proprietary and hard to obtain. Thus, researchers often study networks using synthetic datasets generated via mathematical models. Knowledge of the properties of complex networks is essential in modeling these networks. Additionally, altering the parameters in a network model leads to the generation of datasets with different properties. These datasets can be used for thorough exploration and evaluation of network analysis algorithms (Barabási, A.-L. and R. Albert,1999, Newman, M. E. J., 2003).