2015 International Symposium on Nonlinear Theory and its Applications NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015 A General Framework for Complex Network Applications Liu Xiao Fan † and Chi Kong Tse ‡ † School of Computer Science and Engineering, Southeast University, Nanjing, P. R. China ‡ Department of Electronic and Information Engineering, The Hong Kong Polytechnic University, Hong Kong Email: [email protected], [email protected] Abstract–Complex network theory has been applied to may be more flexible for many complex systems. For solving practical problems from different domains. In this example, in the biological system, each species can be paper, we present a general framework for complex viewed as nodes in the network, while the predator-prey network applications. The keys of a successful application relationship and mutual-dependence relationship shape the are a thorough understanding of the real system and a edges between each species in a food web. In a correct mapping of complex network theory to practical microscopic perspective, each living organic intake food problems in the system. Despite of certain limitations and generates energy through a chemical process called discussed in this paper, complex network theory provides a metabolism. In the metabolism process, chemical foundation on which to develop powerful tools in substances react with each other and transform into new analyzing and optimizing large interconnected systems. chemical substances. In the metabolic networks, the nodes are the chemical substances and the edges are the possible 1. Introduction transformation from one substance to another. Moreover, In the past fifteen years, the underlying network different complex systems can overlap and interfere with structure of complex systems has attracted extensive study each other in real-life, forming a network of networks. For from physics and computer science communities. The example, a social network is a network of people structural properties of complex networks in engineering connected by family ties, collaboration and friendships. In infrastructure, social communities, biological systems, and modern life, people keep up with friends and maintain financial systems are closely examined. Important their social relationships by using the Internet – a network universal properties such as scale-free structure, small- of computer and smart phones. Furthermore, the complex world phenomena, community structure, and dynamical network of electrical transmission supplies the power that processes are found in complex networks from multiple keeps the Internet running. Each of the above mentioned domains [1]. Efforts have also been made to apply networks are closely coupled with each other. Finding the complex network theory to not only describing the underlying network structure poses a great challenge yet topological and dynamical properties of real-world lays the groundwork of applying network theory to systems, but also to solving practical problem and even re- solving practical problems. designing the system for better performance. In this paper, we present a general framework for applying complex 3. Analysis of Network Properties network theory in solving real-world problems. First, we The properties of complex networks can be examined review the network construction process of finding the from different scopes. abstract representation of real-world systems. Then, we review the existing analysis of network properties from 3.1. The Macroscopic View different scopes. Finally, we discuss the feasibility of In the macroscopic view, the statistical properties of the using complex network theory to solve real-world complex networks, such as the degree distribution, the problem, including its capability and its limitation. community structure and the structural robustness, are of particular interest. 2. Construction of Complex Networks The degree distributions of complex networks of real The fundamental pre-requisition of a successful systems have shown a very interesting universal application of complex network theory is finding the characteristic, i.e., they all follow the Zipf’s law, also underlying network structure of complex systems. A known as the power law, , where is known as network is a set of nodes connected by a set of edges. the power-law coefficient. The power-law coefficients of Most complex systems consist of a collection of most complex systems fall between 2 and 3. For example, components which interact with each other. For instance, the Internet is a collection of computational devices for the Internet, ; for the scientific collaboration connected by wires or wireless signals. Here, the devices network, [2]. The power-law degree distribution are the nodes in the network and the physical connections reveals the winner-takes-all nature of the complex system. are the edges in the network. Computers and devices That is, most of the edges in the networks are connected to communicate with each other by exchanging data only a few number of nodes. packages. However, the representation of nodes and edges - 2 - Many complex networks are actually loosely connected can also be used to extend degree centrality [6]. by several densely connected sub-networks. The sub- Betweenness centrality of an importance measure of the networks are called community structure. The detection of node. It is calculated based on the number of shortest community structure in complex networks generally takes paths of all pairs of nodes that include this node. The two different approaches. The first approach is the “top- importance of a node sometimes depends on the down” approach, where algorithms search for the densely importance of its neighbors. Based on this idea, the connected sub-networks in the network, be them cliques PageRank algorithm is proposed to rank the importance of or sets of nodes with maximum modularity. The second webpages. The PageRank algorithm, among many other approach is the “bottom-up” approach, where specific algorithms, are considered related to the eigenvectors and edges, known as the “weak-ties” are removed from the eigenvalues of the adjacency matrix of the complex network, while the remaining disconnected sub-networks networks [7]. are the communities in the network. The weak-ties may refer to the minimum-cut of the network, or edges with 2.3. The Dynamic View largest betweenness centrality [3]. Most complex systems are not static but rather dynamic. In sociology, the term “assortative”, also known as On one hand, the topology of complex networks changes “homophily”, refers to the tendency of individuals with over time. On the other hand, dynamical processes are similar characteristics, e.g., age, nationality, religion, etc., also taking places on the networks. know or interact with each other. In complex network A traditional area of study on topological dynamics of theory, assortative mixing specifically refers to the bias of complex networks is the robustness of the network. By preference that nodes with similar degrees are connected gradually removing random edges from the network, a together. While its opposite term, disassortative mixing, strongly connected network may transform into several exists in biological and technology networks such as the unconnected sub-networks. The critical proportion of Internet and food webs [4]. edges removed in order to disconnect the sub-networks reflects the topological robustness of the network. Study 3.2. The Microscopic View has found that real networks with scale-free structure The analysis of complex network from the microscopic display great robustness against random edge removal. view focuses on single nodes or the combination of a few However, the real networks are more vulnerable to number of nodes. removal of important, rather than random, nodes and In social networks, there is a likelihood that two friends edges. The strongly connected network can be quickly of a person are also friends themselves. In complex disconnected into several sub-networks [8]. network theory, the clustering coefficient is a measure Complex networks play a crucial role in carrying of the likelihood of closed triplets, i.e., three nodes that contents and facilitating communications. For example, are fully connected. Clustering coefficient represents the information spreads on the Internet through social redundancy of edges that keep the network connected. networking services, disease and behavior spreads in Social networks show large clustering coefficients, for social community forming epidemics, etc. Understanding people tend to form a closed society, e.g., family, school, the mechanism of content spreading is the foundation of working environment, etc. While in technological predicting epidemic spreading and identifying super networks and infrastructures, the clustering coefficients spreaders. The traditional model of epidemic spreading is are small, because the redundant links between nodes SIR (susceptible-infected-recovered) model. This model increase the cost of the systems [1]. assumes that a population can transform with a certain Network motifs are defined as recurrent and probability from the susceptible state to the infected state statistically significant small-sized sub-graphs. The and from the infected state to the recovered state. The SIR network motifs are usually related to the functional model is a simplified model of epidemic scenario. Content properties of the network. Despite of this, their detection spreading on complex networks