Complex network

A complex network is within the scope of network research and graph theory a network ( graph ) with non-trivial topological properties, i.e., with properties that are not as lattices or random graphs occur in simple networks. The study of complex networks is a young and active area of current scientific research, which is mainly inspired by the empirical studies of real-world networks such as computer networks or social networks.

Definition

Many social, biological, and computer networks have significant non -trivial topological properties, ie the connections (edges) between their elements (nodes) are neither purely regular nor purely random. Instead, such networks characterized by specific distributions in the occurrence of its elements ( degree distribution, English: degree distribution ), by a high clustering coefficient, a particular community structure (community structure ) or a developed hierarchical structure. Many previously studied mathematical models of networks or graphs, however, do not exhibit these properties.

Two typical classes of complex networks have been studied intensively in the past: Scale -free networks and the so-called small world networks, the discovery and definition are canonical case studies in this area. In scale-free networks, the nodes do not have a typical number of connections, but the distribution of links per node follows a power law. In small -world networks, however, occur on many very short connections between all elements and they have a high clustering coefficient. Due to the rapid advances in current research on complex networks even more important new aspects and insights have been found how changing temporal networks: These networks can change over time (so-called ' evolving networks' ), where nodes and edges with the may arise time new or even disappear. This creates a complex dynamics that can lead to self-organization and stable states. Another important aspect is the occurrence of synchronization.

The study of complex networks is a lively and very active field of research and connects different disciplines such as mathematics, physics, biology, climate research, computer science, sociology, epidemiology, and many more. The concepts of network theory have been included in the analysis of metabolic and regulatory networks in the design of robust and scalable communication networks, in the development of vaccination strategies and in the analysis of climate phenomena. Corresponding research results are regularly published in some of the most well-known scientific journals, are subject to special conferences and have also led to some popular scientific articles and books.

From the research of complex networks, important statements on information and material flows and their optimization and supercritical behavior and stability of the overall system can be learned. As an example, reference is made to the exchange of banknotes, which was investigated by Dirk Brockmann using the theory of complex networks, which attracted worldwide attention found.

Analysis

To create a network, respectively. analyzing a graph, is the importance of the node of interest in many cases. Metrics such as analysis of naive next node level, more complex methods have been proposed. A well-known example is the PageRank, the method that forms the basis for modern search engines. It is closely related to the eigenvector centrality.

Other measures of graphs are:

• Degree (degree centrality ) --- An early developed, simple measure to measure the centrality of a node, is to study the set of all edges incident.
• Proximity ( closeness centrality ) --- The distance of a node to all others is the basis for this measure. This node can be ( automatically ) identified, which are located in the "center " of a network. Usually the reciprocal of the sum of all distances to other nodes is made, in order to achieve, that the value is higher, the higher the perceived centrality of node.

• Betweenness centrality ( betweenness centrality ) --- A node has a high betweenness value if this node is part of a particularly large number of shortest paths and the respective pairs have few other shortest paths on which the node is not included. For each pair of nodes therefore the percentage is calculated on shortest paths between them which contain v. These shares are for all pairs of nodes added up to the betweenness centrality of v to be calculated.
• Eigenvector centrality ( eigenvector centrality ) --- After the eigenvector centrality method, a node is more important, the more important are its neighboring nodes.