The structure and function of complex networks

"The structure and function of complex networks," Newman, 2003.

Generally, there are some terms with regard to a graph, such as vertex, edge, directed/undirected, degree, component, geodesic path, and diameter. Besides, there are also some kinds of networks in the real world. For example, social networks is groups of people with some pattern of interactions between them. The information networks is the network of citations between academic papers or the World Wide Web.
One of the properties of networks is the small-world effect. The effect shows that most pairs of vertices in most networks seem to be connected by a short path through the network, and the information may spread very fast in few steps. Another property is network resilience. Networks vary when vertices are removed or added. That is, the typical path of some paths may increase and the communication between some pairs may become impossible if vertices are removed from the network.
A random graph consists of vertices and the edges with probability. The assigned probabilities are the major study of many papers.