 
Summary: General network theory
réka albert
Pennsylvania State University
complex weblike structures describe a wide variety of systems of high technologi
cal and intellectual importance. For example, the cell is best described as a complex net
work of proteins and small molecules connected by biochemical reactions; the Internet is
a complex network of routers and computers linked by various physical or wireless links;
fads and ideas spread on the social network whose nodes are human beings and edges rep
resent various social relationships; the worldwide web is an enormous virtual network of
webpages connected by hyperlinks.
a system of elements that interact or regulate each other can be represented by a math
ematical object called a graph (Bollobás 1979). here graph is not used in its usual meaning
of `diagram of a functional relationship', but as `a collection of nodes and edges', in other
words, a network. at the simplest level, the system's elements are reduced to graph nodes
(also called vertices) and their interactions are reduced to edges connecting pairs of nodes
(see Figure 1, overleaf). a graph at its simplest is a connection of nodes (a, B, C...) and
edges (aC, BC, CD, CJ...). The node arrangement and length of edges does not mat
ter, only which is connected to which. edges can be either directed, specifying a source
(starting point) and a target (endpoint), or nondirected. Directed edges are suitable for
representing the flow of material or of information, while nondirected edges are used to
