Graph Theory & Small Networks
Introduction
Networks are everywhere. The brain is a sophisticated neural network connected by axons. Society, too, are networks connected by family, friends and professional ties. On a larger scale food webs can be represented as a network of species. Networks have even diffused through our technology such as the World Wide Web where routers and web pages are all interconnected. Even the language we speak today is a network of words connected by syntactic associations. Networks are everywhere.
Yet despite the importance and frequency of such complex networks, little is understood of their properties and structure. How do viruses diffuse so rapidly in communication systems? How do some networks continue to function even after a vast majority of their nodes have failed? Is it possible that everyone is connected by six handshakes?
For much of the last century, scientists treated all complex networks as being completely random. This theory had its roots in the work of two mathematicians, Paul Erdos and Alfred Renyi. Their work suggested that systems such as communications could be effectively modelled by connecting nodes with randomly placed links. Their simple approach revitalised graph theory and led to the emergence of the field of random networks.
An important prediction of random network theory is regardless of the random placement of links most nodes will still have approximately the same number of links. In fact, in a random network the nodes follow a Poisson distribution with a bell shape (see Fig.1). Random networks are also called exponential, because the probability that a node is connected to k other sites decreases exponentially for large k. This is better described by the famous small world networks. It was Watts and Strogatz in 1998 that recognised that a class of random graphs could be categorised as small world networks. They noted that graphs could be classified according to their clustering coefficient and their diameter. Many random
Networks are everywhere. The brain is a sophisticated neural network connected by axons. Society, too, are networks connected by family, friends and professional ties. On a larger scale food webs can be represented as a network of species. Networks have even diffused through our technology such as the World Wide Web where routers and web pages are all interconnected. Even the language we speak today is a network of words connected by syntactic associations. Networks are everywhere.
Yet despite the importance and frequency of such complex networks, little is understood of their properties and structure. How do viruses diffuse so rapidly in communication systems? How do some networks continue to function even after a vast majority of their nodes have failed? Is it possible that everyone is connected by six handshakes?
For much of the last century, scientists treated all complex networks as being completely random. This theory had its roots in the work of two mathematicians, Paul Erdos and Alfred Renyi. Their work suggested that systems such as communications could be effectively modelled by connecting nodes with randomly placed links. Their simple approach revitalised graph theory and led to the emergence of the field of random networks.
An important prediction of random network theory is regardless of the random placement of links most nodes will still have approximately the same number of links. In fact, in a random network the nodes follow a Poisson distribution with a bell shape (see Fig.1). Random networks are also called exponential, because the probability that a node is connected to k other sites decreases exponentially for large k. This is better described by the famous small world networks. It was Watts and Strogatz in 1998 that recognised that a class of random graphs could be categorised as small world networks. They noted that graphs could be classified according to their clustering coefficient and their diameter. Many random
References: Barabási, Albert-Laszlo, (2002): Linked: The New Science of Networks. Perseus Publishing, Barabási, Albert-Laszlo & Bonabeau, Eric, (2003): Scientific American Barabási, Albert-Laszlo, (2003): Linked Networks from Biology to the World Wide Web Seminar