that allows us to capture the information about the relationship between things. But

lets first think about why we might be interested in this at all.

We often describe the world in terms of objects or things and their properties, we talk about

countries and their GDP, people and their age or the color of a car, this type of component

based analysis works well when the system we are interested in is relatively isolated.

But when we turn up the interactions and connectivity between elements within a system it is increasingly

the connections that come to shape the elements and define the system as a whole and thus

we need a model that captures this information about the relationships and allows us to reason

about it, this is where network theory comes in.

Network theory starts with a very simple view of the world as made up of nodes which are

things or objects, like people, cities, computers etc. and the relationships between these things,

called edges, such as friendships, trading partners, cables and so on.

This abstract representation of the world, can be used to model a wide variety of things,

thus we can have social networks, biological networks consisting of interacting creatures

within an ecosystem or logistic networks composed of interacting suppliers and consumers.

Network theory gives us a set of tools for analyzing the individual elements and relations

within these networks, the structure of the network and the properties that these different

networks structures give rise to.

The first set of question we might like to ask about a particular network relate to its

degree of connectivity, that is how connected an individual element or the wholes network

is, this will tell us many things about it such as how quickly a new event could spread

or propagate through the system.

The average degree of connectivity will give us a quick answer to this; this is calculated

by taking the total number of edges and dividing it by the total number of nodes within the

network

We also need to take into account how large the network is, that is to say how far is

it on average from one point to another. This is called the average path length and we can

calculate it by taking the average of all the path length between all the nodes.

Because networks are all about connectivity we often ascribe value to individual nodes

based upon their degree of connectivity, there are various methods for calculating this but

a popular one called Eigenvector centrality, which measures both how many edges a node

has and how connected the nodes it joined to are also.

Popular web search engines use variants of this Eigenvector centrality measure to rank

webpages by calculating both the number of links into a webpage and the degree of connectivity

of the pages that link into them thus gaining an idea of the relative importance of the

website

Next we are interested in talking about the overall structure to the network this will

be largely determined by how the relationship between the nodes was formed.

If the relations between elements was generated randomly we would expect a relatively even

distribution of edges across the network, this type of structure or topology is called

a distributed network as the relative importance of any node is distributed across the entire

network.

A second type of network structure we can get is called decentralized or small world,

this is generated by having local clusters of connections, but also having some random

distant connections.

an example of this might be a group of friends, with some of the friends having distant relatives

in other parts of the world. By using these local connection within the group and distant

connections research has shown that it is possible to connect two random people within

a average of just six steps and thus it is termed small world.

Lastly we have more centralized networks called scale free networks, this is where may nodes

have chosen to connect to the same node giving it a degree of connectivity that greatly exceeds

the average whilst leaving may with a very low level of connectivity.

Many real networks are through to be scale-free, including social, biological and technological

systems such as world-wide web, where very few sites like Wikipedia have a very large

amount of links into them, whilst the vast majority of websites have very few.

These various types of network structures give rise to different properties, a key question

we are interest in asking here is how robust or fragile is a particular type of network

as this will not only help us understand networks better, but will also be of great significance

in how we design and manage them.

For example, think about a country with many small to medium size cities supplying the

population with various public services, if we were to remove one of the cities it would

have a limited effect on the overall system, because the networks has a distributed structure

making it robust to fail of this kind.

In contrary if we take a county with one dominant capital city with the rest of the urban network

dependent upon it for core services, this centralized network may be more efficient

but it is also in what is called a more critical state as effecting this single primary node

would have a large systemic effect.

As we transit from an industrial to information societies, networks are emerging as a new

paradigm in how we structure our systems of organization both social and technological.

Network theory is a young and rapidly growing area that provide us with a set of tools for

designing and managing these new types of organization and more generally understanding

the world around us from a different perspective.