Network Science: A Short Introduction i3 Workshop

1. Network Science: A Short Introductioni3 Workshop Konstantinos Pelechrinis Summer 2014 Figures are taken from: M.E.J. Newman, “Networks: An Introduction”

2. Statistics for real networks

3. Universal properties • It is interesting that regardless of the type of the network there are specific properties that exhibit similar characteristics across networks • Giant component • Short paths • High clustering • Skewed degree distribution

4. Components • What are the component sizes in a real-world network? • Typically there is a large component that fills most of the network • Even more than 90% of the nodes • Rest of the network is divided in many smaller components disconnected from each other • The large components can arise either due to the nature of the net- work (e.g., Internet), or due to the way the network was measured (e.g., Web)

5. The small world effect • In many networks the typical network distances between vertices are surprisingly small • Typically networks have been found to have mean distance less than 20 – or in many cases less than 10 – even though the networks themselves have millions of nodes • Implications such as rumor spread in a social networks, response time in the Internet etc. • The average local clustering coefficient is also “high” • High relatively to the one expected if connections were made at random • These two properties describe the small world effect

6. Skewed degree distribution • A great fraction of networks analyzed exhibit skewed degree distribution • While many nodes have small degrees, there are very few nodes that exhibit very high degrees • Typically this distribution is described through a power-law, where pk is the fraction of nodes in the network with degree k • Skewed distributions are observed in many phenomena • Size of city population • Earthquakes • Use of words of a given language • Income of people…