1. March 3, 2009 Network Analysis Valerie Cardenas Nicolson Assistant Adjunct Professor Department of Radiology and Biomedical Imaging

2. March 3, 2009 What is a network? Complex weblike structures –Cell is network of chemicals connected by chemical reactions –Internet is network of routers and computers linked by physical or wireless links –Social network, nodes are humans and edges are social relationships

3. March 3, 2009 Graph theory Study of complex networks Initially focused on regular graphs –Connections are completely regular, i.e. each node is connected only to nearest neighbors Since 1950s large-scale networks with no apparent design principles were described as random graphs

4. March 3, 2009 N nodes Connect every pair of nodes with probability p Approximately K edges randomly distributed with: Random Graphs Example: p=0.25, N=8 K=7

5. March 3, 2009 But… Are real networks (such as the brain) fundamentally random? Intuitively, complex systems must display some organizing principles, which must be encoded in their topology –arrangement in which the nodes of the network are connected to each other

6. March 3, 2009 Three Concepts in Complex Networks Small worlds Clustering Degree Distribution

7. March 3, 2009 Small Worlds Despite large size, in most networks there is a relatively short path between any two nodes Example: Six degrees of separation –Stanley Milgram (1967) –Path of acquaintances with typical length about six between most pairs of people in the US

8. March 3, 2009 Small World Example p is probability that pair of nodes is rewired From Guye, et al., Curr Opin Neurol 21:393-403.

9. March 3, 2009 Why should we think about the brain as a small world network? Brain is a complex network on multiple spatial and time scales –Connectivity of neurons Brain supports segregated and distributed information processing –Somatosensory and visual systems segregated –Distributed processing, executive functions Brain likely evolved to maximize efficiency and minimize the costs of information processing –Small world topology is associated with high global and local efficiency of parallel information processing, sparse connectivity between nodes, and low wiring costs –Adaptive reconfiguration

10. March 3, 2009 Path Length L i,j := minimal number of edges that must be traversed to form a direct connection between two nodes i and j By definition, for a small world network

11. March 3, 2009 Clustering There are “cliques” or “clusters” where every node is connected to every other node

12. March 3, 2009 Path Length and Clustering Watts and Strogatz, Nature, Vol 393:440-442 C(0) and L(0) are clustering coefficient and path length for regular graph. For small world, C(p)/C(0) < 1 L(p)/L(0) < 1

13. March 3, 2009 Empirical Examples of Small World Networks Watts and Strogatz, Nature, Vol 393:440-442

14. March 3, 2009 Small world metrics

15. March 3, 2009 Degree distribution k i –number of edges connected to a node i –degree of node i Degree distribution of a graph –Probability distribution of k i –In random graph, exponential P(k)  e -  k –WWW, power law P(k)  k -  Existence of few major hubs (google, yahoo) –Transportation, truncated power P(k)  k  –  e k/kc Probability of highly connected hubs greater than in a random graph but smaller than in network such as WWW

16. March 3, 2009