1. Networks Igor Segota Statistical physics presentation

2. Introduction Network / graph = set of nodes connected by edges (lines) The edges can be either undirected or directed (with arrows) Random network = have N nodes and M edges placed between random pairs - simplest mathematical model The mathematical theory of networks originates from 1950’s [Erdos, Renyi] In the last 20 years abundance of data about real networks: – Internet, citation networks, social networks – Biological networks, e.g. protein interaction networks, etc. 1 2 4 6 5 3

3. Introduction Network / graph = set of nodes connected by edges (lines) The edges can be either undirected or directed (with arrows) Random network = have N nodes and M edges placed between random pairs - simplest mathematical model The mathematical theory of networks originates from 1950’s [Erdos, Renyi] In the last 20 years abundance of data about real networks: – Internet, citation networks, social networks – Biological networks, e.g. protein interaction networks, etc. 1 2 4 6 5 3

4. Statistical measures How to systematically analyze a network? Define: Degree: number of neighbors of each node “i”: q i Average degree: [over all nodes] Degree distribution – probability that a randomly chosen node has exactly q neighbors: P(q) Is there a notion of “path” or “distance” on a network? Path length, or node-to-node distance: How many links we need to pass through to travel between two nodes ? Characterizes the compactness of a network 1 2 4 6 5 3

5. “Scale-free” networks If we look at the real world networks, e.g.: a) WWW, b) movie actors, c,d) citation networks, phone calls, metabolic networks, etc.. They aren’t random – the degree distribution follows a power law: P(q) = A q -γ with 2 ≤ γ ≤ 3 They do not arise by chance! Examples: – WWW, publications, citations Can we get an intuitive feeling for the network shape, given some statistical measure?

6. Network comparison

7. NP-complete problems on networks NP-complete problem Problem such that no solution that scales as a polynomial with system size is known. Directed Hamiltonian Path problem – Find a sequence of one-way edges going through each node only once. DNA computation: 1 2 4 6 5 3

8. NP-complete problems on networks NP-complete problem Problem such that no solution that scales as a polynomial with system size is known. Directed Hamiltonian Path problem – Find a sequence of one-way edges going through each node only once. DNA computation: What about the edges ? TATCGGATCGGTATATCCGA GCTATTCGAGCTTAAAGCTA 1 2 = = … [Aldeman; 1994.] 1 2 5 3 4 6

9. NP-complete problems on networks For each pair of nodes, construct a corresponding edge Due to directionality of DNA, edge orientation is preserved and 1->2 is not equal to 2->1 Idea: generate all possible combinations of all possible lengths then filter out the wrong ones CATATAGGCT CGATAAGCGA TATCGGATCGGTATATCCGAGCTATTCGAGCTTAAAGCTA 12

10. NP-complete problems on networks Generate Keep 1… …6 Keep len=6 1246 1235456 31235 23 12354546 124546 4546 123546 1231 1246 1235456 12354546 124546 123546 Keep those containing all 1,2,3,4,5,6 1 2 5 3 4 6 124546

11. Emergent phenomena on networks Critical phenomena: an abrupt emergence of a giant connected cluster [simulation] Analogous to the effect in percolation theory (in fact it is exactly the same effect…)

12. p=0.1

13. p=0.2

14. p=0.3

15. p=0.4

16. p=0.45

17. p=0.47

18. p=0.49

19. p=0.5

20. p=0.51

21. 0.53

22. 0.55

23. p=0.6

24. p=0.7

25. p=0.8

26. p=0.9

27. Network percolation experiments Living neural networks [Breskin et. al., 2006] Nodes = cells, edges = cell extensions + transmitting molecules Rat brain neurons grown in a dish, everyone gets connected Put a chemical that reduces the probability of neuron firing (disables edge) [effectively adjusts the ]