Networks Igor Segota Statistical physics presentation

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

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

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

“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?

Network comparison

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:

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.]

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

NP-complete problems on networks Generate Keep 1… …6 Keep len= Keep those containing all 1,2,3,4,5,

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…)

p=0.1

p=0.2

p=0.3

p=0.4

p=0.45

p=0.47

p=0.49

p=0.5

p=0.51

0.53

0.55

p=0.6

p=0.7

p=0.8

p=0.9

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 ]