Networks Igor Segota Statistical physics presentation.

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 andM edges placed between random pairs- simplest mathematical model

• The mathematical theory of networksoriginates 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.

12

46

53

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 andM edges placed between random pairs- simplest mathematical model

• The mathematical theory of networksoriginates 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.

12

46

53

Statistical measures

• How to systematically analyze a network?Define:

• Degree: number of neighbors of each node “i”: qi

• Average degree: <q> [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

12

46

53

“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 problemProblem such that no solution that scalesas 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

65

3

NP-complete problems on networks

NP-complete problemProblem such that no solution that scalesas 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

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

TATCGGATCGGTATATCCGA GCTATTCGAGCTTAAAGCTA

1 2

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 123546 123546

Keep those containing all 1,2,3,4,5,6

1

2

53 4

6

124546

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 <q>]