# Critical phenomena in complex networks ## Networks

### Statistical physics presentation

Introduction

Network / graph = set of nodes connected by edges (lines)

The edges can be either undirected or directed (with arrows)

3

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]

1

6

5

In the last 20 years abundance of data about real networks:

Internet, citation networks, social networks

Biological networks, e.g. protein interaction networks, etc.

2

4

Introduction

Network / graph = set of nodes connected by edges (lines)

The edges can be either undirected or directed (with arrows)

3

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]

1

6

5

In the last 20 years abundance of data about real networks:

Internet, citation networks, social networks

Biological networks, e.g. protein interaction networks, etc.

2

4

Statistical measures

How to systematically analyze a network?

Define:

Degree: number of neighbors of each node “i”: q

i

Average degree: <q> [over all nodes]

Degree distribution – probability that a randomly chosen node has exactly

q neighbors: P(q)

3

1

6

5

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

2

4

“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

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.

3

DNA computation:

5

1

4

2

6

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.

3

DNA computation:

1

=

TATCGGATCGGTATATCCGA

=

GCTATTCGAGCTTAAAGCTA

2

# …

5

1

4

2

[Aldeman; 1994.]

6

NP-complete problems on networks

CATATAGGCT CGATAAGCGA

TATCGGATCGGTATATCCGA GCTATTCGAGCTTAAAGCTA

1 2

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

NP-complete problems on networks

Generate Keep 1… …6 Keep len=6

Keep those containing all

1,2,3,4,5,6

12354546

1235456

1246

23

124546

31235

1231

123546

4546

12354546

1235456

1246

124546

123546

124546

123546

3

5

123546

1

4

2

6

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>]