Introduction
• Network / graph = set of nodes connected by edges (lines)
1
• The edges can be either undirected or directed (with arrows) 3
5
• 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]
6
• 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)
1
• The edges can be either undirected or directed (with arrows) 3
5
• 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]
6
• 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 -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.
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
• What about the edges ?
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>]