Networks
Igor Segota
Statistical physics presentation
Introduction
• Network / graph = set of nodes connected
1
by edges (lines)
2
• The edges can be either undirected
3
or directed (with arrows)
5
• Random network = have N nodes and
M edges placed between random pairs
4
- simplest mathematical model
6
• 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
1
by edges (lines)
2
• The edges can be either undirected
3
or directed (with arrows)
5
• Random network = have N nodes and
M edges placed between random pairs
4
- simplest mathematical model
6
• 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
3
node “i”: qi
• Average degree: <q> [over all nodes]
• Degree distribution – probability that a
randomly chosen node has exactly
q neighbors: P(q)
1
2
5
4
6
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
3
node only once.
• DNA computation:
1
2
4
5
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
3
node only once.
• DNA computation:
1 = TATCGGATCGGTATATCCGA
1
2
4
5
6
2 = GCTATTCGAGCTTAAAGCTA
…
• What about the edges ?
[Aldeman; 1994.]
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
12354546
12354546
1235456
1235456
1246
1246
Keep those
containing all
1,2,3,4,5,6
Keep len=6
1
2
23
124546
124546
124546
4
3
5
31235
1231
123546
4546
123546
123546
123546
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>]