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.
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46
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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
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“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
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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>]