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**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>]