1. The Erdös-Rényi models
2. Random Graphs
The Erdös-Rényi models

2. Distinguish: Equilibrium random networks
Nonequilibrium random networks

3. Equilibrium random networks
A classical undirected random graph:
the total number of vertices is fixed
connect randomly chosen pairs of vertices

4. Nonequilibrium random network
A classical random graph that grows through simultaneous addition of vertices and links
at each time step a new vertex is added
simultaneously, a pair of randomly chosen vertices is connected

5. Status Graph theory: Physics:
Equilibrium networks with a Poisson degree distribution
Physics:
Nonequilibrium (growing networks), percolation

6. Statistical sense
A particular observed network is only one member of a statistical ensemble of all possible realizations
Random network -> Statistical ensemble
N nodes ->
How should we understand the degree distribution? It determines the the ensemble of the equilibrium random networks
possible graphs

7. The Erdos-Renyi model
Definition: N labeled nodes connected by n links which are chosen randomly from the N(N-1)/2 possible links
There are graphs with N nodes and n links

8. Alternative definition
Binomial model: start with N nodes, every pair of nodes being connected with probability p
The total number of links, n, is a random variable
E(n)=pN(N-1)/2
Probability of generating a graph, G0{N,n}

9. Growing a graph
Sometimes we will study properties of the graph as p increases
Assign a random number qi[0,1] to attach links and then links appear as p is increased p> qi
We are interested in the “static” properties of the graph when N-> and keeping constant p or n

10. N->
Definition: almost every graph has a property Q if the probability of having Q approaches 1 as N-> 
The main goal of Random Graph theory is to determine at what connection probability p a particular property of a graph most likely arises

11. Many important properties appear suddenly:
almost everygraph has the property
almost no graph has it
Usually there exists a critical probability pc(N)
p(N) probability that almost every graph has property Q

12. If p(N) grows slower than pc(N)
If p(N) grows faster than pc(N)

13. Examples
Larger graphs with the same p contain more links since n=pN(N-1)/2
Appearance of cycles can occur for smaller p in large graphs than in smaller ones [pc(N->)->0]
Average degree of the graph
No clar. Segons BA té un valor crític que és independent del tamany del sistema

14. Subgraphs P1 set of nodes, E1 set of links
G1(P1,E1) is a subgraph of G(P,E) if all nodes of G1 belong to G and links too.
Basic subgraphs:
cycles
trees
complete graphs

16. Evolution of the graph (p grows)

17. Subgraph in graph F small graph of k nodes and l links
How many subgraphs like F exist in G?
This expected value depends on p. If N>>k

18. There are no subgraphs like F
If = constant => mean number of subgraphs is a finite number
The critical probability

19. Tree of order k: l=k-1
Cycle of order k: l=k
Complete subgraph l=k(k-1)/2
We can see how the subgraphs appear when increasing p

20. Mean connectivity <k>=pN If then <k> is a constant
If 0< <k> < 1 almost surely all clusters are either trees or clusters containing exactly one cycle
At <k>=1 the structure changes abruptly. Cycles appear and a giant cluster develops

21. Degree distribution
The degree of a node follows a binomial distribution (in a random graph with p)
Probability that a given node has a connectivity k
For large N, Poisson distribution

23. Mean short path Assume that the graph is homogeneous
The number of nodes at distance l are <k>l
How to reach the rest of the nodes?
lrand to reach all nodes => kl=N

25. Clustering coefficient
Probability that two nodes are connected (given that they are connected to a third)?
while it is constant for real networks

27. Spectrum: random matrices
If Aij real, symmetric, NxN uncorrelated random matrix <Aij>=0 and <Aij2>=2
Density of eigenvalues of
Wigner’s or semicircle law (late 50’s)

28. Spectrum: random graph
<Aij>=  not 0
2 =p(1-p)
Plotting
() semicircel law as N increases (p constant)

29. In general,
z<1:
semicircle law
exists an infinite cluster
1 (principal, largest) is isolated, grows like N
z<1: most of the graphs are trees (odd moment vanish). The spectral density contains the weighted sum of the spectral densities of all finite graphs

30. Generalized random graphs
One can construct a graph introducing the degree distribution as an input
How do the properties of the network change with the exponent?
 decreases from  to 0

31. <k>=kmax-+2 (kmax <N, max degree)
The infinite cluster emerges when
There exists a value 0=
>0 disconnected
>0 almost surely connected

32. Exponential cutoff (observed in real world networks)
Normalitzable for any k
>2 disconnected
<2 connected

33. NON-RANDOM aspects of the topology
of real networks

34. Growing networks
See hand-written notes

35. Scaling