1. Network (graph) Models
Lecture 6

2. What is a network (graph) model?

3. Network models We will cover the following network models:
Erdos–Renyi random graphs
Generalized random graphs (with the same degree distribution as the data networks)
Small-world networks
Scale-free networks
Hierarchical model
Geometric random graphs

5. Erdos–Renyi random graphs (ER)
Model a data network G(V,E) with |V|=n and |E|=m
An ER graph that models G is constructed as follows:
It has n nodes
Edges are added between pairs of nodes uniformly at random with the same probability p
Two (equivalent) methods for constructing ER graphs:
Gn,p: pick p so that the resulting model network has m edges
Gn,m: pick randomly m pairs of nodes and add edges between them with probability 1

6. Erdos–Renyi random graphs (ER)
Number of edges, |E|=m, in Gn,p is:
Average degree is:

7. Erdos–Renyi random graphs (ER)
Many properties of ER can be proven theoretically
(See: Bollobas, "Random Graphs," 2002)
Example:
When m=n/2,suddenly the giant component emerges, i.e.:
One connected component of the network has O(n) nodes
The next largest connected component has O(log(n)) nodes

8. DEGREE DISTRIBUTION OF A RANDOM GRAPH
probability of
missing N-1-k
edges
Select k
nodes from N-1
probability of
having k edges
As the network size increases, the distribution becomes increasingly narrow—we are increasingly confident that the degree of a node is in the vicinity of <k>.
Network Science: Random Graphs

9. DEGREE DISTRIBUTION OF A RANDOM GRAPH
For large N and small k, we can use the following approximations:
for
Network Science: Random Graphs

10. POISSON DEGREE DISTRIBUTION
For large N and small k, we arrive to the Poisson distribution:
Network Science: Random Graphs

11. Erdos–Renyi random graphs (ER)
The degree distribution is binomial:
For large n, this can be approximated with Poisson distribution:
where z is the average degree
However, many real world networks have power-law degree distribution

12. Erdos–Renyi random graphs (ER)
Clustering coefficient, C, of ER is low (for low p)
C=p, since probability p of connecting any two nodes in an ER graph is the same, regardless of whether the nodes are neighbors
However, many real world networks have high clustering coefficients

13. Erdos–Renyi random graphs (ER)
Average diameter of ER graphs is small
It is equal to
Real networks also have small average diameters
Summary

14. Generalized random graphs (ER-DD)
Preserve the degree distribution of data
(“ER-DD”)
Constructed as follows:
An ER-DD network has n nodes
(so does the data)
Edges are added between pairs of nodes using the “stubs method” [configuration model discussed earlier]

15. Generalized random graphs (ER-DD)
The “stubs method” for constructing ER-DD graphs:
The number of “stubs” (to be filled by edges) is assigned to each node in the model network according to the degree distribution of the real network to be modeled
Edges are created between pairs of nodes with
“available” stubs picked at random
After an edge is created, the number of stubs left available at the corresponding “end nodes” of the edges is decreased by one
Multiple edges between the same pair of nodes are not allowed

16. Generalized random graphs (ER-DD)
Summary
2 global network properties are matched by ER-DD

17. Small-world networks (SW)
Watts and Strogatz, 1998
Created from regular ring lattices by random rewiring of a small percentage of their edges
E.g.

18. Small-world networks (SW)
SW networks have:
High clustering coefficients – introduced by “ring regularity”
Large average diameters of regular lattices – fixed by randomly re-wiring a small percentage of edges
Summary

19. Scale-free networks (SF)
Power-law degree distributions: P(k) = k−γ
γ > 0; 2 < γ < 3

20. Scale-free networks (SF)
Power-law degree distributions: P(k) = k−γ
γ > 0; 2 < γ < 3

21. Scale-free networks (SF)
Different models exist, e.g.:
A popular one is:
Preferential Attachment Model (SF-BA)
(Barabasi-Albert, 1999)

22. Scale-free networks (SF)
Preferential Attachment Model (SF-BA)
“Growth” model: nodes are added to an existing network
New nodes preferentially attach to existing nodes with probability proportional to the degrees of the existing nodes; e.g.:
This is repeated until the size of SF network matches
the size of the data
“Rich getting richer”

23. Scale-free networks (SF)
Summary

24. Hierarchical model
Preserves network “modularity” via a fractal-like generation of the network

25. Hierarchical model
These graphs do not match any biological data and are highly unlikely to be found in data sets

26. Geometric random graphs
“Uniform” geometric random graphs (GEO)
Take any metric space and, using a uniform random distribution, place nodes within the space
If any nodes are within radius r (calculated via any chosen distance norm for the space), they will be connected
Choose r so that the size of the GEO network matches that of the data
There are many possible metric spaces (e.g., Euclidean space)
There are many possible distance norms
(e.g. the Euclidean distance, the Chessboard distance, and the Manhattan/Taxi Driver distance)

36. Geometric random graphs
“Uniform” geometric random graphs (GEO)
Summary

37. Stochastic Block Models
M matrix (constant p)  Erdos Renyi model
M matrix (not constant)  Erdos Renyi within community; random bipartite across communities.
Vertices within a community are considered exchangeable (i.e. probabilistically equivalent with respect to their interactions with other vertices)

38. SBM: Representation and Instantiation

39. SBM: ASSORTATIVE NETWORKS

40. SBM: Dissortative Networks

41. SBM contd.
A number of other composable models can be viewed as Stochastic Block Models
One may compose/create multiple generative models in this fashion
Can handle directed networks (M is not symmetric in this case)
Can potentially model all three properties of real networks one is often interested in!