Introduction to Network Science 1

Random Models

Introduction to Network Science 2

mean degree in a graph with exactly m edges

Taylor series reminder:

Introduction to Network Science 3

In contrast to the degree distribution in random model …

Introduction to Network Science 4

In contrast to the degree distribution in random model …

Introduction to Network Science 5

Newman, “Random graphs as models of networks”

Introduction to Network Science 6

Q: When p=0 then |gc|=1; when p=1 then |gc|=n. What is the difference between them?

Co-authorship network its largest connected component

Introduction to Network Science 7

Q: When p=0 then |gc|=1; when p=1 then |gc|=n. Is this transition smooth? Is there a point of transition?

i does not belong to gc if for every node j either

a) there is no edge ij, prob =

b) there is edge ij but j is not in giant component, prob =

vertices in giant component

1-p

pu

Introduction to Network Science 8

Newman “Networks, An Introduction” => Demo in Matlab

Introduction to Network Science 9

Introduction to Network Science 10

Introduction to Network Science 11

Conclusion: In the limit of large n, the probability of existence of two separate giant components goes to zero.

large

Introduction to Network Science 12

fraction of nodes in gc

edges in tree edge prob

the component is still tree

Introduction to Network Science 13

Kronecker delta

Introduction to Network Science 14

see handout, pp 412-413

Average size of the small components in a random model does not grow with the number of vertices. Average component size

Introduction to Network Science 15

Distribution of component sizes