Network1

Network2

Network3

• Binary relation
  EDGES between elements of a set ( = VERTICES)

• For example

• Vertices = {A,B,C,D,E}

• Edges = ({A,B},{A,C},{B,C},{C,E})

• It is "better" to draw it:

• If the pairs are unordered, then the graph is undirected

Network4

• Otherwise, it is directed

Network5

• A graph called weighted if both edges and vertices have some weights or values

Network6

• Otherwise, it is unweighted

Network7

• Random Network
  Randomply connecting pairs of nodes with some probabily \(p\)

• Scale-Free Network Few nodes has a large number of links, most nodes have very few.

## Random Graph
g <- erdos.renyi.game(10, 1/10)
plot(g)

degree_distribution(g)

## [1] 0.4 0.4 0.2

## Scale Free
N <- 20
g <- sample_fitness(5*N, sample((1:20)^-2, N, replace=TRUE))
plot(g)

• For classic graph theory and network science
  
• Core functionally is implemented as a C library
  
• High level interfaces from R and Python
• http: //igraph.sf.net

• Vertices are always numbered from zero
• Numbering is continual from 1 to |V|

V = {A,B,C,D,E}
Edges = ((A,B),(A,C),(B,C),(C,E))
A = 1, B = 2, C = 3, D = 4, E = 5

g <- graph(c(1,2, 1,3, 2,3, 3,5),n =5)

g <- graph(c(1,2, 1,3, 2,3, 3,5),n =5)
plot(g)

• igraph object

print(g)

## IGRAPH 356935f D--- 5 4 -- 
## + edges from 356935f:
## [1] 1->2 1->3 2->3 3->5

vcount(g)

## [1] 5

ecount(g)

## [1] 4

\[ \begin{bmatrix} & A & B & C & D & E\\ A & 0 & 1 & 1 & 0 & 0\\ B & 1 & 0 & 1 & 0 & 0\\ C & 1 & 1 & 0 & 0 & 1\\ D & 0 & 0 & 0 & 0 & 0\\ E & 0 & 0 & 1 & 0 & 0 \end{bmatrix} \]

A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph.

– In a directed graph G that may not itself be strongly connected, a pair of vertices u and v are said to be strongly connected to each other if there is a path in each direction between them.

g <- erdos.renyi.game(20, 1/15)
plot.igraph(g)

Either: Just simulate a random graph (n = 10, p = 0.3) and print out its adjacency matrix

Or: Download any network data and print it as a graph.