Introduction to Probability
MSIS 575
Class notes : October 30, 2000
Scribe : Daniel Josiah-Akintonde
Graph Theory
A graph G = (V,E) is set of vertices V, and a set of edges E. There are two types of
graphs: (1) directed graphs (digraphs), and (2) undirected graphs.
1
2
1
2
5
4
3
4
3
Directed graph
5
Undirected graph
Adjacency matrix for a directed graph
1 2 3 4 5
1
2
3
4
5
1
1
0
0
0
1
0
1
0
0
1
0
0
0
0
0
1
1
0
0
0
1
0
1
0
Adjacency matrix for a undirected graph
1 2 3 4 5
1
2
3
4
5
0
1
0
1
0
1
0
1
1
0
0
1
0
1
1
1
1
1
0
1
0
0
1
1
0
Def. A walk in a digraph is a sequence of edges of the form : (a,b), (b,c), (c,d),…..(y,z)
Def. A path is a walk where a vertex is visited at most once.
Examples: (a walk) : 1211324
(s path ) : 1324
Def. A circuit is a walk from a vertex to itself.
Def. A cycle is a path from a vertex to itself.
Notes : All the above definitions apply to both directed and undirected graphs.
Def. An undirected graph is connected if there is a path from any vertex j to any vertex
k.
Def. A directed graph is weakly connected when edges are taken to be undirected. It is
strongly connected if there is a path from any vertex j to any vertex k.
Theorem. Let G = (V,E) be a digraph with adjacency matrix A. Then the number of
m , i.e the jk entry of
Walks of length m from vertex j to vertex k is
[A ]
jk
matrix A raised to power m.
Proof.
By induction hypothesis.
[A ]
For m = 0,
A
For m = 1,
1
=
jk
=
[I ]
jk
A
Assume true for m. Then number of walks of length m from j to k is
m , need to show that number of walks of length m+1 from j to k is
(A )
(A
jk
m 1
)
jk
So, number of walks of length m+1 from j to k
= (# of walks of length m from j to 1)(# of edges from 1 to k) +
(# of walks of length m from j to 2)(# of edges from 2 to k) +
.
.
+ (# of walks of length m from j to n)(# of edges from n to k)
m
m
=
+
+ + ……..
(A ) (A )
1k
j1
=
=
m
(A
(A
A)
(A ) (A )
j2
2k
jk
m 1
)
jk
If there is path from vertex j to vertex k, then one of the matrices
2 ,
3 , ……….
,
A
A
A
Must have a non-zero entry at the jk position. It there is no path, then there is
no walk.
Markov Chains
Represent the [states] possible values of a discrete random variable by the vertices of a
digraph.
1/4
1
1/4
4
1/2
1/2
1/3
1/3
1/6
2/3
2
1/4
3
1/2
1/4
This is a markov chain. The sum of out going probabilities from any node is 1.
Def. Let X 1 , X 2 , X 3 , ……..
X n be a sequence of random variables. The
Sequence is called a markov chain if
Pr[ X n = j |
X
n 1
=
j,X
1
n2
=
j
2
,
,
X
0
=
j
0
] = Pr[[ X n = j |
X
n 1
=
j]
1
Def. A markov chain is homogeneous if:
Pr[[ X n = j |
X
n 1
= k] =
P
jk
Example: In the above example V = {1 , 2 , 3 , 4}. The adjacency matrix is :
P =
1 / 4 1 / 2 0

0 2/3
 0
 0 1/ 4 1/ 2

 0 1/ 3 1/ 6

1/ 4

1/ 3
1/ 4

1 / 2 
The sum of each row is 1.
Pr[ X 1 = 1] = Pr[[ X 1 = 1 | X 0 = 1] . Pr[ X 0 = 1] + Pr[[ X 1 = 1 | X 0 = 2] . Pr[ X 0 = 2]
+ Pr[[ X 1 = 1 | X 0 = 3] . Pr[ X 0 = 3] + Pr[[ X 1 = 1 | X 0 = 4] . Pr[ X 0 = 4]
So,

= 1/16
T
= P
1


In general:
0
T
m
= 0
T
P
m
m = 1, 2, 3, …………
Example: One-dimensional random walk.
1-p
0
p
1
n
1

q
0

0
P= 
0
0

0
0

p
0
0
0
0
0
0
q
p
0
0
p
0
0
0
0
0
0
0
0
q
0
0
q
p
0
0
p
0
0
0
0
0
0
0
0
0
0
0
q
0
0
0
q
0
p
0
q
0

0
0

0
0 
0

p
0 
Example: Urn of Ehrenfert. Molecules move from compartment A to compartment B,
One at a time each with a probability proportional to number of molecules left.
Let X be a random variable = number of molecules in compartment A at any
Given time.
p = k/n
1
q = 1-k/n
1
A
B
n
A
B
1
 0

1 / n 0 1  1 / n

2/n
0
1 2/ n

P= 
3/ n
0
1 3/ n

0














