Internet
Power Grids
City Networks
Communication
Social Networks
Ecological Networks
Anatomical Networks
Molecular Networks

G  V,E

nodes or vertices

V  v1 ,
, vn

edges or links
E  V V
the number of edges incident to the node .
i
ki  3
Leonhard Euler
1707-1783
Euler, L. Solutio Problematis ad Geometriam Situs Pertinentis.
Commentarii Academiae Scientiarium Imperialis Petropolitanae, 8 (1736) 128-40.
3
C
c
g
d
3D
5
A
a
e
b
3
f
B
Theorem: A connected graph is traversible if and only if it has at
most two vertices of odd degree.
Adjacency matrix
A B C
D E
A
0
1
1
0
0
B
1
0
1
1
1
C
1
1
0
0
0
D
0
1
0
0
0
E
0
1
0
0
0
A
B
D
C
E
Adjacency matrix
A B C
D E
A
0
0
1
0
0
B
1
0
1
1
1
C
0
0
0
0
0
D
0
1
0
0
0
E
0
0
0
0
0
A
B
D
C
E
 A
ij
1 iff i ~ j

0 otherwise


A  
Eigenvalues of A: 1  2  3 
Eigenvector of  j :  j   j (1),
 n
,  j (n) 
T
Pn
Cn
Sn
Kn
 j 

 n 1
Pn
 j A   2 cos
j  1,, n
Cn
 2j 
 j A   2 cos

 n 
j  1,, n
Sn
SpA 
Kn
SpA  1  1n 1
n

0n 2

 n

p  0.000
p  0.106
Pál Erdös
1913-1996
Alfréd Rényi
1921-1970
p  0.265
p  1.000
p = 0.01
p = 0.025
Size giant component
p = 0.0075
0
p
1
S  Oln n 
Size giant component
k 1
pn ~1
k 1
0
p
k 1
 
S   n2/ 3
1
a giant component
exists
Number of nodes with k links
Number of links (k)
Number of links (k)
Number of nodes with k links
pu  kv /  k w
w
e k k k
pk  
k!
pk  
pk  ~ e k / k
1
e
2 k


 k k 2

 2 2
k





pk  ~ k 
 n  1 k
n 1 k
pk   
p 1  p 

 k 
n 
pk  
ek k k
k!
Assortative
Disassortative
2
1
 1

m e ki e k j e   m e ki e k j e 
2


r
2
1 2
1
 1

1
2
m e ki e k j e   m e ki e k j e 
2
2


1
r>0
r<0
r  0.118
r  0.304
r  0.129
r  0.277
r = -0.538
r = 0.200


2
1
1E1
2m
r
1
1 E2 1  k A k 
1E1
2m
k Ak 

k

2
k A k  2 P1  4 P2  2 P3  6 C3

1
1E1
2m

2

2 P2
P1
2
 2 P1  4 P2
E
P1 
P2 
C3 
P3 
S1,3 
1 E 2 1  2 P1  10 P2  2 P3  6 S1,3  6 C3
r

P2 P3/2  C  P2/1

3 S1,3  P2 1  P2/1
P1 
P2 
S1,3 


Pr / s  Pr / Ps
C  3 C3 / P2
P3 
C3 
P2 / 1  P3 / 2  C
P2 / 1  P3 / 2  C
r = -0.538
r = 0.200
A B C
D E
A
0
1
1
0
0
2
B
1
0
1
1
1
4
C
1
1
0
0
0
2
D
0
1
0
0
0
1
E
0
1
0
0
0
1
A
B
D
C
E
i, k , j 
BC k   
, i jk
i, j 
i
j

A
B
C
D
E

  A, B, D   1;   A, D   1
  A, B, E   1;   A, E   1
 C , B, D   1;  B, D   1
 C , B, E   1;  C , E   1
 D, B, E   1;  D, E   1
B has a BC of 5
N 1
CC i  
 d i, j 
j
n
A B C
D E
A
0
1
1
2
2
6
B
1
0
1
1
1
4
C
1
1
0
2
2
6
D
2
1
2
0
2
7
E
2
1
2
2
0
7
A
B
D
C
E
 d i, j 
j 1
A
0.67
B
C
1.00
0.67
0.57
D
E
0.57
xi 
1
n
A x


x
j 1
1

ij
Ax
j
1 0.500
2 0.238
3 0.238
5
6
7
8
0.354
0.354
0.168
0.168