CSCI 5352 Lecture 2 (supplement) - Santa Fe Institutetuvalu.santafe.edu/~aaronc/courses/5352/csci5352_2017_L2... · 2017-09-07 · Number of nodes, n 101 102 103 104 105 106 107 Mean

CSCI 5352Lecture 2 (supplement)

Aaron Clauset@aaronclausetComputer Science Dept. & BioFrontiers InstituteUniversity of Colorado, BoulderExternal Faculty, Santa Fe Institute

© 2017 Aaron Clauset

Kansuke Ikehara Ellen Tucker Matthias SainzAnna Broido

counts as of 7 Sept 2017

mean path length

h`i ⇠ O(log n)

Number of nodes, n101 102 103 104 105 106 107

Mea

n ge

odes

ic le

ngth

, l

0

5

10

15

20

25

30Number of graphs = 1056

mean path length

Number of nodes, n101 102 103 104 105 106 107

Mea

n ge

odes

ic le

ngth

, l

0

5

10

15

20

25

30Number of graphs = 1056

Non-BiologicalBiological (non-fungal)

h`i ⇠ 1.02 log n

mean path length

Number of nodes, n101 102 103 104 105 106 107

Mea

n ge

odes

ic le

ngth

, l

0

5

10

15

20

25

30Number of graphs = 1056

Non-SocialSocial

h`i ⇠ 0.52 log n

mean path length

Number of nodes, n101 102 103 104 105 106 107

Mea

n ge

odes

ic le

ngth

, l

0

5

10

15

20

25

30Number of graphs = 1056

Non-TechnologicalTechnological

h`i ⇠ 1.52 log n

mean path length

Number of nodes, n101 102 103 104 105 106 107

Mea

n ge

odes

ic le

ngth

, l

0

5

10

15

20

25

30Number of graphs = 1056

Non-InformationalInformational

h`i ⇠ 1.16 log n

mean path length

Number of nodes, n101 102 103 104 105 106 107

Mea

n ge

odes

ic le

ngth

, l0

5

10

15

20

25

30Number of graphs = 1056

socialbiological

technologicalinformation

mean path length

}h`i ⇠ 1.16 log n

h`i ⇠ 1.52 log n

h`i ⇠ 0.52 log n

h`i ⇠ 1.02 log n

random graph theory

O(log n)

Molloy & Reed. Random Structures Algorithms 6, 61 (1995)Chung & Lu Proc. Natl. Acad. Sci. USA 99(25), 15879-15882 (2002)

clustering coefficient

c ⇠ n�1

c ⇠ n� = n7�3↵↵�1

Erdös-Rényi and similarpower-law random graphs

Number of nodes, n101 102 103 104 105 106 107

Clu

ster

ing

coef

ficie

nt, c

10-3

10-2

10-1

100 Number of graphs = 920

clustering coefficient

Number of nodes, n101 102 103 104 105 106 107

Clu

ster

ing

coef

ficie

nt, c

10-3

10-2

10-1

100 Number of graphs = 920

SocialNon-Social

c ⇠ n�0.23

clustering coefficient

Number of nodes, n101 102 103 104 105 106 107

Clu

ster

ing

coef

ficie

nt, c

10-3

10-2

10-1

100 Number of graphs = 920

BiologicalNon-Biological

c ⇠ n�0.14

clustering coefficient

Number of nodes, n101 102 103 104 105 106 107

Clu

ster

ing

coef

ficie

nt, c

10-3

10-2

10-1

100 Number of graphs = 920

TechnologicalNon-Technological

c ⇠ n�0.38

clustering coefficient

Number of nodes, n101 102 103 104 105 106 107

Clu

ster

ing

coef

ficie

nt, c

10-3

10-2

10-1

100 Number of graphs = 920

InformationalNon-Informational

c ⇠ n�0.34

clustering coefficient

Number of nodes, n101 102 103 104 105 106 107

Clu

ster

ing

coef

ficie

nt, c

10-3

10-2

10-1

100 Number of graphs = 920

c ⇠ n�0.34

socialbiological

technologicalinformation

c ⇠ n�0.38

c ⇠ n�0.14

c ⇠ n�0.23

clustering coefficient

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