csci 5352 lecture 2 (supplement) - santa fe...
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![Page 1: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/1.jpg)
CSCI 5352Lecture 2 (supplement)
Aaron Clauset@aaronclausetComputer Science Dept. & BioFrontiers InstituteUniversity of Colorado, BoulderExternal Faculty, Santa Fe Institute
© 2017 Aaron Clauset
![Page 2: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/2.jpg)
![Page 3: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/3.jpg)
Kansuke Ikehara Ellen Tucker Matthias SainzAnna Broido
![Page 4: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/4.jpg)
![Page 5: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/5.jpg)
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![Page 7: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/7.jpg)
counts as of 7 Sept 2017
![Page 8: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/8.jpg)
![Page 9: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/9.jpg)
mean path length
h`i ⇠ O(log n)
![Page 10: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/10.jpg)
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
![Page 11: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/11.jpg)
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
![Page 12: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/12.jpg)
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
![Page 13: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/13.jpg)
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
![Page 14: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/15.jpg)
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)
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clustering coefficient
c ⇠ n�1
c ⇠ n� = n7�3↵↵�1
Erdös-Rényi and similarpower-law random graphs
![Page 17: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/18.jpg)
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
![Page 19: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/19.jpg)
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
![Page 20: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/20.jpg)
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
![Page 21: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/21.jpg)
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
![Page 22: 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](https://reader034.vdocuments.net/reader034/viewer/2022042805/5f63f347f1a99e68250e9c29/html5/thumbnails/22.jpg)
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