enpm809g random networks, power laws, and...
TRANSCRIPT
R A N D O M N E T W O R K S , P O W E R L A W S , A N D M I X I N G
E N P M 8 0 9 G
C O D Y B U N TA I N @ C O D Y B U N TA I N
C B U N TA I N @ C S . U M D . E D U
H O M E W O R K Q U E S T I O N S ?
E N P M 8 0 9 G
I N - C L A S S E X E R C I S EE N P M 8 0 9 G
I T E R AT E D G R A P H B U I L D I N G
• Objective: Minimize your maximal shortest path
• Group 1: Everyone evenly divisible by 3
• Group 2: Remaining even numbers
• Group 3: Remaining odd numbers
#1
#28
#9
#20
R A N D O M G R A P H M O D E L S
E N P M 8 0 9 G
E R D O S - R E N Y I G R A P H S
D - R E G U L A R R I N G L AT T I C E S
W AT T S - S T R O G AT Z M O D E L S
J U P Y T E R D E M O
M O D E L S U M M A R Y
I F N O T B I N O M I A L , T H E N W H AT ?
E N P M 8 0 9 G
D E G R E E D I S T R I B U T I O N S I N R E A L - W O R L D G R A P H S
P O I S S O N V S . R E A L D I S T R I B U T I O N
Poisson Real
H U B S
Hub Regime
B A R A B A S I A N D A L B E R T A N D T H E W O R L D W I D E W E B
D Y N A M I C G E N E R AT I V E M O D E L
“ R I C H G E T R I C H E R ”
P O W E R L A W : G E N E R A L F O R M
Pr(k) = C k��
T H E I N T E R N E T ’ S P O W E R L A W
Pr(k) = 2m2t k�3
Analytic Solution:
Pr(k) = C k��
Data:
C = ⇣(�), � = 3.42
S P O T T I N G P O W E R L A W S
G R A P H I C A L LY
A L G E B R A I C A L LY
J U P Y T E R D E M O
H U B S
kmax ~ (n
-1)
A L S O C A L L E D “ S C A L E - F R E E ” N E T W O R K S
P O W E R L A W - A B I D I N G N E T W O R K S
D O E S 𝛾= 3 A LW AY S T R U E ?
E X A M P L E S
G E N E R A L LY, 2≤ 𝛾≤ 3
N O I S E , R E G I M E C H A N G E , A N D N O R M A L I Z AT I O N
N O I S E AT T H E E N D S
K M I N A N D C H A N G E P O I N T S
J U P Y T E R D E M O
N O R M A L I Z I N G P ( K )
F I N D I N G E X P O N E N T S
C A N D O O L S O N L O G - L O G P L O T
C A N D O O L S O N L O G - L O G P L O T… ( B U T D O N ’ T )
� = 1 + |V |✓X
i
lnki
kmin � 12
◆�1
J U P Y T E R D E M O
M O M E N T S I N A P O W E R L A W G R A P H
< kn >=X
knp(k)
Nth Moment
W H Y “ S C A L E - F R E E ” ?
AV E R A G E D E G R E E I S N ’ T A G O O D “ S C A L E ” F O R T H E G R A P H
M I X I N G A N D C O M M U N I T Y S T R U C T U R E S
E N P M 8 0 9 G
N E T W O R K - V S . V E R T E X -M E A S U R E S
C O N F I G U R AT I O N M O D E L S
J U P Y T E R D E M O
M I X I N G A N D L O C A L S T R U C T U R E
H O M O G E N E O U S M I X I N G
H E T E R O G E N E O U S M I X I N G
M I X I N G I N K N O W N G R A P H M O D E L S
C O R O L L A R Y: L I M I T E D C O M M U N I T Y S T R U C T U R E I N T H E S E M O D E L S
T Y P E S O F H E T E R O G E N E O U S M I X I N G
A S S O R TAT I V E , A K A A S S O C I AT I V I T Y
D I S A S S O R TAT I V E , A K A D I S A S S O C I AT I V I T Y
S T R U C T U R E S I N A S S O C I AT I V I T Y
M O D U L A R S T R U C T U R E
H I E R A R C H I C A L S T R U C T U R E
C O R E - P E R I P H E R Y S T R U C T U R E
O R D E R E D S T R U C T U R E
A S S O C I AT I V I T Y A N D H O M O P H I LY
M E A S U R I N G A S S O C I AT I V I T Y W I T H M O D U L A R I T Y
M O D U L A R I T Y E Q U AT I O N
Q =X
ij
"✓1
2mAij �
ki2m
⇥ kj2m
◆�(xi, xj)
#
M O D U L A R I T Y E Q U AT I O N V 2
Q =X
ij
"✓1
2mAij �
ki2m
⇥ kj2m
◆�(xi, xj)
#
M O D U L A R I T Y E X A M P L E
M O D U L A R I T Y E X A M P L E
Q = 5/14 = 0.357 Q = 6/49 = 0.122
M O D U L A R I T Y I N G E P H I
• Figures reproduced from:
• Albert-Laszlo, Barabasi. NETWORK SCIENCE: THE SCALE-FREE PROPERTY
• Aaron Clauset’s CSCI5352 Lecture Notes, Lecture 5, http://tuvalu.santafe.edu/~aaronc/courses/5352/csci5352_2017_L5.pdf