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