WolframAlpha.com
WolframCloud.com
All Sites & Public Resources...
Products & Services
Wolfram|One
Mathematica
Wolfram|Alpha Notebook Edition
Finance Platform
System Modeler
Wolfram Player
Wolfram Engine
WolframScript
Enterprise Private Cloud
Application Server
Enterprise Mathematica
Wolfram|Alpha Appliance
Enterprise Solutions
Corporate Consulting
Technical Consulting
Wolfram|Alpha Business Solutions
Resource System
Data Repository
Neural Net Repository
Function Repository
Wolfram|Alpha
Wolfram|Alpha Pro
Problem Generator
API
Data Drop
Products for Education
Mobile Apps
Wolfram Player
Wolfram Cloud App
Wolfram|Alpha for Mobile
Wolfram|Alpha-Powered Apps
Services
Paid Project Support
Wolfram U
Summer Programs
All Products & Services »
Technologies
Wolfram Language
Revolutionary knowledge-based programming language.
Wolfram Cloud
Central infrastructure for Wolfram's cloud products & services.
Wolfram Science
Technology-enabling science of the computational universe.
Wolfram Notebooks
The preeminent environment for any technical workflows.
Wolfram Engine
Software engine implementing the Wolfram Language.
Wolfram Natural Language Understanding System
Knowledge-based broadly deployed natural language.
Wolfram Data Framework
Semantic framework for real-world data.
Wolfram Universal Deployment System
Instant deployment across cloud, desktop, mobile, and more.
Wolfram Knowledgebase
Curated computable knowledge powering Wolfram|Alpha.
All Technologies »
Solutions
Engineering, R&D
Aerospace & Defense
Chemical Engineering
Control Systems
Electrical Engineering
Image Processing
Industrial Engineering
Mechanical Engineering
Operations Research
More...
Finance, Statistics & Business Analysis
Actuarial Sciences
Bioinformatics
Data Science
Econometrics
Financial Risk Management
Statistics
More...
Education
All Solutions for Education
Trends
Machine Learning
Multiparadigm Data Science
Internet of Things
High-Performance Computing
Hackathons
Software & Web
Software Development
Authoring & Publishing
Interface Development
Web Development
Sciences
Astronomy
Biology
Chemistry
More...
All Solutions »
Learning & Support
Learning
Wolfram Language Documentation
Fast Introduction for Programmers
Wolfram U
Videos & Screencasts
Wolfram Language Introductory Book
Webinars & Training
Summer Programs
Books
Need Help?
Support FAQ
Wolfram Community
Contact Support
Premium Support
Paid Project Support
Technical Consulting
All Learning & Support »
Company
About
Company Background
Wolfram Blog
Events
Contact Us
Work with Us
Careers at Wolfram
Internships
Other Wolfram Language Jobs
Initiatives
Wolfram Foundation
MathWorld
Computer-Based Math
A New Kind of Science
Wolfram Technology for Hackathons
Student Ambassador Program
Wolfram for Startups
Demonstrations Project
Wolfram Innovator Awards
Wolfram + Raspberry Pi
Summer Programs
More...
All Company »
Search
WOLFRAM COMMUNITY
Connect with users of Wolfram technologies to learn, solve problems and share ideas
Join
Sign In
Dashboard
Groups
People
Message Boards
Answer
(
Unmark
)
Mark as an Answer
GROUPS:
Staff Picks
Wolfram Science
Mathematics
Discrete Mathematics
Geometry
Graphs and Networks
Wolfram Language
Statistics and Probability
Wolfram Summer School
2
Yousof Mardoukhi
[WWS21] Alexander Orbach Relation and Random Walks on Hypergraphs
Yousof Mardoukhi, University of Potsdam
Posted
1 year ago
2392 Views
|
1 Reply
|
2 Total Likes
Follow this post
|
Introduction
T
h
e
t
h
e
o
r
y
o
f
r
a
n
d
o
m
w
a
l
k
s
p
r
o
v
i
d
e
s
o
n
e
w
i
t
h
a
t
o
o
l
t
o
u
n
d
e
r
s
t
a
n
d
a
n
d
i
n
v
e
s
t
i
g
a
t
e
t
h
e
g
e
o
m
e
t
r
i
c
a
l
a
n
d
d
y
n
a
m
i
c
a
l
p
r
o
p
e
r
t
i
e
s
o
f
t
h
e
u
n
d
e
r
l
y
i
n
g
s
t
r
u
c
t
u
r
e
o
n
w
h
i
c
h
r
a
n
d
o
m
w
a
l
k
s
t
a
k
e
s
p
l
a
c
e
.
F
o
r
i
n
s
t
a
n
c
e
i
f
t
h
e
u
n
d
e
r
l
y
i
n
g
s
t
r
u
c
t
u
r
e
i
s
a
g
r
a
p
h
,
b
y
i
n
v
e
s
t
i
g
a
t
i
n
g
h
o
w
f
a
r
a
r
a
n
d
o
m
w
a
l
k
p
r
o
c
e
s
s
h
a
s
g
o
n
e
w
i
t
h
r
e
s
p
e
c
t
t
o
i
t
s
i
n
i
t
i
a
l
p
o
s
i
t
i
o
n
,
o
n
e
c
a
n
e
x
t
r
a
c
t
u
s
e
f
u
l
i
n
f
o
r
m
a
t
i
o
n
s
u
c
h
a
s
t
h
e
c
o
n
n
e
c
t
i
v
i
t
y
b
e
t
w
e
e
n
t
h
e
n
o
d
e
s
o
f
t
h
e
g
r
a
p
h
.
W
o
l
f
r
a
m
M
o
d
e
l
s
w
h
i
c
h
h
a
v
e
h
y
p
e
r
g
r
a
p
h
r
e
p
r
e
s
e
n
t
a
t
i
o
n
,
a
r
e
i
n
t
e
r
e
s
t
i
n
g
c
a
n
d
i
d
a
t
e
s
t
o
i
n
v
e
s
t
i
g
a
t
e
t
h
e
d
y
n
a
m
i
c
s
o
f
s
u
c
h
a
p
r
o
c
e
s
s
.
T
h
i
s
i
s
o
f
u
t
m
o
s
t
i
m
p
o
r
t
a
n
c
e
,
s
i
n
c
e
t
h
e
W
o
l
f
r
a
m
M
o
d
e
l
s
h
a
v
e
c
a
u
s
a
l
s
e
t
r
e
p
r
e
s
e
n
t
a
t
i
o
n
s
.
T
h
i
s
c
a
u
s
a
l
s
e
t
s
t
h
u
s
c
a
n
b
e
f
a
i
t
h
f
u
l
l
y
e
m
b
e
d
d
e
d
o
n
L
o
r
e
n
t
z
i
a
n
m
a
n
i
f
o
l
d
s
.
T
h
u
s
i
n
t
h
e
c
o
n
t
i
n
u
u
m
l
i
m
i
t
,
t
h
e
g
e
o
m
e
t
r
i
c
a
l
p
r
o
p
e
r
t
i
e
s
t
h
e
m
a
n
i
f
o
l
d
c
a
n
b
e
e
x
t
r
a
c
t
e
d
f
r
o
m
t
h
e
e
m
b
e
d
d
e
d
c
a
u
s
a
l
s
e
t
a
n
d
i
t
s
a
l
g
e
b
r
a
i
c
s
t
r
u
c
t
u
r
e
a
n
d
v
i
c
e
v
e
r
s
a
[
4
]
.
M
a
n
y
p
r
o
p
e
r
t
i
e
s
o
f
t
h
e
r
a
n
d
o
m
w
a
l
k
p
r
o
c
e
s
s
a
r
e
u
n
d
e
r
s
t
o
o
d
i
n
t
e
r
m
s
o
f
t
h
e
s
i
t
e
o
c
c
u
p
a
t
i
o
n
a
n
d
f
i
r
s
t
-
p
a
s
s
a
g
e
p
r
o
b
a
b
i
l
i
t
i
e
s
;
w
i
t
h
c
o
m
b
i
n
a
t
o
r
i
c
s
a
n
d
p
r
o
b
a
b
i
l
i
t
y
g
e
n
e
r
a
t
i
n
g
f
u
n
c
t
i
o
n
o
f
t
h
e
s
e
q
u
a
n
t
i
t
i
e
s
a
n
d
k
n
o
w
i
n
g
t
h
e
p
r
o
b
a
b
i
l
i
t
y
t
r
a
n
s
i
t
i
o
n
m
a
t
r
i
x
,
s
u
c
h
p
r
o
p
e
r
t
i
e
s
a
r
e
s
t
u
d
i
e
d
f
o
r
d
i
f
f
e
r
e
n
t
v
a
r
i
a
t
i
o
n
s
o
f
r
a
n
d
o
m
w
a
l
k
p
r
o
c
e
s
s
e
s
.
H
e
r
e
i
n
t
h
i
s
p
r
o
j
e
c
t
t
h
e
f
o
c
u
s
w
i
l
l
b
e
o
n
t
h
e
m
o
s
t
g
e
n
e
r
i
c
t
y
p
e
o
f
r
a
n
d
o
m
w
a
l
k
p
r
o
c
e
s
s
c
a
l
l
e
d
P
ó
l
y
a
’
s
w
a
l
k
.
P
ó
l
y
a
’
s
w
a
l
k
i
s
a
r
a
n
d
o
m
w
a
l
k
o
n
a
d
-
d
i
m
e
n
s
i
o
n
a
l
l
a
t
t
i
c
e
w
h
i
c
h
i
s
h
o
m
o
g
e
n
e
o
u
s
,
w
h
i
c
h
i
m
p
l
i
e
s
t
h
a
t
t
h
e
l
a
t
t
i
c
e
i
s
t
r
a
n
s
l
a
t
i
o
n
a
l
l
y
i
n
v
a
r
i
a
n
t
.
M
o
r
e
o
v
e
r
,
t
h
e
P
ó
l
y
a
’
s
w
a
l
k
i
s
u
n
b
i
a
s
e
d
w
h
i
c
h
i
m
p
l
i
e
s
t
h
e
f
a
c
t
t
h
a
t
,
t
h
e
p
r
o
b
a
b
i
l
i
t
y
o
f
m
o
v
i
n
g
f
r
o
m
o
n
e
s
i
t
e
t
o
i
t
s
n
e
i
g
h
b
o
u
r
i
n
g
s
i
t
e
i
s
t
h
e
s
a
m
e
a
s
i
f
t
h
e
w
a
l
k
w
a
s
t
a
k
i
n
g
p
l
a
c
e
t
h
e
o
t
h
e
r
w
a
y
a
r
o
u
n
d
.
T
h
e
f
o
r
m
a
l
i
s
m
c
a
n
b
e
e
a
s
i
l
y
e
x
t
e
n
d
e
d
t
o
t
h
e
c
a
s
e
w
h
e
r
e
t
h
e
u
n
d
e
r
l
y
i
n
g
s
t
r
u
c
t
u
r
e
i
s
g
r
a
p
h
.
I
t
i
s
w
o
r
t
h
t
o
m
e
n
t
i
o
n
t
h
a
t
,
i
n
t
h
e
c
a
s
e
o
f
g
r
a
p
h
s
,
t
h
e
d
e
g
r
e
e
o
f
n
o
d
e
s
d
o
e
s
n
o
t
n
e
e
d
n
e
c
e
s
s
a
r
i
l
y
t
o
b
e
t
h
e
s
a
m
e
a
s
l
o
n
g
a
s
t
h
e
t
r
a
n
s
l
a
t
i
o
n
a
l
i
n
v
a
r
i
a
n
c
e
i
s
n
o
t
v
i
o
l
a
t
e
d
[
1
]
.
T
h
e
r
a
n
d
o
m
w
a
l
k
t
h
e
o
r
y
n
o
t
o
n
l
y
p
r
o
v
i
d
e
s
o
n
e
w
i
t
h
i
n
s
i
g
h
t
s
a
b
o
u
t
t
h
e
g
e
o
m
e
t
r
y
o
f
t
h
e
u
n
d
e
r
l
y
i
n
g
s
t
r
u
c
t
u
r
e
a
n
d
h
o
w
a
d
y
n
a
m
i
c
a
l
p
r
o
c
e
s
s
w
o
u
l
d
b
e
h
a
v
e
o
n
s
u
c
h
a
s
t
r
u
c
t
u
r
e
,
b
u
t
n
a
t
u
r
a
l
l
y
e
s
t
a
b
l
i
s
h
e
s
a
c
o
n
n
e
c
t
i
o
n
b
e
t
w
e
e
n
t
h
o
s
e
p
r
o
p
e
r
t
i
e
s
.
O
n
e
o
f
s
u
c
h
r
e
l
a
t
i
o
n
s
i
s
t
h
e
A
l
e
x
a
n
d
e
r
-
O
r
b
a
c
h
(
A
O
)
r
e
l
a
t
i
o
n
w
h
i
c
h
e
s
t
a
b
l
i
s
h
e
s
a
c
o
n
n
e
c
t
i
o
n
b
e
t
w
e
e
n
t
h
e
t
h
e
H
a
r
m
o
n
i
c
(
s
p
e
c
t
r
a
l
)
d
i
m
e
n
s
i
o
n
H
,
t
h
e
d
y
n
a
m
i
c
a
l
e
x
p
o
n
e
n
t
ν
w
a
n
d
t
h
e
H
a
u
s
d
o
r
f
f
d
i
m
e
n
s
i
o
n
D
f
o
r
a
r
a
n
d
o
m
w
a
l
k
p
r
o
c
e
s
s
o
n
a
c
o
n
n
e
c
t
e
d
g
r
a
p
h
w
h
i
c
h
i
s
e
m
b
e
d
d
e
d
i
n
a
R
i
e
m
a
n
n
i
a
n
m
a
n
i
f
o
l
d
.
T
h
e
r
e
l
a
t
i
o
n
a
s
s
e
r
t
s
t
h
a
t
[
1
,
2
,
3
]
H
=
2
D
ν
w
.
(
1
)
I
t
i
s
n
o
t
e
w
o
r
t
h
y
t
o
m
e
n
t
i
o
n
t
h
a
t
t
h
e
r
e
l
a
t
i
o
n
a
b
o
v
e
d
o
e
s
o
n
l
y
h
o
l
d
t
r
u
e
u
n
d
e
r
c
o
n
d
i
t
i
o
n
s
m
e
n
t
i
o
n
e
d
a
b
o
v
e
.
F
o
r
i
n
s
t
a
n
c
e
,
f
o
r
t
h
e
K
o
c
h
c
u
r
v
e
,
t
h
e
A
O
r
e
l
a
t
i
o
n
r
e
m
a
i
n
s
v
a
l
i
d
o
n
t
h
e
c
o
n
t
r
a
r
y
,
i
t
i
s
v
i
o
l
a
t
e
d
f
o
r
t
h
e
r
a
n
d
o
m
w
a
l
k
o
n
C
a
n
t
o
r
d
u
s
t
s
.
T
h
e
i
n
t
e
n
t
i
o
n
h
e
r
e
i
s
t
o
a
d
d
r
e
s
s
t
h
e
v
a
l
i
d
i
t
y
o
f
A
O
r
e
l
a
t
i
o
n
f
o
r
h
y
p
e
r
g
r
a
p
h
s
a
n
d
f
u
r
t
h
e
r
m
o
r
e
,
i
f
i
t
i
s
v
i
o
l
a
t
e
d
w
h
a
t
a
r
e
t
h
e
p
o
s
s
i
b
l
e
e
x
t
e
n
s
i
o
n
s
o
r
c
o
n
s
i
d
e
r
a
t
i
o
n
s
t
h
a
t
h
a
v
e
t
o
b
e
t
a
k
e
n
i
n
t
o
a
c
c
o
u
n
t
t
o
p
r
o
p
o
s
e
a
g
e
n
e
r
a
l
i
s
e
d
r
e
l
a
t
i
o
n
s
u
c
h
t
h
a
t
i
t
c
a
n
e
s
t
a
b
l
i
s
h
c
o
n
n
e
c
t
i
o
n
b
e
t
w
e
e
n
t
h
e
g
e
o
m
e
t
r
y
o
f
t
h
e
h
y
p
e
r
g
r
a
p
h
s
a
n
d
t
h
e
d
y
n
a
m
i
c
s
o
f
t
h
e
r
a
n
d
o
m
w
a
l
k
p
r
o
c
e
s
s
o
n
t
h
e
m
.
T
h
i
s
c
o
u
l
d
p
o
t
e
n
t
i
a
l
l
y
p
r
o
v
i
d
e
u
s
e
f
u
l
i
n
s
i
g
h
t
s
t
o
e
s
t
a
b
l
i
s
h
i
s
o
m
o
r
p
h
i
s
m
b
e
t
w
e
e
n
t
h
e
c
a
u
s
a
l
s
e
t
s
a
n
d
t
h
e
L
o
r
e
n
t
z
i
a
n
m
a
n
i
f
o
l
d
o
n
w
h
i
c
h
t
h
e
y
a
r
e
e
m
b
e
d
d
e
d
.
Preliminaries
A
r
a
n
d
o
m
w
a
l
k
o
f
P
ó
l
y
a
t
y
p
e
o
n
a
h
y
p
e
r
g
r
a
p
h
H
=
(
V
,
E
)
,
w
h
e
r
e
V
i
s
t
h
e
s
e
t
o
f
a
l
l
n
o
d
e
s
a
n
d
E
i
s
t
h
e
s
e
t
o
f
h
y
p
e
r
e
d
g
e
s
,
i
s
d
e
f
i
n
e
d
a
s
r
a
n
d
o
m
m
o
v
e
f
r
o
m
a
g
i
v
e
n
n
o
d
e
′
v
t
o
a
n
o
d
e
v
a
t
t
h
e
s
t
e
p
n
+
1
d
e
s
c
r
i
b
e
d
v
i
a
t
h
e
f
o
l
l
o
w
i
n
g
P
n
+
1
(
v
|
v
0
)
=
Σ
′
v
p
(
v
|
′
v
)
P
n
(
′
v
|
v
0
)
(
2
)
w
h
e
r
e
P
n
(
v
|
v
0
)
i
s
t
h
e
o
c
c
u
p
a
t
i
o
n
p
r
o
b
a
b
i
l
i
t
y
o
f
b
e
i
n
g
a
t
n
o
d
e
v
o
n
t
h
e
t
h
n
s
t
e
p
g
i
v
e
n
t
h
a
t
t
h
e
r
a
n
d
o
m
w
a
l
k
h
a
s
s
t
a
r
t
e
d
f
r
o
m
t
h
e
i
n
i
t
i
a
l
n
o
d
e
v
0
,
a
n
d
p
(
v
|
′
v
)
i
s
t
h
e
p
r
o
b
a
b
i
l
i
t
y
t
r
a
n
s
i
t
i
o
n
f
r
o
m
t
h
e
n
o
d
e
′
v
t
o
t
h
e
n
o
d
e
v
.
T
h
e
p
r
o
b
a
b
i
l
i
t
y
t
r
a
n
s
i
t
i
o
n
m
u
s
t
b
y
s
y
m
m
e
t
r
i
c
a
n
d
m
o
r
e
o
v
e
r
,
i
t
m
u
s
t
b
e
e
q
u
a
l
l
y
d
i
s
t
r
i
b
u
t
e
d
a
m
o
n
g
s
t
t
h
e
n
e
i
g
h
b
o
u
r
s
o
f
v
,
k
n
o
w
n
a
s
t
h
e
d
e
g
r
e
e
o
f
t
h
e
n
o
d
e
.
H
e
n
c
e
,
p
(
v
|
′
v
)
=
1
d
e
g
(
v
)
i
f
′
v
i
s
a
n
e
i
g
h
b
o
u
r
o
f
v
;
w
e
d
e
n
o
t
e
t
h
e
d
e
g
r
e
e
o
f
v
w
i
t
h
d
e
g
(
v
)
.
T
h
e
w
a
l
k
i
s
s
a
i
d
t
o
b
e
h
o
m
o
g
e
n
e
o
u
s
(
t
h
e
g
r
a
p
h
i
s
t
r
a
n
s
l
a
t
i
o
n
a
l
l
y
i
n
v
a
r
i
a
n
t
)
i
f
f
o
r
e
v
e
r
y
v
i
n
V
,
P
n
(
v
|
v
)
(
w
h
i
c
h
m
e
a
n
s
t
h
e
p
r
o
b
a
b
i
l
i
t
y
o
f
b
e
i
n
g
a
t
n
o
d
e
v
a
f
t
e
r
n
s
t
e
p
g
i
v
e
n
t
h
a
t
t
h
e
r
a
n
d
o
m
w
a
l
k
h
a
s
s
t
a
r
t
e
d
f
r
o
m
t
h
e
s
a
m
e
n
o
d
e
)
i
s
t
h
e
s
a
m
e
.
T
h
i
s
r
e
a
d
i
l
y
i
m
p
l
i
e
s
t
h
a
t
n
o
n
o
d
e
i
s
p
r
e
f
e
r
r
e
d
t
o
t
h
e
o
t
h
e
r
n
o
d
e
s
.
A
h
y
p
e
r
e
d
g
e
i
s
a
s
e
t
o
f
n
o
d
e
s
t
h
a
t
c
o
n
n
e
c
t
s
t
h
e
m
.
T
h
e
n
u
m
b
e
r
o
f
n
o
d
e
s
w
i
t
h
i
n
e
a
c
h
h
y
p
e
r
e
d
g
e
i
s
r
e
f
e
r
r
e
d
t
o
a
s
t
h
e
d
e
g
r
e
e
o
f
t
h
a
t
h
y
p
e
r
e
d
g
e
.
T
h
i
s
i
s
a
g
e
n
e
r
a
l
i
s
a
t
i
o
n
o
f
t
h
e
c
o
n
c
e
p
t
o
f
e
d
g
e
s
i
n
g
r
a
p
h
s
t
h
a
t
a
r
e
i
n
c
i
d
e
n
t
o
n
o
n
l
y
t
w
o
n
o
d
e
s
,
e
x
a
c
t
l
y
.
F
o
r
i
n
s
t
a
n
c
e
a
h
y
p
e
r
e
d
g
e
o
f
n
o
d
e
s
{
1
,
2
,
3
}
h
a
s
d
e
g
r
e
e
o
f
t
h
r
e
e
.
T
o
m
a
k
e
i
t
c
l
e
a
r
e
r
f
o
r
t
h
e
r
e
a
d
e
r
,
c
o
n
s
i
d
e
r
t
h
e
f
o
l
l
o
w
i
n
g
h
y
p
e
r
g
r
a
p
h
s
E
(
H
1
)
=
{
{
1
,
2
,
3
}
,
{
2
,
3
,
4
,
5
}
}
E
(
H
2
)
=
{
{
1
,
2
}
,
{
1
,
3
}
,
{
2
,
3
}
,
{
2
,
4
}
,
{
2
,
5
}
,
{
3
,
4
}
,
{
3
,
5
}
,
{
4
,
5
}
}
T
h
e
h
y
p
e
r
g
r
a
p
h
H
2
i
s
a
n
o
r
m
a
l
g
r
a
p
h
.
I
n
o
t
h
e
r
w
o
r
d
s
,
e
v
e
r
y
h
y
p
e
r
g
r
a
p
h
w
h
e
r
e
t
h
e
f
o
r
e
v
e
r
y
h
y
p
e
r
e
d
g
e
w
i
t
h
i
n
t
h
e
s
e
t
o
f
h
y
p
e
r
e
d
g
e
s
t
h
e
d
e
g
r
e
e
i
s
t
w
o
,
t
h
a
t
h
y
p
e
r
g
r
a
p
h
i
s
a
g
r
a
p
h
.
B
e
l
o
w
a
r
e
t
h
e
p
l
o
t
s
o
f
H
1
o
n
t
h
e
l
e
f
t
h
a
n
d
s
i
d
e
a
n
d
H
2
o
n
t
h
e
r
i
g
h
t
h
a
n
d
s
i
d
e
.
h
y
p
g
1
=
{
{
1
,
2
,
3
}
,
{
2
,
3
,
4
,
5
}
}
;
h
y
p
g
2
=
{
{
1
,
2
}
,
{
1
,
3
}
,
{
2
,
3
}
,
{
2
,
4
}
,
{
2
,
5
}
,
{
3
,
4
}
,
{
3
,
5
}
,
{
4
,
5
}
}
;
G
r
a
p
h
i
c
s
R
o
w
[
{
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
H
y
p
e
r
g
r
a
p
h
P
l
o
t
"
]
[
h
y
p
g
1
,
"
S
u
b
s
e
t
B
o
u
n
d
a
r
y
"
F
a
l
s
e
]
,
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
H
y
p
e
r
g
r
a
p
h
P
l
o
t
"
]
[
h
y
p
g
2
,
"
S
u
b
s
e
t
B
o
u
n
d
a
r
y
"
F
a
l
s
e
]
}
]
O
u
t
[
]
=
Their corresponding
projected graphs
are shown in the following figure. Note that the graph representation of these two hypergraphs are isomorphic.
g
r
a
p
h
1
=
U
n
d
i
r
e
c
t
e
d
G
r
a
p
h
[
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
H
y
p
e
r
g
r
a
p
h
T
o
G
r
a
p
h
"
]
[
h
y
p
g
1
]
]
;
g
r
a
p
h
2
=
U
n
d
i
r
e
c
t
e
d
G
r
a
p
h
[
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
H
y
p
e
r
g
r
a
p
h
T
o
G
r
a
p
h
"
]
[
h
y
p
g
2
]
]
;
G
r
a
p
h
i
c
s
R
o
w
[
{
G
r
a
p
h
P
l
o
t
[
g
r
a
p
h
1
,
V
e
r
t
e
x
L
a
b
e
l
s
A
u
t
o
m
a
t
i
c
,
P
l
o
t
L
a
b
e
l
M
a
T
e
X
[
"
\
\
m
a
t
h
c
a
l
{
G
}
_
1
"
]
]
,
G
r
a
p
h
P
l
o
t
[
g
r
a
p
h
2
,
V
e
r
t
e
x
L
a
b
e
l
s
A
u
t
o
m
a
t
i
c
,
P
l
o
t
L
a
b
e
l
M
a
T
e
X
[
"
\
\
m
a
t
h
c
a
l
{
G
}
_
2
"
]
]
}
]
O
u
t
[
]
=
Thus, the random walk process on the projected graph of hypergraphs does not carry the same information, as during the transformation of the hypergraphs to their projected graphs, the information related to the hyperedges is lost. Therefore, a consideration has to be sought such that the structure of the hypergraphs are encoded in the probability transition matrix.
The natural distance on a graph is defined by the shortest path between two given nodes. It is denoted by
d(u,v)
for every pair of nodes
u
and
v
in
V.
If there is no path between the two nodes then
d(u,v)=
∞
.
Since in this project we are dealing with connected (hyper)graphs, the distance between every two given nodes is finite. Obviously
d(v, v) = 0.
Probability Transition Matrix
I
n
o
r
d
e
r
t
o
t
a
k
e
i
n
t
o
a
c
c
o
u
n
t
t
h
e
s
t
r
u
c
t
u
r
e
o
f
t
h
e
h
y
p
e
r
g
r
a
p
h
s
,
i
t
i
s
r
e
q
u
i
r
e
d
t
o
u
n
d
e
r
s
t
a
n
d
t
h
e
c
o
n
n
e
c
t
i
v
i
t
y
b
e
t
w
e
e
n
t
h
e
n
o
d
e
s
t
h
a
t
b
e
l
o
n
g
t
o
t
h
e
s
a
m
e
h
y
p
e
r
e
d
g
e
s
e
t
a
n
d
t
h
o
s
e
n
o
d
e
s
t
h
a
t
a
r
e
c
o
n
n
e
c
t
e
d
v
i
a
t
h
e
o
v
e
r
l
a
p
o
f
o
t
h
e
r
h
y
p
e
r
e
d
g
e
s
.
A
s
s
u
m
e
t
h
e
h
y
p
e
r
g
r
a
p
h
H
1
,
n
o
t
i
c
e
t
h
a
t
f
o
r
t
h
e
n
o
d
e
1
,
t
h
e
p
r
o
b
a
b
i
l
i
t
y
o
f
m
o
v
i
n
g
t
o
t
h
e
e
i
t
h
e
r
n
o
d
e
2
o
r
3
i
s
1
2
.
T
h
a
t
i
s
a
l
s
o
r
e
f
l
e
c
t
e
d
o
n
t
h
e
p
r
o
j
e
c
t
e
d
g
r
a
p
h
G
1
.
Y
e
t
,
t
h
e
n
o
d
e
3
b
e
l
o
n
g
s
t
o
t
w
o
d
i
f
f
e
r
e
n
t
h
y
p
e
r
e
d
g
e
s
;
i
n
t
h
e
h
y
p
e
r
e
d
g
e
{
1
,
2
,
3
}
,
t
h
e
p
r
o
b
a
b
i
l
i
t
y
t
o
m
o
v
e
t
o
e
i
t
h
e
r
n
o
d
e
1
o
r
2
i
s
a
g
a
i
n
1
2
w
h
i
l
e
i
n
t
h
e
o
t
h
e
r
h
y
p
e
r
e
d
g
e
{
2
,
3
,
4
,
5
}
t
h
e
p
r
o
b
a
b
i
l
i
t
y
t
o
m
o
v
e
t
o
o
t
h
e
r
t
h
r
e
e
n
o
d
e
s
2
,
3
a
n
d
5
i
s
1
3
.
N
o
t
s
u
r
p
r
i
s
i
n
g
l
y
,
i
t
i
s
s
e
e
n
t
h
a
t
i
n
G
1
,
n
o
d
e
3
i
s
c
o
n
n
e
c
t
e
d
t
o
a
l
l
o
t
h
e
r
n
o
d
e
s
a
n
d
t
h
e
p
r
o
b
a
b
i
l
i
t
y
o
f
m
o
v
i
n
g
t
o
e
a
c
h
o
n
e
o
f
t
h
e
m
i
s
1
4
.
I
n
o
r
d
e
r
t
o
i
n
c
o
r
p
o
r
a
t
e
t
h
e
s
t
r
u
c
t
u
r
e
o
f
t
h
e
h
y
p
e
r
g
r
a
p
h
s
,
i
n
i
t
i
a
l
l
y
i
t
i
s
r
e
q
u
i
r
e
d
t
o
d
e
f
i
n
e
t
h
e
i
n
c
i
d
e
n
c
e
m
a
t
r
i
x
[
7
]
.
A
s
s
u
m
i
n
g
t
h
e
c
o
l
u
m
n
s
r
e
p
r
e
s
e
n
t
i
n
g
t
h
e
h
y
p
e
r
e
d
g
e
s
a
n
d
t
h
e
r
o
w
s
t
h
e
n
o
d
e
s
,
t
h
e
i
n
c
i
d
e
n
c
e
m
a
t
r
i
x
s
h
o
w
s
w
h
e
t
h
e
r
a
n
o
d
e
b
e
l
o
n
g
s
t
o
a
s
p
e
c
i
f
i
c
h
y
p
e
r
e
d
g
e
o
r
n
o
t
.
I
t
i
s
s
a
i
d
t
h
a
t
a
h
y
p
e
r
e
d
g
e
i
s
i
n
c
i
d
e
n
t
o
n
a
n
o
d
e
i
f
t
h
a
t
n
o
d
e
b
e
l
o
n
g
s
t
o
t
h
a
t
g
i
v
e
n
h
y
p
e
r
e
d
g
e
.
T
h
e
r
e
f
o
r
e
,
t
h
e
i
n
c
i
d
e
n
c
e
m
a
t
r
i
x
i
s
a
n
n
m
m
a
t
r
i
x
w
h
e
r
e
n
i
s
t
h
e
n
u
m
b
e
r
o
f
n
o
d
e
s
i
n
t
h
e
s
e
t
V
a
n
d
m
i
s
t
h
e
n
u
m
b
e
r
o
f
h
y
p
e
r
e
d
g
e
s
i
n
E
.
T
h
e
n
c
e
,
o
n
e
c
a
n
r
e
a
d
i
l
y
d
e
f
i
n
e
t
h
e
a
d
j
a
c
e
n
c
y
m
a
t
r
i
x
a
n
d
t
h
e
h
y
p
e
r
e
d
g
e
m
a
t
r
i
x
b
y
m
u
l
t
i
p
l
y
i
n
g
t
h
e
t
r
a
n
s
p
o
s
e
o
f
t
h
e
i
n
c
i
d
e
n
c
e
m
a
t
r
i
x
o
n
t
o
i
t
s
e
l
f
f
r
o
m
r
i
g
h
t
a
n
d
l
e
f
t
r
e
s
p
e
c
t
i
v
e
l
y
[
7
]
.
F
o
r
i
n
s
t
a
n
c
e
t
h
e
i
n
c
i
d
e
n
c
e
m
a
t
r
i
c
e
s
f
o
r
t
h
e
h
y
p
e
r
g
r
a
p
h
s
H
1
a
n
d
H
2
a
r
e
g
i
v
e
n
b
y
t
h
e
f
o
l
l
o
w
i
n
g
s
i
n
c
M
1
=
T
r
a
n
s
p
o
s
e
[
{
{
1
,
1
,
1
,
0
,
0
}
,
{
0
,
1
,
1
,
1
,
1
}
}
]
/
/
M
a
t
r
i
x
F
o
r
m
i
n
c
M
2
=
T
r
a
n
s
p
o
s
e
[
{
{
1
,
1
,
0
,
0
,
0
}
,
{
1
,
0
,
1
,
0
,
0
}
,
{
0
,
1
,
1
,
0
,
0
}
,
{
0
,
1
,
0
,
1
,
0
}
,
{
0
,
1
,
0
,
0
,
1
}
,
{
0
,
0
,
1
,
1
,
0
}
,
{
0
,
0
,
1
,
0
,
1
}
,
{
0
,
0
,
0
,
1
,
1
}
}
]
/
/
M
a
t
r
i
x
F
o
r
m
O
u
t
[
]
/
/
M
a
t
r
i
x
F
o
r
m
=
1
0
1
1
1
1
0
1
0
1
O
u
t
[
]
/
/
M
a
t
r
i
x
F
o
r
m
=
1
1
0
0
0
0
0
0
1
0
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
0
0
1
0
1
1
The
probability transition matrix
is achieved by knowing the adjacency matrix and the hyperedge matrix. For the hypergraph
H
1
and
H
2
the followings are their respective probability transition matrix. Note that, they are indeed different and the first matrix reflects the hyperedge structure of the hypergraph.
O
u
t
[
]
/
/
M
a
t
r
i
x
F
o
r
m
=
0
1
2
1
2
0
0
2
1
3
0
5
1
3
3
1
3
3
1
3
2
1
3
5
1
3
0
3
1
3
3
1
3
0
1
3
1
3
0
1
3
0
1
3
1
3
1
3
0
O
u
t
[
]
/
/
M
a
t
r
i
x
F
o
r
m
=
0
1
2
1
2
0
0
1
4
0
1
4
1
4
1
4
1
4
1
4
0
1
4
1
4
0
1
3
1
3
0
1
3
0
1
3
1
3
1
3
0
Random Walks on wm7714 WolfromWorld
The wm7714 WolframModel is chosen to investigate the validity of the AO relation.
h
y
p
g
r
a
p
h
=
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
W
o
l
f
r
a
m
M
o
d
e
l
"
]
[
{
{
{
1
,
2
,
2
}
,
{
3
,
1
,
4
}
}
{
{
2
,
5
,
2
}
,
{
2
,
3
,
5
}
,
{
4
,
5
,
5
}
}
}
,
{
{
1
,
1
,
1
}
,
{
1
,
1
,
1
}
}
,
1
0
0
0
,
"
F
i
n
a
l
S
t
a
t
e
"
]
;
g
r
a
p
h
=
U
n
d
i
r
e
c
t
e
d
G
r
a
p
h
[
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
H
y
p
e
r
g
r
a
p
h
T
o
G
r
a
p
h
"
]
[
h
y
p
g
r
a
p
h
]
]
;
d
i
j
=
G
r
a
p
h
D
i
s
t
a
n
c
e
M
a
t
r
i
x
[
g
r
a
p
h
]
;
H
y
p
e
r
g
r
a
p
h
P
l
o
t
[
h
y
p
g
r
a
p
h
]
G
r
a
p
h
P
l
o
t
[
g
r
a
p
h
]
O
u
t
[
]
=
O
u
t
[
]
=
The figure depicted at top is the hypergraph representation of wm7714 whilst the one at the bottom demonstrates its undirected projected graph which preserves the geodesic distance between the nodes. After arriving at the probability transition matrix associated with the hypergraph representation of wm7714, it is straight forward to simulate random walk process on the undirected projected graph using
discrete Markov process
, needless to say that a random walk of Polya’s type is a Markov process. Here the initial condition is chosen such that the random walk process starts at node 1 and takes a path of length 1000. An ensemble of 500 of such walks is sampled.
i
n
i
t
i
a
l
S
t
a
t
e
=
C
o
n
s
t
a
n
t
A
r
r
a
y
[
0
,
{
L
e
n
g
t
h
[
v
e
r
t
i
c
e
s
]
}
]
;
i
n
i
t
i
a
l
S
t
a
t
e
[
[
1
]
]
=
1
;
m
a
r
k
o
v
P
=
D
i
s
c
r
e
t
e
M
a
r
k
o
v
P
r
o
c
e
s
s
[
i
n
i
t
i
a
l
S
t
a
t
e
,
h
y
p
e
r
T
r
a
n
s
i
t
i
o
n
M
a
t
r
i
x
]
;
d
a
t
a
=
R
a
n
d
o
m
F
u
n
c
t
i
o
n
[
m
a
r
k
o
v
P
,
{
0
,
1
0
0
0
}
,
5
0
0
]
;
Spectral Dimension and the Distinct Visited Nodes
I
n
t
h
e
d
y
n
a
m
i
c
a
l
t
h
e
o
r
y
o
f
r
a
n
d
o
m
w
a
l
k
,
t
h
e
n
u
m
b
e
r
o
f
d
i
s
t
i
n
c
t
v
i
s
i
t
e
d
n
o
d
e
s
i
s
r
e
l
a
t
e
d
t
o
t
h
e
s
p
e
c
t
r
a
l
d
i
m
e
n
s
i
o
n
H
v
i
a
t
h
e
f
o
l
l
o
w
i
n
g
r
e
l
a
t
i
o
n
[
2
]
<
N
>
H
/
2
n
(
3
)
w
h
e
r
e
N
i
s
t
h
e
n
u
m
b
e
r
o
f
d
i
s
t
i
n
c
t
v
i
s
i
t
e
d
n
o
d
e
s
u
p
t
o
n
d
i
s
c
r
e
t
e
t
i
m
e
s
t
e
p
a
n
d
H
i
s
t
h
e
s
p
e
c
t
r
a
l
d
i
m
e
n
s
i
o
n
m
e
n
t
i
o
n
e
d
a
b
o
v
e
.
T
h
e
b
a
r
c
k
e
t
n
o
t
a
t
i
o
n
i
s
t
o
b
e
u
n
d
e
r
s
t
o
o
d
a
s
t
h
e
e
n
s
e
m
b
l
e
a
v
e
r
a
g
e
.
T
h
e
s
i
m
u
l
a
t
i
o
n
s
h
o
w
s
t
h
e
f
o
l
l
o
w
i
n
g
I
n
[
]
:
=
d
s
L
o
g
L
i
n
e
=
P
a
r
a
l
l
e
l
T
a
b
l
e
[
d
i
s
t
i
n
c
t
V
i
s
i
t
e
d
S
i
t
e
s
[
d
a
t
a
,
i
]
,
{
i
,
1
,
5
0
0
}
]
;
d
s
p
o
w
e
r
l
a
w
f
i
t
=
N
o
n
l
i
n
e
a
r
M
o
d
e
l
F
i
t
[
d
s
L
o
g
L
i
n
e
,
a
*
b
x
,
{
a
,
b
}
,
x
]
;
d
s
p
o
w
e
r
l
a
w
f
i
t
d
a
t
a
=
P
a
r
a
l
l
e
l
T
a
b
l
e
[
d
s
p
o
w
e
r
l
a
w
f
i
t
[
i
]
,
{
i
,
1
,
5
0
0
}
]
;
L
i
s
t
L
o
g
L
o
g
P
l
o
t
[
{
d
s
L
o
g
L
i
n
e
,
d
s
p
o
w
e
r
l
a
w
f
i
t
d
a
t
a
}
,
A
x
e
s
T
r
u
e
,
A
x
e
s
L
a
b
e
l
{
"
n
"
,
"
<
N
>
"
}
,
P
l
o
t
L
e
g
e
n
d
s
{
"
S
i
m
u
l
a
t
i
o
n
"
,
"
P
o
w
e
r
-
l
a
w
F
i
t
"
}
]
d
s
p
o
w
e
r
l
a
w
f
i
t
[
"
P
a
r
a
m
e
t
e
r
T
a
b
l
e
"
]
O
u
t
[
]
=
S
i
m
u
l
a
t
i
o
n
P
o
w
e
r
-
l
a
w
F
i
t
O
u
t
[
]
=
E
s
t
i
m
a
t
e
S
t
a
n
d
a
r
d
E
r
r
o
r
t
-
S
t
a
t
i
s
t
i
c
P
-
V
a
l
u
e
a
0
.
7
6
6
5
4
6
0
.
0
0
3
4
0
8
2
2
4
.
9
2
6
0
.
b
0
.
8
6
0
0
1
8
0
.
0
0
0
7
5
8
7
5
2
1
1
3
3
.
4
6
0
.
which shows that the half of the spectral dimension has the value 0.86 and consequently the value 1.72 for
H.
Dynamical Exponent
I
n
o
r
d
e
r
t
o
e
s
t
i
m
a
t
e
t
h
e
d
y
n
a
m
i
c
a
l
e
x
p
o
n
e
n
t
,
t
h
e
d
y
n
a
m
i
c
a
l
t
h
e
o
r
y
o
f
r
a
n
d
o
m
w
a
l
k
s
u
g
g
e
s
t
s
t
h
a
t
t
h
e
d
i
s
t
a
n
c
e
t
h
a
t
a
r
a
n
d
o
m
w
a
l
k
p
r
o
c
e
s
s
h
a
s
r
e
a
c
h
e
d
a
t
s
t
e
p
n
,
e
x
h
i
b
i
t
s
t
h
e
f
o
l
l
o
w
i
n
g
b
e
h
a
v
i
o
u
r
[
2
]
<
|
r
(
n
)
-
r
(
0
)
|
>
∼
2
ν
w
n
(
4
)
w
h
e
r
e
2
ν
w
i
s
t
h
e
d
y
n
a
m
i
c
a
l
e
x
p
o
n
e
n
t
a
n
d
r
(
n
)
i
s
t
h
e
d
i
s
t
a
n
c
e
t
h
e
r
a
n
d
o
m
w
a
l
k
p
r
o
c
e
s
s
h
a
s
r
e
a
c
h
e
d
w
i
t
h
r
e
s
p
e
c
t
t
o
t
h
e
i
n
i
t
i
a
l
p
o
s
i
t
i
o
n
i
t
h
a
d
a
t
n
=
0
.
T
h
e
s
i
m
u
l
a
t
i
o
n
r
e
s
u
l
t
s
s
h
o
w
s
t
h
e
f
o
l
l
o
w
i
n
g
t
r
e
n
d
i
n
t
h
e
l
o
g
a
r
i
t
h
m
i
c
s
c
a
l
e
I
n
[
]
:
=
m
e
a
n
s
q
=
P
a
r
a
l
l
e
l
T
a
b
l
e
[
m
e
a
n
S
q
u
a
r
e
d
D
i
s
p
l
a
c
e
m
e
n
t
[
d
a
t
a
,
d
i
j
,
i
]
,
{
i
,
3
0
0
}
]
;
m
s
d
p
o
w
e
r
l
a
w
f
i
t
=
N
o
n
l
i
n
e
a
r
M
o
d
e
l
F
i
t
[
m
e
a
n
s
q
,
a
*
b
x
,
{
a
,
b
}
,
x
]
;
m
s
d
p
o
w
e
r
l
a
w
f
i
t
d
a
t
a
=
P
a
r
a
l
l
e
l
T
a
b
l
e
[
m
s
d
p
o
w
e
r
l
a
w
f
i
t
[
i
]
,
{
i
,
1
,
3
0
0
}
]
;
L
i
s
t
L
o
g
L
o
g
P
l
o
t
[
{
m
e
a
n
s
q
,
m
s
d
p
o
w
e
r
l
a
w
f
i
t
d
a
t
a
}
]
m
s
d
p
o
w
e
r
l
a
w
f
i
t
[
"
P
a
r
a
m
e
t
e
r
T
a
b
l
e
"
]
O
u
t
[
]
=
O
u
t
[
]
=
E
s
t
i
m
a
t
e
S
t
a
n
d
a
r
d
E
r
r
o
r
t
-
S
t
a
t
i
s
t
i
c
P
-
V
a
l
u
e
a
0
.
8
9
3
3
5
4
0
.
0
0
6
5
6
3
2
2
1
3
6
.
1
1
5
2
.
4
5
2
2
1
×
-
2
7
0
1
0
b
0
.
5
0
1
7
0
6
0
.
0
0
1
4
0
4
6
8
3
5
7
.
1
6
7
0
.
which yields a value of 0.50 for
2
ν
w
.
Hausdorff Dimension
I
n
o
r
d
e
r
t
o
c
a
l
c
u
l
a
t
e
t
h
e
H
a
u
s
d
o
r
f
f
d
i
m
e
n
s
i
o
n
,
i
t
i
s
r
e
q
u
i
r
e
d
t
o
d
e
f
i
n
e
t
h
e
g
e
o
d
e
s
i
c
b
a
l
l
.
T
h
e
n
a
t
u
r
a
l
m
e
t
r
i
c
d
e
f
i
n
e
d
o
n
h
y
p
e
r
g
r
a
p
h
i
s
t
h
e
s
a
m
e
a
s
i
t
i
s
d
e
f
i
n
e
d
f
o
r
g
r
a
p
h
s
a
n
d
i
t
i
s
d
i
s
c
u
s
s
e
d
i
n
t
h
e
i
n
t
r
o
d
u
c
t
i
o
n
.
T
h
e
r
e
i
s
a
d
e
b
a
t
e
w
h
e
t
h
e
r
t
h
e
d
e
f
i
n
i
t
i
o
n
o
f
t
h
e
H
a
u
s
d
o
r
f
f
d
i
m
e
n
s
i
o
n
h
a
s
t
o
b
e
a
l
o
c
a
l
o
r
g
l
o
b
a
l
p
r
o
p
e
r
t
y
o
f
t
h
e
s
t
r
u
c
t
u
r
e
.
B
e
a
r
i
n
g
t
h
i
s
i
n
m
i
n
d
,
t
h
e
p
r
o
c
e
d
u
r
e
b
e
l
o
w
y
i
e
l
d
s
a
g
l
o
b
a
l
d
i
m
e
n
s
i
o
n
f
o
r
t
h
e
H
a
u
s
d
o
r
f
f
d
i
m
e
n
s
i
o
n
.
A
s
s
u
m
e
a
r
a
n
d
o
m
n
o
d
e
o
f
t
h
e
h
y
p
e
r
g
r
a
p
h
.
T
h
e
i
m
m
e
d
i
a
t
e
n
e
i
g
h
b
o
u
r
s
o
f
t
h
i
s
n
o
d
e
a
r
e
t
h
e
o
n
e
s
t
h
a
t
h
a
v
e
d
i
s
t
a
n
c
e
1
w
i
t
h
r
e
s
p
e
c
t
t
o
t
h
i
s
v
e
r
y
n
o
d
e
;
t
h
e
y
a
r
e
c
a
l
l
e
d
t
h
e
n
e
a
r
e
s
t
n
e
i
g
h
b
o
u
r
s
.
N
o
w
t
a
k
e
t
h
e
u
n
i
o
n
o
f
t
h
e
n
e
a
r
e
s
t
n
e
i
g
h
b
o
u
r
s
a
n
d
t
h
e
i
n
i
t
i
a
l
n
o
d
e
a
n
d
s
e
e
k
t
h
e
n
e
a
r
e
s
t
n
e
i
g
h
b
o
u
r
s
o
f
t
h
i
s
n
e
w
s
e
t
.
I
t
e
r
a
t
i
v
e
l
y
c
o
n
t
i
n
u
e
t
h
i
s
u
n
t
i
l
a
l
l
t
h
e
n
o
d
e
s
o
f
t
h
e
h
y
p
e
r
g
r
a
p
h
a
r
e
i
n
t
h
e
s
e
t
w
h
i
c
h
i
s
f
o
r
m
e
d
b
y
s
u
c
c
e
s
s
i
v
e
u
n
i
o
n
o
p
e
r
a
t
i
o
n
.
D
e
n
o
t
e
t
h
e
s
e
t
o
f
n
o
d
e
s
a
t
e
a
c
h
i
t
e
r
a
t
i
o
n
w
i
t
h
S
.
T
h
e
d
i
a
m
e
t
e
r
o
f
S
d
e
n
o
t
e
d
b
y
d
i
a
m
(
S
)
i
s
t
h
e
l
a
r
g
e
s
t
d
i
s
t
a
n
c
e
b
e
t
w
e
e
n
t
h
e
n
o
d
e
s
t
h
a
t
b
e
l
o
n
g
t
o
t
h
i
s
s
e
t
.
C
o
n
s
e
q
u
e
n
t
l
y
,
w
e
c
a
n
d
e
f
i
n
e
t
h
e
m
a
s
s
d
i
s
t
r
i
b
u
t
i
o
n
a
s
t
h
e
n
u
m
b
e
r
o
f
n
o
d
e
s
t
h
a
t
b
e
l
o
n
g
t
o
t
h
e
g
e
o
d
e
s
i
c
b
a
l
l
o
f
d
i
a
m
(
S
)
.
T
h
u
s
[
2
,
3
,
5
]
,
ρ
∼
d
i
a
m
D
(
S
)
(
5
)
w
h
e
r
e
ρ
d
e
n
o
t
e
s
t
h
e
n
u
m
b
e
r
o
f
n
o
d
e
s
.
T
h
e
s
i
m
u
l
a
t
i
o
n
y
i
e
l
d
s
t
h
e
f
o
l
l
o
w
i
n
g
r
e
s
u
l
t
g
i
v
e
n
t
h
a
t
t
h
e
i
n
i
t
i
a
l
n
o
d
e
w
a
s
c
h
o
s
e
n
t
o
b
e
8
.
I
n
[
]
:
=
v
e
r
t
e
x
s
e
t
=
{
8
}
;
t
u
p
l
e
a
r
r
a
y
=
{
}
;
F
o
r
[
i
=
0
,
i
<
5
0
0
,
i
+
+
,
A
p
p
e
n
d
T
o
[
t
u
p
l
e
a
r
r
a
y
,
{
M
a
x
[
G
r
a
p
h
D
i
s
t
a
n
c
e
M
a
t
r
i
x
[
N
e
i
g
h
b
o
r
h
o
o
d
G
r
a
p
h
[
g
r
a
p
h
,
v
e
r
t
e
x
s
e
t
]
]
]
,
L
e
n
g
t
h
[
v
e
r
t
e
x
s
e
t
]
}
]
;
v
e
r
t
e
x
s
e
t
=
V
e
r
t
e
x
L
i
s
t
[
N
e
i
g
h
b
o
r
h
o
o
d
G
r
a
p
h
[
g
r
a
p
h
,
v
e
r
t
e
x
s
e
t
]
]
;
I
f
[
L
e
n
g
t
h
[
v
e
r
t
e
x
s
e
t
]
L
e
n
g
t
h
[
V
e
r
t
e
x
L
i
s
t
[
g
r
a
p
h
]
]
,
B
r
e
a
k
[
]
]
;
]
d
f
p
o
w
e
r
l
a
w
f
i
t
=
N
o
n
l
i
n
e
a
r
M
o
d
e
l
F
i
t
[
t
u
p
l
e
a
r
r
a
y
,
a
*
b
x
,
{
a
,
b
}
,
x
]
;
d
f
p
o
w
e
r
l
a
w
f
i
t
d
a
t
a
=
T
a
b
l
e
[
{
j
,
d
f
p
o
w
e
r
l
a
w
f
i
t
[
j
]
}
,
{
j
,
T
a
b
l
e
[
t
u
p
l
e
a
r
r
a
y
[
[
i
]
]
[
[
1
]
]
,
{
i
,
1
,
L
e
n
g
t
h
[
t
u
p
l
e
a
r
r
a
y
]
}
]
}
]
;
L
i
s
t
L
o
g
L
o
g
P
l
o
t
[
{
t
u
p
l
e
a
r
r
a
y
,
d
f
p
o
w
e
r
l
a
w
f
i
t
d
a
t
a
}
]
d
f
p
o
w
e
r
l
a
w
f
i
t
[
"
P
a
r
a
m
e
t
e
r
T
a
b
l
e
"
]
O
u
t
[
]
=
O
u
t
[
]
=
E
s
t
i
m
a
t
e
S
t
a
n
d
a
r
d
E
r
r
o
r
t
-
S
t
a
t
i
s
t
i
c
P
-
V
a
l
u
e
a
0
.
5
0
2
1
2
8