advanced data mining - datalab.snu.ac.krukang/courses/20f-adm/l2-graphbasic.pdf · protein...

U Kang 1

Advanced Data Mining

Graph Basics and Diameter

U KangSeoul National University

U Kang 2

In This Lecture

Basic definitions in graph mining Small World Phenomenon Diameter over time

U Kang 3

Outline

Basic DefinitionSmall World PhenomenonDiameter over TimeConclusion

U Kang 4

Basic Definition

Graph: a way of specifying relationships among a collection of items. Nodes (or vertices) Edges

U Kang 5

Basic Definition

Types of graph Directed vs. Undirected graph

Directed Undirected

U Kang 6

Basic Definition

Types of graph Weighted vs. Unweighted graph

Weighted Unweighted

1.2

0.1

2.5

U Kang 7

Basic Definition

Types of graph Simple vs. Attributed graph

Simple Attributed

CEO

Research ManagerAssistant

Marketing Manager

U Kang 8

Examples of graph

Arpanet, DEC 19701) Social network2)

1) UCLA and BBN, ARPANET in December 1970, 1970,

https://commons.wikimedia.org/wiki/File:ARPANET_1970_Map.png

2) MRFerocius, Social Network, 2011,

https://stackoverflow.com/questions/4594962/social-network-directed-graph-library-for-net

facebooktwitter

U Kang 9

More Examples

Protein Interactions1) World Wide Web Document Network2)

PatentDBLP

1) Wikipedia, Schizophrenia PPI, 2016,

https://en.wikipedia.org/wiki/Protein%E2%80%93protein_interaction

2) Widipedia, Logo of the English Wikipedia, 2001,

https://en.wikipedia.org/wiki/English_Wikipedia

U Kang 10

Even more examples

Call graph (who calls whom) Email graph (who emailed whom) Movie-actor database from IMDB (more examples?)

U Kang 11

More definitions

Two vertices are adjacent if they share a common edge

Two adjacent vertices are neighbors An edge is incident with another edge if they share

a vertex An edge is incident with two vertices A degree of a node is the number of neighbors of it

Directed graph: in-degree, out-degree

U Kang 12

Connection

Connected graph

Disconnected graph

U Kang 13

Connected Component

Disconnected Component

Disconnected Component

C

DE

A

B

Giant Connected Component

F

G

H

I

K

M

L

U Kang 14

Special Families of Graph

Star graph Equation for |E| (=m) as a function of |V| (=n)?

Chain graph Equation for |E| (=m) as a function of |V| (=n)?

Complete graph = full graph = clique graph Equation for |E| (=m) as a function of |V| (=n)?

U Kang 15

Special Families of Graph

Bipartite graph

Complete bipartite graph

U Kang 16

Path, Cycle, Walk

Path : a sequence of connected vertices Cycle : a path whose start and end vertices are the

same Simple path : no repeated vertices Simple cycle : no repeated vertices, except the

start vertex (=end vertex)

Some authors use ‘path’ for simple path (no repetition), and ‘walk’ for path (repetition allowed)

U Kang 17

Subgraph

Subgraph A subset of the graph

Induced subgraph A subgraph induced from a set of vertices

U Kang 18

Outline


U Kang 19

Distance in graph

Length of a path : # of steps from beginning to end

You

U Kang 20

Distance in graph


You

Distance 1: “friends”

U Kang 21

Distance in graph


You

Distance 2: “friends of friends”

U Kang 22

Distance in graph


You

Distance 3: “friends of friends of friends”

U Kang 23

Algorithm: Breadth First Search

Find one step neighbors. Find two step neighbors. Continue…

You

U Kang 24

Radius and Diameter

Radius of a node : the longest shortest distance to the all other nodes

Diameter of a graph : the maximum radius

v

U Kang 25

Small World Phenomenon

Milgram’s experiment

Six degrees of separation

The median value is “6”N

UMBE

RO

F CH

AIN

S

NUMBER OF INTERMEDIARIES

N = 64

- Jeffrey Travers and Stanley Milgram, An Experimental Study of the Small World Problem, Sociometry, Vol. 32, No. 4, 1969, pp. 432

U Kang 26


Erdos Number Paul Erdos : published ~1500 papers Erdos number : the distance to Paul Erdos in co-

authorship graph

1) Billy and Grace Tao, Paul Erdos with Terence Tao, 1985, https://commons.wikimedia.org/wiki/File:Paul_Erdos_with_Terence_Tao.jpg 2) H2g2bob, Erdosnumber, 2006, https://commons.wikimedia.org/wiki/File:Erdosnumber.png

1) 2)

U Kang 27


Erdos Number Albert Einstein : 2 Enrico Fermi : 3 Noam Chomski : 4 Most mathematicians have Erdos numbers at most 4 or 5

U Kang 28


Kevin Bacon number Distance to the Kevin Bacon in actor-actor graph from

Internet Movie DataBase (IMDB) at 1994 Average Bacon number of actors : 2.9

- GabboT, Kevin Bacon TIFF 2015, 2015, https://commons.wikimedia.org/wiki/File:Kevin_Bacon_TIFF_2015.jpg

U Kang 29

Outline

Basic DefinitionSmall World PhenomenonDiameter over Time

ObservationModel

Conclusion

Materials based on Jure Leskovec’s slideJ. Leskovec, J. Kleinberg, C. Faloutsos. “Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations”, ACM SIGKDD, 2005

U Kang 30

Graph Evolution

How do the graphs evolve over time?

Application Network simulation Network prediction

U Kang 31

Graph Evolution

How do the graphs evolve over time?

Conventional Wisdom Constant average degree: the number of edges grows

linearly with the number of nodes Slowly growing diameter

New findings Densification power law: networks are becoming denser

over time Shrinking diameter

U Kang 32

Temporal Evolution of Graphs

Densification Power Law Networks are becoming denser over time Average degree is increasing

orequivalently

U Kang 33

Graph Densification – A closer look

Densification Power Law

Densification exponent: 1 ≤ a ≤ 2: a=1: linear growth – constant out-degree

(assumed in the literature so far) a=2: quadratic growth – clique

What are the exponents a for real graphs?

U Kang 34

Densification – Physics Citations

Citations among physics papers

1992: 1,293 papers,

2,717 citations 2003:

29,555 papers, 352,807 citations

For each month M, create a graph of all citations up to month M

N(t)

E(t)

1.69


U Kang 35

Densification – Patent Citations

Citations among patents granted

1975 334,000 nodes 676,000 edges

1999 2.9 million nodes 16.5 million edges

Each year is a data point

N(t)

E(t)

1.66


U Kang 36

Densification – Autonomous Systems

Graph of Internet 1997

3,000 nodes 10,000 edges

2000 6,000 nodes 26,000 edges

One graph per dayN(t)

E(t)

1.18


U Kang 37

Densification – Affiliation Network

Authors linked to their publications

1992 318 nodes 272 edges

2002 60,000 nodes

20,000 authors 38,000 papers

133,000 edges N(t)

E(t)

1.15


U Kang 38

Graph Densification – Summary

The traditional constant out-degree assumption does not hold

Real world graphs:

The average degree is increasing

U Kang 39

Evolution of the Diameter

Prior work on Power Law graphs hints atSlowly growing diameter: diameter ~ O(log N) diameter ~ O(log log N)

What is happening in real data?

Diameter shrinks over time As the network grows the distances between nodes

slowly decrease

U Kang 40

Diameter – ArXiv citation graph

Citations among physics papers

1992 –2003 One graph per year

time [years]

diameter


U Kang 41

Diameter – “Autonomous Systems”

Graph of Internet One graph per day 1997 – 2000

number of nodes

diameter


U Kang 42

Diameter – “Affiliation Network”

Graph of collaborations in physics – authors linked to papers

10 years of data

time [years]

diameter


U Kang 43

Diameter – “Patents”

Patent citation network

25 years of data

time [years]

diameter


U Kang 44

Outline

Basic DefinitionSmall World PhenomenonDiameter over Time

ObservationModel

Conclusion

U Kang 45

Models

Existing graph generation models do not capture Densification Power Law and Shrinking diameters

Can we find a simple model of local behavior, which naturally leads to observed phenomena?

U Kang 46

“Forest Fire” model

How do people make friends in a new environment?

1. Find first a person and make friends2. Follow one of his friends3. Continue recursively4. From time to time get introduced to a new person

Forest Fire model imitates exactly this process

U Kang 47

“Forest Fire” – the Model

A node arrives Randomly chooses an “ambassador” Starts burning nodes (with probability p) and

adds links to burned nodes “Fire” spreads recursively

U Kang 48

Forest Fire in Action

Forest Fire generates graphs that Densify and have Shrinking Diameter


U Kang 49

Outline


U Kang 50

Conclusion

Definitions: make sure you know them Small World Phenomenon

Six degrees of separation Diameter over time

Shrinking diameter

U Kang 51

Questions?

advanced data mining - datalab.snu.ac.krukang/courses/20f-adm/l2-graphbasic.pdf · protein...

Documents