Evolution of Random Threshold...

54
α-bins and Random Threshold Graphs Sudipta Saha Indian Institute of Technology Kharagpur 721302, India

Transcript of Evolution of Random Threshold...

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α-bins and

Random Threshold Graphs

Sudipta Saha

Indian Institute of Technology

Kharagpur – 721302, India

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Plan of the discussion Alpha-bipartite networks

Introduction

Edges Creation technique and meaning

Our aim Special structure - One-mode

projection

Modeling real life constraints Thresholds

Works already done

Further research issues Understanding the structure

New approach Random threshold graphs

Graphs as binary strings

Evolution of random threshold graphs

Present status of the work

Discussion

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Alpha-bipartite networks (α-bins)

Bipartite Network o Two partitions

A special property o Fixed partition o Growing partition

Alpha-Bipartite Network (a name originated from linguistics)

Examples o Word - letter o Gene - codon o User - group o Railway - station o Protein – amino-acid o People - place o Event - person o Movie - actor o Paper - author

Growing partition

Fixed partition

Many real life systems comprise of two distinct entity sets

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Alpha-bipartite networks (α-bins)

……

..

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

G A V L I S T C M P H R N Q E D F W Y K

English alphabet

Set of users/ people

Set of 64 codons

Set of 20 amino acids

Set of genes

Set of English words/ sentences Groups / Places

of interest

Set of Proteins

Word-letter People-place / User-group

Protein-amino-acid

Gene-codon

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Interpretation of edges in α-bins Member of a growing set may connect to one or many nodes in

the fixed set

Subject / topic similarity o Groups in user-group network

(online social network)

o Places in people-place network

o Authors in paper-author network

Co-occurrence o Codons in gene-codon network

o Amino-acids in protein-amino-acid network

Information flow o Groups in user-group network in

online social networks by posting / sharing of information

o Among places in people-place network

Flow of disease / opinion

Meaning / implication of multiple connections

Set of places

Visiting a place by a person

Information flow

One person

……

……

……

… …

.…

….

Set of people

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Alpha-bipartite networks (α-bins)

Selection of an element of the fixed set by an element of the growing set Fully Preferential

The probability that a node is selected at time step t is proportional to the number of connections it has already acquired before t

Gained higher popularity

Growing partition

Fixed partition

• dt(i) = degree of the ith element in the fixed set after t number of

nodes have joined the growing set

• N = cardinality of the fixed set

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Alpha-bipartite networks (α-bins)

Popularity / attractiveness A random variable - denoted by θi

Asymptotic growth rate of the degree of a node in the bipartite network

Can be sampled from Beta distribution / Dirichlet distribution

Gained higher popularity

Growing partition

Fixed partition θ

Pro

bab

ility

0,0

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Alpha-bipartite networks (α-bins)

Selection of an element of the fixed set by an element of the growing set Fully Random selection

• dt(i) = degree of the ith element in the fixed set after t number of

nodes have joined the growing set

• N = cardinality of the fixed set

Growing partition

Fixed partition

All the nodes acquired

almost same number of

connection

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Alpha-bipartite networks (α-bins)

Selection of an element of the fixed set by an element of the growing set Fully preferential and fully Random

selection can be combined in a single formula

• dt(i) = degree of the ith element in the fixed set after t number of

nodes have joined the growing set

• N = cardinality of the fixed set

δ is a regulating parameter = 0, Fully preferential case = Very large number, Fully random case

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One-mode projection of α-bins

Focus - understanding the connectivity / relationships among the nodes in the fixed set – indirectly the growing set Special structure – one-mode projection

o An induced uni-partite graph on the fixed partition

Formation – one possible way o An edge : both nodes are connected to some common element of growing set o Weight of an edge : count of the distinct paths of length 2 via growing partition

Set of places

……

.…

….

Set of people Set of places

.…

….

Joining of a node in the growing set A clique in the one-mode projection

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Construction of one-mode projection

Fixed set U Growing set V

1

2

3

4

Projection – multiple edges

1

2 3

4

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Construction of one-mode projection

Fixed set U Growing set V

1 a

2

3

4

1

2 3

4

Projection – multiple edges

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Construction of one-mode projection

Fixed set U Growing set V

1 a

2

3

4

1

2 3

4

b

Projection – multiple edges

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Construction of one-mode projection

Fixed set U Growing set V

1 a

2

3

4

1

2 3

4

b

c

Projection – multiple edges

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Construction of one-mode projection

Fixed set U Growing set V

1 a

2

3

4

Weighted projection

1

2 3

4

b

c

2 2

2

1 1

1

Degree of a node = sum of weights of the edges

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Construction of one-mode projection

Different ways to form one-mode projection (induce a graph on one partition) Multiple connections to a single node in the fixed set can be

assumed to be a single connection

But in one-mode projection formation we may ignore and consider only single connection

Agents may connect multiple times to the same node in the fixed set

An user in the online social network may post different messages in a single group multiple times

A single protein may be expressed by a certain gene multiple times

Examples

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Construction of one-mode projection

Groups Users

1

a

2

3

4

1

2 3

4

1

2 3

4

Group-activity by users

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Various ways to form one-mode projection

Different other ways to form one-mode projection Probabilistic formation of the edges

In one mode projection we can reflect this by using a probability distribution to connect any two persons coming to the same party

People-party bipartite network o Coming to a specific event by

two different persons

o May not lead to relationship formation between them

Similarly – movie - actor bipartite network

Visiting / connecting two distinct places may not deterministically lead to a connection formation between the two nodes in the fixed set

Examples

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Various ways to form one-mode projection

Persons Party / Gathering

1

a

2

3

4

1

2 3

4

1

2 3

4

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One-mode projection after large growth

Previous studies in this direction yielded o Asymptotic degree distribution *

o Effect of randomness in the attachment policy *

o Many other interesting properties and observations

14

0 123

* Degree Distributions of Evolving Alphabetic Bipartite Networks and their Projections, Niloy Ganguly, Saptarshi Ghosh, Tyll Krueger, Ajitesh Srivastava, Theoretical Computer Science, (In Press)

Understanding the connectivity / relationships among the nodes in the fixed set after a large growth of the growing set Asymptotic status depends on the initial condition Attachment policy etc

o It usually becomes a complete weighted graph

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Constraints and thresholded one-mode projection

Real life constraints Generic constraints - Time and

Energy - with respect to message dissemination in various distributed systems Mobile networks User groups

Specific – buffer time o Buffer-equipped DTN Four persons are

there who visit both of these two places frequently

Only one person visits these two places frequently

From the perspective of dissemination of message under constrained resource, the probability that a

message will pass from Place A to B by some person, is higher than that between both A and C or B and C

Place A Place B

Place C

A simple scenario of distributed communication among different places

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Constraints and thresholded one-mode projection

Thresholded one-mode projection o A threshold value

o Prune those edges from the one-mode projection (complete graph) whose edge weights are below the threshold value

o Un-weighted graph

10

8

7

5

2

2 2

8

Threshold = 4

10 8

8

Threshold = 7 10 8

7

5

8

Real life constraints can be modeled by thresholds on the one-mode projection

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Works done on thresholded one-mode projection

Modeling information dissemination dynamics in delay tolerant network through

evolving bipartite networks, Sudipta Saha, Niloy Ganguly and Animesh Mukherjee (Under preparation for submission in IEEE Transaction on Autonomous and Adaptive Systems) o Threshold represents a specific constraint `Message buffering time’ in

places for enhancing indirect communication in DTN o Yields the notion of optimal buffer time

Understanding evolution of inter-group relationships using bipartite networks,

Saptarshi Ghosh, Sudipta Saha, Ajitesh Srivastava, Tyll Krueger, Niloy Ganguly and Animesh Mukherjee (Accepted in IEEE Journal on Selected Areas in Communications, 2012) o Detailed analysis of the thresholded one mode projection o Understanding the communication / information flow between user-groups o Comparing with real data – youtube, flickr and livejournal

Previous and ongoing works in this direction

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Research issues

? ?

?

?

?

?

?

Detailed analysis of the thresholding process For threshold, Δ=0

o Complete Graph

For sufficiently large value of threshold o All nodes are isolated

Graph configurations for other threshold values Have not been investigated yet

Significance o It is essential to understand structure of the

graph at various threshold value

A new approach : Thresholded one-mode projections as random threshold graphs

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Threshold Graphs (Chvátal and Hammer, 1977)

Nodes o Associated with a real number o Example– popularity of an actor, richness

(wealth) of a person Edge between two nodes

o Sum of the node weights crosses a certain threshold Δ • Wi + Wj > Δ

o “Rich persons always know each other”

Threshold graphs and random threshold graphs

Random Threshold Graphs (Diaconis et al, 2009)

Weights are chosen as independent and identically distributed random variables (i.i.d.) from some probability distribution

240

310 3

100

240

310 3

100

Δ = 300

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Thresholded-one-mode projection as threshold Graphs

Previous analysis on one-mode projection yielded The edge weight between two

nodes to be cθi θj o Where θi and θj – are two real

numbers associated with the two nodes i and j respectively (Popularity / attractiveness)

o θi‘s are random variables following beta distribution

o c is a constant

Thresholded one-mode projection An edge exists between node i

and j if c θi θj > Δ

Generalized threshold function

Any monotonically increasing

and symmetric function can be used to define a threshold graph

Product can be easily transformed to addition by taking log

Any thresholded one mode projection can be thought of as a Random Threshold Graph associated with a specific threshold value All existing theories of random threshold graphs are applicable

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Applicability - detailed study of the alpha-bins Under various constraints

Largest connected component

Structural information

Example : - Information flow

o Delay Tolerant Network

o User-group network

Flow of opinion

Protein/Gene o Stickiness

o co-expression

Threshold graphs and random threshold graphs

Constraint is modeled by

some value of threshold

Which configuration is

more likely ?

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Construction of threshold graph

Any threshold graph can be built by sequentially inserting nodes of two types Either isolated

o Denoted by 0

Dominating o Denoted by 1

One-to-one correspondence between a Random Threshold Graph with N nodes and a binary string of length N-1 The first vertex can be denoted by

either 0 or 1 Easier to deal with strings than

graphs

0 1 1 0 1 1

An isolated vertex

A dominating vertex

N = 5

Example : A possible threshold graph with 6 nodes represented by 6 or 5 bit binary string

0 1 1 0 1 1 and 1 1 1 0 1 1 represent the same

thereshold graph A hierarchical structure

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Construction of threshold graph

0 A

1

The first node can be denoted by 0 or 1

Equivalent binary string A threshold graph

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Construction of threshold graph

0 0 A

1 0

An isolated node (0) is added

Equivalent binary string A threshold graph

B

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Construction of threshold graph

0 0 1 A

1 0 1

A dominating node (1) is added

Equivalent binary string A threshold graph

B

C

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Construction of threshold graph

0 0 1 0 A

1 0 1 0

An isolated node (0) is added

Equivalent binary string A threshold graph

B

C D

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Construction of threshold graph

0 0 1 0 1 A

1 0 1 0 1

A dominating node (1) is added

Equivalent binary string A threshold graph

B

C D E

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Construction of threshold graph

0 0 1 0 1 0 A

1 0 1 0 1 0

An isolated node (0) is added

Equivalent binary string A threshold graph

B

C D E

F

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Construction of threshold graph

0 0 1 0 1 0 0 A

1 0 1 0 1 0 0

An isolated node (0) is added

Equivalent binary string A threshold graph

B

C D E

F

G

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Construction of threshold graph

0 0 1 0 1 0 0 0 A

1 0 1 0 1 0 0 0

An isolated node (0) is added

Equivalent binary string A threshold graph

B

C D E

F

G

H

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Construction of threshold graph

0 0 1 0 1 0 0 0 1 A

1 0 1 0 1 0 0 0 1

A dominating node (1) is added

Equivalent binary string A threshold graph

B

C D E

F

G

H

I

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Construction of threshold graph

0 0 1 0 1 0 0 0 1 0 A

1 0 1 0 1 0 0 0 1 0

An isolated node (0) is added

Equivalent binary string A threshold graph

B

C D E

F

G

H

I

J

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Construction of threshold graph

0 0 1 0 1 0 0 0 1 0 0 A

1 0 1 0 1 0 0 0 1 0 0

An isolated node (0) is added

Equivalent binary string A threshold graph

B

C D E

F

G

H

I

J

K

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Construction of threshold graph

0 0 1 0 1 0 0 0 1 0 0 0 A

1 0 1 0 1 0 0 0 1 0 0 0

An isolated node (0) is added

Equivalent binary string A threshold graph

B

C D E

F

G

H

I

J

K

L

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Construction of threshold graph

A

B

C D E

F

G

H

I

J

K

L

1 0 1 0 1 0 0 0 1 0 0 0

0 0 1 0 1 0 0 0 1 0 0 0

There can be total 2N-1 distinct threshold graphs from N number of nodes

Ignoring first bit 0 1 0 1 0 0 0 1 0 0 0

A 12 node threshold graph is represented by

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Construction of threshold graph

A

B

C D

E 1 1 1 1

0 0 0 0

The status at Δ =0 for N = 5

i.e., a 5 node complete graph can be represented by

A

B

C D

E

The status at very high Δ i.e., fully isolated status can be represented by

To understand the configuration/binary string for a given threshold value, we need to identify the full ascending order of the edge weights

Edges are deleted following the ascending order of the

edge weights

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Deletion of edge in threshold graph

Degree of a node is related to its ‘attractiveness’

The degree of `0’ node increases from right to left

The degree of `1’ node increases from left to right

0 … 0

1 … 1

1 … 0

No edge possible

Edge with higher weight

No edge possible

0 … 1 Edges with least possible weights

Higher degree denotes higher attractiveness value

(Higher node weight)

It can be shown that the ‘01’ sequences represent the least weight edges

0 1 0 1 1 0 0 0 1 0 1

Highest degree isolated node

Least degree dominating node

Highest degree dominating node

Least degree isolated node

Possible edges for deletion

Thresholding implies deletion of edges

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Deletion of edge in threshold graph

0 1 0 1 1 0 0 0 1 0 1 Let us take the second occurrence

0 1 0 1 1 0 0 0 1 0 1

A dominating vertex; Has degree = 6

An isolated vertex; Has degree = 4

0 1 1 0 1 0 0 0 1 0 1

An isolated vertex; Has degree = 3

A dominating vertex; Has degree = 5

Interchange the order of insertion of the dominating vertex and the isolated vertex

Each of the vertices will loose one degree which implies deletion of an edge between them

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Deletion of edge in threshold graph

0 1 0 1 1 0 0 0 1 0 1

But, there can be many occurrences of ‘01’, Which one represents the

least weight edge ?

For fixed values of the node weights – we can easily determine this

But this is the most difficult job if the node weights are i.i.d. random variables from some distribution

Consider N = 4, nodes with weights (attractiveness) - θ1 , θ2 , θ3 and θ4 which are i.i.d. random variables

Consider the order (θ1 < θ2 < θ3 < θ4) Relationship between θ1xθ4 and θ2xθ3 is

not easily known

For 4 variables finding the probability P{θ1xθ4 > θ2xθ3} is

solvable; but becomes very

difficult for large N

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Evolution of threshold graphs Representing by - Evolution graphs

Arranging all possible configurations in course of thresholding of a one-mode projection in the form of a graph where o A node : A possible graph configuration o Edge : One configuration is possible to

achieve from the other configuration by adding or deleting some edge

Height =

Number of nodes = N

Starting from a complete graph with N nodes needs levels of thresholding o Hight of the graph is o Number of nodes is 2N-1 o But what is the width of a graph ?

2

N

2

N

2

N

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Evolution graph

The starting configuration 1111

The left most bit does not matter

– make it 0 always

Thus we proceed through the pruning the least weight edge

Whenever there are more than one such possibility, we generate all possible configurations

Example for a 4 node graph

θ4

θ3

θ2

θ1

θ4 θ3

θ2 θ1

θ4

θ3

θ2

θ1

θ4

θ3

θ2

θ1

θ4

θ3

θ2

θ1

θ4

θ3

θ2

θ1

θ4

θ3

θ2

θ1

θ4

θ3

θ2 θ1

Fully connected

Isolated

θ2 x θ3 > θ1 x θ4 θ1 x θ4 > θ2 x θ3

θ4 > θ3 > θ2 > θ1

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Evolution graph

θ4 θ3

θ2 θ1

θ4

θ3

θ2

θ1

θ4

θ3

θ2

θ1

θ4

θ3

θ2

θ1

θ4

θ3

θ2

θ1

θ4

θ3

θ2 θ1

θ4

θ3

θ2

θ1

Fully connected

Fully Isolated

θ2 x θ3 > θ1 x θ4 θ1 x θ4 > θ2 x θ3

θ4 > θ3 > θ2 > θ1

θ4

θ3

θ2

θ1

0011

0100

0101

0010

0001 0110

0111

0000

N=4 N=4

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Evolution graph 011111

001111

010111

000111

001011

001101

010101

011001

000110

001010

010010

000000

001000

010000

011011

011101

000010

000100

010100

011000

010011

000011

000101

001001

010001

000001

011110

001110

010110

011010

011100

001100

N=6 Height is = 15 Number of nodes = 32

E=15

E=14

E=13

E=12

E=11

E=10

E=9

E=8

E=7

E=6

E=5

E=4

E=3

E=2

E=1

E=0

No of edges

2

6

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Width of the evolution graph

We have N nodes in the projection

Each node (ith) can be either dominating or isolated (0 or 1)

If dominating then

Number of edges created at the time of its insertion is same as its position – 1 i.e., (i - 1)

If isolated then

Brings no edge (0)

For N nodes we have the following Generating function

This gives us NC2 coefficients each of which corresponds to the number of possible graphs for a given number of edges (i.e. at a certain level of the evolution graph)

The problem is similar to finding how many ways we can partition an integer N by the integers from 1 to N-1

For N=4, the coefficients are

1 1 1 2 1 1 1

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Coefficients (log-linear plots)

Value of the Threshold (Degree of specific constraint)

Nu

mb

er

of

po

ssib

le g

rap

h c

on

figu

rati

on

s /

Wid

th o

f th

e e

volu

tio

n g

rap

h

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Coefficients (rescaled linear plots)

Rescaled value of the Threshold (Degree of specific constraint)

Re

scal

ed

nu

mb

er

of

po

ssib

le g

rap

h c

on

figu

rati

on

s /

Wid

th o

f th

e e

volu

tio

n g

rap

h

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Evolution graphs

Problems yet to be solved All graph configurations at

a certain level do not have equal probability

Thus all paths also do not have equal probability

Finding out the typical paths in the evolution graphs o Finding the full order of the

pair-wise products of N random variables – for few possible distributions of their weights

• Beta, power-law

011111

001111

010111

000111

001011

001101

010101

011001

000110

001010

010010

000000

001000

010000

011011

011101

000010

000100

010100

011000

010011

000011

000101

001001

010001

000001

011110

001110

010110

011010

011100

001100

P1 P2

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Sudipta Saha [email protected]

http://cse.iitkgp.ac.in/~sudiptas