Discover a Network by Discover a Network by Walking on it!Walking on it!

Discover a Network by Discover a Network by Walking on it!Walking on it!

Andrea Asztalos & Zoltán ToroczkaiDepartment of PhysicsDepartment of Physics

University of Notre DameUniversity of Notre DameDepartment of PhysicsDepartment of Physics

University of Notre DameUniversity of Notre Dame

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MotivationMotivationMotivationMotivation What is the structure of a graph?

Nodes, Edges, Loops

Knowing the network, and some features of the walk

What will be the network seen by the walker?

- example: gradient network

How can one optimize an exploration?

Exploring the NetworkExploring the NetworkExploring the NetworkExploring the Network

• Connected graph

)'|( ssp - transition probabilities define the walk on the graph

EXPLORATION – recording the set of nodes and edges which have been visited

RANDOM WALK RANDOM WALK

• Jumping only to adjacent sites

The underlying, “unexplored” network NetSci07NetSci07

Graph exploration algorithm:

Mathematical ApproachMathematical ApproachMathematical ApproachMathematical Approach)|( 0ssPn

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- probability of being at site s on the nth step, given that the walk started from site s0 - probability of being at site s for the first time on the nth step, given that the walk started from site s0

)|( 0ssFn

N

snn ssPsspssP

1'001 )|'()'|()|(

0,00 )|(, ssssP Evolution law:

nS

nX - number of visited distinct edges up to n steps, averaged over many realizations of the walk

- number of visited distinct sites up to n steps, averaged over many realizations of the walk

,)(0

n

nnAA

1|| )(2

11

Ad

iA

nn

Generating Function Formalism

Generating function of An asymptotic behavior of An

Discovering the NodesDiscovering the NodesDiscovering the NodesDiscovering the Nodes,

0

n

jnn IS

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Valid for arbitrary graph

*B.D. Hughes, Random Walks and Random Environments,Vol. 1, Oxford (1995)

s ssP

ssPS

);|(

);|(

1

1)( 0*

• Introduce an indicator function In

• Using the first-passage probability distribution for the nodes:

0

)|(}1{Pr 0ss

nnn ssFIobI

Discovering the EdgesDiscovering the EdgesDiscovering the EdgesDiscovering the Edges- probability of arriving at edge e for the first time on the nth step, given that the walk started at site s0

)|( 0seFn First - passage probabilities for

the EDGES?

????

Generating function for <Sn>

Discovering the Edges …Discovering the Edges …Discovering the Edges …Discovering the Edges …

);|();(1

)( 0

ssPsWXs

'

2)()(1

)(1),(

s bcadda

cdsW

);|'()(),;'|'()(

);|()(),;'|()(

)'|(),|'(

ssPddssPcc

ssPbbssPaa

sspssp

z – an auxiliary node placed on the edge (s,s’)

zGEXTENDED GRAPH

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Generating function for <Xn>

Valid for arbitrary graph

Notation:

Complete GraphComplete GraphComplete GraphComplete Graph

)1()|'()'|( ',sspsspssp 1

1

N

p

);|();|( 00 ssPssP Homogeneous walk: No directional bias:

p

pNS

n

n

)1( ))(1(

)1(

))(1(

)1(

2

)1(

2122

221

2111

121

qqqq

pqqq

qqqq

pqqqNNX

nnn

)(),( 2211 pqqpqq

Estimate of steps needed to explore the majority of

nodes

edges NNn ln2

NNn ln

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1d lattice and Trees1d lattice and Trees1d lattice and Trees1d lattice and Trees

)(2

1)( 1,1, lllp

nasn

Sn2

1* )

8(~

nasn

X n2

1

)8(~

Homogeneous, without any directional bias Translationally invariant walk

Step distribution:

N

N

U

US

)(1

)(1

)1(

1)(

2/3

)1)(1

)(11(

)1(

1)(

2/3

N

N

U

UX

1|)(| U

1 nn SX

1D infinite lattice 1D finite lattice

For trees and 1d lattices:

NetSci07NetSci07*B.D. Hughes, Random Walks and Random Environments,

Vol. 1, Oxford (1995)

d>1 Infinite Latticesd>1 Infinite Latticesd>1 Infinite Latticesd>1 Infinite Lattices

dtd

tIeP

d

j

t

j)();(

1||

0

*

ll

d

jjjd

p1

,, )(2

1)( elell

nasn

nSn )8ln(

~*

nasn

nX n )8ln(23

4~

nasPd

dnX n )1;(212

2~

0

dtd

tIeP dt )()1;( 0

01lim

0

Homogeneous, without any directional bias Translationally invariant walk

Step distribution:

d=2 d>2

Generating function:

*B.D. Hughes, Random Walks and Random Environments,Vol. 1, Oxford (1995) NetSci07NetSci07

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Simulation results for three different graphs of the same size N=210

• In the case of ER graph with high degree the walkermakes fewer steps in whichit does not discover new nodes.

• In the case of SF graphthere are many small nodes, the probability to go backwards increases, thus the discovering process is slow.

ER Random Graphs & Scale-free ER Random Graphs & Scale-free GraphsGraphs

ER Random Graphs & Scale-free ER Random Graphs & Scale-free GraphsGraphs

ER Random Graphs & Scale-free ER Random Graphs & Scale-free GraphsGraphs

ER Random Graphs & Scale-free ER Random Graphs & Scale-free GraphsGraphs

Simulation results for three different graphs of the same size N=210

Sn=1Xn=0

Sn=2Xn=1Sn=1

Xn=0

Sn=3Xn=2

Sn=2Xn=1Sn=1

Xn=0

Sn=4Xn=3

Sn=3Xn=2

Sn=2Xn=1Sn=1

Xn=0

Sn=5Xn=4 Sn=4

Xn=3

Sn=3Xn=2

Sn=2Xn=1Sn=1

Xn=0

Sn=5Xn=4 Sn=4

Xn=3

Sn=5Xn=5

Sn=2Xn=1Sn=1

Xn=0

Ln((<X(t)>+1)/<S(t)>) - a measure of discovering loops in the graph

This quantity only grows when a new edge is discovered,

which means a new loop in the graph. In a SF graph, this quantity

grows faster than in an ER graph.

SummarySummarySummarySummary

• Exploring graphs via a general class of random walk

• Increase of the set of revealed nodes as a function of time

• Counting edges by introducing an auxiliary node, thus extending the original graph

• Deriving expressions for particular cases: complete graph, infinite and finite hypercubic lattices, analyze random and scale-free graphs

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