Discover a Network by Walking on it! Andrea Asztalos & Zoltán Toroczkai Department of Physics...
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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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