school of information university of michigan si 614 search in random networks lecture 16
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School of InformationUniversity of Michigan
SI 614Search in random networks
Lecture 16
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MotivationPower-law (PL) networks, social and P2P
Analysis of scaling of search strategies in PL networks
Simulationartificial power-law topologies, real Gnutella networks
Comparison with existing P2P search strategiesReflector, Morpheus
Path finding
Directed SearchFreenet
Search in random networks
2
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Mary
Bob
Jane
Who couldintroduce me toRichard Gere?
How do we search?
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AT&T Call Graph
Aiello et al. STOC ‘00
# o
f te
lep
hon
e n
um
be
rsfr
om
wh
ich
ca
lls w
ere
ma
de
# of telephone numbers called
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100
101
100
101
102
number of neighbors
pro
po
rtio
n o
f n
od
es
datapower-law fit = 2.07
Gnutella network
power-law link distribution
summer 2000,data provided by Clip2
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Preferential attachment model
Nodes join at different times
The more connections a node has, the more likely it is to acquirenew connections
Growth process produces power-law network
host cache
pingping
ping
ping
ping
ping ping
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file sharing w/o a central index
queries broadcast to every node within radius ttl as network grows, encounter a bandwidth barrier (dial up modems cannot keep up with query traffic, fragmenting the network)
Gnutella and the bandwidth barrier
Clip 2 reportGnutella: To the Bandwidth Barrier and Beyondhttp://www.clip2.com/gnutella.html#q17
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1
6
54
63
67
2
94
number ofnodes found
power-law graph
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93
number ofnodes found
13
711
1519
Poisson graph
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Search with knowledge of 2nd neighbors
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Outline of search strategy
pass query onto only one neighbor at each step
requires that nodes sign query- avoid passing message onto a node twice
requires knowledge of one’s neighbors degree- pass to the highest degree node
requires knowledge of one’s neighbors neighbors- route to 2nd degree neighbors
OPTIONS
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Generating functions
M.E.J. Newman, S.H. Strogatz, and D.J. Watts ‘Random graphs with arbitrary degree distributions and their
applications’, PRE, cond-mat/0007235
Generating functions for degree distributions
Useful for computing moments of degree distribution, component sizes, and average pathlengths
00
( ) kk
k
G x p x
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Fun with generating functions
normalization condition: probabilities sum to 1
1)1(0
0
kkpG
derivatives: the generating function contains all the information of the degree distribution
00
!
1
xk
k
k x
G
kp
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Fun with generating functions (cont’d)
Expected degree of a randomly chosen vertex
'0(1)k
k
k kp G
Higher moments of degree distribution
1
0 )(
x
n
kk
nn xGdx
dxpkk
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Example: Poisson distribution
Let p = z/N be the probability of an edge existing between two vertices (z is the average degree)
kkNkN
k
xppk
NxG
)1()(
00
)1(~)1( xzN epxp for large N
)1(0 )(' xzzexG
zG )1('0
zkxk
k
k ezkx
G
kp
!
1
!
10
0 just the regular Poisson
distribution
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Introducing cutoffs
max 1k N a node cannot have more connections than there are other nodes
This is important for exponents close to 2
1 1
11kp C
x 2 2
6C
1000
( 1000, 2) ~ 0.001kp k p
Probability that none of the nodes in a 1,000 node graph has 1000 or more neighbors:
1000(1 ( 1000, 2)) ~ 0.36p k
without a cutoff, for = 2have > 50% chance of observing a node with more neighbors than there are nodes
for = 2.1, have a 25% chance
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# of sites linking to the site
prop
ortio
n of
site
s w
/ so
man
y lin
ks
1000
Selecting from a variety of cutoffs
Nk max1.
2. /k
k eCkp Newman et al.
3.
0
Ckpk
1CNk
otherwise
Generating Function
kCN
k
xkCxG
1
10
Aiello et al.
1 million websites (~ 1997)
N
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Aiello’s ‘conservative’ vs. Havlin’s ‘natural’ cutoff
1
1
* 1
~
kN p
Ck N
k N
cutoff where expectednumber of nodes of degreek is 1
k
n(k)
k
n(k)
1
1
cutoff so thatexpected number of nodesof degree > k is 1
max
max
1
1 1max
1
1max
* 1
~
~
~
kk k
k k
N p
ck N
k N
k N
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The imposed cutoff can have a dramaticeffect on the properties of the graph
degrees drawn at random, for = 2, and N = 1000
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00
( ) kk
k
G x p x
is the probability that a randomlychosen vertex has degree k
~kp k
is a generating function
'0(1)k
k
k kp G is the expected degree of a randomlychosen vertex
'0
1 '0 1
G xG x
G is the distribution of remaining
outgoing edges following and edge
assuming neighbors don’t share edges
11 '1
'02 GGz is the expected number of second
degree neighbors
22
2
2 2
2
1
11
Generating functions for degree distributions Random graphs with arbitrary degree distributions and their applications by Newman, Strogatz & Watts
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search with knowledge of first neighbors
max
max
maxmax
max
max
01
' 1 10 0
1
' 1 1 20 max
1 1
'' 1 101 ' '
10 0
1 2'
20
3 2' max1 '
0
( )
( ) ( )
1(1) 1
2
( )( )
(1) (1)
( 1)(1)
( 2) 21(1)
(1)
kk
kk
kk
kk
kk
G x c k x
G x G x c k xx
G k c k k dk k
G x cG x k x
G G x
ck k x
G
kG
G
2max( 1) (3 )
( 2)(3 )
k
Generating function with cutoff
Average degree of vertex
constant in Nfor 2<<3, and kmax~Na, decreaseswith N
Average number of neighborsfollowing an edge
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search with knowledge of first neighbors (cont’d)
3 3' 3max max
1 1 max' 20 max
1 2(1)
(1) (3 ) 1 (3 )B
k kz G k
G k
In the limit t->2, ' max1
max
(1)log( )
kG
k
Let’s for the moment ignore the fact that as we do a random walk, we encounter neighborsthat we’ve seen before
s = number of steps =1B
N
z
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Search time with different cutoffs
0.18(2.1)s N
3 3m
2
ax
( ) ,2 3N N
sk N
NIf kmax = N,
max
max
log(log( )
, 2)N k
s Nk
2
21
33max 1
( ) ,2 3NN N
sk
N
If kmax = N1/(-1),
max
max
log( )(2) log( )
N ks
kN
0.1(2.1) Ns
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search with knowledge of first neighbors (cont’d)
2 3 /13
3max
,2 3
( )
N Ns N
kN
If kmax = N,
So the best we can do is for exponents close to 2 N
2nd neighbor random walk, ignoring overlap:
15.0~1.2, NNS 213~, NNS
2
2
~
s B
B
n z N
NS
z N
232' max
2 1 1 1 21 max
2( ( )) (1)
1 (3 )Bx
kz G G x G
x k
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Following the degree sequence
a ~ s = # of steps taken
1.0deg 1.2, NNS
2nd neighbors, ignoring overlap:
Go to highest degree node, then next highest, … etc.
max
max
1 11 max~
k
D k az Nk dk Nak
max
max
' 2(2 )1 1
2( 2) 2 4 /
( ) ~
~ ~
Dz G x Nak
s k N
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0 10 20 30 40 50 60 70 80 90 100
1
= 2.00 = 2.25 = 2.50 = 2.75 = 3.00 = 3.25 = 3.50 = 3.75
degree of node
deg
ree
of
nei
gh
bo
r -
1d
egre
e o
f n
od
e
2
20
10
5
Ratio of the degree of a node to the expected degree of its highest degree neighbor for 10,000 node power-law graphs of varying exponents
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Actor collaboration graph (imdb database)
~ 2.0-2.2
Exponents close to 2 required to search effectively
World Wide Web, ~ 2.0-2.3, high degree nodes: directories, search engines
Social networks, AT&T call graph ~ 2.1
Gnutella
100 101 102 103 104100
101
102
103
104
105
number of costars
num
ber
of a
ctor
s/ac
tres
ses actors, = 2
actresses, = 2.1
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15
50
18 17
8
10
9
6
Following the degree sequence
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Complications
Should not visit same node more than once
Many neighbors of current node being visited were also neighbors of previously visited nodes, and there is a bias toward high degree nodes being ‘seen’ over and over again
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0 100 200 300 400 500 6000
5
10
15
20
25
30
step
deg
ree
of
no
de
not visitedvisitedneighbors visited
Status and degree of node visited
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1
10-2
0.1
1
step
pro
po
rtio
n o
f n
od
es
fo
un
d a
t s
tep
random walk
10 102
103
104
105
106
10-3
10-4
0 20 40 60 80 100
0.2
0.4
0.6
0.8
1
step
cum
ula
tive
no
des
fo
un
d a
t st
ep
random walk degree sequence
seeking high degree nodesspeeds up the search process
about 50% of a 10,000 node graphis explored in the first 12 steps
Progress of exploration in a 10,000 node graph knowing2nd degree neighbors
12
degree sequence
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101
102
103
104
105
100
101
102
103
size of graph
cove
rtim
e fo
r h
alf
the
no
des
random walk = 0.37 fit
degree sequence = 0.24 fit
Scaling of search time with size of graph
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Comparison with a Poisson graph
100
101
102
103
100
101
102
de
gre
e o
f c
urr
en
t n
od
e
step
Poisson power-law
101
102
104
106
100
101
102
103
104
105
number of nodes in graph
co
ve
r ti
me
fo
r 1
/2 o
f g
rap
h
constant av. deg. = 3.4
= 1.0 fit
10
xzexG
xGxGz
xxG 001
expected degree and expecteddegree following a link are equal
scaling is linear
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0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
step
cum
ula
tive
no
des
fo
un
d a
t st
ep
high degree seeking 1st neighborshigh degree seeking 2nd neighbors
50% of the files in a 700 node network can be found in < 8 steps
Gnutella network
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• Maintain a list of files in their neighborhood
• Check query against list.
• Periodically contact neighbors to maintain list
• Append ID to each query processed
Required modifications to nodes
Tradeoff
storage/cpu(available)
bandwidth(limited)
for
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• localized indexing• traffic routed to high degree nodes
Partial implementation:
Theory vs. reality:
• overloading high degree nodes but no worse than original scenario where all nodes handle all traffic
assume high degree -> high bandwidth so can carry the traffic load
• fewer nodes used for routing, system is more susceptible to maliciousattack
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Clip2 Distributed Search Solutionshttp://dss.clip2.com© Clip2.com, Inc.
Broadband user runningReflector
Broadband user runningGnutella
Dial-up user runningGnutella
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LimeWire, BearShare:drop connections to unresponsive hosts drives slower hosts to have fewer connections &move to edge of network
Kazaa, BearShare defender, Morpheus SuperNodes
from Clip2: Morpheus out of the Underworldhttp://www.openp2p.com/pub/a/p2p/2001/07/02/morpheus.html
Connection-preferencing rules
Supernodes
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Conclusions
Search is faster and scales in power-law networks
Networks intended to be searched, such as Gnutella,have a favorable P-L topology
High degree strategy has partially been implemented in existing p2pclients, such as BearShare, Kazaa & Morpheus
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A PL link distribution shortens the average shortest path
1
1
1
21
1 zz
zzz
r
rr
Poisson: = z1
PL: > z1
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1 1.5 2 2.5 3 3.5 4 4.5 510
0
101
102
103
104
105
106
radius
neig
hbor
s at
rad
ius
power-law =2.5Poisson =1.0
102
104
106
0
2
4
6
N
PLPS
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A
B
What about the shortest path discovered along the way?
B.J. Kim et al. ‘Path finding strategies in scale-free networks’, PRE (65) 027103.
each node passes message to highest degree neighbor it hasn’t passed the message to previously
‘cut off’ loops
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102
103
104
105
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
N
av
. pa
th le
ng
th f
ou
nd
PL high degree0.72*ln(N)
A high degree seeking strategy finds shortest paths whose average scales logarithmically with the size of the graph
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102
103
104
105
101
102
N
av
. pa
th le
ng
th f
ou
nd
PLPoisson
N0.46
N0.48
Scaling of the path length found using a• random strategy on a PL graph• high-degree strategy on a Poisson graph
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102
103
104
105
100
101
102
103
104
N
me
dia
n s
ea
rch
co
st
PL high degreePL randPoisson high degree
But…Search costs are prohibitive, might as well do a BFS
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Freenet
Queries are passed to one peer at a time.
Queries routed to high degree nodes.
Has a power-law topology Theodore Hong, ‘Performance’ chapter in O’Reilley’s “Peer-to-Peer, Harnessing the Power of Disruptive Technologies”
Scales as N0.275 with the size of the network, N.
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Theodore Hong, power - law link distribution of a simulated Freenet network
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Theodore Hong, scaling of mean search time
on a simulated Freenet network
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Node specialization key to Freenet’s speed
Each node forwards query to node with “closest” hash key
Node passing back a match remembers the address the data came from
Results in nodes developing a bias towards a part of the keyspace
112659 ?356?
340388
396 135214
356340388
396 135214
Queries are naturally routed to high degree nodesUse keys for orientation
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Applications to peer to peer networks
Adriana Iamnitchi, Matei Ripeanu, Ian Foster “Small-World File-Sharing Communities”, http://arxiv.org/abs/cs.DC/0307036create localized indeces for peers with similar download patterns
Foreseer:Proposed P2P architecture with friend & neighbor overlay friend: has shared a file neighbor: short ping time
Fletcher, George , Sheth, Hardik and Börner, Katy. (2004). Unstructured Peer-to-Peer Networks: Topological Properties and Search Performance. Third International Joint Conference on Autonomous Agents and MUlti-Agent Systems. W6: Agents and Peer-to-Peer Computing, Moro, Gianluca, Bergmanschi, Sonia and Aberer, Karl, Eds., New York, July 19-23, pp. 2-13.http://ella.slis.indiana.edu/~katy/paper/04-fletcher.pdf
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How do networks become navigable?Aaron Clauset and Cris Moore
arxiv.org/abs/cond-mat/0309415
In the limit N->long range link distribution becomes 1/r,r = lattice distance betweennodes