on the robustness of power law random graphs
DESCRIPTION
On the robustness of power law random graphs. Hannu Reittu in collaboration with Ilkka Norros, Technical Research Centre of Finland (Valtion Teknillinen Tutkimuskeskus, VTT). Content. Model definition Asymptotic architecture The core Robustness of the core - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/1.jpg)
ABI March 1. 2007, Espoo 1
On the robustness of power law random graphs
Hannu Reittu in collaboration with Ilkka Norros,
Technical Research Centre of Finland
(Valtion Teknillinen Tutkimuskeskus, VTT)
![Page 2: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/2.jpg)
ABI March 1. 2007, Espoo 2
Content
Model definition Asymptotic architecture The core Robustness of the core Main result and a sketch of proof Corollaries Conjecture Resume
![Page 3: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/3.jpg)
ABI March 1. 2007, Espoo 3
References
Norros & Reittu, Advances in Applied Prob. 38, pp.59-75, March 2006
Related models and review:
Janson-Bollobás-Riordan, http://www.arxiv.org/PS_cache/math/pdf/0504/0504589.pdf
R Hofstad: http://www.win.tue.nl/~rhofstad/NotesRGCN.pdf
![Page 4: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/4.jpg)
ABI March 1. 2007, Espoo 4
Classical random graph ( )
Independent edges with equal probability (pN)
pN
pN 1-pN
NpG
![Page 5: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/5.jpg)
ABI March 1. 2007, Espoo 5
However,
=> degrees ~ Bin(N-1, pN) ≈ Poisson(NpN)
Internets autonomous systems graph (and many others) have power law degrees
Pr(d>k) ~ k-
With 2 < < 3
![Page 6: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/6.jpg)
ABI March 1. 2007, Espoo 6
![Page 7: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/7.jpg)
ABI March 1. 2007, Espoo 7
Conditionally Poissonian random graph model
Sequence of i.i.d., >0,r.v.
(the ‘capacities’)
number of edges between nodes i and j:
,...),( 21
_
),( jiEN
Ni
iNL1
![Page 8: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/8.jpg)
ABI March 1. 2007, Espoo 8
Properties, conditionally on :
(i)
(ii)
(iii) The number of edges between disjoint pairs of nodes are independent
,~),(
N
jiN L
PoissonjiE
N
jiN L
jiEE
)|),((_
)(~),()(1
iNj
NN PoissonjiEiD
iN iDE )|)((_
_
![Page 9: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/9.jpg)
ABI March 1. 2007, Espoo 9
Assume:1)Pr( xx
32
![Page 10: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/10.jpg)
ABI March 1. 2007, Espoo 10
Theorem (Chung&Lu; Norros&Reittu):
a.a.s. has a giant component distance in giant component has the upper
bound: , almost surely for large N
,)2log(
loglog)(**
N
Nkk
NG
))1(1)((2 * oNk
![Page 11: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/11.jpg)
ABI March 1. 2007, Espoo 11
Asymptotic architecture
Hierarchical layers:
0},:{)( jNiNU j
ij
)},({)( *0 NiNU
)(,...,1,0),(1
)2()( * NkjNcN j
j
j
0log/)(,log/)(,log
)()( 34 NNlNNl
N
NlN
*,...,2,1,0 kj
![Page 12: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/12.jpg)
ABI March 1. 2007, Espoo 12
The ‘core’:
}:{)( )()(*
*NlN
ikeNNiNUC k
![Page 13: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/13.jpg)
ABI March 1. 2007, Espoo 13
‘Tiers’:Short (loglog N) paths:
Routing in the core: next step to largest degree neighbour
...2,1,1 jUUW jjj
....}{...... 21*
121 WWiWWWW jjj
![Page 14: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/14.jpg)
ABI March 1. 2007, Espoo 14
The core
‘Achilles heel’?
![Page 15: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/15.jpg)
ABI March 1. 2007, Espoo 15
Typical path in the ‘core’
Wj
Wj-1
Wj-2
i*
![Page 16: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/16.jpg)
ABI March 1. 2007, Espoo 16
Uj-1 is destroyed
Wj
Wj-1
Wj-2
i*
XX
X
![Page 17: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/17.jpg)
ABI March 1. 2007, Espoo 17
Hypothesis:
has a sub graph, a classical random graph
with constant diameter, jW
jd
![Page 18: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/18.jpg)
ABI March 1. 2007, Espoo 18
Back up
Wj
Wj-1
Wj-2
i*
XX
X
![Page 19: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/19.jpg)
ABI March 1. 2007, Espoo 19
hop counts:
a.a.s.
Wj
jNk )(*
jNk )(*
jdd jj 2, jdk j 22 * }
![Page 20: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/20.jpg)
ABI March 1. 2007, Espoo 20
dj is a constant => asymptotically, the same distance ( )*2k
![Page 21: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/21.jpg)
ABI March 1. 2007, Espoo 21
Proposition:
Fix integer j>0
a.a.s., diam(Wj)
j
jjd
)3(
)1(1
3
))2(1)(1(
)2(
1 j
j
![Page 22: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/22.jpg)
ABI March 1. 2007, Espoo 22
Remarks
Back up path in Wj has at most dj hops
However, in classical random graph, short paths are hard to find
Wj is connected sub graph ('peering')
![Page 23: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/23.jpg)
ABI March 1. 2007, Espoo 23
Sketch of proof:
Use the following result (see: Bollobás, Random Graphs, 2nd Ed. p 263, 10.12)
Suppose that functions and
satisfy
and
Then a.e. (cl. random graph) has diameter d
3)( ndd
1)(0 npp
nnp
ndndd log2
loglog3/)(log1 nnp dd log221
pG
![Page 24: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/24.jpg)
ABI March 1. 2007, Espoo 24
We have:
)3( jNpn
)1(1)3()( jjdd
Nn
pn
![Page 25: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/25.jpg)
ABI March 1. 2007, Espoo 25
Find such d:
and
=> the claim follows
0)1(1)3( jjd
0)1(1)3()1( jjd
![Page 26: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/26.jpg)
ABI March 1. 2007, Espoo 26
Corollaries
Nodes with are removed =>
extra steps (u.b.). More precisely:
10, N
1)( d
1
3
))2)(1(1()(
1
)2)(3(
))1(1(d
![Page 27: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/27.jpg)
ABI March 1. 2007, Espoo 27
Can we proceed:
0)( N
*)(k
N
![Page 28: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/28.jpg)
ABI March 1. 2007, Espoo 28
Yes and no
If goes to 0 no quicker that:
With this speed
3
3,
log
loglogc
N
Nc
N
Nd
loglog
log)(
![Page 29: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/29.jpg)
ABI March 1. 2007, Espoo 29
but
Is too quick! These tiers are not connected because degrees
are too low.
NNlNk
log/)()( *
![Page 30: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/30.jpg)
ABI March 1. 2007, Espoo 30
Conjecture
However, has a giant component And degrees => Diameter of g.c. (Chung and Lu 2000), yields u.b.
*kW
)(NN
).(/log NlN
![Page 31: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/31.jpg)
ABI March 1. 2007, Espoo 31
Resume
Removal of ‘large nodes’ has, eventually, no effect on asymptotic distance up to some point
We can imagine graceful growth in path lengths: Core ( ) is important! Although:
in cl. random graphs, such events do not matter
)(/logloglog/logloglog ?* NlNNNNk
0N
C
C
![Page 32: On the robustness of power law random graphs](https://reader036.vdocument.in/reader036/viewer/2022062722/568139b8550346895da157a2/html5/thumbnails/32.jpg)
ABI March 1. 2007, Espoo 32
Thank You!