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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)
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
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
ABI March 1. 2007, Espoo 4
Classical random graph ( )
Independent edges with equal probability (pN)
pN
pN 1-pN
NpG
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
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
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 )|)((_
_
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
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
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
ABI March 1. 2007, Espoo 17
Hypothesis:
has a sub graph, a classical random graph
with constant diameter, jW
jd
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
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')
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
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
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)(
ABI March 1. 2007, Espoo 29
but
Is too quick! These tiers are not connected because degrees
are too low.
NNlNk
log/)()( *
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
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