On the robustness of power law random graphs

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)


Content

Model definition Asymptotic architecture The core Robustness of the core Main result and a sketch of proof Corollaries Conjecture Resume


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


Classical random graph ( )

Independent edges with equal probability (pN)

pN

pN 1-pN

NpG


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


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


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 )|)((_

_


Assume:1)Pr( xx

32


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


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


The ‘core’:

}:{)( )()(*

*NlN

ikeNNiNUC k


‘Tiers’:Short (loglog N) paths:

Routing in the core: next step to largest degree neighbour

...2,1,1 jUUW jjj

....}{...... 21*

121 WWiWWWW jjj


The core

‘Achilles heel’?


Typical path in the ‘core’

Wj

Wj-1

Wj-2

i*


Uj-1 is destroyed

Wj

Wj-1

Wj-2

i*

XX

X


Hypothesis:

has a sub graph, a classical random graph

with constant diameter, jW

jd


Back up

Wj

Wj-1

Wj-2

i*

XX

X


hop counts:

a.a.s.

Wj

jNk )(*

jNk )(*

jdd jj 2, jdk j 22 * }


dj is a constant => asymptotically, the same distance ( )*2k


Proposition:

Fix integer j>0

a.a.s., diam(Wj)

j

jjd

)3(

)1(1

3

))2(1)(1(

)2(

1 j

j


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')


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


We have:

)3( jNpn

)1(1)3()( jjdd

Nn

pn


Find such d:

and

=> the claim follows

0)1(1)3( jjd

0)1(1)3()1( jjd


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


Can we proceed:

0)( N

*)(k

N


Yes and no

If goes to 0 no quicker that:

With this speed

3

3,

log

loglogc

N

Nc

N

Nd

loglog

log)(


but

Is too quick! These tiers are not connected because degrees

are too low.

NNlNk

log/)()( *


Conjecture

However, has a giant component And degrees => Diameter of g.c. (Chung and Lu 2000), yields u.b.

*kW

)(NN

).(/log NlN


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


Thank You!

On the robustness of power law random graphs

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