Analyzing the Vulnerability of Superpeer Networks Against Attack

B. Mitra (Dept. of CSE, IIT Kharagpur, India), F. Peruani(ZIH, Technical University of Dresden, Germany),

S. Ghose, N. Ganguly(Dept. of CSE, IIT Kharagpur, India)

Junction

Outline

• Problem Definition• Environment Definition• Development of the analytical framework• Stability of Superpeer Networks against Attack

Outline

• Problem Definition• Environment Definition• Development of the analytical framework• Stability of Superpeer Networks against Attack

Problem Definition• P2P network architecture

– All peers act as both clients and servers– No centralized data source– File sharing and other applications like IP telephony,

distributed storage, publish subscribe system etc

NodeNode

Node Node

NodeInternet

Problem Definition• Overlay network

– An overlay network is built on top of physical network – Nodes are connected by virtual or logical links– Underlying physical network becomes unimportant – Interested in the complex graph structure of overlay

Problem Definition• Dynamicity of overlay networks

– Peers in the p2p system leave network randomly without any central coordination

– Important peers are targeted for attack• DoS attack drown important nodes in fastidious

computation– Fail to provide services to other peers

• Importance of a node is defined by centrality measures– Like degree centrality, betweenness centraltiy etc

• Makes overlay structures highly dynamic in nature • Frequently it partitions the network into smaller

fragments• Communication between peers become impossible

Problem Definition• Investigating stability of the networks against the churn and

attackNetwork Topology + Attack = How (long) stable

• Developing an analytical framework• Examining the impact of different structural parameters upon

stability– Peer contribution – degree of peers, superpeers – their individual fractions

• Modeling of– Overlay topologies (pure p2p networks, superpeer networks, hybrid networks)– Various kinds of attacks

• Defining stability metric• Validation through simulation

Outline

• Problem Definition• Environment Definition

– Modeling superpeer network– Different kind of attack models– Stability metric

• Development of the analytical framework• Stability of Superpeer Networks against Attack

Environment Definition

• Modeling superpeer networks– Simple model : strict bimodal structure

• A large fraction (r) of peer nodes with small degree kl

• Few superpeer nodes (1-r) with high degree km

if k = kl, km

otherwise

pkl = r and pkm = 1-r

,0,0

k

k

pp

Environment Definition• Different kinds of attack models

– Deterministic attack• Nodes having high degrees are progressively removed• qk : the probability that a node of degree k survives after attack• qk = 0, when k > kmax

0 < qk < 1, when k = kmax

qk = 1, when k < kmax

– Degree dependent attack• Nodes having higher degrees are more likely to be removed• Probability of removal of a node having degree k is proportional to kr where r

> 0 is a real number• With proper normalization , C is a normalizing constant• The fraction of nodes having degree k which survives after this kind of attack is

Ckfr

k

Ckfqr

kk 11

Environment Definition• Stability metric

– Percolation threshold : • disintegrates the network into large number of small,

disconnected components by removing certain fraction of nodes (fc)

• Higher values indicate greater stability against attack

Stability Matric

• Percolation Threshold

Nodes in the network are connected and form a single component

Initially all the nodes in the network are connected Forms a single giant componentSize of the giant component is the order of the network sizeGiant component carries the structural properties of the entire network

Stability Matric

• Percolation Threshold

f fraction of nodes removed

Initial single connected component Giant

component still exists

Stability Metric

• Percolation Threshold

Initial single connected component

f fraction of nodes removed

Giant component still exists

fc fraction of nodes removed

The entire graph breaks into smaller fragments Therefore fc =1-qc becomes the percolation

threshold

Percolation Threshold• Remove a fraction of nodes ft from the network in step t and

check whether reach the percolation point•

– s : size of the components formed– ns : number of componets of size s – CSt(s) : the normalized component size distribution at step t

s sst snsnsCS /)(

Initial : only single giant component of size 500

Intermediate:Bimodal character (a large component along with a set of small components)

Percolation point(tn)percolation threshold (ftn)monotonically decreasing function

Outline

• Problem Definition• Environment Definition• Development of the analytical framework

– Generating function• Stability of Superpeer Networks against Attack

Development of the analytical framework• Generating Function:

– Formal power series whose coefficients encode information

Here encode information about a sequence– Used to understand different properties of the graph– generates probability distribution of the

vertex degrees.– Average degree

0

33

2210 .........)(

k

kk xaxaxaxaaxP

,.....),,( 210 aaa

0

0 )(k

kk xpxG

)1('0Gkz

Vertex

Edge

Degree = 5

Development of the analytical framework– specifies the probability of a node having degree k

to be present in the network after (1-qk) fraction of nodes removed.

– becomes the corresponding generating

function.

– Distribution of the outgoing edges of first neighbor of a randomly chosen node

kk qp .

0

0 )(k

kkk xqpxF

(1-qk) fraction of nodes removed

zxF

kp

xqkpxF

kk

k

kkk )()( 0

1

1

Random node

First neighbor

Development of the analytical framework

– H1(x) generates the distribution of the size of the components that are reached through random edge

– H1(x) satisfies the following condition

F1(x) : the probability of finding a node following a random edge=> 1 - F1(x) : the probability of following a randomly chosen edge that leads to a zero size component.

The rest condition reached through random edge, which satisfies a Self-consistency condition.

Development of the analytical framework

– generates distribution for the component size to which a randomly selected node belongs to

– Average size of the components

– Average component size becomes infinity when– theoretically ‘infinite’ size graph reduces to the ‘finite’ size

components

)(0 xH

)1(1

)1()1()1()1(1

1000

F

FFFH

0)1(1 1 F

Development of the analytical framework– Average component size becomes infinity when– With the help of generating function, we derive the

following critical condition for the stability of giant component

– The critical condition is applicable• For any kind of topology (modeled by pk) • Undergoing any kind of dynamics (modeled by 1-qk)

0)1(1 1 F

0

0)1(k

kkk qkqkp

Degree distribution Peer dynamics

Outline

• Problem Definition• Environment Definition• Development of the analytical framework• Stability of Superpeer Networks against Attack

– Simulation result

Stability of Superpeer Networks against Attack

• Theoretically derived results & simulation– Deterministic attack– Degree dependent attack

• Network Generation– Represented by a simple undirected graph– Bimodal degree distribution– Graphs with 5000 nodes

An undirected arc is an edge that has no arrow. Both ends of an undirected arc are equivalent--there is no head or tail. Undirected

graphDirected

graph

Deterministic Attack• Two cases may arise in the deterministic attack

– 1. The removal of a fraction of superpeers is sufficient to disintegrate the network

– 2. The removal of all the superpeers is not sufficient to disintegrate the network. Therefore we need to remove some of the peer nodes along with the superpeers.

Recall : when , the critical condition for the stability

ml kkk

kk kqpkk,

)1(

1)1('1 F

l mkk kk

kkkk kqpkkqpkk )1()1(

Deterministic Attack• Case 1:

– fsp : the critical fraction of superpeer nodes, removal of which disintegrates the giant component

– qk = 1 for k = kl

qk = 1 – fsp for k = km

• Case 2:– fp : fraction of peer to be removed along with all the superpeers to

breack down the betwork– qk = 1 - fp for k = kl

qk = 0 for k = km sptar

kmm

kllsp

kk kkspkk

kk kkkkkk

frf

pkkpkkk

f

kfpkkpkk

kqpkkqpkk

m

l

l m

l m

)1(

)1()1(

1

)1()1()1(

)1()1(

)1(

)1(1

)1()1(

)1()1(

rrff

pkkk

f

kfpkk

kqpkkqpkk

ptar

kllp

kkpk

kk kkkkkk

l

l

l m

Deterministic Attack

• Parameter– Average degree <k> = 10– Superpeer degree km = 50

– Increase the peer degree kl gradually (the peer fraction changes accordingly) and observe the change in the percolation threshold ftar

Peer degree kl=1,2,3, the removal of only a fraction of superpeers causes breakdown of the network

The increase of peer degree from 1 to 2 and 3 further reduces the fraction of superpeers in the network It is not large enough to form effective connections within themselves

A fraction of peers is reqired to be removed.

The high degree peers connect among themselves and they are not entirely dependent on superpeers for connectivity.

The steep increase of stability with peer degree > 5

Deterministic Attack

• Peer contribution: – controls the total bandwidth contributed by the peers which

determines the amount of influence superpeer nodes exerts on the network

– two factors: peer degree & fraction of peers in the network

mll

r krrkkwherekrKP

C)1(,

Peer degree kl=1 can be disintegrated without attacking peers at all

Prc<0.2 does not have any impact upon the stability of the network no mater what peer degree is.

For kl=5, at Prc=0.3, a fraction of peers is required to be removed to disintegrate the networks.

The impact of high degree peers upon the stability of the network becomes more eminent as peer contribution Prc > 0.5.

For kl=1, 3, ftar gradually reduces, since increase in peer contribution decreases superpeer contribution, it decreases stability of these networks also.

For kl =5, peers are strongly connected among themselves, hence stability is more dependent on peer contribution.

Degree Dependent Attack• Probability of a node of degree k is directly proportional to kγ where γ > 0

is a real number.– Probability of survival of a node having degree k after a degree

dependent attack is – Critical condition for the stability of the giant component :

Ckfq kk

11

)2)(()1()1()1(

1,:

)1(

11

,

kkkkkCkkrkrk

kkkk

rpkkkk

rpapply

kqpkk

mlmmmll

lm

lk

lm

mk

kkkkk

ml

ml

Degree Dependent Attack• Probability of removal of a node is directly proportional to its

degree, hence • Minimum value • This yields an inequality

– The solution set of the above inequality can be• either bounded • either bounded

mkC

Ckfk

)2)(()1()1()1( 11 kkkkkkkkrkrk mlmmmmll

)0( bdcc

)0( c

Degree Dependent Attack

– Obtaining minimum value of C, each γc results in the corresponding normalizing constant

percolation threshold becomes

c

c

c

ccc

Ckr

Ckrfrrff ml

sppc

)1()1(

kkkkkkkrkrkC

mlm

mmllcc

c 2)()1()1()1( 1

)2)(()1()1()1( 11 kkkkkCkkrkrk mlmmmll

Degree Dependent Attack• The breakdown of the network can be due to one of the three

situations and reasons noted below:– 1: The removal of all the superpeers along with a fraction of peers.

• Networks having a bounded solution set Src where exhibit this kind of behavior at the maximum value of the solution .

• Here the fraction of superpeers removed becomes = 1• and fraction of peers removed

– 2: The removal of only a fraction of superpeers.• Some networks have an open solution set Src where • At converges to 0 and converges to some x where

0<x<1.– 3: The removal of some fraction of both superpeers and peers.

• Intermediate critical exponents signifies the fractional removal of both peers and superpeers.

bdcc 0

bdcc

bdc

spf

bdc

bdcbd

c

Ckf l

p

c0cpc f , c

spf

cSc

Case 2 of deterministic attack

Degree Dependent Attack• Two superpeer degrees km=25, 50

fixed average degree <k> = 10• Behavior of peer contribution Prc due to the change in peer

degree kl

In order to keep the average degree and peer constant, the network with higher superpeer degree results higher fraction of peer which increases the peer contribution.

Degree Depend Attack• Behavior of boundary critical exponent due to the change in

peer degree

Γcbd remains unbounded :

peer degree kl < 3 with superpeer degree km = 50

Γcbd remains ubounded :

peer degree kl < 4 with superpeer degree km = 25

Removal of only a fraction of superpeers disintegrate these networks: the low peer degree -> low peer contribution -> high superpeer contribution

Case 1 of deterministic attack

Degree Depend Attack• Fraction of peers and superpeers required to be removed to

breakdown the network and its impact upon percolation threshold fc.

The gradual increase in peer degree increases the peer contribution -> the higher peer contribution ensures the necessity to remove a fraction of them to breakdown the network.

Peer contribution has profound impact on the stability of the network specially with the networks having high peer degree kl.

Degree Depend Attack• Case study 1: The removal of all the superpeers along with a

fraction of peers.

(a) Behavior of γcbd with respect to

the change in superpeer fraction (b) Fraction of peers and superpeers required to be removed to breakdown the network and its impact upon percolation threshold fc

Peer degrees kl =3,4; Average degree <k>=5

Kl=3, spth=1.9

Kl=4, spth=4.1

1. Impact upon the fraction of peers removed:*The increase of superpeer fraction slowly increases γc

bd *Which in turn gradually decreases the fraction of peers removed fp γcbd

*higher degree peers -> higher values of fp γcbd to removed

2. Impact upon the fraction of peers removed:*recall : two factors

Depending upon the weightage of influence, fp γcbd either decreases or increases slowly when the fraction of superpeers is lass than spth.

)1()(

)1()1(

)1(

2121

)2(11

21

)2(1)2()2(1

rrf

rrfrfrf

rff

bdc

bdc

bdc

bdc

bdc

bdc

p

ppc

pc

Degree Depend Attack• Case study 2: The removal of only a fraction of superpeers.

– Superpeers degree km = 25, average degree <k> = 5, peer degree kl =2– Initially remove a fraction of superpeers fsp

rc and then start removing peers gradually

γ

*The fraction of peers removed gradually decreases with the increase of critical exponent γc which in turn decreases the value of fc

rc.

*As , with where(0<x<1) and eventually reach some steady value.

c 0cpf xf c

sp

*removal of only a fraction of superpeers is sufficient to any network with peer degree kl =1, 2, irrespective of superpeer degree and its fraction since the solution set Src becomes unbounded.

Degree Depend Attack• Case study 3: The removal of some fraction of both superpeers

and peers.– Superpeer degree km=5, average degree <k>=5, peer degree kl=3

Removal of any combination of (fp

rc, fsprc) where 0<rc<rc

bd, results in the breakdown of the network.

γcbd = 1.171

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