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
Node
Node
Node
Internet
Node
Node
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
attack
Network 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
 pk  0,

 pk  0,
if k = kl, km
otherwise
pkl = r and pkm = 1-r
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 r
k
• With proper normalization f k 
, C is a normalizing constant
C
• The fraction of nodes having degreerk which survives after this
kind of attack is qk  1  f k  1  k
C
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
Initially all the nodes in the
network are connected
Forms a single giant component
Size of the giant component is the
order of the network size
Nodes in the network
are connected and
form a single
component
Giant 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
f fraction of
nodes
removed
Initial single
connected
component
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
• CSt (s)  sns / s sns
– s : size of the components formed
– ns : number of componets of size s
– CSt(s) : the normalized component size distribution at step t
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

a x
Here (a0 , a1 , a2 ,.....) encode information about a sequence
– Used to understand different properties of the graph
– G ( x)   p x generates probability distribution of the
vertex degrees.
Edge
– Average degree z   k   G0 ' (1)
P ( x )  a0  a1 x  a2 x
2
 a3 x  ......... 
3

k
0
k 0
k
Vertex
Degree = 5
k
k 0
k
Development of the analytical framework
– pk .qk specifies the probability of a node having degree k
to be present in the network after (1-qk) fraction of nodes
removed.

– F0 ( x)   pk qk x k becomes the corresponding generating
k 0
function
.
(1-qk)
fraction of nodes
removed
– Distribution of the outgoing edges of first neighbor of a
randomly chosen node
 kp q x
F ( x) 
 kp
k
k
k
1
k
k
k 1

F0 ( x)

z
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
– H 0 ( x) generates distribution for the component size to
which a randomly selected node belongs to
– Average size of the components

F

0 (1) F1 (1)
H 0 (1)  F0 (1) 

1  F1 (1)
– Average component size becomes infinity when 1  F1(1)  0
– theoretically ‘infinite’ size graph reduces to the ‘finite’ size
components
Development of the analytical framework
– Average component size becomes infinity when 1  F1(1)  0
– With the help of generating function, we derive the
following critical condition for the stability of giant
component

 kpk (kqk  qk  1)  0
k 0
Degree distribution
Peer dynamics
– The critical condition is applicable
• For any kind of topology (modeled by pk)
• Undergoing any kind of dynamics (modeled by 1-qk)
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
Undirected
Directed
graph
An undirected arc is an edge that has no
arrow. Both ends of an undirected arc are
equivalent--there is no head or tail.
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 F1 ' (1)  1 , the critical condition for the stability
 k (k  1) p
k
qk  k
 k (k 1) p q
k

k  kl , k m
k  kl
k
 k (k 1) p q
k km
k
k
 k
Deterministic Attack
• Case 1:  k (k  1) pk qk   k (k  1) pk qk
 k
kl
k km
– fsp : thek critical
fraction of superpeer
nodes, removal of which
disintegrates
k (kthe
 1giant
) pk (1component
 fp)  k
– qk = 1 k  k l
for k = kl
qk = 1 – fsp for k = km k

 f p  1
kl (kl  1) pkl
kf(tar
 (1 r ) k ( k  1) p q
k 1rf
) pp q


k  kl
k
k
k km
k
k
 k
• Case 2: k (k  1) p   k (k  1) p (1  f ) 
k
k
sp
k
k  kl
k  removed
km
– fp : fraction
of peer to be
along with all the superpeers to
breack down the betwork
k  kl ( kl  1) pkl
1 k = k
– qk = 1
- fp f sp for
kl m ( k m  1) pk m
qk = 0
for k = km
 f tar  (1  r ) f sp
Deterministic Attack
• Parameter
degree kl=1,2,3,
–Peer
Average
degree <k> = 10
the removal of only a
–fraction
Superpeer
degree km = 50
of superpeers
fraction of peers is reqired
breakdown
of
–causes
Increase
the peer
degree kl gradually A(the
peer fraction
to
be
removed.
the network
changes accordingly) and observe the change in the
The high degree peers
percolation threshold ftar
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
connect among themselves
and they are not entirely
dependent on superpeers for
connectivity.
The steep increase of stability
with peer degree > 5
Deterministic Attack
•
For kl=1, 3, ftar gradually reduces,
since increase
in peer contribution
Peer
contribution:
decreases superpeer contribution,
–it decreases
controlsstability
the total
bandwidth contributed by the peers which
of these
For kl=5, at Prc=0.3, a
networks
also.
determines the amount
influence
superpeer nodes exerts
fraction ofof
peers
is
on the network required to be removed to
disintegrate the networks. For kl =5, peers are strongly
Prc<0.2 does not have
–
two
factors:
peer
degree & fraction ofconnected
peers inamong
the network
any impact upon the
themselves,
stability of the
network no mater
what peer degree is.
hence stability is more
rK l
PrC 
, where k  rkl  (1dependent
 r )km on peer
k
contribution.
Peer degree kl=1 can be disintegrated
without attacking peers at all
The impact of high degree
peers upon the stability of
the network becomes
more eminent as peer
contribution Prc > 0.5.
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
k
dependent attack is qk  1  f k  1 
C
– Critical condition for the stability of the giant component :
 k (k  1) p q
k
k  kl , k m
k
apply : pkl  r 
 rk l
 1
 k
km  k
k m  kl
, pk m  1  r 
k  kl
k m  kl
(kl  1)  (1  r )k m 1 (k m  1)  C ( k (k m  kl )  k m  2 k )
Degree Dependent Attack
• Probability of removal of a node is directly proportional to its
degree, hence C  km 
k
f

• Minimum value k
C
• This yields an inequality
 1
 1

rkl (kl 1)  (1  r )km (km 1)  km (k (km  kl )  km  2k )
– The solution set of the above inequality can be
bd
(
0




• either bounded
c
c )
• either bounded (0   c )
Degree Dependent Attack
rkl
 1
(kl 1)  (1  r )km
 1
(km 1)  C(k (km  kl )  km  2k )
– Obtaining minimum value of C, each γc results in the
corresponding normalizing constant
rk l c (kl  1)  (1  r )k m c 1 (k m  1)
C c 
k ( k m  kl )  k m  2 k
percolation threshold becomes
f c  rf p c  (1  r ) f sp c
kl c
km c
r
 (1  r )
C c
C c
Degree Dependent Attack
Case to
2 ofone
deterministic
attack
• The breakdown of the network can be due
of the three
situations and reasons noted below:
– 1: The removal of all the superpeers along with a fraction of peers.
bd
• Networks having a bounded solution set Src where 0   c   c
exhibit this kind of behavior at the maximum value of the
bd
solution  c   c .
 cb d
• Here the fraction of superpeers removed
becomes f sp = 1

k
• and fraction of peers removed f p  l
C
– 2: The removal of only a fraction of superpeers.
• Some networks have an open solution set Src where 0   c  


• At  c  , f p c converges to 0 and f spc converges to some x where
0<x<1.
– 3: The removal of some fraction of both superpeers and peers.
• Intermediate critical exponents  c  S c signifies the fractional
removal of both peers and superpeers.
bd
c
bd
c
bd
c
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 ubounded :
peer degree kl < 4 with
superpeer degree km = 25
Case 1 of deterministic attack
Γcbd remains unbounded :
peer degree kl < 3 with
superpeer degree km = 50
Removal of only a fraction of superpeers disintegrate these networks:
the low peer degree -> low peer contribution -> high superpeer contribution
Degree Depend Attack
• Fraction of peers and superpeers required to be removed to
breakdown the network and its impact upon percolation
threshold fc.
Peer contribution has profound
impact on the stability of the network
specially with the networks having
high peer degree kl.
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.
Degree Depend Attack
• Case study 1: The removal of all the superpeers along with a
fraction of peers.
Peer degrees kl =3,4; Average degree <k>=5
Kl=4,
spth=4.1
Kl=3,
th=1.9
2. Impactspupon
the fraction of peers removed:
*recall : two factors
f c1(c2 )   1( 2 ) f p1c( 2 )  (1  r1( 2 ) )
bd
bd
f c c  r1 f p1c  r2 f p2c  (1  r1 )  (1  r2 )
bd
bd

bd
 ( rf p c )   (1  r )
bd
Depending upon the weightage
of influence,
bd with respect to
(b) Fraction of peers and superpeers
(a)
Behavior
of
γ
cor increases slowly
fp γcbd either
decreases
Impact
the fraction
required to be removed to breakdown
the 1.
change
in upon
superpeer
fractionof peers removed:
when the *The
fraction
of
superpeers
is
lass
than
bd
increase
of
superpeer
fraction
slowly
increases
γ
c
the
network
and
its impact upon
spth.
γcbd
*Which in turn gradually decreases the percolation
fraction of peers
removed
threshold
fc fp
*higher degree peers -> higher values of fp γcbd to removed
Degree
Depend
Attack
*The fraction of peers removed gradually
decreases
the 2:
increase
of critical of only a fraction of superpeers.
• Case with
study
The removal
exponent γc which in turn decreases the
– Superpeers degree km = 25, average degree <k> = 5, peer degree kl =2
value of fcrc.
– Initially remove a fraction of superpeers fsprc and then start removing


f


peers
gradually
*As c
, p  0 with f sp  x
c
c
where(0<x<1) and eventually reach some
steady value.
*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.
γ
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 (fprc, fsprc) where 0<rc<rcbd,
results in the breakdown of
the network.
γcbd = 1.171
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