1. 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

2. Outline Problem Definition Environment Definition Development of the analytical framework Stability of Superpeer Networks against Attack

3. Outline Problem Definition Environment Definition Development of the analytical framework Stability of Superpeer Networks against Attack

4. 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 Internet

5. 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

6. Problem Definition Dynamicity of overlay networks – Peers in the p2p system leave network randomly without any central coordination – Important peers are targeted for attack 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

7. 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

8. 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

9. Environment Definition Modeling superpeer networks – Simple model : strict bimodal structure A large fraction (r) of peer nodes with small degree k l Few superpeer nodes (1-r) with high degree k m if k = k l, k m otherwise p k l = r and p km = 1-r

10. Environment Definition Different kinds of attack models – Deterministic attack Nodes having high degrees are progressively removed q k : the probability that a node of degree k survives after attack q k = 0, when k > k max 0 < q k < 1, when k = k max q k = 1, when k < k max – 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 k r 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

11. Environment Definition Stability metric – Percolation threshold : disintegrates the network into large number of small, disconnected components by removing certain fraction of nodes (f c ) Higher values indicate greater stability against attack

12. 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 component Size of the giant component is the order of the network size Giant component carries the structural properties of the entire network

13. Stability Matric Percolation Threshold f fraction of nodes removed Initial single connected component Giant component still exists

14. Stability Metric Percolation Threshold Initial single connected component f fraction of nodes removed Giant component still exists f c fraction of nodes removed The entire graph breaks into smaller fragments Therefore f c =1-q c becomes the percolation threshold

15. Percolation Threshold Remove a fraction of nodes f t from the network in step t and check whether reach the percolation point – s : size of the components formed – n s : number of componets of size s – CS t (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

16. Outline Problem Definition Environment Definition Development of the analytical framework – Generating function Stability of Superpeer Networks against Attack

17. 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 Vertex Edge Degree = 5

18. Development of the analytical framework – specifies the probability of a node having degree k to be present in the network after (1-q k ) fraction of nodes removed. – becomes the corresponding generating function. – Distribution of the outgoing edges of first neighbor of a randomly chosen node ( 1-q k ) fraction of nodes removed Random node First neighbor

19. Development of the analytical framework – H 1 (x) generates the distribution of the size of the components that are reached through random edge – H 1 (x) satisfies the following condition F 1 (x) : the probability of finding a node following a random edge => 1 - F 1 (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.

20. 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

21. 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) Degree distribution Peer dynamics

22. Outline Problem Definition Environment Definition Development of the analytical framework Stability of Superpeer Networks against Attack – Simulation result

23. 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 graph Directed graph

24. 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

25. Deterministic Attack Case 1: – f sp : the critical fraction of superpeer nodes, removal of which disintegrates the giant component – q k = 1 for k = k l q k = 1 – f sp for k = k m Case 2: – f p : fraction of peer to be removed along with all the superpeers to breack down the betwork – q k = 1 - f p for k = k l q k = 0 for k = k m

26. Deterministic Attack Parameter – Average degree = 10 – Superpeer degree k m = 50 – Increase the peer degree k l gradually (the peer fraction changes accordingly) and observe the change in the percolation threshold f tar 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

27. 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 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.

28. Degree Dependent Attack Probability of a node of degree k is directly proportional to k γ 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 :

29. 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

30. Degree Dependent Attack – Obtaining minimum value of C, each γ c results in the corresponding normalizing constant percolation threshold becomes

31. 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 S rc 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 S rc 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. Case 2 of deterministic attack

32. Degree Dependent Attack Two superpeer degrees km=25, 50 fixed average degree = 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.

33. Degree Depend Attack Behavior of boundary critical exponent due to the change in peer degree Γ c bd remains unbounded : peer degree kl < 3 with superpeer degree km = 50 Γ c bd 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

34. 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.

35. Degree Depend Attack Case study 1: The removal of all the superpeers along with a fraction of peers. (a) Behavior of γ c bd 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 =5 Kl=3, sp th =1.9 Kl=4, sp th =4.1 1. Impact upon the fraction of peers removed: *The increase of superpeer fraction slowly increases γ c bd *Which in turn gradually decreases the fraction of peers removed f p γcbd *higher degree peers -> higher values of f p γcbd to removed 2. Impact upon the fraction of peers removed: *recall : two factors Depending upon the weightage of influence, f p γcbd either decreases or increases slowly when the fraction of superpeers is lass than sp th.

36. Degree Depend Attack Case study 2: The removal of only a fraction of superpeers. – Superpeers degree km = 25, average degree = 5, peer degree kl =2 – Initially remove a fraction of superpeers f sp 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 f c rc. *As, with 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.

37. Degree Depend Attack Case study 3: The removal of some fraction of both superpeers and peers. – Superpeer degree km=5, average degree =5, peer degree kl=3 Removal of any combination of (f p rc, f sp rc ) where 0<r c <r c bd, results in the breakdown of the network. γ c bd = 1.171

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