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Farnoush Banaei-Kashani and Cyrus Shahabi Criticality-based Analysis and Design of Unstructured P2P Networks as “ Complex Systems ” Mohammad Al-Rifai

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2 December 2003Mohammad Al Rifai Outline Introduction Motivation Flooding search Probabilistic Flooding Percolation Theory TTL selection policy Summary Questions

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2 December 2003Mohammad Al Rifai Introduction Motivation - improving scalability of flooding search applied in unstructured P2P networks (Gnutella) Proposed approach - recognizing P2P networks as Complex Systems, and exploiting the accurate statistical models used to characterize them for formal analysis and efficient design of P2P networks.

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2 December 2003Mohammad Al Rifai Introduction Flooding search Each query is flooded through the entire network

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2 December 2003Mohammad Al Rifai Introduction Flooding search Each query is flooded through the entire network Algorithm: – a node initiates a query, sets TTL value, sends the query to all of its neighbors.

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2 December 2003Mohammad Al Rifai Introduction Flooding search Each query is flooded through the entire network Algorithm: – a node initiates a query, sets TTL value, sends the query to all of its neighbors. – each receiver of the query decrements TTL by one, forwards t he query to its neighbors in turn, and so on

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2 December 2003Mohammad Al Rifai Introduction Flooding search Each query is flooded through the entire network Algorithm: – a node initiates a query, sets TTL value, sends the query to all of its neighbors. – each receiver of the query decrements TTL by one, forwards t he query to its neighbors in turn, and so on – the flooding continues till the object is found.

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2 December 2003Mohammad Al Rifai Introduction Flooding search Each query is flooded through the entire network Algorithm: – a node initiates a query, sets TTL value, sends the query to all of its neighbors. – each receiver of the query decrements TTL by one, forwards t he query to its neighbors in turn, and so on – the flooding continues till the object is found.

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2 December 2003Mohammad Al Rifai Introduction Flooding search Problems : – extra overhead through duplicated queries - initial TTL is set regardless of the size of the network

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2 December 2003Mohammad Al Rifai Introduction Flooding search Problems : – extra overhead through duplicated queries - initial TTL is set regardless of the size of the network does not scale

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2 December 2003Mohammad Al Rifai Introduction Flooding search Problems : – extra overhead through duplicated queries - initial TTL is set regardless of the size of the network does not scale Proposed solutions: 1- Probabilistic flooding search 2- TTL self selection policy

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2 December 2003Mohammad Al Rifai I- Probabilistic Flooding Each node forwards the query to its neighbors with probability p, and drops the query with probability (1 – p ). The normal flooding search is an extreme case of probabilistic flooding with p =1.

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2 December 2003Mohammad Al Rifai Each node forwards the query to its neighbors with probability p, and drops the query with probability (1 – p ). The normal flooding search is an extreme case of probabilistic flooding with p =1. I- Probabilistic Flooding By decreasing the value of p, the probabilistic flooding cuts some paths (not only redundant ones).

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2 December 2003Mohammad Al Rifai Each node forwards the query to its neighbors with probability p, and drops the query with probability (1 – p ). The normal flooding search is an extreme case of probabilistic flooding with p =1. I- Probabilistic Flooding By decreasing the value of p, the probabilistic flooding cuts some paths (not only redundant ones).

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2 December 2003Mohammad Al Rifai Each node forwards the query to its neighbors with probability p, and drops the query with probability (1 – p ). The normal flooding search is an extreme case of probabilistic flooding with p =1. decreasing the value of p furthermore towards 0, cuts more and more paths, and turns out law reachability, thus an inefficient search. I- Probabilistic Flooding

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2 December 2003Mohammad Al Rifai Each node forwards the query to its neighbors with probability p, and drops the query with probability (1 – p ). The normal flooding search is an extreme case of probabilistic flooding with p =1. decreasing the value of p furthermore towards 0, cuts more and more paths, and turns out law reachability, thus an inefficient search. I- Probabilistic Flooding

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2 December 2003Mohammad Al Rifai Goal: all redundant paths must be cut effectively to eliminate duplicated queries and avoid the overhead cost, while full reachability must be preserved. How? p must be tuned to an optimal (critical) operating point p c. to achieve that, the system must be formally modeled. I- Probabilistic Flooding

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2 December 2003Mohammad Al Rifai Formalizing and modeling the P2P networks unstructured P2P networks are large-scale, dynamic, and self-configure systems, which are the main characteristics of Complex Systems. Hence, P2P networks can be recognized as Complex Systems, and theoretical and statistical models applied on Complex Systems can be exploited with P2P networks. Percolation Theory is one of the most important theories applied on Complex Systems that can help to find the critical value p c. I- Probabilistic Flooding

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2 December 2003Mohammad Al Rifai Given a 2D lattice of some sites (dots) and bonds (lines) connecting neighboring sites as shown I- Probabilistic Flooding – Percolation Theory

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2 December 2003Mohammad Al Rifai Given a 2D lattice of some sites (dots) and bonds (lines) connecting neighboring sites as shown I- Probabilistic Flooding – Percolation Theory (in terms of P2P networks, sites are nodes and bonds are links between them)

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2 December 2003Mohammad Al Rifai Given a 2D lattice of some sites (dots) and bonds (lines) connecting neighboring sites as shown Assuming that each bond can be open with probability p, or closed with probability (1 – p ). depending on p, some clusters (sites connected by open bonds) starts to appear. I- Probabilistic Flooding – Percolation Theory (in terms of P2P networks, sites are nodes and bonds are links between them)

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2 December 2003Mohammad Al Rifai Given a 2D lattice of some sites (dots) and bonds (lines) connecting neighboring sites as shown. I- Probabilistic Flooding – Percolation Theory (in terms of P2P networks, sites are nodes and bonds are links between them) The larger the value of p, the larger the size of clusters is. Assuming that each bond can be open with probability p, or closed with probability (1 – p ).

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2 December 2003Mohammad Al Rifai Given a 2D lattice of some sites (dots) and bonds (lines) connecting neighboring sites as shown. Giant cluster I- Probabilistic Flooding – Percolation Theory Due to Percolation Theory: above a threshold probability p c, a giant cluster spanning the whole lattice starts to appear. (in terms of P2P networks, sites are nodes and bonds are links between them) Assuming that each bond can be open with probability p, or closed with probability (1 – p ).

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2 December 2003Mohammad Al Rifai I- Probabilistic Flooding – Percolation Theory - Unstructured P2P networks are random graphs of size N ∞, with connectivity distribution P(k). - nodes and links between them may be thought of as sites and bonds respectively in terms of Percolation Theory.

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2 December 2003Mohammad Al Rifai - Unstructured P2P networks are random graphs of size N ∞, with connectivity distribution P(k). - nodes and links between them may be thought of as sites and bonds respectively in terms of Percolation Theory. I- Probabilistic Flooding – Percolation Theory Percolation Theory verifies that once probabilistic flooding is applied, above a threshold p c the giant cluster spans the whole network with minimum connectivity.

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2 December 2003Mohammad Al Rifai - Unstructured P2P networks are random graphs of size N ∞, with connectivity distribution P(k). - nodes and links between them may be thought of as sites and bonds respectively in terms of Percolation Theory. I- Probabilistic Flooding – Percolation Theory Percolation Theory verifies that once probabilistic flooding is applied, above a threshold p c the giant cluster spans the whole network with minimum connectivity. How could p c be computed ?

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2 December 2003Mohammad Al Rifai Analysis: The following assumption has been made: “ percolation threshold takes place when each node i connected to a node j in the spanning cluster, is also connected to at least one other node” I- Probabilistic Flooding i j

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2 December 2003Mohammad Al Rifai Analysis: this criterion can be written as follows: I- Probabilistic Flooding

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2 December 2003Mohammad Al Rifai Analysis: this criterion can be written as follows: I- Probabilistic Flooding k i : the degree of node i Expected value of k i

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2 December 2003Mohammad Al Rifai Analysis: this criterion can be written as follows: I- Probabilistic Flooding

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2 December 2003Mohammad Al Rifai Analysis: this criterion can be written as follows: I- Probabilistic Flooding Conditional probability of a node i having k i degree, given that it is connected to j

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2 December 2003Mohammad Al Rifai Analysis: this criterion can be written as follows: I- Probabilistic Flooding But due to Bayes rule,

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2 December 2003Mohammad Al Rifai Analysis: this criterion can be written as follows: I- Probabilistic Flooding where, But due to Bayes rule,

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2 December 2003Mohammad Al Rifai Analysis: this criterion can be written as follows: I- Probabilistic Flooding where, But due to Bayes rule, N : total number of nodes

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2 December 2003Mohammad Al Rifai Analysis: this criterion can be written as follows: I- Probabilistic Flooding where, But due to Bayes rule, Thus, at criticality:

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding Given the connectivity distribution of the network using probability flooding results in the effective connectivity distribution as follows:

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding (2) Given the connectivity distribution of the network using probability flooding results in the effective connectivity distribution as follows:

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding (2) Given the connectivity distribution of the network using probability flooding results in the effective connectivity distribution as follows:

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding (2) Given the connectivity distribution of the network using probability flooding results in the effective connectivity distribution as follows:

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding …(3)

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding …(4)

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding …(5) from (3) and (4) the ratio of the second to first moment is: is the ratio of the second to first moment of the actual graph.

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding …(5) from (3) and (4) the ratio of the second to first moment is: is the ratio of the second to first moment of the actual graph. Gnutella network follows power-law connectivity distribution C is a normalization factor Power-law exponent (6) Exponential cutoff factor required for representing real-world networks

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding the ratio α is computed from equation (6), (7) Hence, p c is a factor of cutoff-index v and τ

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding the ratio α is computed from equation (6), (7) Hence, p c is a factor of cutoff-index v and τ Li τ ( x ) : τ-th Ploylogarithm of x

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2 December 2003Mohammad Al Rifai Analysis: I- Probabilistic Flooding the ratio α is computed from equation (6), Hence, p c is a factor of cutoff-index v and τ (7) For Gnutella, the power-law exponent is estimated as low as 1.4 and as high as 2.3 in different times, and v is in the range of 100 to 1000.

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2 December 2003Mohammad Al Rifai 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 Power-law Exponent τ = 2.3 Power-law Exponent τ = 1.4 100 200 300 400 500 600 700 800 900 1000 pcpc Cut-off index v Critical probability can be less than 0.01 Hence, flooding cost is reduced by more than 99% without losing reachability I- Probabilistic Flooding v

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2 December 2003Mohammad Al Rifai 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 Power-law Exponent τ = 2.3 Power-law Exponent τ = 1.4 100 200 300 400 500 600 700 800 900 1000 pcpc Cut-off index v Critical probability can be less than 0.01 Hence, flooding cost is reduced by more than 99% without losing reachability I- Probabilistic Flooding v i.e. scalable search

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2 December 2003Mohammad Al Rifai II- TTL selection policy Problem: in normal flooding search TTL is restricted to the initial value set by the search originator regardless of the actual size of the network. i.e. not scalable

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2 December 2003Mohammad Al Rifai II- TTL selection policy Solution: selection policy is based on the typical length λ of the shortest path between two randomly chosen nodes on any random graph, which is provided by Newman as follows: Problem: in normal flooding search TTL is restricted to the initial value set by the search originator regardless of the actual size of the network. i.e. not scalable

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2 December 2003Mohammad Al Rifai Problem: in normal flooding search TTL is restricted to the initial value set by the search originator regardless of the actual size of the network. i.e. not scalable Solution: selection policy is based on the typical length λ of the shortest path between two randomly chosen nodes on any random graph, which is provided by Newman as follows: II- TTL selection policy N Average number of active nodes is not heavily variant in short time-intervals z1 number of neighbors which are one hop away z2 number of neighbors which are two hops away

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2 December 2003Mohammad Al Rifai II- TTL selection policy Solution: Each node estimates z 1 and z 2 periodically with local ping packets, and sets TTL of its query to the estimated typical length of path between two nodes λ. Problem: in normal flooding search TTL is restricted to the initial value set by the search originator regardless of the actual size of the network. i.e. not scalable

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2 December 2003Mohammad Al Rifai II- TTL selection policy Solution: Each node estimates z 1 and z 2 periodically with local ping packets, and sets TTL of its query to the estimated typical length of path between two nodes λ. TTL is adapted based on information collected locally, hence: scalable TTL selection Problem: in normal flooding search TTL is restricted to the initial value set by the search originator regardless of the actual size of the network. i.e. not scalable

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2 December 2003Mohammad Al Rifai Summary Flooding search scalability is improved by employing probabilistic flooding search and adopting new TTL selection policy. Percolation Theory is used to formally analyze P2P networks at critical operation points. Conclusion: theoretical and statistical models applied on Complex Systems can be exploited effectively to formally model and analyzes unstructured P2P networks.

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2 December 2003Mohammad Al Rifai Questions ?

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