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Hybrid Search Schemes for Unstructured
Peer-to-Peer NetworksRandom Walks in
Peer-to-Peer Networks

• Christos Gkantsidis, Milena Mihail, Amin Saberi
• Presented by Paul Bogdan
• February 28th, 2007

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Hybrid Search Schemes for Unstructured
Peer-to-Peer Networks
Christos Gkantsidis, Milena Mihail, Amin Saberi
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Outline

• Random Graph Models
• Flooding and Normalization
• Random Walks and Replication
• Generalized Search Schemes
• Experimental evaluation

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Motivation

• Flooding small time-to-live (TTL) performs well
  in regular graphs
• Performance metric number of exchanged
  messages/distinct response
• Its performance decreases when TTL increases or
  for irregular networks
• Random Walk performs better than flooding
• scalability, granularity
• Hybrid Generalized search schemes
• Random Walks with lookahead, Random Walks with
  1-step replication

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Contribution

• Random walks (RW) with shallow flooding offer
  good performance (analytic justification)
• R1 In a random graph model with O(n) nodes of
  constant degree and
• O(n1/2) nodes of degree O(n1/2) the expected time
  to discover O(n) is O(n1/2).
• R2 Random Walks with look-ahead 1 or 1-step
  replication perform better
• when there is discrepancy on the degrees of the
  underlying topology.
• Normalized Flooding (NF) solution
• R3 NF achieves comparable performance to
  flooding in regular graphs.
• R4 NF with 1-step replication achieves
  performance comparable to RW
• with 1-step replication.
• R5 Local information of the network (nodes
  degree) offers global benefit.
• Generalized Search Schemes

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Random Graph Models

• Random Regular Graphs Gn,d
• Gn,d represents a graph with n nodes and each
  node is of degree d.
• Gn,d has a sum of degree D nd .
• Random Graphs with super-nodes - Gn,d,a,ß
• Given a and ß constants, Gn,d,a,ß denotes a
  graphs with an1/2 of degree ßn1/2 (i.e. large
  vertices) and the remaining nodes of degree d
  (i.e. small vertices).
• Gn,d,a,ß has a sum of degree D (aßd)n.

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Flooding and Normalization

• Theorem 3.1. Let us consider Gn,d random regular
  graph, flooding scenario from node v with
  time-to-live t, S the number of distinct nodes
  queried by flooding with S V / 2
• Claims
• 
  
  (1)
• 
  
  (2)
• 
  
  (3)

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• 
  
• 
  (1)
• Proof

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• 
  (2)
• Proof

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(No Transcript)
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• 
  
  
  (3)
• Proof

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Flooding and Normalization

• Theorem 3.2. Let Gn,d,a,ß be a random graph with
  supernodes and a flooding scenario from node v of
  degree d with time-to-live t.
• Claim For some t O(log log n), the number of
  distinct responses is O(n).
• Proof
• Consider flooding with t c logd-1(log n)1 and
  vertices visited with TTL t-1.
• Assumption this set (of visited nodes) doesnt
  contain a large degree vertex.
• From d-regular graphs we know that this set
  contains at least (d - 1)t-1 edges.
• The probability that no vertex in G(St-1(v)) is
  bounded by (d/(daß))(d - 1)(t-1)
  (d/(daß))clog n so within the first O(loglog n)
  steps we see a large vertex.

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Flooding and Normalization

• Theorem 3.3. Let Gn,d,a,ß be a random graph
  with supernodes, a normalized
• flooding scenario from node v with TTL
  . Then the number of distinct
• responses is O((d - 1)t-1) and the number of
  messages per response is O(1).
• Proof
• From Theorem 3.1. the number of minigroups seen
  is (d - 1)t-1
• The expected number of small vertices is Q (d
  (d - 1)t-1)/(daß)
• Let Xi, i 1,,N be random variables with P
  Xi1pi and PXi01-pi
• Using the above Chernoff bound the probability
  that less than Q/2 are seen is
• vanishingly small.

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Random Walks and Replication

• Random Walk with Look-Ahead
• a random walk with shallow flooding on each step
  of the walk
• RW with lookahead 1 visits O(n) nodes with
  response O(n(1/2))
• Theorem 4.2. Let Gn,d,a,ß be a random graph with
  supernodes and consider a
• random walk from a node v. Then, in 1-step
  replication scenario, the expected
• number of messages and response time to obtain
  distinct
• responses is

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• Theorem 4.3. Let Gn,d,a,ß be a random graph with
  supernodes and consider
• Normalized flooding from v with TTL t (log
  n)/(2log(d-1)). Then, in 1-step
• replication scenario, the number of distinct
  responses is at least
• and the number of messages is at most
• Proof
• The number of minigroups seen is (d - 1)t 1 and
  using the Chernoff bounds
• there will be
  minigroups corresponding to large vertices.

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Generalized Search Schemes

• Searching procedure
• A node of degree d initiates a search based on a
  budget k
• budget number of messages that are propageted
  in the network
• Among its d neighbors the node picks certain
  quantities k1,k2,,kd such that k1 k2 kd
  k
• For every neighbor i the master node forwards the
  message with budget ki ( for ki 0 the message
  is not transmitted)
• Each neighbor i reduces the budget by 1 unit and
  repeat the process until the budget is greater
  than 0
• Every node that receives the message for the
  second yime from another neighbor forwards the
  message with the corresponding budget
• Random Walks Flooding

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Experimental Evaluation

• Methodology
• Performance Metrics
• Median and Mean number of distinct peers
  discovered (hits)
• Minimum, Maximum, Standard Deviation of the
  number of hits
• Number of messages
• Granularity of number of messages
• Response time
• Topologies
• Random d-Regular Graphs
• Power Law Graphs
• Bimodal topologies
• Clustered topologies

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Normalized Flooding (NF)

• Mean number of unique peers discovered as a
  function of the initial TTL
• NF and Standard Flooding behave similarly in
  Regular Graphs
• NF controls the number of messages and provides
  higher efficiency

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Normalized Flooding (NF)

• The number of unique peers increases
  exponentially with TTL in NF case
• The number of peers increases faster than
  exponentially with TTL in topologies with high
  degrees

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Random Walk with 1-step replication
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Random Walk with LookAhead (RWLA)

• RWLA performance is similar to long RW without
  lookahead (in terms of unique peers discovered)
• RWLA response time is much smaller compared to
  standard RW

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Edge Criticality Searching with weights

• Generalized Searching performs similarly to
  Standard Flooding in regular graphs
• Generalized Searching behaves similarly to
  Standard Flooding in other topologies if
  normalized edge criticality is used.

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Conclusions

• Normalized Flooding (NF) could substitute the
  Standard Flooding in irregular graphs
• RW with 1-step replication performs better than
  RW and NF in irregular graphs
• Open for improvements
• Generalized schemes (analytic investigation)
• Quantifying Directional flooding

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Random Walks in Peer-to-Peer (P2P)
Networks

• Christos Gkantsidis, Milena Mihail, Amin Saberi

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Outline

• Motivation
• Statistical Estimation and Random Walks (RW)
• Searching
• Methodology and Topologies importance
• Construction and Summary

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Motivation

• Random Walks (RW) were proposed for constructing
  searching and topology maintenance protocols in
  P2P networks
• RW improve searching performance as compared to
  flooding (Cao et al., 2002)
• A RW approach to constructing and maintaining
  unstructured topologies provides good
  connectivity properties (i.e. constant degree,
  constant expansion)
• Claim RW approach is a good candidate
• to simulate uniform sampling
• the number of simulation steps required can be as
  low as the number of samples in independent
  uniform sampling
• Searching and Overlay Topology Construction
• RW searching performs better than flooding for
  the same number of messages and for cluster and
  slow dynamic topologies
• Construction of P2P networks by random walks

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Statistical Estimation Random Walks

• Coupon collection and Chernoff bounds
• n - type of coupons each time one is drawn
  (uniformly distributed)
• Tn - time by which we extracted coupons belonging
  to all n types
• Tan - time by which we encountered an distinct
  types, 0 lt a lt 1
• X1,,Xk independent Bernoulli trials, PXi1pi
  and PXi01-pi
• p - probability that a random drawn object has a
  particular property
• the probability that the property is found in
  substantially fewer draws than its frequency in
  the search space and the quality of the estimator
  X/k are bounded by

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Statistical Estimation Random Walks

• Random Walks (RW), Convergence and Cover Time
• G (V,E) undirected graph, V n, and di-
  degree of vertex I
• Aij - adjacency matrix, P - transition matrix
  which satisfies
• f V?0,1 which satisfies
• Convergence rate metric - the rate at which the
  RW approaches the stationary distribution
• Cover time metric - the time by which all nodes
  were visited
• Trajectory sample average - the rate at which the
  value of f averaged over successive vertices of
  the RW trajectory approaches p

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Statistical Estimation Random Walks

• Convergence rate is related to the second
  eigenvalue of P
• 
  (1)
• yt the vertex that the RW visited at time t
• Cover time
• 
  (2)
• Trajectory sample average
• 
  (3)

(1) 11, (2) 12, 13 , (3) 3, 4, 5, 6
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Statistical Estimation Random Walks

• Second Eigenvalue, Expansion and Conductance
• S subset of V, C(S) cutset of V (i.e. edges with
  one point in S and the other one in V\S), vol(S)
  (i.e. the sum of degrees of vertices in S)
• Expansion
• Conductance
• Known bound

11, 14, 15, 16, 17, 18, 19
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Searching

• Performance metrics for Flooding and RW
• average number of distinct copies of an item
  located in the search
• number of messages used by the searching
  algorithm
• RW performs better than flooding if
• multiple search requests for the same item with
  slow-changing topology
• peer clustering ( see 20, 21, 22, 23, 24, 25
  for details)
• Searching analysis
• Methodology
• Flat topologies with Uniformly Distributed
  Content
• Topologies with Peer Clustering
• Re-issuing the Same Query
• Real topologies

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Searching - Methodology

• Performance Metrics
• mean of the number of distinct copies (i.e. Mean)
• discrepancy around the mean (i.e. Std) and the
  failure probability
• Cost
• number of messages or queries performed during
  search
• Peer-to-peer topologies ( 1 million nodes)
• Flat regular expanders, Two tier topologies with
  clustering, Power law graphs, Samples from real
  topologies
• Dynamic topologies
• rewiring
• Content placement
• Content clustering affects the performance of
  searching

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Searching Flat Topologies

• Experiment
• one request in a network of 500K peers
• Mean hits, Minimum of hits and Std are similar
  for Flooding and RW
• the entire distribution of hits is similar for
  Flooding and RW

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Searching -Topologies with Peer Clustering

• Cluster topology consists of
• 5 flat regular graphs of size 40K from each one
  pick randomly 1000 nodes to construct another
  flat regular graph
• Number of hits for RW is more concentrated around
  the mean compared to Flooding

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Searching - Reissuing the Same Query

• Experiment setup repeat 4 times the below
  procedure
• each peer sends a request and waits for response
• between requests 2 of the links are rewired
• each peer initiates a new searching
• RW have better performance than Flooding
• Mean Hits and Failure Probability

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Searching - Reissuing the Same Query

• Performance of successive searches depends
• on the number of topology changes considered
  between consecutive searches
• Performance of Flooding increases as the rate of
  topological changes increases
• RW Performance remains the same for small
  variations

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Searching Real Topologies

• The number of hits for RW is more concentrated
  around the mean than in Flooding
• P2P have good expansion properties

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Construction

• P2P network construction concerns with
• peers arrive and leave the network dynamically
• strong and weak decentralization
• low network overhead per addition or deletion

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Baseline Construction of Expander Graphs

• ABASE (undirected graph) consists of
• n vertices where each one chooses randomly d
  vertices
• total number of edges nd and expected vertex
  degree 2d
• Theorem 4.1. Let G(V,E) a graph constructed by
  ABASE.
• Then, G is an expander with high probability and
  for positive
• constant a lt 1

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Baseline Construction of Expander Graphs with
Constant Overhead in Random Bits

• ABASE construction algorithm
• start a RW at a random vertex on H (constant
  degree expander graph)
• when ABASE needs a random number this is taken
  from the RW on H
• Theorem 4.2. Let G(V,E) a graph constructed by
  ABASE.
• There are positive constants a, 0 lt ß lt 0.5 such
  that any
• subset S of at least ßV and at most 0.5V has
  cutset
• expansion a almost surely.

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Distributed Construction of Expanders with
Constant Overhead on Network Resources

• AH construction
• d daemons , one for each Hamilton cycle
• a new arriving node, it contacts the daemon
  associated with the i-th Hamilton cycle
• it attaches after c number of steps between the
  peer that currently hosts daemon i and one of its
  neighbors in the cycle i

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Distributed Construction of Expanders with
Constant Overhead on Network Resources

• AM construction
• d daemons , one for each Hamilton cycle
• the arrival of a new arriving node consists of
  two X and Y nodes X and Y contact the central
  server to discover the location of the d daemons
• X becomes the neighbor of daemon i and Y the
  neighbor of the initial daemons neighbor

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Summary

• For Searching
• Random Walks (RW) are superior to Flooding
• For Construction
• RW add new peers with constant overhead
• Open Problems
• Strong Decentralized Construction algorithm
• Can we handle better deletions and expansions of
  small sets?
• How the P2P network parameters (e.g. capacities)
  affect the performance of RW?