Recourse Allocation
In P2P Framework
Yoni Peleg
Outline
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Previous and current work on resource
allocation in MAS.
• P2P Framework, Multi-Hop Cellular Networks:
– Overview & Motivation
– Problems in these models
– Resource Allocation in these models
• Solving the difficulties with:
– Cryptographic methods
– Random graph theory
Resource Allocation Overview
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Allowing an agent to use other agents’
resources in order to complete its task.
• Works in resource allocation can be divided
into 2 main areas:
– Economical (Marketplace)
– Matching
Economics Resource Allocation
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Assumes a central marketplace (usually
doesn’t refer to its implementation).
• Sellers and buyers meet in the marketplace in
order to trade resources.
• Solves questions about the price of the
resources:
– Will an equilibrium be reached?
– How long does it take to reach the equilibrium?
– What is the price of the resources in the
equilibrium point?
Economics Resource Allocation (cont.)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• A lot of sellers (easy to find an agent with an
available resource).
• All the available resources are identical.
Market
Place
Matching Resources
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Resources are rare
– Only a few agents have resources to share.
• Resources are not necessarily identical
– For example, using the resource of a closer agent
might be preferable due to communication costs.
• Cooperative / Non-Cooperative Environments
– Agents might not share their resource for free.
• Task: Match a supplier to each consumer.
The Contact Net
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• A fundamental work in resource allocation:
– Each consumer starts an auction for a resource.
– All suppliers participate in the auction.
– The consumer chooses the best possible offer.
• Big disadvantage: Using broadcast.
Task Announcement
Manager
Centralized Match-Mark
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• We’ll have a special new agent (a matchmark)
that will:
– Hold a list of all consumers / suppliers (will get a
message from the other agents anytime they
supply / need a resource).
– Match appropriate <consumer, supplier> couples.
• Disadvantages:
– Hot Spot
– Inefficient (not scalable for systems with a big
number of agents)
Distributed Match-Mark
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• The (suppliers, consumers) list is distributed
between many agents.
• Each agent can also be a matchmark.
Distributed Match-Mark – lower bounds
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Lower bound is known if the matchmarks
can’t pass information between each other:
– One match-mark that is familiar with
O(
) agents exists.
– An agent that is familiar with O(
)
matchmarks exists.
• Can be improved to O(
) if the max
distance in a graph between a supplier and a
consumer is k.
• Can we do better by allowing matchmarks to
pass information between each other?
P2P Framework properties
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Symmetric
– All the agents have the same contribution to the
matching process.
• Distributed Control
– The control of the system is distributed between
all the agents.
– System is more robust, no hot spot.
P2P Framework properties (cont.)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Simple agents.
• Each agent has connections only to a
constant number of other agents.
Algorithm For A Cooperative Environment
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Parameters:
– TTL – How deep in the network the message will
arrive.
– TimeOut – How long to wait for a response.
– d – To how many neighbors to send a message
when searching for a resource.
Algorithm For A Cooperative Environment
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Algorithm (Vanzin 2006):
– When an agent ‘a’ needs a resource, it sends a
message to d of its neighbors. ‘a’ attaches to the
message the TTL parameter.
– Each agent that receives the message, reduces 1
from the TTL parameter, and passes the message
forward (with the new TTL).
Algorithm For A Cooperative Environment
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Algorithm (Vanzin 2006):
– An agent that can supply the resource will contact
‘a’ directly.
– Stop passing the message when TTL reach 0.
– If ‘a’ didn’t receive a resource by TimeOut it
sends another request with a bigger TTL / d.
Algorithm For A Cooperative Environment
Resource Allocation
Cryptographic
Graph Theory
Open subjects
TTL = 3
d=2
s
t
Algorithm For A Cooperative Environment
Resource Allocation
Cryptographic
Graph Theory
Open subjects
TTL = 32
d=2
s
t
Algorithm For A Cooperative Environment
Resource Allocation
Cryptographic
Graph Theory
Open subjects
TTL = 321
d=2
s
t
Algorithm For A Cooperative Environment
Resource Allocation
TTL = 3210
Cryptographic
Graph Theory
Open subjects
Algorithm Failed!
d=2
s
t
Algorithm For A Cooperative Environment
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Analysis of the algorithm (Vanzin 2006):
– If d=1
• Random walk
• Will find the resource in O(log(n)) (average case)
• Long time to find a resource
– d is large
• Find the resource in shorter time
• Big message complexity
Non-Cooperative Environment
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• We’ll assume that agents only care about
increasing their personal utility
– We’ll ignore attacks like “denial of service” that
only try to hurt the system.
• We’ll have to pay an agent in order to use its
resource
– Auctions (like in the CNET).
• We’ll have to pay other agents if we want
them to pass our ‘seek for resource’ message.
Difficulties In Non-Cooperative Environment
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• An agent may try to:
– Get paid without passing a message.
– Avoid paying other agents that pass its message.
– Block bids of distant-agents
• “Man in the middle” attack
– Get paid for passing messages to a DECOY agent.
– Register many copies of itself to the system.
– Whitewash
Multi Hop Cellular Phone Network
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Regular cellular phone:
– All calls are transferred through a central location.
Multi Hop Cellular Phone Network
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• A possible problem in case of multiple calls in
a small area.
• Also relevant to sensors communications.
Multi Hop Cellular Phone Network
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• An alternative: the calls will be routed by the
cell-phones themselves.
T
S
Difficulties
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Problem: keeping the cell-phone on and
passing other users messages costs money
(battery).
• The greedy user may keep his cell-phone off,
and just take advantage of other users for
passing his messages (free ride).
Non-Economic Solutions
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Hardware Solution:
– Add hardware (like an OTP) to the cell-phones
that will transfer the messages.
– Home users can’t change the hardware (criminal
organizations might be able to do so).
– Task: the hardware should be as simple as
possible.
– Won’t work in a general P2P framework.
Non-Economic Solutions
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Repetitive solution:
– Assume that any cell-phone can “listen” to the
outgoing messages of the cell-phones close to it
• Can’t understand the content of the message
• Can tell if a message was sent
– No user has incentive to send “fake” messages
• They also cost buttery power
– If a user didn’t forward your message – punish it
in the future.
– Doesn’t fit a general P2P model.
Economic Solutions
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Central Bank Solution (CBS):
– A trusted (centralized) third party
– All the agents have its public key
• All the agents can verify that the bank has ‘signed’ a
message
Economic Solutions (CBS)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Naïve algorithm:
– The consumer connects to the bank, and sends it
the search message.
– The bank signs this message and charges the
consumer.
– Any agent in the path:
• Checks that the message is legitimate (the signature on
the message is valid).
• Send the bank a ‘receipt’ – a proof that it got the
message (cryptographic hash signature), and the
names of the previous and the next agents in the
search path.
Economic Solutions (CBS)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Naïve algorithm (cont.):
– The bank will pay both the agent that sends the
confirmation and the agents previous and after it
in the search path.
• This way, each agent will have incentive both to pass
the message forward and to send the receipt.
• Also, the bank can track the path and check that it’s
legitimate.
Economic Solutions (CBS)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Problem of the naïve algorithm: the
entire control of the message distribution
process passes through the bank
– Back to centralized recourse allocation
Economic Solutions (Sprite, Zhong 2003)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Possible distributed algorithm?
– We can let the last agent on each search path
serve as the bank.
– It has no incentive to lie (it’ll only take money
from the consumer and give it to the agents in
the path).
• Big problems:
– Coalitions
– Fake identities
Micro-payment Scheme (Jakobsson 2003)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Use lottery tickets
– When entering the system, each agent gets from
the bank a secret personal random-number.
– There exists a global function F: N X N -> {0,1}
(known to all the agents in the system) that
represents the lottery ticket.
• The function returns ‘1’ (a win) with probability ‘p’
(parameter we can choose).
• Cryptographic hash function
Micro-payment Scheme (Jakobsson 2003)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Use lottery tickets (cont.)
– The consumer sends the central bank the
message it wants to distributes.
– The bank charges the consumer for the
distribution, adds to the message a random
number and signs it.
– Each middle-agent checks ‘F’ with the random
number and its personal key to see if it has won.
Micro-payment Scheme (Jakobsson 2003)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Use lottery tickets (cont.)
– An agent connects to the central bank only
if it won
• In that case, also the previous agent and the next
agents in the search path get paid.
– Winning is rare (probability ‘p’)
• The central bank is connected only at the beginning of
a lookup for a resource and in a winning case.
Micro-payment Scheme (Jakobsson 2003)
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Use lottery tickets (cont.)
– Agent has incentive to pass forward messages
even if it didn’t win
• The next agent might win and then it’ll get paid.
– Possible attack: lie about the previous / next
agent’s identity – can be solved with statistical
methods.
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Two different scenarios:
– Regular P2P Framework:
• An agent that wants to participate in the auction, may
avoid forwarding the auction message to other agents
in order to increase its probability of winning.
– it’ll loose the payment for forwarding the bid.
• An agent might publish “a bid” of its own in order to
check what the other agents offer
– cost money.
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Two different scenarios (cont.):
– Multi-hop cellular phones framework
• A distant agent may not be able to connect to the
manager of the auction directly.
• A middle agent can decide to avoid passing a bid only if
it’s a better offer than its own bid.
– it’ll loose the payment for passing the bid.
– The value of the bid can be sealed from the middle-agent
using cryptographic methods.
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Regular P2P framework
Hi, my name
is 32.13.1.6 and
I’m looking for a
resource X
32.13.1.6
56.91.0.1
22.2.8.3
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
Hi, my name
is 56.91.0.1 and
I’m looking for a
resource X
56.91.0.1
32.13.1.6
22.2.8.3
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
I better
bid bellow
900NIS
56.91.0.1
Resource X
For 900 NIS
From 22.2.8.3
32.13.1.6
22.2.8.3
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
Resource X
For 899 NIS
From 56.91.0.1
56.91.0.1
32.13.1.6
22.2.8.3
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• The solution:
– Look at the following network:
Bob
– All the agents want to
participate in the auction.
Avi
– Bob and Cain need
to choose (independently)
whether to follow the protocol
and pass David messages.
David
Cain
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• The solution:
– Can be viewed as the following game table:
Bob
Cain
Cooperate
(follow the
protocol)
Bob
Avi
David
Cain
Defect
(block
David)
Cooperate
(follow the
protocol)
Defect
(block
David)
x/3
y+x/3
y+x/3
y+x/3
0.5x
y+x/3
x/3
0.5x
x – payment if winning the auction
y – payment for following the protocol
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• The game has two Nash equilibriums –
Cooperate-Cooperate and Defect-Defect.
• If there are k paths from David to Avi, it’s
enough that the agents in one path cooperate
to cause all the agents that defected to loose
the game.
• If an agent doesn’t participate in the auction,
it has a dominate strategy to cooperate.
The “middle man” attack problem
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• A reminder: d is the parameter that
represents the number of neighbors each
agent forward the auction message to
• We would like to analyze the influence of the
parameter d on the probability that there
won’t be a single agent that appears in all the
paths between two random agents.
Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• The probability space:
– A graph with n nodes, each node represents
an agent.
– Each vertex has d outgoing-edges (Gd-out graph).
– The edges are drawn uniformly.
• Reasonable model for P2P frameworks (for
example – BitTorrent).
• The following calculations will be done for d=3.
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• What is the expected number of edges that
enter the target agent?
Probability that a random edge will enter the
target vertex: 1
n 1
Probability that all the d edges don’t enter the
d
1

target vertex: 1 



n 1 
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• What is the expected number of edges that
enter the target agent?
Probability that a vertex has an outgoing edge to
d
the target vertex: 1  1  1 



n 1 
Expected number of edges that will enter the
target vertex:
d

 d2 
 
1  
d
O
  d
 n  1 1  1 
    n  1 1  1 
2
  n  1  
n 1
  n 1  



Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Assume that we know the number of vertexes
with distance of up do j from t:
s
t
• What is the expected number of vertexes of
distance j+1 from t?
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Assume that we know the number of vertexes
with distance of up do j from t, What is the
expected number of vertexes of distance j+1
from t?
s
X–
#Distance
t
j+1
Y–
#Distance j
Z - #Distance
less than j
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
Probability that a vertex has an outgoing edge to
d
a vertex in Y:
Y 

1  1 

n

Z


Expected value of X:
d
 
Y  
 n  Y  Z  1  1 
 
  nZ  
X – #dist j+1, Y – #dist j, Z – #dist<j
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• We would like to see until what stage we can
expect the number of vertexes of distance j
to be at least twice the number of vertexes of
distance j-1.
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
Lemma: Let {T1,T2,…,Tn} be a sequence of
numbers such that for each i, Ti>1.5*Ti-1.
Then:
i 1
2i n : 2Ti   T j
j 1
Proof: By induction.
i 2
2Ti  2*1.5* Ti 1  Ti 1  2Ti 1  Ti 1   T j
j 1
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
If we assume that in each step the number of
vertexes of distance j is at least 1.5 times the
number of vertexes of distance j+1 we’ll get that:
1  Z  1.5Y
Thus by our assumption the expected value of X:
  Y 3 
E  X    n  3Y  1  1   
  n 


X – #dist j+1, Y – #dist j, Z – #dist<j
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• What is the probability that the number of
vertexes doesn’t multiply by 1.5 (at least)?
Chernoff bound (for Y=O(n)):
Pr  X  1.5Y   e
Y 
 
 48 
For small Y the situation is less promising – 5% of
the vertexes in the graph don’t have incoming
edges!
Still we can get reasonable results using the fact
that Y is an integer.
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Conclusions:
– if the number of vertexes of distance j from t is
smaller than 0.1n then the number of vertexes of
distance j+1 from t is at least 1.5 times the
previous distance (with very high probability).
– Best strategy for d=3 is TTL=log(0.1n).
Analysis Of The Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Conclusions (cont.):
– Choosing TTL=log(0.1n)+2 will promise (with
good probability) that no agent will be placed on
all the paths between s to a random vertex t.
– We can repeat the calculations for other values of
d and their appropriate TTL.
– Solution is robust against coalitions with
constant size.
• Results are supported by simulations
Non Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• The probability space:
– A graph with n nodes, each node represents
an agent.
– Agents are located uniformly on L X L square
• L – a parameter.
– Each vertex has d outgoing-edges (Gd-out graph).
Non Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• The probability space (cont.):
– Edge between vertexes u and v is drawn with
probability:
e  dist u ,v 
Pr   u, v   E  
– Where:
  
dist u ,v  X u  X v

2
D

 Yu Yv

2
,D 
 e distu ,v
vV
• Reasonable model for cellular phones network
(in reality – using different parameters).
Non Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• In order to reach all the vertexes in a search
we need to set TTL=O(n) in worst case.
…
• Turns out that this also holds for our uniform
model…
Non Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• First, bound E(D):
– Get maximum when an agent is located in the
middle of the square, and minimum on the edge.
• With high probability:
0.6n  D   n
• Next, we would like to calculate the
probability of a connection between vertexes
s and t that goes through TTL vertexes.
  
dist u ,v  X u  X v

2

 Yu Yv

2
,D 
 e distu ,v
vV
Non Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Draw rectangles around the straight line
between s and t.
s
t
• Since the probability for an edge drops
exponentially, consider only paths from ‘s’
through vertexes inside the rectangles to ‘t’.
  
dist u ,v  X u  X v

2

 Yu Yv

2
,D 
 e distu ,v
vV
Non Uniform Model
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Calculate the probability by:
– Calculate the expected number of vertexes in
each rectangle.
– Calculate the probability for a path to any of the
vertexes in the rectangle.
• Problem: miss paths:
s
t
  
dist u ,v  X u  X v

2

 Yu Yv

2
,D 
 e distu ,v
vV
Open Subjects
Resource Allocation
Cryptographic
Graph Theory
Open subjects
• Correct the analysis of the non-uniform model
– Try to use circles that represent the error rate
instead of rectangles.
• Formalize the robustness of the model against
random coalitions.
• Gather formal simulation results.
• Try to deal with the ability of an agent to
conduct an auction for predicting the current
best offer.