recourse allocation in p2p framework yoni peleg. outline previous and current work on resource...

Recourse AllocationIn P2P FrameworkYoni Peleg

Outline

• 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 AllocationCryptographic Graph Theory Open subjects

Resource Allocation Overview

• 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

Cryptographic Graph Theory Open subjectsResource Allocation

Economics Resource Allocation

• 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.)

• A lot of sellers (easy to find an agent with an available resource).

• All the available resources are identical.


MarketPlace

Matching Resources

• 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

• 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.


Manager

Task Announcement

Centralized Match-Mark

• 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

• The (suppliers, consumers) list is distributed between many agents.

• Each agent can also be a matchmark.


Distributed Match-Mark – lower bounds• 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

• 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.)

• Simple agents.

• Each agent has connections only to a

constant number of other agents.


Algorithm For A Cooperative Environment

• 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 (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 (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 EnvironmentCryptographic Graph Theory Open subjectsResource Allocation

t

s

TTL = 3

d = 2

s


t

s

TTL = 32

d = 2


t

s

TTL = 321

d = 2


t

s

TTL = 3210

d = 2Algorithm Failed!


• 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

• 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

• 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

• Regular cellular phone:

– All calls are transferred through a central

location.



• A possible problem in case of multiple calls in a small area.

• Also relevant to sensors communications.



• An alternative: the calls will be routed by the cell-phones themselves.


S

T

Difficulties

• 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

• 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

• 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

• 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)

• 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.



• 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.



• 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)

• 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)

• 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



• 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.




– 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.




– 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

• 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.

Graph Theory Open subjectsResource AllocationCryptographic


• 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.



• Regular P2P framework


Hi, my nameis 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 problemGraph Theory Open subjectsResource AllocationCryptographic

32.13.1.6

56.91.0.1

22.2.8.3

Hi, my nameis 56.91.0.1

and I’m looking for a resource X


32.13.1.6

56.91.0.1

22.2.8.3

Resource XFor 900 NIS

From 22.2.8.3

I better bid bellow

900NIS


Resource XFor 899 NIS

From 56.91.0.1

32.13.1.6

56.91.0.1

22.2.8.3


• The solution: – Look at the following network:

– All the agents want toparticipate in the auction.

– Bob and Cain needto choose (independently)whether to follow the protocoland pass David messages.


Avi

Bob

David

Cain


• The solution: – Can be viewed as the following game table:


Avi

Bob

David

Cain

Bob

Cain

Cooperate(follow the protocol)

Defect(block David)

Cooperate(follow the protocol)

Defect(block David)

y+x/3

y+x/3

y+x/3

x/3

x/3

y+x/3

0.5x

0.5x

x – payment if winning the auctiony – payment for following the protocol


• 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.



• 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

• 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

• What is the expected number of edges that enter the target agent?


1

1n

Probability that a random edge will enter the target vertex:

Probability that all the d edges don’t enter the target vertex:11

1

d

n


• What is the expected number of edges that enter the target agent?


11 1

1

d

n

Probability that a vertex has an outgoing edge to the target vertex:

Expected number of edges that will enter the target vertex:

2

2

11 1 1 1 1 1

1 1 1

dd d

n n O dn n n


• 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?


ts


• 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?


t

Z - #Distance less than j

s

Y –#Distance j

X –#Distance

j+1

Analysis Of The Uniform ModelGraph Theory Open subjectsResource AllocationCryptographic

1 1d

Y

n Z

Probability that a vertex has an outgoing edge to a vertex in Y:

Expected value of X:

1 1d

Yn Y Z

n Z

X – #dist j+1, Y – #dist j, Z – #dist<j


• 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.



1

21

: 2i

i n i jj

T T

Lemma: Let {T1,T2,…,Tn} be a sequence of

numbers such that for each i, Ti>1.5*Ti-1.

Then:

Proof: By induction.2

1 1 1 11

2 2*1.5* 2i

i i i i i jj

T T T T T T


1 1.5Z Y

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:

Thus by our assumption the expected value of X:

3

3 1 1Y

E X n Yn

X – #dist j+1, Y – #dist j, Z – #dist<j


• What is the probability that the number of vertexes doesn’t multiply by 1.5 (at least)?


Chernoff bound (for Y=O(n)):

48Pr 1.5Y

X Y e

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.


• 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).



• 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 withconstant size.

• Results are supported by simulations


Non Uniform Model

• 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

• The probability space (cont.):– Edge between vertexes u and v is drawn

with probability:

– Where:

• Reasonable model for cellular phones network (in reality – using different parameters).


,

Pr ,dist u ve

u v ED

2 2,, , dist u v

u v u vv V

dist u v X X Y Y D e

Non Uniform Model

• 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

• 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:

• Next, we would like to calculate the probability of a connection between vertexes s and t that goes through TTL vertexes.


0.6n D n

2 2,, , dist u v

u v u vv V


Non Uniform Model

• Draw rectangles around the straight line between s and t.


2 2,, , dist u v

u v u vv V


s

t

• Since the probability for an edge drops exponentially, consider only paths from ‘s’ through vertexes inside the rectangles to ‘t’.

Non Uniform Model

• 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:


2 2,, , dist u v

u v u vv V


s

t

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.

Open subjectsResource AllocationCryptographic Graph Theory

recourse allocation in p2p framework yoni peleg. outline previous and current work on resource...

Documents

economics resource allocation

resource allocation

models resource allocation

recourse allocation

agents resources

matching resources resources

o agents

available resources