consensus networks as agreement mechanism for autonomous agents in water markets
DESCRIPTION
TRANSCRIPT
Outline Consensus Networks Water distribution Conclusions
Consensus Networks as Agreement Mechanismfor Autonomous Agents in Water Markets
M. Rebollo, A. Palomares and C. Carrascosa
Dept. Sistemas Informáticos y ComputaciónUniv. Politécnica de Valencia (Spain)
Math. Models of Addictive Behav., Medicine & EngineeringValencia, September 2010
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution Problem
Motivation
Water management is a complex task
centralised solutions trend to fail: low implication of users
WUA valid for small and medium domains
pure market solutions result in unfair distribution
agreements related with natural resources involve complexnegotiations
Social-ecological systems (SES) suggest self-organized solutions asthe most sustainable in the long term (Ostrom, 2009)
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Our Proposal
The challenge
Design a procedure that allows a set of self-organised agents toachieve agreements
What is needed. . .
to obtain a theoretical model of agreement
to define protocol to achieve agreements by consensus
to design a self-regulated system to deal with waterdistribution problems
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Outline
1 Outline
2 Consensus Networks as Agreement Mechanism
3 Water Distribution as a Consensus Problem
4 Conclusions and Future Work
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Consensus Networks as Agreement Mechanism
Consensus networks
Let (G ,X ) be the state of a network with value X and topology G ,where X = (x1, . . . , xn) ∈ R
n, where xi is a real value associatedwith the node Ei .
a b c
d e f g
h i
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Consensus Networks as Agreement Mechanism
Theoretical Model (Olfaty, 2004)
The consensus problem can be formulated as:1
xi(k + 1) = xi(k) + ε∑
j∈Ni
(aij(xj(k) − xi(k))),
The collective dynamics of the network for this algorithm can bewritten as
x(k + 1) = Px(k)
where P = I − εL is the Perron matrix of a graph with parameter ε
1Agents with discrete-time model
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Consensus Networks as Agreement Mechanism
Simple Consensus Protocol with Initiator
Initiator: Facilitator Participant-i
request
not-understood
refuse
inform-disagree
inform-agree
consensus value
calculation
n
n
n
Participant-j
inform-value
inform-value
k
k'
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Consensus Networks as Agreement Mechanism
Consensus Protocol for Agreement Spaces
But sometimes we do not need to know a common value
just the existence of a possible consensus is needed
definition of an agreement space
So the process can be interrupted when some conditions are met
deliberation time is over
one agent leaves the process
a percentage of agents leave the network
a threshold has been reached
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Consensus Networks as Agreement Mechanism
Additional considerations
weighted agents: weights in consensus networks canrepresent concepts as reputation or trust, so the most relevantagents can have higher importance in the consensus processand they can influence the final consensus value.
stubborn agents: if an agent does not change the value forthe dimension the final value of the consensus clearly convergeto this value, distorting the result.
the behavior of stubborn agents can be used to create somekind of decentralized control (for example, fulfillment ofnorms)
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Problem Description
The market consist of n entities (agents) Ei , i = 1, . . . , n
Ei = {Ri , R̃i ,Pi , P̃i}
where
Ri rights that Ei owns
R̃i rights that entity Ei desires
Pi initial price proposed byEi
P̃i upper/lower price bound for Ei .2 This parameter is private
2It depends on been a buyer or a seller
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Problem Dynamics
PSi (k + 1) = PS
i (k) + ε∑
j∈Ni
(Bj(PBj (k)− PS
i (k))),
PBi (k + 1) = PB
i (k) + ε∑
j∈Ni
(Sj(PSj (k)− PB
i (k)))
where the index S and B denotes seller and buyers respectively.Agents will disconnect from the network if
PSi (k) < P̃S
i for seller agents.
PBi (k) > P̃B
i for buyer agents.
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Model Reformulation
Detected problem: convergence of water rights
PSi (k + 1) = PS
i (k) + ε∑
j∈Ni
(Bj(PBj (k) − PS
i (k) + Ci(k))),
PBi (k + 1) = PB
i (k) + ε∑
j∈Ni
(Sj(PSj (k) − PB
i (k) + Ci (k)))
where the added term Ci(k) is proportional to rights bought andsold by agents in each iteration, and is calculated as follows:
Ci (k) = δ ·
∑j∈Ni
Bj∑j∈Ni
Sj
where δ > 0. In this experiment the algorithm converges and stopswhen the mean prices of buyers and sellers are approximately equal.
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Experiments Desgin
Parameter Exp. 1 Exp. 2 Exp. 3
All
n 2000 2000 2000Rmax 4 4 4RT 4000 4000 4000
R̃max 4 4 4
R̃T 4000 4000 4000ε 0.01 0.01 0.01δ 0 1 1
SellersP̃
S
10 10 10σS 0.2 0.2 0.2F S 1.25 1.25 1.25
BuyersλB 0.5 0.5 0.5
P̃B
12 12 6
F B 0.9 0.9 0.9
Table: Parameters used in the experiments.
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Experiment 1: full connected, fixed topology
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Experiment 2: scale free α = 2.5, fixed topology
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Experiment 3: full connected, switching topology, unbiased
rights
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Experiment 4:full connected, switching topology, biased
rights
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Experiment 5: scale free α = 2.5, switching topology,
unbiased rights
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Experiment 6: scale free α = 2.5, switching topology,
biased rights
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Experiment 7: scale free α = 2.5, switching topology,
biased rights
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Water Distribution as a Consensus Problem
Experiment 8: scale free α = 2.5, switching topology,
biased rights
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Conclusions and Future Work
What we have done
test theoretical consensus model
design a protocol that allow intelligent agents to achieveagreements based on consensus
model a self-regulated, water rights ’market’
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets
Outline Consensus Networks Water distribution Conclusions
Conclusions and Future Work
Future Work
multidimensional
time delay
re-entry of agents
group identification
study the impact of other network models
M. Rebollo et al. DSIC-UPV
Consensus Networks as Agreement Mechanism for Autonomous Agents in Water Markets