Introduction AHP Decentralized Group AHP Application Example Conclusions

Decentralized Group Analytical HierarchicalProcess on Multilayer Networks by Consensus

M. Rebollo, A. Palomares, C. CarrascosaUniversitat Politècnica de València

PAAMS 2016

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Problem

Analytic Hierarchical Process (AHP)How a group of people can take a complex decision?

optimization processmulti-criteriacomplete knowledge

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

The Proposal

Combination of consensus and gradient descent over a multilayernetwork

decentralizedpersonal, private preferencespeople connected in a networklocally calculated (bounded rationality)layers capture the criteria

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

AHP decision scenario [Saaty, 2008]

Choose a candidate.Select the most suitablecandidate based on 4 criteria

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

AHP decision scenario [Saaty, 2008]

Choose a candidate.Criteria are weighteddepending on its importance.

p∑α=1

wα = 1

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Scale for Pairwise comparisons

Importance Definition Explanation1 equal imp. 2 elements contribute equally3 moderate imp. preference moderately in favor of one

element5 strong imp. preference strongly in favor of one el-

ement7 very strong imp. strong preference, demonstrate in

practice9 extreme imp. highest possible evidence

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Pairwise matrix

For each criterion, apairwise matrix thatcompares all thealternatives is defined

aij = 1aji

Tom Dick Harry L.p. (lαi )Tom 1 1/4 4Dick 4 1 9Harry 1/4 1/9 1

Experience

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Pairwise matrix

The local priority iscalculated as thevalues of the principalright eigenvector ofthe matrix

Tom Dick Harry L.p. (lαi )Tom 1 1/4 4 0.217Dick 4 1 9 0.717Harry 1/4 1/9 1 0.066

Experience

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Making a decision

The final priorities are calculated as the weighted average

pi =∑α

wαlαi

Candidate Exp Edu Char Age G.p. (pi)Tom 0.119 0.024 0.201 0.015 0.358Dick 0.392 0.010 0.052 0.038 0.492Harry 0.036 0.093 0.017 0.004 0.149

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Group AHP

Participants have their own (private) weights for the criteria

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Main idea

Each criterion is negotiated ina layer of a multiplex network

consensus process (fi)executed in each layer αdeviations from individualpreferences compensatedwith a gradient ascent(gi) among layers

xαi (t + 1) = xαi (t) + fi(xα1 (t), . . . , xαn (t))+ gi(x1

i (t), . . . , xpi (t))

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Consensus [Olfati, 2004]

Gossiping process

xi(t+1) = xi(t)+ ε

wi

∑j∈Ni

[xj(t)− xi(t)]

converges to the weighted average ofthe initial values xi(0)

limt→∞

xi(t) =∑

i wixi(0)∑i wi

∀i

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Individual preferences as utility functions

Desired behaviormax. value in the local prioritylαihigher weight → faster decay

Local utility defined for each criterionas a renormalized multi-dimensionalgaussian with ui(lαi ) = 1.

uαi (xαi ) = e− 1

2

(xαi −lαi1−wα

i

)2

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Global utility functionThe final purpose of the system is to maximize the global utility Udefined as the sum of the individual properties

ui(xi) =∏α

uαi (xαi ) U(x) =∑

iui(xi)

This function U is never calculated nor known by anyone@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Multidimensional Networked Decision Process

Two-step process1 consensus in each layer2 individual gradient ascent crossing layers

xαi (t + 1) = xαi +

fi︷ ︸︸ ︷ε

wαi

∑j∈Nα

i

(xαj (t)− xαi (t)) +

+ϕ∇ui(x1i (t), . . . , xp

i (t))︸ ︷︷ ︸gi

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Gradient calculation

In the case of the chosen utility functions (normal distributions),

∇ui(xi) =(∂ui(xi)∂x1

i, . . . ,

∂ui(xi)∂xp

i

)

and each one of the terms of ∇ui

∂ui(xi)∂xαi

= −xαi (t)− lαi(1− wα

i )2 ui(xi)

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Convergence of the gradient

The convergence of this method depends on the stepsize ϕ

ϕ ≤ mini

1Lui

where Lui is the Lipschitz constant of the each utility function

Normal distribution the maximum value of the derivative appearsin its inflection point xαi ± (1− wα

i ).∂ui(xαi − (1− wα

i ))∂xαi

= 11− wα

ie−p/2

Lui =(∑

α

e−p/2

1− wαi

)1/2

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Final model

Complete consensus and gradient equation

xαi (t + 1) = xαi + ε

wαi

∑j∈Nα

i

(xαj (t)− xαi (t))−

− 1maxi ||∇ui(xi)||2

· xαi (t)− lαi(1− wα

i )2 ui(xi)

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Initial conditions

9 nodes2 criteriaconnection by proximity of preferences

—————–@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Evolution of the group decision

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Evolution of the priority values

The group obtain common priorities for both criteria

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Counterexample: local maximumIf some participants have ui = 0 in the solution space, it notconverges to the global optimum value.

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Solution: break linksBreak links with undesired neighbors is allowed.

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Group identification

The networks is split into separated components

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Consensus process

The group obtain common priorities for both criteria

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Performance. Network topology, size and criteria

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Performance. Execution time

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

Introduction AHP Decentralized Group AHP Application Example Conclusions

Conclusions

Conclusionssolve group AHP in a network with private priorities andbounded communicationcombination of consensus and gradient ascent processbreak links to avoid a local optimum

Future workextend to networks of preferences (ANP)extend to dynamic networks that evolve during the process

@mrebollo UPVDecentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus