Talk in PAAMS Conference (Sevilla, 2016). Use of a combined process of consensus and gradient ascent in multiplex networks in order to solve multi criteria optimization problems using Analytical Hierarchical Process (AHP)

1. Introduction AHP Decentralized Group AHP Application Example Conclusions
Decentralized Group Analytical Hierarchical
Process on Multilayer Networks by Consensus
M. Rebollo, A. Palomares, C. Carrascosa
Universitat Politècnica de València
PAAMS 2016
@mrebollo UPV
Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

4. Introduction AHP Decentralized Group AHP Application Example Conclusions
Problem
Analytic Hierarchical Process (AHP)
How a group of people can take a complex decision?
optimization process
multi-criteria
complete knowledge
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

5. Introduction AHP Decentralized Group AHP Application Example Conclusions
The Proposal
Combination of consensus and gradient descent over a multilayer
network
decentralized
personal, private preferences
people connected in a network
locally calculated (bounded rationality)
layers capture the criteria
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

6. Introduction AHP Decentralized Group AHP Application Example Conclusions
AHP decision scenario [Saaty, 2008]
Choose a candidate.
Select the most suitable
candidate based on 4 criteria
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

7. Introduction AHP Decentralized Group AHP Application Example Conclusions
AHP decision scenario [Saaty, 2008]
Choose a candidate.
Criteria are weighted
depending on its importance.
p
α=1
wα
= 1
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

8. Introduction AHP Decentralized Group AHP Application Example Conclusions
Scale for Pairwise comparisons
Importance Definition Explanation
1 equal imp. 2 elements contribute equally
3 moderate imp. preference moderately in favor of one
element
5 strong imp. preference strongly in favor of one el-
ement
7 very strong imp. strong preference, demonstrate in
practice
9 extreme imp. highest possible evidence
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

9. Introduction AHP Decentralized Group AHP Application Example Conclusions
Pairwise matrix
For each criterion, a
pairwise matrix that
compares all the
alternatives is defined
aij =
1
aji
Tom Dick Harry L.p. (lα
i )
Tom 1 1/4 4
Dick 4 1 9
Harry 1/4 1/9 1
Experience
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

10. Introduction AHP Decentralized Group AHP Application Example Conclusions
Pairwise matrix
The local priority is
calculated as the
values of the principal
right eigenvector of
the matrix
Tom Dick Harry L.p. (lα
i )
Tom 1 1/4 4 0.217
Dick 4 1 9 0.717
Harry 1/4 1/9 1 0.066
Experience
@mrebollo UPV
Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

11. Introduction AHP Decentralized Group AHP Application Example Conclusions
Making a decision
The final priorities are calculated as the weighted average
pi =
α
wα
lα
i
Candidate Exp Edu Char Age G.p. (pi )
Tom 0.119 0.024 0.201 0.015 0.358
Dick 0.392 0.010 0.052 0.038 0.492
Harry 0.036 0.093 0.017 0.004 0.149
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

12. Introduction AHP Decentralized Group AHP Application Example Conclusions
Group AHP
Participants have their own (private) weights for the criteria
@mrebollo UPV
Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

13. Introduction AHP Decentralized Group AHP Application Example Conclusions
Main idea
Each criterion is negotiated in
a layer of a multiplex network
consensus process (fi )
executed in each layer α
deviations from individual
preferences compensated
with a gradient ascent
(gi ) among layers
xα
i (t + 1) = xα
i (t) + fi (xα
1 (t), . . . , xα
n (t))
+ gi (x1
i (t), . . . , xp
i (t))
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

14. 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 of
the initial values xi (0)
lim
t→∞
xi (t) = i wi xi (0)
i wi
∀i
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

15. Introduction AHP Decentralized Group AHP Application Example Conclusions
Individual preferences as utility functions
Desired behavior
max. value in the local priority
lα
i
higher weight → faster decay
Local utility defined for each criterion
as a renormalized multi-dimensional
gaussian with ui (lα
i ) = 1.
uα
i (xα
i ) = e
−1
2
xα
i
−lα
i
1−wα
i
2
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

16. Introduction AHP Decentralized Group AHP Application Example Conclusions
Global utility function
The final purpose of the system is to maximize the global utility U
defined as the sum of the individual properties
ui (xi ) =
α
uα
i (xα
i ) U(x) =
i
ui (xi )
This function U is never calculated nor known by anyone
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

17. Introduction AHP Decentralized Group AHP Application Example Conclusions
Multidimensional Networked Decision Process
Two-step process
1 consensus in each layer
2 individual gradient ascent crossing layers
xα
i (t + 1) = xα
i +
fi
ε
wα
i j∈Nα
i
(xα
j (t) − xα
i (t)) +
+ϕ ui (x1
i (t), . . . , xp
i (t))
gi
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

18. 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 )
∂xα
i
= −
xα
i (t) − lα
i
(1 − wα
i )2
ui (xi )
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

19. Introduction AHP Decentralized Group AHP Application Example Conclusions
Convergence of the gradient
The convergence of this method depends on the stepsize ϕ
ϕ ≤ min
i
1
Lui
where Lui is the Lipschitz constant of the each utility function
Normal distribution the maximum value of the derivative appears
in its inflection point xα
i ± (1 − wα
i ).
∂ui (xα
i − (1 − wα
i ))
∂xα
i
=
1
1 − wα
i
e−p/2
Lui =
α
e−p/2
1 − wα
i
1/2
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

20. Introduction AHP Decentralized Group AHP Application Example Conclusions
Final model
Complete consensus and gradient equation
xα
i (t + 1) = xα
i +
ε
wα
i j∈Nα
i
(xα
j (t) − xα
i (t)) −
−
1
maxi || ui (xi )||2
·
xα
i (t) − lα
i
(1 − wα
i )2
ui (xi )
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

21. Introduction AHP Decentralized Group AHP Application Example Conclusions
Initial conditions
9 nodes
2 criteria
connection by proximity of preferences
—————–
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

22. Introduction AHP Decentralized Group AHP Application Example Conclusions
Evolution of the group decision
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

23. Introduction AHP Decentralized Group AHP Application Example Conclusions
Evolution of the priority values
The group obtain common priorities for both criteria
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

24. Introduction AHP Decentralized Group AHP Application Example Conclusions
Counterexample: local maximum
If some participants have ui = 0 in the solution space, it not
converges to the global optimum value.
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

25. Introduction AHP Decentralized Group AHP Application Example Conclusions
Solution: break links
Break links with undesired neighbors is allowed.
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

26. Introduction AHP Decentralized Group AHP Application Example Conclusions
Group identification
The networks is split into separated components
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

27. Introduction AHP Decentralized Group AHP Application Example Conclusions
Consensus process
The group obtain common priorities for both criteria
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

28. Introduction AHP Decentralized Group AHP Application Example Conclusions
Performance. Network topology, size and criteria
@mrebollo UPV
Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

29. Introduction AHP Decentralized Group AHP Application Example Conclusions
Performance. Execution time
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Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus

30. Introduction AHP Decentralized Group AHP Application Example Conclusions
Conclusions
Conclusions
solve group AHP in a network with private priorities and
bounded communication
combination of consensus and gradient ascent process
break links to avoid a local optimum
Future work
extend to networks of preferences (ANP)
extend to dynamic networks that evolve during the process
@mrebollo UPV
Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus