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- 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
- 2. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 3. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 4. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 5. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 6. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 7. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 8. 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
- 9. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 10. 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
- 11. 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)) @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 12. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 13. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 14. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 15. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 16. 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 ) @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 17. 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 @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 18. 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 ) @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 19. Introduction AHP Decentralized Group AHP Application Example Conclusions Initial conditions 9 nodes 2 criteria connection by proximity of preferences —————– @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 20. Introduction AHP Decentralized Group AHP Application Example Conclusions Evolution of the group decision @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 21. Introduction AHP Decentralized Group AHP Application Example Conclusions Evolution of the priority values The group obtain common priorities for both criteria @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 22. 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. @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 23. Introduction AHP Decentralized Group AHP Application Example Conclusions Solution: break links Break links with undesired neighbors is allowed. @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 24. Introduction AHP Decentralized Group AHP Application Example Conclusions Group identiﬁcation The networks is split into separated components @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 25. Introduction AHP Decentralized Group AHP Application Example Conclusions Consensus process The group obtain common priorities for both criteria @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 26. 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
- 27. Introduction AHP Decentralized Group AHP Application Example Conclusions Performance. Execution time @mrebollo UPV Decentralized Group Analytical Hierarchical Process on Multilayer Networks by Consensus
- 28. 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

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