Markov and Consensus Processes

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Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundIntroduction: Markov vs. Consensus Processes and Opinion DynamicsExample: Opinion Dynamics under BoundedConfidencex(0) ∈ R nx(t + 1) = A(x(t), ε)x(t){1a ij :=#I(i,x(t))if j ∈ I(i, x(t))0 otherwiseI(i, x) := {j | |x i (t) − x j (t)| ≤ ε}


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundIntroduction: Markov vs. Consensus Processes and Opinion DynamicsExample: Opinion Dynamics under BoundedConfidence1ε = 0.10.8opinion space0.60.40.21 2 3 4 5 6 7 8 9 10t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundIntroduction: Markov vs. Consensus Processes and Opinion DynamicsExample: Opinion Dynamics under BoundedConfidence1ε = 0.20.8ε = 0.2opinion space0.60.40.21 2 3 4 5 6 7 8 9 10t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundIntroduction: Markov vs. Consensus Processes and Opinion DynamicsExample: Opinion Dynamics under BoundedConfidence1ε = 0.30.8ε = 0.3opinion space0.60.40.21 2 3 4 5 6 7 8 9 10 11 12 13 14 15t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundIntroduction: Markov vs. Consensus Processes and Opinion DynamicsExample: Opinion Dynamics under BoundedConfidence1ε = 0.1 (45 agents), 0.3 (5 agents)opinion space0.80.60.4ε = 0.3ε = 0.10.210 20 30 40 50 60 70 80 90 100t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundIntroduction: Markov vs. Consensus Processes and Opinion DynamicsExample: Opinion Dynamics under BoundedConfidence1ε = 0.1 (45 agents), 0.3 (5 agents)opinion space0.80.60.4ε = 0.1ε = 0.3ε = 0.10.210 20 30 40 50t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundIntroduction: Markov vs. Consensus Processes and Opinion DynamicsThe QuestionA(0), A(1), . . . sequence of row-stochastic matrices n × nwith positive diagonalsfor s < t defineforward accumulation A(s, t) = A(s) . . . A(t − 1)backward accumulation A(t, s) = A(t − 1) . . . A(s)Does lim t→∞ A(t, 0) converge?(And what about lim t→∞ A(0, t)?)


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundIntroduction: Markov vs. Consensus Processes and Opinion DynamicsA First Observationrow-stochastic K which rank is 1 is called consensus matrixbecauseKx has equal entriesSuppose that A(t) := K is a consensus matrix. It is easy to seethat for all u ≥ t it holdsA(u, 0) = A(u) . . . A(t + 1)KA(t − 1) . . . A(1)A(0) = KA(0, u) = A(0)A(1) . . . A(t − 1)KA(t + 1) . . . A(u) is aconsensus matrix but may change with u.


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundThe positive diagonalZero Pattern Convergence 2TheoremLet (A(t)) t∈N be a sequence of nonnegative matrices withpositive diagonals. Then for the backward accumulation thereexists a sequence of natural numbers 0 < t 0 < t 1 < . . . suchthat. . . , A(t i+1 , t i ), . . . , A(t 2 , t 1 ), A(t 1 , t 0 ) have the same zero patternAll A(t i+1 , t i ) can be brought to the same Gantmacher form witheach block either positive or zero.2 J. Lorenz 2003


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundThe positive diagonalGantmacher’s canonical formA can be brought to⎡⎤A 1 0. ..0 A gA g+1,1 . . . A g+1,g A g+1⎢⎣... . ..⎥⎦A p,1 . . . A p,g A p,g+1 . . . A pby simultaneous row and column permutations.


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundThe positive diagonalProofBA = (B diag + B offdiag )Ahas at least the same entries positive as A


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundThe positive diagonalProofBA = (B diag + B offdiag )Ahas at least the same entries positive as A


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundThe positive diagonalProofBA = (B diag + B offdiag )Ahas at least the same entries positive as A


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundThe positive diagonalProofBA = (B diag + B offdiag )Ahas at least the same entries positive as A


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundThe positive diagonalProofBA = (B diag + B offdiag )Ahas at least the same entries positive as A


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundThe positive diagonalProofBA = (B diag + B offdiag )Ahas at least the same entries positive as A


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceCoefficient of ErgodicityWhat about convergence of the Gantmacher diagonal blocks?coefficient of ergodicity of Aτ(A) := 1 − minIt holds submultiplicativityi,j∈nk=1n∑min{a ik , a jk }.τ(A i · · · A 1 A 0 ) ≤ τ(A i ) · · · τ(A 1 )τ(A 0 ).


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceConvergence TheoremTheorem(A(t)) t∈N0 row-stochastic matrices with positive diagonals0 < t 0 < t 1 < . . . and Gantmacher form of first theoremIf for all i ∈ N 0 it holds min + A(t i+1 , t i ) ≥ δ i and ∑ ∞i=1 δ i = ∞,then⎡lim A(t, 0) = ⎢t→∞ ⎣K 1 , . . . , K g consensus matricesK 1 0 0. .. .0 K g 0not converging 0⎤⎥⎦ A(t 0, 0)


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(2,1)0.250.8100.2opinion space0.60.420300.150.10.21 3 5 7 9t405010 20 30 40 500.050


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(3,1)0.140.8100.12opinion space0.60.420300.10.080.060.2405010 20 30 40 500.040.021 3 5 7 9t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(6,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(7,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(8,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(9,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(10,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(11,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(12,1)0.8100.120.1opinion space0.60.420300.080.060.2405010 20 30 40 500.040.021 3 5 7 9 11 13 15 17 19t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(13,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9 11 13 15 17 19t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(14,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9 11 13 15 17 19t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(15,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9 11 13 15 17 19t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(16,1)0.8100.120.1opinion space0.60.420300.080.060.2405010 20 30 40 500.040.021 3 5 7 9 11 13 15 17 19t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(17,1)0.8100.120.1opinion space0.60.420300.080.060.2405010 20 30 40 500.040.021 3 5 7 9 11 13 15 17 19t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(18,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9 11 13 15 17 19t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(19,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9 11 13 15 17 19t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(20,1)0.8100.120.1opinion space0.60.420300.080.060.21 3 5 7 9 11 13 15 17 19t405010 20 30 40 500.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceOpinion Dynamics with Changing Confidence12 essential classes, 3 inessential classA(21,1)0.8100.120.1opinion space0.60.420300.080.060.2405010 20 30 40 500.040.021 3 5 7 9 11 13 15 17 19t


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,2)0.4infinite to the left, A(2,1)0.40.350.3570.370.3140.25140.25210.20.15210.20.15280.1280.1357 14 21 28 350.050357 14 21 28 350.050


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,3)0.2infinite to the left, A(3,1)0.27140.157140.15210.1210.1280.05280.05357 14 21 28 350357 14 21 28 35


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the left714212835infinite to the right, A(1,4)7 14 21 28 350.20.180.160.140.120.10.080.060.040.02714212835infinite to the left, A(4,1)7 14 21 28 350.180.160.140.120.10.080.060.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the left714212835infinite to the right, A(1,5)7 14 21 28 350.180.160.140.120.10.080.060.040.02714212835infinite to the left, A(5,1)7 14 21 28 350.180.160.140.120.10.080.060.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the left714212835infinite to the right, A(1,6)7 14 21 28 350.180.160.140.120.10.080.060.040.020714212835infinite to the left, A(6,1)7 14 21 28 350.180.160.140.120.10.080.060.040.02


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,7)0.2infinite to the left, A(7,1)0.16714210.150.1714210.140.120.10.08280.05280.060.04357 14 21 28 350357 14 21 28 350.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the left714infinite to the right, A(1,8)0.180.160.140.120.1714infinite to the left, A(8,1)0.160.140.120.1210.08210.08280.060.04280.060.04357 14 21 28 350.020357 14 21 28 350.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the left7142128infinite to the right, A(1,9)0.180.160.140.120.10.080.060.047142128infinite to the left, A(9,1)0.160.140.120.10.080.060.04357 14 21 28 350.020357 14 21 28 350.02


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the left714212835infinite to the right, A(1,10)7 14 21 28 350.180.160.140.120.10.080.060.040.020714212835infinite to the left, A(10,1)7 14 21 28 350.160.140.120.10.080.060.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,11)0.2infinite to the left, A(11,1)0.167140.157140.140.120.1210.1210.08280.05280.060.04357 14 21 28 350357 14 21 28 350.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the left7infinite to the right, A(1,12)0.20.180.167infinite to the left, A(12,1)0.160.14142128357 14 21 28 350.140.120.10.080.060.040.02142128357 14 21 28 350.120.10.080.060.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,13)0.2infinite to the left, A(13,1)0.167140.157140.140.120.1210.1210.08280.05280.060.04357 14 21 28 350357 14 21 28 350.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,14)0.2infinite to the left, A(14,1)0.16714210.150.1714210.140.120.10.08280.05280.060.04357 14 21 28 350357 14 21 28 350.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,15)0.2infinite to the left, A(15,1)0.16770.14140.15140.120.1210.1210.08280.05280.060.04357 14 21 28 350357 14 21 28 350.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,16)0.2infinite to the left, A(16,1)0.16714210.150.1714210.140.120.10.08280.05280.060.04357 14 21 28 350357 14 21 28 350.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,17)0.2infinite to the left, A(17,1)0.167140.157140.140.120.1210.1210.08280.05280.060.04357 14 21 28 350357 14 21 28 350.02


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,18)0.2infinite to the left, A(18,1)0.16714210.150.1714210.140.120.10.08280.05280.060.04357 14 21 28 350357 14 21 28 350.02


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,19)0.20.18infinite to the left, A(19,1)0.167142128357 14 21 28 350.160.140.120.10.080.060.040.027142128357 14 21 28 350.140.120.10.080.060.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConvergenceAccumulations to the right and to the leftinfinite to the right, A(1,20)0.20.18infinite to the left, A(20,1)0.167142128357 14 21 28 350.160.140.120.10.080.060.040.027142128357 14 21 28 350.140.120.10.080.060.040.020


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundA Small ImprovementQuestionsWhen does min + A(t i+1 , t i ) ≥ δ i with ∑ δ i = ∞ hold?What do we need to assume for min + A(t)?First idea: min + A(t) > δ uniform for all t. It holdsmin + (A t · · · A s ) ≥ min + A t · · · min + A s > δ t−sBut this is not enough.


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundA Small ImprovementQuestionsWhen does min + A(t i+1 , t i ) ≥ δ i with ∑ δ i = ∞ hold?What do we need to assume for min + A(t)?First idea: min + A(t) > δ uniform for all t. It holdsmin + (A t · · · A s ) ≥ min + A t · · · min + A s > δ t−sBut this is not enough.


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundA Small ImprovementQuestionsWhen does min + A(t i+1 , t i ) ≥ δ i with ∑ δ i = ∞ hold?What do we need to assume for min + A(t)?First idea: min + A(t) > δ uniform for all t. It holdsmin + (A t · · · A s ) ≥ min + A t · · · min + A s > δ t−sBut this is not enough.


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundA Small ImprovementCounter-Example of Moreau 33 Stability of Multiagent Systems With Time-Dependent CommunicationLinks, IEEE Transactions on Automatic Control, 2005


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundA Small ImprovementConditions to ensure convergence 4If min + A(t) > δ we need eithersymmetry of the zero pattern for all A(t) orbounded intercommunication intervals t i+1 − t i < N for alli ∈ N 0to ensure min + A(t i+1 , t i ) ≥ ˆδ i with ∑ ∞ ˆδ i=1 i = ∞But a small improvement is possible.4 J. Lorenz 2003 ,Moreau 2005,Hendrickx and Blondel 2005


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundA Small ImprovementConditions to ensure convergence 4If min + A(t) > δ we need eithersymmetry of the zero pattern for all A(t) orbounded intercommunication intervals t i+1 − t i < N for alli ∈ N 0to ensure min + A(t i+1 , t i ) ≥ ˆδ i with ∑ ∞ ˆδ i=1 i = ∞But a small improvement is possible.4 J. Lorenz 2003 ,Moreau 2005,Hendrickx and Blondel 2005


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundA Small ImprovementSmall growing of intercommunication boundsPropositionLet 0 < δ < 1 and a ∈ R >0 then∞∑δ a log(n) = ∞ ⇐⇒ δ ≥ e −1 .n=1PropositionLet 0 < δ < 1 and a ∈ R >0 then∞∑δ a log(log(n)) = ∞.n=37060504030200 200 400 600 800 1000


Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics backgroundConclusionThe take-awayThe positive diagonal delivers strong result about theconvergence of the zero structure.Assuming that the positive minimum in all confidencematrices is bigger than δ, a growing of the time to reachmaximal connectivity t i+1 − t i as quick as log(log(i)) isacceptable to ensure convergence.But this acceptable growing is really very slow.More information on opinion dynamics: www.janlo.de

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