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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

**and** the opinion dynamics backgroundThe positive diagonalProofBA = (B diag + B offdiag )Ahas at least the same entries positive as A

**and** the opinion dynamics backgroundThe positive diagonalProofBA = (B diag + B offdiag )Ahas at least the same entries positive as A

**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