Opinion Dynamics on Multiple Interdependent Topics: Modeling and - PowerPoint PPT Presentation

Multiple Interdependent Topics Modeling for the update of Hyosung’s opinions Game: S_g = 0.7 vs. H_g = 0.3 0.7-0.3 Positive effect (coupling) vs. negative coupling + S_g – H_g ? S_b – H_b S_c – H_c Magnitude: inverse relationship of abs(S_g – H_g) Positive effect: sign(S_b- H_b) Magnitude: or, proportional to abs(S_g – H_g) Negative effect: -sign(S_b- H_b) Positive coupling (S_g = 0.7, H_g = 0.3) vs. (S_g = 0.49, H_g = 0.51) (Less close) Almost agreement (close)

Multiple Interdependent Topics Modeling for the update of Hyosung’s opinions It may be complicated! Positive effect (coupling) vs. negative coupling S_g – H_g ? S_b – H_b S_c – H_c Magnitude: inverse relationship of abs(S_g – H_g) Positive effect: sign(S_b- H_b) Magnitude: or, proportional to abs(S_g – H_g) Negative effect: -sign(S_b- H_b)

Multiple Interdependent Topics What happens?  Interdependent ? ? ? S_g – H_g S_b – H_b (t,x) ….. S_c – H_c ? Time and state dependent…. general matrix…

Multiple Interdependent Topics What happens?  Interdependent ? ? ? S_g – H_g S_b – H_b (t,x) ….. S_c – H_c ? Time and state dependent…. general matrix… Nomin inal m l model o l or lin lineariz izatio ion or so some sp specif ific ic for orms…

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 S_g – H_g S_b – H_b >0 S_c – H_c >0 Fixed matrix elements  Linea earized ed i inter erdep epen enden ent model el ar around a n a nominal al ( (temporal al-in instant) s socia ial l opin inio ion network!

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 S_g – H_g S_b – H_b >0 S_c – H_c >0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning?

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Case 1: S_g – H_g <0, S_b- H_b <0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Anti (or non)-cooperative Case 1: S_g – H_g <0, S_b- H_b <0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Anti (or non)-cooperative Case 1: S_g – H_g <0, S_b- H_b <0 Case 2: S_g – H_g <0, S_b- H_b >0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Anti (or non)-cooperative Case 1: S_g – H_g <0, S_b- H_b <0 Case 2: S_g – H_g <0, S_b- H_b >0 cooperative

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Anti (or non)-cooperative Case 1: S_g – H_g <0, S_b- H_b <0 Case 2: S_g – H_g <0, S_b- H_b >0 cooperative cooperative Case 3: S_g – H_g >0, S_b- H_b <0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Anti (or non)-cooperative Case 1: S_g – H_g <0, S_b- H_b <0 Case 2: S_g – H_g <0, S_b- H_b >0 cooperative cooperative Case 3: S_g – H_g >0, S_b- H_b <0 Anti (or non)-cooperative Case 4: S_g – H_g >0, S_b- H_b >0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 >0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning?

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 >0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Case 1: S_g – H_g <0, S_b- H_b <0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 >0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Cooperative Case 1: S_g – H_g <0, S_b- H_b <0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 >0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Cooperative Case 1: S_g – H_g <0, S_b- H_b <0 Case 2: S_g – H_g <0, S_b- H_b >0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 >0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Cooperative Case 1: S_g – H_g <0, S_b- H_b <0 Case 2: S_g – H_g <0, S_b- H_b >0 Anti (or non)-cooperative

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 >0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Cooperative Case 1: S_g – H_g <0, S_b- H_b <0 Case 2: S_g – H_g <0, S_b- H_b >0 Anti (or non)-cooperative Anti (or non)-cooperative Case 3: S_g – H_g >0, S_b- H_b <0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 >0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 Ex. -1  what is the physical meaning? Cooperative Case 1: S_g – H_g <0, S_b- H_b <0 Case 2: S_g – H_g <0, S_b- H_b >0 Anti (or non)-cooperative Anti (or non)-cooperative Case 3: S_g – H_g >0, S_b- H_b <0 Cooperative Case 4: S_g – H_g >0, S_b- H_b >0

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 What is the optimal way (ex. change minimum number of elements) for a consensus (or cluster consensus)?

Multiple Interdependent Topics Deterministic model – Static case! What happens?  Interdependent >0 <0 <0 How (what values) to S_g – H_g design the elements of S_b – H_b matrix weights for a >0 perfect consensus (or S_c – H_c cluster consensus)? >0 What is the optimal way (ex. change minimum number of For example, chang nge yo your ur m mind nd f for the g e game e for or elements) for a a c complet ete e consen ensus..^^ consensus (or cluster consensus)?

Independent Update Topics-1 Topics-2 Independent update x : Topics- m

Multiple Interdependent Topics x

Multiple Interdependent Topics x

Multiple Interdependent Topics Interdependent update x

Scalar vs. Matrix vs.

Scalar vs. Matrix vs. Connected? - Positive connected - Semi-positive connected

Scalar vs. Matrix vs. More g general, l, r realis listic ic, but complic licated differ eren ent p phen enomen enon Connected? - Positive connected - Semi-positive connected

Scalar vs. Matrix vs. More g general, l, r realis listic ic, but complic licated differ eren ent p phen enomen enon Connected? - Positive connected Clusters - Semi-positive connected

Physical Meaning of P .D and P .S.D football Junkfod paper baseball North Korea drama football homework football game exercise drink Brushing teeth conference chocolate Father(advisor)

Physical Meaning of P .D and P .S.D Not many common interests football Junkfod paper baseball North Korea drama football homework football game exercise drink Brushing teeth conference chocolate Father(advisor)

Physical Meaning of P .D and P .S.D Not many common interests  Posit sitiv ive se semi-def efinite ! e ! football Junkfod paper baseball North Korea drama football homework football game exercise drink Brushing teeth conference chocolate Father(advisor)

Physical Meaning of P .D and P .S.D drama drama exercise paper homework homework drama homework exercise drama drama homework drink exercise homework homework homework

Physical Meaning of P .D and P .S.D Many common interests drama drama exercise paper homework homework drama homework exercise drama drama homework drink exercise homework homework homework

Physical Meaning of P .D and P .S.D Many common interests  Posit sitiv ive defin init ite ! drama drama exercise paper homework homework drama homework exercise drama drama homework drink exercise homework homework homework

Part-2: Analysis Problem 1-Fixed Matrices (Typical consensus-based ideas- Linearized/ Nominal or, positive & negative mixed)

Model 1- static case

Model 1- static case

Model 1- static case

Model 1- static case Def.: Clusters & cluster consensus (clustered opinions)

Model 1- static case Def.: Clusters & cluster consensus (clustered opinions)

Model 1- static case x x x o o x x o o Def.: Clusters & cluster consensus (clustered opinions) o > > > > > > >

Model 1- static case x x x o o x x o o Def.: Clusters & cluster consensus (clustered opinions) o > > > > > > >

Model 1- static case x x x x o o x x o o Def.: Clusters & cluster consensus (clustered opinions) o o > > > > > > > >

Model 1- static case Def.: Clusters & cluster consensus (clustered opinions) Property: Null space of Laplacian

Model 1- static case Def.: Clusters & cluster consensus (clustered opinions) Property: Null space of Laplacian

Model 1- static case Def.: Clusters & cluster consensus (clustered opinions) A sole null space of scalar consensus Property: Null space of Laplacian

Model 1- static case Def.: Clusters & cluster consensus (clustered opinions) A sole null space of scalar consensus Property: Null space of Laplacian Additional null space !

Model 1- static case Def.: Positive (semi-) connected Def.: Clusters & cluster consensus (clustered opinions) Property: Null space of Laplacian

Model 1- static case Def.: Positive (semi-) connected Def.: Clusters & cluster consensus (clustered opinions) Property: Null space of Laplacian

Model 1- static case Def.: Positive (semi-) connected Def.: Clusters & cluster consensus (clustered opinions) Thm.: Exact condition for a consensus Property: Null space of Laplacian

Model 1- static case Def.: Positive (semi-) connected Def.: Clusters & cluster consensus (clustered opinions) Thm.: Exact condition for a consensus Property: Null space of Laplacian No other null space!

Model 1- static case Def.: Positive (semi-) connected Def.: Clusters & cluster consensus (clustered opinions) Thm.: Exact condition for a consensus Property: Null space of Laplacian

Model 1- static case Def.: Positive (semi-) connected Def.: Clusters & cluster consensus (clustered opinions) Thm.: Exact condition for a consensus Property: Null space of Laplacian

Model 1- static case Def.: Positive (semi-) connected Def.: Clusters & cluster consensus (clustered opinions) Connected ! Thm.: Exact condition for a consensus Property: Null space of Laplacian

Model 1- static case Def.: Positive (semi-) connected Def.: Clusters & cluster consensus (clustered opinions) Thm.: Exact condition for a consensus Property: Null space of Laplacian

Model 1- static case Def.: Positive (semi-) connected Def.: Clusters & cluster consensus (clustered opinions) Thm.: Exact condition for a consensus Property: Null space of Laplacian Clusters!

Model 1- static case EX-1:

Model 1- static case EX-1: Positive semidefinite

Model 1- static case EX-1: Positive semidefinite

Model 1- static case EX-1: Path 1 Path 2 Positive semidefinite

Model 1- static case EX-1: Path 1 Path 2 Path 1 + path 2 = positive path

Model 1- static case EX-1: EX-3: EX-2:

Model 1- static case EX-1: EX-3: EX-2: EX-4:

Model 1- static case EX-1: EX-3: EX-2: EX-4: