What Is the Role of Node j in the Consensus Equation for Agent i?

[asd1249jf]

http://img112.imageshack.us/img112/2433/consensusequationdv9.jpg

I am having the most trouble understanding this equation. Why is the limit of summation, declared as a node j in a member of the neighbors of agents i (Which is a set of nodes and links, taken from disk graphs)? Why is it that by the term $$x_j(t)$$, a derivative of X(of the i index) must be taken? Can anyone explain how this equation works?

I am sure that this is a very difficult question and may take time to answer it, but I would heavily appreciate it if anyone can explain it well, as my research is at stake here.

More information about this equation can be found on slide #11 of powerpoint presentation.

http://www.piaggio.ccii.unipi.it/Bertinoro%202007/Materiale%20Didattico/EgerstedtTalk.pdf

 

[HallsofIvy]

If i=1, N₁= {1} so $\overdot{x}$₁= x₁- x₁= 0.
If i= 2, N₂= {1, 2} so $\overdot{x}$₂= (x₃₂- x₁)+ (x₂- x₂= x₂- x₁
If i= , N₃= {1,2,3} so $\overdot{x}$= (x₃- x₁)+ (x₃)+ (x₃- x₂)+ x₃- x₁)= 2x₃- x₁-x₂.
etc.

 

[asd1249jf]

Ok, I stared at what you said for 10 minutes but failed to understand it. Can you please provide additional details?

 

[asd1249jf]

anyone?

 

[asd1249jf]

:( anyone?

 

[asd1249jf]

Please?

 

[EnumaElish]

http://img112.imageshack.us/img112/2433/consensusequationdv9.jpg

I am having the most trouble understanding this equation. Why is the limit of summation, declared as a node j in a member of the neighbors of agents i (Which is a set of nodes and links, taken from disk graphs)? Why is it that by the term $$x_j(t)$$, a derivative of X(of the i index) must be taken? Can anyone explain how this equation works?

I am sure that this is a very difficult question and may take time to answer it, but I would heavily appreciate it if anyone can explain it well, as my research is at stake here.

More information about this equation can be found on slide #11 of powerpoint presentation.

http://www.piaggio.ccii.unipi.it/Bertinoro%202007/Materiale%20Didattico/EgerstedtTalk.pdf

My non-technical understanding is that the change in node i is an average of the distances between i and its neighbors. So, if node i is agnostic in its beliefs, and its distance to node 1_i (who is atheist) is x1_i - x_i = d1_i, and its distance to node 2_i (who is religious) is x2_i - x_i = d2_i, then i's beliefs will change in proportion to the average distance to its neighbors: (d1_i + d2_i)/2. If i's initial distance to 1_i is greater than its initial distance to 2_i, d1_i > d2_i, then in the next period (or iteration) i's beliefs will get closer to 1_i.