Question

From Jon Kleinberg's "Networks, Crowds, and Markets":

19.8 #5.) Continuing with the diffusion model from Chapter 19, recall that the threshold q was derived from a coordination game that each node plays with each of its neighbors.

Specifically, if nodes v and w are each trying to decide whether to choose behaviors A and B, then:

• if v and w both adopt behavior A , they each get a payoff of a > 0;

• if they both adopt B, they each get a payoff of b > 0; and

• if they adopt opposite behaviors, they each get a payoff of 0.

The total payoff for any one node is determined by adding up the payoffs it gets from the coordination game with each neighbor. Let’s now consider a slightly more general version of the model, in which the payoff for choosing opposite behaviors is not 0, but some small positive number x. Specifically, suppose we replace the third point above with:

• if they adopt opposite behaviors, they each get a payoff of x , where x is a positive number that is less than both a and b.

Here’s the question: in this variant of the model with these more general payoffs, is each node’s decision still based on a threshold rule? Specifically, is it possible to write down a formula for a threshold q , in terms of the three quantities a , b , and x , so that each node v will adopt behavior A if at least a q fraction of its neighbors are adopting A , and it will adopt B otherwise?

In your answer, either provide a formula for a threshold q in terms of a, b, and x; or else explain why in this more general model, a node's decision can't be expressed as a threshold in this way.

Answer 1

ANSWER:

The General formula for Network Coordinate game for the equation to diffusion model theory with Graph

• G= (V, W).
• Where V=>rows and W=>Columns
• Now PayOff Matrix can be rewritten as follows for a, b and c
• Coordinationg Game G=(A,B) with respect to v rows and w columns

  

(a,a)	(0,0)
(0,0)	(b,b)

• In this given set of rows v and columns w , we can consider following behaviours as Neighbours adopt A and for some adopt B.
• The relation between the payoff matrices values a and b are:

• In this representation of that a p fraction of V neighbors have behavior A and a (1?p) fraction have behavior B

  i.e. if V has neighbors then pd adopt A and (1?p)d adopt B

• in this methods, for a given constraints, which gets payoff of pda value and if its choosen as B which gets a payoff of (1?p)db pda ? (1?p)db i.e. p?b/a+b

• Cascading behavior : In a typical Network wide coordinated Game like this, there are two possibilities either every one adopts A and on the other hand remaining adopts B. from the origin point of network till the the end in order to have a equilibrium between the coexistence of A and B in some parts of As adoption B is also adopted..

• In this methods, for a given contains, which gets payoff, pda value and nits choose as B which gets a payoff of (1?p)db pda (1?p)db i.e. P?b/a +b

• Cascading behavior: In a typical Network wide coordinated Game like this, then e are two possibilities either every one adopts A and on the other hand remaining adopts B. from the origin point of network till the end in or Oder to have a equilibrium between the coexistence of A and B in some parts of As adoption B is also adopted..

• hence by considering all set of Initial adopter(A,B) which will start a new behavior A while every other nodes will starts with other one i.e. behavior B. these nodes are repeatedly evaluate the decision to switch from B to A using threshold of q.