5. Continuing with the diffusion model from Chapter 19, recall that the threshol
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Question
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.
19.8. EXERCISES 607 2 9 8 6 7 4 12 10 15 13 14 16 Figure 19.30: A social network on which a new behavior diffuses. In your answer, either provide a formula for a threshold q in terms of a, b, and ; or else explain why in this more general model, a node's decision can't be expressed as a threshold in this way.Explanation / Answer
Defination:
The Diffusion model can be obtained by Game theory using Information social network analysis node behaviors. The following are set of rules & regulations for implementations with different values of nodes generated in the social network analysis. The Information graphical representation techniques describes nodes behaviors and provides details study about Information & diffusion model basis interactions & evolutionary graph theory. It Consists of the following set of techinical representations.
Algorthim Execises:
Line 1: INPUT - Enter the total number of nodes;
Line 2: Get the nodes behavoris v & w;
Line 3: If v and w both adopt behavior A, they each get a payoff of a > 0;
Line 4 : If they both adopt B, they each get a payoff of b > 0;
Line 5: If they adopt opposite behaviors, they each get a payoff of 0 ;
Line 6 : 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;
Line 7: OUTPUT - It provide a equational formula for a threshold q in terms of a, b, and x ;
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Description:
Networked Coordination Game general formula for the equation to diffusion model theory with graph G = (V, W). With respect to 'v' rows & 'w' columns.
Payoff Matrix can be written has follows for (a) , (b) & (c)
Coordinating Game G = (A,B) w.r.t. ' v' rows & 'w' columns.
In given set of rows 'v' and columns 'w' we can consiered the following behaviors, Neighbors adopt A, and some adopt B.
The relation between the payoff matrics values a and b.
In representation of that a p fraction of 'v' neighbors have behavior A, and a (1 ? p) fraction have behavior B, that is, if v has d neighbors, then pd adopt A and (1 ? p)d adopt B.
In given methods that gets a payoff of pda value and if it chooses B, it gets a payoff of (1 ? p)db
pda ? (1 ? p)db
p ? b /a + b
Cascading Behavior : In any network, there are two obvious equilibria to this networkwide coordination game: one in which everyone adopts A, and another in which everyone adopts B. From the to “tip” the network from one of these equilibria to the other. Thus to “intermediate” equilibria look like — states of coexistence where A is adopted in some parts of the network and B is adopted.
Thus, we consider the following type of situation. If that everyone in the network is initially using B as a default behavior. Then, a small set of “initial adopters” all decide to use A. We will assume that the initial adopters have switched to A for some reason outside the definition of the coordination game — they have somehow switched due to a belief in A’s superiority, rather than by following payoffs — but we’ll assume that all other nodes continue to evaluate their payoffs using the coordination game. Given the fact that the initial adopters are now using A, some of their neighbors may decide to switch to A & B, and then some of their neighbors might, In a potentially cascading techniques. It does this result in every node generation in the entire network eventually switching over to A. when this isn’t the result, what causes the spread of A to stop. The answer will depend on the network structure, the choice of initial adopters, and the value of the threshold q that nodes use for deciding whether to switch to A.
By consider a set of initial adopters (A,B) who start with a new behavior A, while every other node starts with behavior B. Nodes then repeatedly evaluate the decision to switch from B to A using a threshold of q.
Consider Example: The coordination game is set up so that a = 3 and b = 2; that is, the payoff to nodes interacting using behavior A is 3/2 times what it is with behavior B. Using the threshold formula, we see that nodes will switch from B to A. if at least a q = 2/(3 + 2) = 2/5 fraction of their neighbors are using A.
A social Network diffuse behavior
Claim: Consider a set of initial adopters of behavior A, with a threshold of q for nodes in the remaining network to adopt behavior A.
(i) If the remaining network contains a cluster of density greater than 1 ? q, then the set of initial adopters will not cause a complete cascade.
(ii) Moreover, whenever a set of initial adopters does not cause a complete cascade with threshold q, the remaining network must contain a cluster of density greater than 1 ? q.
(a,a) (0,0) (0,0) (b,b)Related Questions
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