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Fig. 1 | Computational Social Networks

Fig. 1

From: Spheres of legislation: polarization and most influential nodes in behavioral context

Fig. 1

LIG example. A four-node LIG is shown here. The directed edges are labeled with influence levels. Any absence of a directed edge implies an influence level of 0. Threshold values of 0 (for simplicity) are shown with a connector to each node. We assume binary actions \(\{1, -1\}\). In this game, nodes A and B playing 1 and nodes C and D playing \(-1\) is a pure strategy Nash equilibrium (PSNE). To see this, consider node A first. We add up the incoming influences from those nodes (in this case, B) that are playing 1 and then subtract from it the influences coming from nodes (in this case, C and D) playing \(-1\). We get \(1 - (-2 -1.5) = 4.5\), which is basically the total weighted influence on A. Since 4.5 is greater than A’s threshold of 0, A’s best response is 1. Similarly, it can be shown that B, C, and D’s best responses are \(1, -1, -1\), respectively, and therefore, this is a PSNE. Similarly, nodes A and B playing \(-1\) and C and D playing 1 is another PSNE. As a negative example, all nodes playing 1 is not a PSNE. To see this, consider node A. The total weighted influence on A is \(1 + (-2) + (-1.5) = -2.5\), which is less than A’s threshold of 0. Therefore, A’s best response is to play \(-1\), which violates the mutual best response condition for PSNE

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