From 29a6a98bbda1f33aad5017d15bcaf72e2f12e28e Mon Sep 17 00:00:00 2001 From: Prabhat Agarwal Date: Tue, 26 Nov 2019 16:27:16 -0800 Subject: [PATCH] Add notes for decision based models (#1) --- assets/img/decision_model_1.png | Bin 0 -> 10559 bytes assets/img/decision_model_2.png | Bin 0 -> 60680 bytes assets/img/decision_model_3.png | Bin 0 -> 11229 bytes assets/img/decision_model_4.png | Bin 0 -> 54152 bytes index.md | 2 +- .../network-effects-and-cascading-behavior.md | 62 ++++++++++++++++++ 6 files changed, 63 insertions(+), 1 deletion(-) create mode 100644 assets/img/decision_model_1.png create mode 100644 assets/img/decision_model_2.png create mode 100644 assets/img/decision_model_3.png create mode 100644 assets/img/decision_model_4.png create mode 100644 network-methods/network-effects-and-cascading-behavior.md diff --git a/assets/img/decision_model_1.png b/assets/img/decision_model_1.png new file mode 100644 index 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z1(=>>ck7r+?+5Yti^n?*X3fT4?c-u&lc~$c<%B+Fl4|91q-WNK&6<(xK>B&xvbnrC7Ma8qGtCS=e)t74d{NsvNKn3GMP-jFC@nok5JVOgD{OWtM0C3>{xtrK&cWc8!l;dpqW34(Qcxe zqGx6GJ>M*z7^3{w`UA3>2KwWOyYVgN6J)gE(n1kU%6~V(AAM;??l{oNYOM99{OEMp zOGOiWf5jUs`AqJMKUaRFu64UpM)_!s*YB(P61E}7^c%VjMHpPEQGr5!G%k%8+MiW^ zinTj`wW`;JJwX*U;%anhp+c$hA#35F(OrjkYAjFm%sxW`RQ3F32x=uIXKiJ>QsOUW z!Y_&)m-}rBRyFp=*)G`hoCFEdqY6UsgExX+s~FYZft+>dTB!`tcr)hoQM$YKGJciI zUo+T~wa3sDMGqHDc^=!rx|3PB^oLUgzlaJ?zVZ#e5#;s%l_d)*f+ b(1-p)d \implies p > \frac{b}{a+b} = q$$(threshold) + +### Case Study: [Modelling Protest Recruitment on social networks](https://arxiv.org/abs/1111.5595) +Key Insights: +- Uniform activation threhold for users, with two peaks +- Most cascades are short +- Successful cascades are started by central users + +#### Note: +**k-core decomposition**: biggest connected subgraph where every node has at least degree k (iteratively remove nodes with degree less than k) + +### Multiple Contagion Model +There are two contagions $$A$$ and $$B$$ in the network and initially every node has behavior $$B$$. In this case a node can have both behavior $$A$$ and $$B$$ at a total cost of $$c$$ (over all interactions). The payoff matrix is given as: + +| | A | B | AB | +|---|---|---|----| +| A | a | 0 | a | +| B | 0 | b | b | +| AB| a | b | max(a,b)| + +### Example: Infinite Line graph +**Case 1**:**A-w-B** +![decision_case_1](../assets/img/decision_model_1.png?style=centerme) + +Payoffs for $$w$$: $$A: a$$, $$B: 1$$, $$AB: a+1-c$$ + +![decision_case_2](../assets/img/decision_model_2.png?style=centerme) + +**Case 1**: **AB-w-B** +![decision_case_3](../assets/img/decision_model_3.png?style=centerme) + +Payoffs for $$w$$: $$A: a$$, $$B: 1$$, $$AB: max(a, 1) + 1 -c$$ + +![decision_case_4](../assets/img/decision_model_4.png?style=centerme) +