Diffusion and Propagation Social and Economic Networks Jafar Habibi - PowerPoint PPT Presentation

Diffusion and Propagation Social and Economic Networks Jafar Habibi MohammadAmin Fazli Social and Economic Networks 1

ToC • Diffusion and Propagation • Information Cascades • Cascading Behavior • Epidemics • Readings: • Chapter 7 from the Jackson book • Chapter 16 from the Kleinberg book • Chapter 19 from the Kleinberg book • Chapter 21 from the Kleinberg book Social and Economic Networks 2

Diffusion • Why we follow the crowd? How people influence each other? • Random • Direct benefit • Epidemics • Information cascade • … • How network structure affects the diffusion? • Many models are proposed to study diffusion behavior in social networks Social and Economic Networks 3

Information Cascade • An experiment: • Consider we have an urn containing 3 marbles colored red or blue. • With probability 50% it contains 2 red and 1 blue (majority-red) and with 50% 1 red and 2 blue (majority-blue) • People should sequentially draws a marble from the urn and look at the color and place it back without showing it to others. • Then they should guess whether the urn is majority-blue or majority-red and publicly announce their guess. • The first person: Pr 𝑛𝑏𝑘𝑝𝑠𝑗𝑢𝑧 − 𝑐𝑚𝑣𝑓 𝑐𝑚𝑣𝑓 = 2 3 • The second person: Pr 𝑛𝑏𝑘𝑝𝑠𝑗𝑢𝑧 − 𝑐𝑚𝑣𝑓 𝑐𝑚𝑣𝑓, 𝑐𝑚𝑣𝑓 = 4 5 Social and Economic Networks 4

Information Cascade • The Model (Wit Bayes ’ Rule): • Consider a group of people (numbered 1 , 2 , 3 , . . . ) who will sequentially make decisions about accepting or rejecting an option (an idea) • The world has a state G (good) or B (Bad) Pr 𝐻 = 𝑞, Pr 𝐶 = 1 − 𝑞 • If the option is a good idea, the payoff from accepting it is 𝑤 𝑕 > 0 , and if the option is a bad idea, accepting it has payoff of 𝑤 𝑐 < 0 . The payoff of rejection is always 0 𝑞𝑤 𝑕 + 1 − 𝑞 𝑤 𝑐 = 0 , • People receive signals about the state of the world: H (high) or L (low) Pr 𝐼 𝐻 = Pr 𝑀 𝐶 = 𝑟 > 1 2 Social and Economic Networks 5

Information Cascades • Individual decisions: • Suppose that a person gets a high signal. This shifts their expected payoff from 𝑤 𝑕 Pr 𝐻 + 𝑤 𝑐 Pr 𝐶 = 0 to 𝑤 𝑕 Pr 𝐻 𝐼 + 𝑤 𝑐 Pr[𝐶|𝐼] Social and Economic Networks 6

Information Cascades • Decisions with multiple signals: • When a person receives the signal set S with a high signals and b low signals: • the posterior probability Pr [G | S] is greater than the prior Pr [G] when a > b; • the posterior Pr [G | S] is less than the prior Pr [G] when a < b • the two probabilities Pr [G | S] and Pr [G] are equal when a = b • For comparison we replace 1 − 𝑞 𝑟 𝑏 1 − 𝑟 𝑐 and get Pr 𝐻 𝑇 = 𝑞 . The question is this replacement makes it larger or smaller? Social and Economic Networks 7

Information Cascade • Cascades can be wrong • Cascades can be based on very little information • Cascades are fragile Social and Economic Networks 8

Modeling Diffusion Through a Network • A networked coordination game: • A is the better choice if 𝑞𝑒𝑏 ≥ 1 − 𝑞 𝑒𝑐 𝑐 𝑞 ≥ 𝑏 + 𝑐 Social and Economic Networks 9

Modeling Diffusion Through a Network Social and Economic Networks 10

Modeling Diffusion Through a Network • Cascading Behavior: • Initially all nodes adopt B • A set of initially adopters all decide to use A and don ’ t change their decision to the end • All other players continue to evaluate their payoffs using the coordination game • Complete Cascades: • If the resulting cascade of adoptions of A eventually causes every node to switch from B to A, then we say that the set of initial adopters causes a complete cascade at threshold q Social and Economic Networks 11

Modeling Diffusion Through a Network • Cascades & Clusters: • We say that a cluster of density p is a set of nodes such that each node in the set has at least a p fraction of its network neighbors in the set. • Theorem: Consider a set of initial adopters of behavior A, with a threshold of q for nodes in the remaining network to adopt behavior A. • 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. • 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 • Proof: see the blackboard. Social and Economic Networks 12

The Role of Weak Ties • Weak ties are powerful ways to convey awareness of new things, but they are weak at transmitting behaviors that are in some way risky or costly to adopt Social and Economic Networks 13

Modeling Diffusion Through a Network • An extension: Heterogeneous Thresholds Social and Economic Networks 14

Modeling Diffusion Through a Network • An extension: The bilingual option • A game about learning languages • Learning each language benefits us according to the table • Learning an additional language costs us c • Example: a = 3, b=2, c =1 Social and Economic Networks 15

The Cascade Capacity • Defined for infinite networks • The cascade capacity of the network is the largest value of the threshold q for which some finite set of early adopters can cause a complete cascade. Social and Economic Networks 16

The Cascade Capacity • Theorem: There is no network in which the cascade capacity exceeds 1 2 • Proof: See on the blackboard Social and Economic Networks 17

The Cascade Capacity • The cascade capacity in the bilingual model: • Let ’ s study the infinite path graph • WLOG we assume that b = 1 • Two typical moves in the evolution from B to A or AB Social and Economic Networks 18

The Cascade Capacity • The cascade capacity in the bilingual model: • Two typical moves in the evolution from B to A or AB Social and Economic Networks 19

The Cascade Capacity • The cascade capacity in the bilingual model: • In overall we have: Social and Economic Networks 20

Epidemics • A branching process for diffusion of diseases: • Each person meets k another persons • Each infected person transmits his disease to each person he meets independently with a probability of p Social and Economic Networks 21

Analysis of Branching Process • The basic reproductive number: 𝑆 0 = 𝑙𝑞 • Theorem: If R 0 < 1, then with probability 1, the disease dies out after a finite number of waves. If R 0 > 1, then with probability greater than 0 the disease persists by infecting at least one person in each wave. • Proof: see the blackboard • If q n denotes the probability that the epidemic survives for at least n waves, X n denotes the number of infected nodes in the n ’ th wave, we have: 𝐹 𝑌 𝑜 = 𝑞 𝑜 𝑙 𝑜 = 𝑆 0 𝑜 𝐹 𝑌 𝑜 = Pr 𝑌 𝑜 ≥ 1 + Pr 𝑌 𝑜 ≥ 2 + ⋯ 𝐹 𝑌 𝑜 ≥ lim 𝑟 𝑜 = 𝑟 ∗ 𝑟 ∗ ≤ 𝑆 0 𝑜 Social and Economic Networks 22

Analysis of Branching Process • The previous theorem may not work for non-tree structures: 2 • 𝑞 = 3 4 • 𝑆 0 = 𝑙𝑞 = 3 > 1 • We can easily see that the process will terminate • The probability that a wave do not get inf Social and Economic Networks 23

Analysis of Branching Process • A formula for q n : • 𝑟 𝑜 = 1 − 1 − 𝑞𝑟 𝑜−1 𝑙 • 𝑔 𝑦 = 1 − 1 − 𝑞𝑦 𝑙 • We have the following sequence for q n 1, 𝑔 1 , 𝑔 2 1 , 𝑔 3 1 , … . Social and Economic Networks 24

The SIR Epidemic Model • 3 states for each node: • Susceptible (S): Before the node has caught the disease, it is susceptible to infection from its neighbors. • Infectious (I): Once the node has caught the disease, it is infectious and has some probability of infecting each of its susceptible neighbors. • Removed (R): After a particular node has experienced the full infectious period, this node is removed from consideration, since it no longer poses a threat of future infection. • The total process • Initially, some nodes are in the I state and all others are in the S state. • Each node v that enters the I state remains infectious for a fixed number of steps t I • During each of these t I steps, v has a probability p of passing the disease to each of its susceptible neighbors. • After t I steps, node v is no longer infectious or susceptible to further bouts of the disease Social and Economic Networks 25

The SIR Epidemic Model Social and Economic Networks 26