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
• ption is a bad idea, accepting it has payoff of 𝑤𝑐 < 0. The payoff of rejection

is always 0 𝑞𝑤𝑕 + 1 − 𝑞 𝑤𝑐 = 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 + 𝑤𝑐 Pr 𝐶 = 0 to 𝑤𝑕 Pr 𝐻 𝐼 + 𝑤𝑐Pr[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 𝑟𝑏 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
• n 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 𝑞 ≥ 𝑐 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
• ption
• 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

• f 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: 𝑆0 = 𝑙𝑞
• Theorem: If R0 < 1, then with probability 1, the disease dies out after

a finite number of waves. If R0 > 1, then with probability greater than 0 the disease persists by infecting at least one person in each wave.

• Proof: see the blackboard
• If qn denotes the probability that the epidemic survives for at least n waves,

Xn denotes the number of infected nodes in the n’th wave, we have: 𝐹 𝑌𝑜 = 𝑞𝑜𝑙𝑜 = 𝑆0

𝑜

𝐹 𝑌𝑜 = Pr 𝑌𝑜 ≥ 1 + Pr 𝑌𝑜 ≥ 2 + ⋯ 𝐹 𝑌𝑜 ≥ lim 𝑟𝑜 = 𝑟∗ 𝑟∗ ≤ 𝑆0

𝑜

Social and Economic Networks 22

Analysis of Branching Process

• The previous theorem may not work for non-tree structures:
• 𝑞 =

2 3

• 𝑆0 = 𝑙𝑞 =

4 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 qn:
• 𝑟𝑜 = 1 − 1 − 𝑞𝑟𝑜−1 𝑙
• 𝑔 𝑦 = 1 − 1 − 𝑞𝑦 𝑙
• We have the following sequence for qn

1, 𝑔 1 , 𝑔2 1 , 𝑔3 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 tI
• During each of these tI steps, v has a probability p of passing the disease to each of

its susceptible neighbors.

• After tI 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

Percolation

• Percolation theory in physics: A canonical scenario from percolation

theory deals with some substance, which is porous, and a central question is whether a liquid on one side of the substance will penetrate to the other side.

• The same question in the branching process
• We can think the SIR epidemic as a percolation scenario:
• Node v has one chance to infect w and it succeeds with probability p
• We can view the outcome of this random event as being determined by flipping a

coin that has a probability p

• It clearly does not matter whether the coin was flipped at the moment that v first

became infectious. Therefore we can flip a coin for each edge to decide to include the edge for transmission or not.

• A node v will become infected during the epidemic if and only if there is a path to v

from one of the initially infected nodes that consists entirely of open edges

Social and Economic Networks 27

Percolation

• Percolation and SIR model:

Social and Economic Networks 28

Percolation

• Assume that 𝜌 percent of (randomly chosen) people are immune to

infection

• We remove them from the network and study the giant component
• The emergence of the giant component is where

𝑒2 𝜌 = 2 𝑒 𝜌

• For the degree distribution we have:

Social and Economic Networks 29

Percolation

• With some calculation we have:

𝑒 𝜌 = 𝑒 1 − 𝜌 , 𝑒2 𝜌 = 𝑒2 1 − 𝜌 2 + 𝑒 𝜌(1 − 𝜌)

• Thus, we have:

𝜌 = 𝑒2 − 2 𝑒 𝑒2 − 𝑒

• For d-regular networks:
• For d = 2, the graph needs no immunity
• For d = 3, 𝜌 =

1 2, thus if less than half of people get immune the many of them

will get infected

Social and Economic Networks 30

Percolation

• Thus, we have:

𝜌 = 𝑒2 − 2 𝑒 𝑒2 − 𝑒

• For Poisson networks:
• 𝜌 = 1 −

1 𝑜−1 𝑞

• For scale-free networks:
• For 𝛿 < 3, 𝑒2 is diverging and thus 𝜌 = 1, thus every node should get

immune

Social and Economic Networks 31

Percolation

• We can calculate the size of

the giant component from what we learned 1 − 𝑟 =

𝑒

1 − 𝑟 𝑒𝑄

𝜌(𝑒)

Social and Economic Networks 32

Percolation

• Percolation & SIR model (revisit):
• Choosing a random node as an initially infected node
• Remove links from the paths emanating from this initial node, each with

probability (1-t)

• We have:

Which is analogous to the previous distribution with t in place of 1 − 𝜌

Social and Economic Networks 33

Percolation

• Assume that high degree nodes get immune, i.e.

𝑒 𝜌 is such that

𝑒=1 𝑒 𝜌

𝑄 𝑒 = 1 − 𝜌

• The new degree distribution is simply

𝑄 𝑒 1−𝜌 for degrees 0 to

𝑒 𝜌

• We have removed a much larger fraction of links, namely

𝑔 𝜌 = 𝑒=

𝑒 𝜌 +1 ∞

𝑄 𝑒 𝑒 𝑒

Social and Economic Networks 34

Percolation

• Thus the new degree distribution is of the form
• By the same reasoning we have in the previous slides we have:

Social and Economic Networks 35

The SIS Epidemic Model

• Two states for each node: Susceptible (S), Infectious (I)
• 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 tI.

• During each of these tI steps, v has a probability p of passing the disease to

each of its susceptible neighbors.

• After tI steps, node v is no longer infectious, and it returns to the S state.

Social and Economic Networks 36

The SIS Epidemic Model

• The linkage between SIS and SIR:

Social and Economic Networks 37

The SIS Epidemic Model

• Study the SIS model with Mean Field Approximation:
• The probability that a meeting of individual i is with an individual who has

degree d:

𝑄 𝑒 𝑒 𝑒

• Suppose that the fraction of individuals of degree d who are infected is

currently ρ(d). The chance that a given interaction is with an infected individual, denoted 𝜄, can be calculated: 𝜄 = 𝑒 𝑄 𝑒 𝜍 𝑒 𝑒 𝑒

• Consider that this is different from the average infection rate in the

population which is 𝜍 =

𝑒

𝑄 𝑒 𝜍(𝑒)

Social and Economic Networks 38

The SIS Epidemic Model

• Study the SIS model with Mean Field Approximation:
• The chance that a given susceptible individual who has degree d becomes

infected in a given period when faced with a probability 𝜄 that any given meeting is with an infected individual is 𝜉𝜄𝑒

• If we assume that infected nodes become susceptible with rate 𝜀, for the

steady state we have: 1 − 𝜍 𝑒 𝜉𝜄𝑒 = 𝜍 𝑒 𝜀

• Thus we have:

𝜍 𝑒 = 𝜇𝜄𝑒 𝜇𝜄𝑒 + 1 For 𝜇 =

𝜉 𝜄

Social and Economic Networks 39

The SIS Epidemic Model

• Study the SIS model with Mean Field Approximation:
• From 𝜄 =

𝑒 𝑄 𝑒 𝜍 𝑒 𝑒 𝑒

and 𝜍 𝑒 =

𝜇𝜄𝑒 𝜇𝜄𝑒+1, we have:

𝜄 =

𝑒

𝑄 𝑒 𝜇𝜄𝑒2 𝑒 (𝜇𝜄𝑒 + 1)

• This is very hard to find the steady points from the above equality, but we can

explore it for special kinds of graphs

Social and Economic Networks 40

The SIS Epidemic Model

• Study the SIS model with Mean Field Approximation:

𝜄 =

𝑒

𝑄 𝑒 𝜇𝜄𝑒2 𝑒 (𝜇𝜄𝑒 + 1)

• For d-regular graphs:

𝜄 = 𝑒 𝜇𝜄 𝑒 𝜇𝜄 + 1

• And thus two solutions 𝜄 = 0, 𝜄 = 1 −

1 𝜇 𝑒

• For scale free networks, for example for 𝑄 𝑒 = 2𝑒−3 by integrating the rhs:

𝜄 = 1 𝜇(𝑓

1 𝜇 − 1)

Social and Economic Networks 41

The SIS Epidemic Model

• Study the SIS model with Mean Field Approximation:
• Let assume that:

𝐼 𝜄 =

𝑒

𝑄 𝑒 𝜇𝜄𝑒2 𝑒 (𝜇𝜄𝑒 + 1)

• Some general observations:

Social and Economic Networks 42

The SIS Epidemic Model

• Study the SIS model with Mean Field Approximation:
• We have:

𝐼′ 𝜄 = 𝑄 𝑒 𝑒 𝑒 ( 𝜇𝑒 𝜇𝜄𝑒 + 1 2) and thus 𝐼′ 0 = 𝜇 𝑒2 𝑒

Social and Economic Networks 43

The SIRS Epidemic Model

• 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 tI.

• During each of these tI steps, v has a probability p of passing the

disease to each of its susceptible neighbors.

• After tI steps, node v is no longer infectious. It then enters the R state

for a fixed number of steps tR. During this time, it cannot be infected with the disease, nor does it transmit the disease to other nodes. After tR steps in the R state, node v returns to the S state

Social and Economic Networks 44

Synchronization

• When immunity is temporary, the

state of nodes get oscillation shape in very localized parts of the network, with patches of immunity following large number of infections in a concentrated area

• If the number of long-distance

edges grows, more synchronization among nodes’ states can be seen

Social and Economic Networks 45

Transient Contacts

• Sometimes it is important to enrich models with time-related

properties such as a period of time.

• Two study how concurrent activities can affect the diffusion behavior
• Example: HIV epidemic

Social and Economic Networks 46

Single Parent Ancestry Model

Social and Economic Networks 47

Single Parent Ancestry Model

Social and Economic Networks 48

• Where it applies:
• Most directly, it can be used to model species that engage in asexual

reproduction, with each organism arising from a single parent.

• It can be used to model single-parent inheritance even in sexually

reproducing populations, including the inheritance of mitochondrial DNA among women as in our discussion above.

• It can be used to model purely “social” forms of inheritance, such as master

apprentice relationships. For example, if you receive a Ph.D. in an academic field, you generally have a single primary advisor.

• And …

Single Parent Ancestry Model

Social and Economic Networks 49

• The Probability that Lineages Collide in One Step:
• For j lineages the probability of no-collision:

1 − 1 𝑂 1 − 2 𝑂 1 − 3 𝑂 … 1 − 𝑘 − 1 𝑂 = 1 − 1 + 2 + 3 + ⋯ + 𝑘 − 1 𝑂 + 𝑕 𝑘 𝑂2 ≈ 1 − 𝑘 𝑘 − 1 2𝑂 .

• Thus the probability of a collision is

𝑘 𝑘−1 2𝑂

Single Parent Ancestry Model

Social and Economic Networks 50

• The expected time until coalescence
• If we have coin that comes up “head” with

probability p, the expected number of coin flips to see a head is equal to Pr 𝑌 ≥ 1 + Pr 𝑌 ≥ 2 + ⋯ = 1 + 1 − 𝑞 + 1 − 𝑞 2 + ⋯ = 1 1 − 1 − 𝑞 = 1 𝑞

• Thus the expected time until coalescence

2𝑂 2 2 − 1 + 2𝑂 3 3 − 2 + ⋯ + 2𝑂 𝑙 𝑙 − 1 = 2𝑂(1 − 1 2 + 1 2 − 1 3 + ⋯ + 1 𝑙 − 1 − 1 𝑙 = 2𝑂 1 − 1 𝑙