The document discusses various models of diffusion, or the spread of social contagions between people. It begins by defining diffusion and providing examples of social contagions like diseases, behaviors, ideas, and attitudes. It then describes several types of diffusion models, including epidemic models, threshold models, and weighted averaging models. Threshold models in particular account for the idea that people must be exposed to a contagion by enough friends before adopting it themselves. The document also discusses issues in distinguishing social influence from other factors like selection bias and issues in modeling non-biological contagion processes. It proposes the social space diffusion model to address limitations in observed networks by modeling diffusion over latent social spaces.

1. Applications of networks: diffusion
Jake Fisher
Social Networks & Health | May 25, 2017

2. What is diffusion?
Diffusion is the process of social contagions spreading between people

3. What is diffusion?
Social contagion means anything that people could transmit to one another
Diseases
• HIV (Merli et al. 2006)
Behaviors
• Binge drinking (Kreager & Haynie 2011)
• Obesity (Christakis & Fowler 2007, but see Lyons 2010)
• Exercise (Aral & Nicolaides 2017)
• Use of a new medicine (Coleman, Katz, Menzel 1966) or an innovation generally
(Rogers 1962)
Ideas / Attitudes
• Beliefs about concurrency (Yamanis et al. 2015)
• Beliefs about smoking (Fisher 2017)

4. Types of diffusion models: epidemic model
Compartment models imply that people interact randomly
Population
Both infected: nothing happens
Both susceptible: nothing happens
Discordant: susceptible person gets
infected with a given probability

5. Types of diffusion models: epidemic model
Random mixing models can be represented by a set of differential eqs.
S I
𝛽
R
𝛾
𝑑𝑆
𝑑𝑡
= −𝛽𝑆𝐼
𝑑𝐼
𝑑𝑡
= 𝛽𝑆𝐼 − 𝛾𝐼
𝑑𝑅
𝑑𝑡
= 𝛾𝐼

6. Types of diffusion models: epidemic model
Disease models lead to a logistic growth curve

7. Types of diffusion models: epidemic model
Network models generalize by allowing any specific contact pattern
Discordant edge:
susceptible person is
infected with some
probability
Both susceptible:
nothing happens

8. Types of diffusion models: epidemic model
Network models generalize by allowing any specific contact pattern
Direct transmission
no longer possible!

9. Relationship timing
Time ordering of ties matters!
Time 1 Time 2
Time 2 Time 1

10. Relationship timing
The “network” conflates three relevant networks…
Who can “A” reach?
(1) Contact network: A can reach everyone, it is a connected component

11. Relationship timing
The “network” conflates three relevant networks…
Who can “A” reach?
(2) Exposure network: node “A” could reach up to 8 others

12. Relationship timing
The “network” conflates three relevant networks…
Who can “A” reach?
(3) Transmission network: upper limit is 8 through the exposure links (dark blue).
Transmission is path dependent: if no transmission to B, then also none to {K,L,O,J,M}
Exposable Link (from A’s p.o.v.)
Contact

13. Issues in diffusion
Concurrent relationships create more paths in the exposure network…

14. Issues in diffusion
…and in turn increases the extent of the transmission network

15. Issues in diffusion
…and in turn increases the extent of the transmission network

16. Types of diffusion models: threshold models
In threshold models, people behave differently if enough of their friends do
Threshold = 90% of friends
…nothing happens

17. Types of diffusion models: threshold models
Changing the threshold can lead to different levels of adoption
Threshold = 75% of friends

18. Types of diffusion models: threshold models
Threshold = 50% of friends
Changing the threshold can lead to different levels of adoption

19. Types of diffusion models: threshold models
Threshold models can be written as a matrix multiplication problem
𝑌 𝑡+1
= 𝑰 𝑊𝑌 𝑡
> 𝜏 =
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 1 0 1 1 0 0 0
0 0 1 0 1 0 0 0
0 0 1 1 0 1 1 1
0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0
0
0
0
1
0
0
1
1
Needs to be row-normalized

20. Types of diffusion models: threshold models
Threshold models can be written as a matrix multiplication problem
𝑌 𝑡+1 = 𝑰 𝑊𝑌 𝑡 > 𝜏 =
𝑰
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 1 3 0 1 3 1 3 0 0 0
0 0 1 2 0 1 2 0 0 0
0 0 0 1 4 0 1 4 1 4 1 4
0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0
0
0
0
1
0
0
1
1
> 𝜏

21. Types of diffusion models: threshold models
Threshold models can be written as
a matrix multiplication problem
𝑌 𝑡+1
= 𝑰 𝑊𝑌 𝑡
> 𝜏 =
𝑰
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 1 3 0 1 3 1 3 0 0 0
0 0 1 2 0 1 2 0 0 0
0 0 0 1 4 0 1 4 1 4 1 4
0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 1 0 0 0
0
0
0
1
0
0
1
1
> 𝜏
𝐈
0
0
1 3
0
0
3 4
0
0
> 𝜏 =
0
0
0
0
0
1
0
0
Note the influence of
non-adopters, who are
driving abandonment

22. Types of diffusion models: threshold models
Threshold models can be written as
a matrix multiplication problem
𝑌 𝑡+1
= 𝑰 𝑊𝑌 𝑡
> 𝜏 =
𝑰
1 1 0 0 0 0 0 0
0 1 1 0 0 0 0 0
0 1 3 1 1 3 1 3 0 0 0
0 0 1 2 1 1 2 0 0 0
0 0 0 1 4 1 1 4 1 4 1 4
0 0 0 0 1 1 0 0
0 0 0 0 1 0 1 0
0 0 0 0 1 0 0 1
0
0
0
1
0
0
1
1
> 𝜏
𝐈
0
0
1 3
1
0
3 4
1
1
> 𝜏 =
0
0
0
1
0
1
1
1
Setting the diagonal to
1 forces adopters to
stay adopted, because 𝜏
is less than 1.

23. Types of diffusion models: threshold models
The diffusion of innovations has a typology for who adopts when

24. Types of diffusion models: threshold models
Variation in thresholds can lead to large differences in outcomes

25. Types of diffusion models: threshold models
Complex contagions require multiple people to convince someone to adopt
Simple contagion:
One friend who has adopted
is enough to convince you to
adopt
Complex contagion:
Two or more friends need to
adopt before you adopt
(costlier activities)

26. Types of diffusion models: threshold models
Complex contagions spread within denser regions of the network
Simple contagion:
Even a few “shortcuts” can let an
innovation spread across the network
Complex contagion:
Only larger bridges allow innovations to
jump between regions of the network

27. Types of diffusion models: threshold models
Health studies increasingly think about complex contagions…

28. Types of diffusion models: threshold models
…although a recent study (exercise contagion) found little support for them

29. Types of diffusion models: weighted averaging models
Weighted averaging models treat people as the average of their friends
Strongly disagree (1)
A/D: Smoking makes you look cool
Slightly disagree (2)
Strongly agree (5)
Prediction:
(1 + 3 + 5) / 3 = 2.67
(slightly disagree –
neither agree nor
disagree)

30. Types of diffusion models: weighted averaging models
Weighted averaging models can also be written as matrix multiplication
𝐴 𝑡+1 = 𝑊𝐴 𝑡 =
0 1 3 1 3 1 3
0 0 0 0
0 0 0 0
0 0 0 0
0
1
2
5
Strongly disagree (1)
A/D: Smoking makes you look cool
Slightly disagree (2)
Strongly agree (5)

31. Types of diffusion models: weighted averaging models
Weighted averaging models can also be written as matrix multiplication
𝐴 𝑡 = 𝛼𝑊𝐴 𝑡−1 + 1 − 𝛼 𝐴 1
Vector of predicted
attitude values
Social influence Person’s existing
beliefs
Scalar between 0 and 1,
which indicates what
percent of the weight to
give to others
Weighted average of
others’ beliefs at
previous time point
Self-weight value
Initial attitudes

32. Types of diffusion models: weighted averaging models
Weighted averaging models converge to a consensus

33. Types of diffusion models: weighted averaging models
Weighted averaging models converge to a consensus

34. Types of diffusion models: weighted averaging models
Weighted averaging models converge to a consensus

35. Types of diffusion models: weighted averaging models
Weighted averaging models converge to a consensus

36. Types of diffusion models: weighted averaging models
Weighted averaging models converge to a consensus

37. Types of diffusion models: weighted averaging models
Weighted averaging models converge to a consensus

38. Types of diffusion models: weighted averaging models
Weighted averaging models converge to a consensus

39. Types of diffusion models: weighted averaging models
Weighted averaging models converge to a consensus

40. Types of diffusion models: weighted averaging models
Weighted averaging models converge to a consensus

41. Application of the weighted averaging model
Diffusion models underlie the concept of “opinion leaders”

42. Current issues in diffusion
Influence is very, very hard to distinguish from selection

43. Current issues in diffusion
Suppose that there are two friends named Ian and Joey, and Ian’s parents ask him the classic hypothetical of
social influence: “If your friend Joey jumped off a bridge, would you jump too?” Why might Ian answer “yes”?
1. because Joey’s example inspired Ian (social contagion/influence);
2. because Joey infected Ian with a parasite that suppresses fear of falling (biological contagion);
3. because Joey and Ian are friends on account of their shared fondness for jumping off bridges (manifest
homophily, on the characteristic of interest);
4. because Joey and Ian became friends through a thrill-seeking club, whose membership rolls are publicly
available (secondary homophily, on a different yet observed characteristic);
5. because Joey and Ian became friends through their shared fondness for roller-coasters, which was caused by
their common thrill-seeking propensity, which also leads them to jump off bridges (latent homophily, on an
unobserved characteristic);
6. because Joey and Ian both happen to be on the Tacoma Narrows Bridge in November 1940, and jumping is
safer than staying on a bridge that is tearing itself apart (common external causation).
The distinctions between these mechanisms—and others that no doubt occur to the reader—are all ones that
make causal differences. In particular, if there is any sort of contagion, then measures that specifically prevent
Joey from jumping off the bridge (e.g., restraining him) will also have the effect of tending to keep Ian from doing
so; this is not the case if contagion is absent. However, the crucial question is whether these distinctions make
differences in the purely observational setting, since we are usually not able to conduct an experiment in which
we push Joey off the bridge and see whether Ian jumps (let alone repeated trials.)

44. Current issues in diffusion
Experimental designs may alleviate this problem
Exogenously decide peers via natural experiment…

45. Current issues in diffusion
Experimental designs may alleviate this problem
… or by developing an online world where you can define the ties

46. Current issues in diffusion
Volunteer science is a new tool for developing online world experiments

47. Current issues in diffusion
Abandonment of innovations is poorly understood
Is this the diffusion process in reverse?
How does experimentation play in to this?

48. Current issues in diffusion
Transmission process for non-biological contagions is less well understood

49. Current issues in diffusion
Transmission process for non-biological contagions is less well understood

50. Social space diffusion model
The observed network is measured with error, and doesn’t capture all of the social
influence pathways.
• Networks are measured with error. (Paik and
Sanchagrin 2013; Eagle and Proeschold-Bell 2015)
• People are influenced by people that they aren’t
directly connected to. (e.g., Fujimoto and Valente
2012)
• Friendships experience routine churn. (Cairns and
Cairns 1994; Branje et al. 2007)
Solution: smooth the observed network by modeling the predicted probability of a tie
between each pair of people, and simulate diffusion over the model results.

51. Social space diffusion model
Visually: In symbols:
Latent space models estimate the probability of each tie conditionally independent of the
others, given each person’s position in a latent space.
Person i Person j
Person k
𝑧𝑗 − 𝑧 𝑘
𝑧𝑖 − 𝑧 𝑘
Let 𝑦𝑖𝑗 = 1 if 𝑖 → 𝑗 and 0 otherwise, 𝑖 = 1, … , 𝑛, 𝑗 = 1, … , 𝑛.
Ties (𝑌) are conditionally independent, given positions (𝑍):
Pr 𝑌 Β, 𝑍 =
𝑖≠𝑗
Pr 𝑦𝑖𝑗|𝛽0, 𝑧𝑖, 𝑧𝑗
Ties are modeled in a logistic regression:
logit Pr(𝑦𝑖𝑗 = 1|𝛽0, 𝑧𝑖, 𝑧𝑗 = 𝜂𝑖𝑗 = 𝛽0 − 𝑧𝑖 − 𝑧𝑗
Β and 𝑍 are given minimally informative priors centered at 0
(not shown), pulling vertices towards the center of the latent
space.
𝑧𝑖 − 𝑧𝑗

52. Social space diffusion model: predicted probabilities
Model people incorporating information from everyone else in the network using the
posterior probabilities of ties as weights.
Steps:
Get the predicted probability of a tie for each of the 𝑘 = 1, … , 𝐾 draws
from the posterior.
𝑝𝑖𝑗
𝑘
= exp 𝜂𝑖𝑗
𝑘
∕ 1 + exp 𝜂𝑖𝑗
𝑘
Set the weight matrix to the predicted probability of a tie:
𝑊 𝑘
= 𝑤𝑖𝑗
𝑘
= 𝑝𝑖𝑗
𝑘
Simulate the diffusion process:
𝐴 𝑘,𝑡+1
= 𝑊 𝑘
𝐴 𝑘,𝑡
For several iterations, starting from the initial values, this becomes:
𝐴 𝑘,𝑡+1 = 𝑊 𝑘 𝑡
𝐴 1

53. Social space diffusion model: posterior predictive distribution
Model routine churn using the posterior predictive distribution (simulated networks).
Steps:
Get the predicted probability of a tie for each of the 𝑘 = 1, … , 𝐾 draws
from the posterior.
𝑝𝑖𝑗
𝑘
= exp 𝜂𝑖𝑗
𝑘
∕ 1 + exp 𝜂𝑖𝑗
𝑘
Draw a simulated network for each of the posterior draws:
𝑦𝑖𝑗
𝑘,𝑙
∼ 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 𝑝𝑖𝑗
𝑘
Set the weight matrix to the simulated network
𝑊 𝑘,𝑙
= 𝑌 𝑘,𝑙
Simulate the diffusion process:
𝐴 𝑘,𝑡+1
= 𝑊 𝑘,𝑙
𝐴 𝑘,𝑡
Draw a new simulated network from the same posterior draw 𝑘 for each
iteration 𝑙 = 1, … , 𝐿:
𝐴 𝑘,𝑡+1 = 𝑊 𝑘,𝐿 … 𝑊 𝑘,2 𝑊 𝑘,1 𝐴 1

54. Social space diffusion model
Initial conditions of School 212 friendship network. Vertices (students) are colored by their answers to the question “How wrong is it for someone your age
to smoke cigarettes?” where 1=Not at all wrong, 2=A little bit wrong, 3=Fairly wrong, 4=Very Wrong. Colors from www.ColorBrewer2.org by Cynthia A.
Brewer, Geography, Pennsylvania State University.

55. Social space diffusion model
Predicted attitudes after one simulated time step. Colors represent attitude rounded to the nearest 0.25; color values are interpolated between color values
from colorbrewer2.org.
Isolated vertices don’t
change attitudes
Nearly completely homogeneous
attitudes after one time step
Isolated vertices reconnected probabilistically;
can influence and be influenced
Still heterogeneous after one step

56. Social space diffusion model
Isolates still unchanged in
network-only model
Models with smoothing converge
to a single consensus value