Chapter 7

Question 1

What are the 3 possible outcomes in the 2-word speakers game?

Answer 1

There are three possible outcomes: (1) If the initial proportion of A is greater than B, A wins; (2) if the initial proportion of B is greater than A, B wins; and (3) if these proportions are exactly equal, all three options A, B, and AB coexist, but this equilibrium is unstable.

Question 2

At what level of intolerance does segregation occur in schellings model?

Answer 2

Near B = 30 (=agents stay at their location of 30% of their 8 neighbors are the same type)

Question 3

What is B in the schelling model?

Answer 3

Agents stay at their location if a certain percentage of their eight neighbors, B, are of the same type as the agent. So, if B = 0%, nobody moves, if B = 100% everybody moves.

Question 4

What 3 order parameters for the schelling model does the book suggest?

Answer 4

• % of similar neighbors
• a segregation coefficient, which requires the identification of clusters
• the density of unwanted locations, which better distinguishes between the three states of the system.

Question 5

What happens in the language gam assuming that there is only one object to name and that the social network is fully connected?

Answer 5

After a phase in which agents use lots of different words, a language consisting of just a single word emerges abruptly.

Question 6

What do the three terms in the equation for dX(A)/dt mean and what to they represent together? (the 1t equation of the language game with only two words)

Answer 6

1st term: -X(B)X(A) A-agents becoming AB-agents from listening to B-agents
2nd term: 1/2 X(AB)X(AB): AB-agents becoming A-agents from listening to AB-agents (½ because equal random chance of the speaker choosing to use B-word -> listener becomes B-agent instead)
3rd: X(A)X(AB) = AB-agents becoming A-agents from listening to A-agents
Together: = change in number of A-agents

Question 7

What is homophily?

Answer 7

The tendency or preference for individuals to associate or connect with others who are similar to themselves.

Question 8

What is the probability that one of the mismatching features of one of the two agents is set equal to that of the other, in Axlrods model of cultural diffusion?

Answer 8

The number of shared features divided by F

Question 9

What are F and Q in axlrods model of cultural diffusion and what does the choice of these parameters determine.

Answer 9

• F = number of features
• Q = number of nominal states of these features
• whether the system converges to a single culture or a state of diversity

Question 10

Why does the cultural diffusion model often result in a global convergence towards a single culture?

Answer 10

The combination of interaction and homophily creates a self-reinforcing dynamic

Question 11

What is the answer to why differences between people do not disappear in Axlrods model and in continuous opinions models, respectively?

Answer 11

Axlrod: Selective interaction
Continuous-opinion models: bounded confidence

Question 12

What is “bounded confidence”?

Answer 12

The concept that individuals are influenced by the opinions of others only when those opinions fall within a certain range of their own opinions.

Question 13

What are the basic building blocks of social contagion models?

Answer 13

1. topology to the social network.
2. interaction rules
3. defintion of opinion

Question 14

Why do many social contagion models use fully connected networks, and what other network structures are used?

Answer 14

• because they allow an analytical (mean-field) approach
• e.g random, small-world, lattice

Question 15

What is a majot contention in social contagion models regarding how opinions are defined?

Answer 15

whether they are discrete or continuous

Question 16

In the simplest voter model, with only two possible opinions (-1, 1), two connected agents A and B meet, and A simply copies B’s opinion. What determines what happens in this system?

Answer 16

The typology of the network:
* it’s dimensions; i.e d = 1 (line), d = 2 (lattice) or d>2
* it’s size, N

Question 17

In which cases of the simplest voter model does the system NOT converge to a single opinion?

Answer 17

d > 2 and N = infinite

Question 18

How long does it take for opinions to converge in the simplest voter model, and for which typology does it take the longest

Answer 18

• if d = 1, convergence time = N^2
• if d = 2, convergence time = NlnN
• if d > 2, convergence time = N
• Thus, slowest convergence for d = 1 (agents connected in a line)

Question 19

What is the probability of ending up in the +1 state in the simplest case of the voter model?

Answer 19

same as the initial probability of +1s (i.e if intial probability of +1 is 50%, probability of the system ending up in this state is 50%)

Question 20

What is the heterogeneous voter model?

Answer 20

A copies the opinion of agent B with probability ri (agents with low ri - zealots- are more stubborn)

Question 21

In which social network typology is voting consesus generally more easily reached?

Answer 21

Scale-free networks with broad degree-distributions

Question 22

What is the basic idea of the CODA model?

Answer 22

Agents act discretely but update their continuous opinions based on observations of other agents’ discrete actions

Question 23

How does an agent choose between ption A and B in the CODA model?

Answer 23

It has a subjective probability pi that A is the best option, and 1-pi that B is the best option - and choice is made according to sgn(pi-0.5). So A is chosen when pi > 0.5

Question 24

How is v(i), agent i’s subjective (log-odds) probability that A is the best option, updated in the CODA model?

Answer 24

Using bayestheorem we can update it as follows:
* if agent j chooses A: v(i) = v(i) + alpha,
* if agent j chooses B: v(i) = v(i) - alpha

Question 25

What outcomes do we see when integrating CODA with the voter model?

Answer 25

A strongly bimodal distribution, i.e extreme forms of polarization

Question 26

What are the temperature variable and the external field variable in the ising model of opinions?

Answer 26

* temperature = randomness * external field = external social field

Question 27

What is the basic idea of the majority model

Answer 27

1. random group of voters is selected 2. all voters in this group adopt the local majority opinion 3. process can be repeated until convergence to one opinion is reached (which will always happen in a finite population)

Question 28

What are the 4 majority type models mentioned and which one of these corresponds to an ising model with 0 temperature?

Answer 28

* hierarchical * only one voter could be influenced by the local majority (this one corresponds to ising with 0 temp) * q-voter model * snzadj model

Question 29

what is q in the q-voter model and for which value of q is it the same as the standard voter model?

Answer 29

agents change their opinions only if all q voters selected from the neighborhood agree on the opinion. If q = 1, this reduces to standard voter model

Question 30

how does the q-voter model differ from the basic voter model "in practice"?

Answer 30

generally allows for ghigher degree of opinion diversity

Question 31

Explain how the basic Snzadj model works?

Answer 31

* agents are placed on a line * 2 neighbors with the same opinion spread this to their own neighbors * if they disagree they enforce their disagreement on the neighbors. Thus (?,1,1,?) -> (1,1,1,1) and (?,-1,1,?) -> (1,-1,1,-1)

Question 32

What are the possible result states in the snzadj model?

Answer 32

* all 1s * all -1s * a sequence of 1 and -1 pairs

Question 33

What is the possibility of the basic sznadj model converging to a sequence of 1 and -1 pairs?

Answer 33

0.5

Question 34

In the social impact theory, opinion X is either -1 or 1 and opinion change depends on social impact I. what is social impact a function of? what modifiable parameter does the function have? And what additional parameter beyond I affects opinion?

Answer 34

* p(i) = the persuasiceness of opponents (connected agents with opposite opinions) * s(i) = the supportiveness of suppoerters (with same opinions) * d(ij) = the distance o these agents * alpha = modifiable parameter determining effect of distance. Higher alpha -> less influence of agents far away * H = external field, additional param

Question 35

Without individual fields, the social impact model ends up with an infinite number of stationary opinion states, one of which is usually dominantl. How does the behavior change if individual fields are added and what can this explain?

Answer 35

* some minority opinions can become metastable. * These smaller minority clusters can also persist for a long time before shrinking again, and the process repeats itself, -> staircase behavior *can explain why small minority groups (e.g flat earth) often persist for a long time, against all odds

Question 36

Discrete opinion type models

Answer 36

voter models ( standard voter model + coda), majority type models, social impact model

Question 37

Continuous opinion models

Answer 37

classical continuous opinion models (DeGroot model, friedkin-jognson model), bounded confidence models (deffaunt, hegelmann-krause)

Question 38

how does the DeGroot model (classic continuous opinion model) work?

Answer 38

* weighted network. * At each iteration, an agent’s opinion is set equal to the weighted average of all connected agents in the network. In this way, opinions tend to converge

Question 39

What is the friedkin-johnson model?

Answer 39

* extension of the degroot model that includes a confidence level for each agent. More confidence in own opinion -> reduced effect of others

Question 40

In which cases can clustering or polarization occur in classic linear models such as degroot and friedkin-johnson models?

Answer 40

Only if parts of the network are unconnected

Question 41

What mechanism can be added to the classical continuous model so that polarization can also occur in connected networks?

Answer 41

Bias mechanism in which confirming evidence is weighted more heavily relative to disconfirming ecidence

Question 42

What does the bounded confidence mechanism assume about individuals?

Answer 42

that they have a limited willingness to accept and consider opinions that differ from their own and will only update their opinions if they are within a certain range or “bound” of similarity.

Question 43

What is the deffuant model?

Answer 43

A continuous opinion model with bounded confidence, in which: *initial opinions of n agents are randomly set to values between 1 and 2 * at each time step, two agents i and j meet. Only if Xi(t)-Xj(t) < epsilon, opinions are exchanged bonus facts: drawback - converges slowly; topology does not make much difference

Question 44

How do the parameters epsilon and upsilon change the behavior in a deffuant model?

Answer 44

* if u = 0.5, they find each other in the middle, if u = 1, they take each other's position (as in voter model). Does not make much different, but model converges faster with u = 0.5 * choice of e makes big difference. If e = 0, agents stick to their positions; if e > 0.5, they all converge to X = 0.5; for intermediate values, different forms of clustering occur

Question 45

How can you reduce polarization in the deffuant model? How can you increase it? How can you produce hysteresis?

Answer 45

* Reduce polarization: Add some noise to X at each time step * Increase polarization: lower bound with the number of interactions * hysteresis: increasing the bound after polarization emerged for a low bound

Question 46

How does the Hegselman-Krause model differ from the deffaunt model?

Answer 46

* Instead of communicating with 1 other agent, they communicate with all connected agents - but only if the difference in opinion with these agents is sufficiently small. * Thus, agents average the opinion of all connected agents for which the difference in opinion is less than the bound.

Question 47

What is the main issue in regards to the relationship between theoretical models of opinion and the empirical data?

Answer 47

* these data do not discriminate between models. Most of the data fit all opinion models (supporting the general modeling approach but not specific models). *This relates to the point that current opinion models are difficult to falsify because they lack specificity and are too flexibility.

Question 48

What is HIOM?

Answer 48

Hierarchical Ising opinion model = * Ising-type social network in which each agent is a cusp. * opinion of agent i at time t changes according to the cusp equation with information (external field)and attention (inverse temperature) as control variables * Interactions affect information & attention-> changes in opinion

Question 49

How is the HIAM similar to Sobkowicz's model? How does it differ and what is the reasoning behind this?

Answer 49

Similar: cusp model for the individual agent + interactions change both opinions and the control variables (S: emotion & information, HIAM: information & involvement/attnetion) Different: S reduces the cusp dynamics to a three-state system (opinions = -1, 0 or 1) * HIAM doesn't adopt this simplification bc much of the interesting dynamics (hysteresis within agents) are lost

Question 50

Is opinion discrete or continuous in HIAM?

Answer 50

Depends on another continuous variable (attention). Low attention = continuous, high attention = discrete

Question 51

What assumptons do HIOM inherit from the Ising attitude model?

Answer 51

1, attutyde bides are binary (-1,1) 2. undirected pairwise interactions between nodes 3. attitude networks should be reasonably balanced

Question 52

What 3 assumptions does HIOM make about interactions?

Answer 52

1. agents initiate interaction based on involvement 2. attention/involvement slowly decreases over time 3. attention increases again through social interactions

Question 53

What assumption does HIAM make about information? How is this formalized?

Answer 53

* it is an averaging process weighted by attention. If agent i is less attentive than agent j, agent i will move more to the information position of j than j will move to i. * formalized as r (resistance) - a logistic function of the difference in attention

Question 54

what do the variables p and r(min) do in HIAM?

Answer 54

p = steepness of the resistance logistic function = persuation - determines the strength of the effect of attention difference between agents r(min) = minimal value of r. if high, r will be high -> agents stick to the information state

Question 55

Why is decay(A) a special variable in HIAM?

Answer 55

*Instead of being fixed, it depends on the difference between the percentage of agents in the “active” set and the desired percentage of active agents * This allows us to manipulate the general interest (attention) in the opinion object.

Question 56

What is the persuasion paradox and how does this happen in the HIAM model?

Answer 56

* describes how efforts to persuade can backfire, causing the other person to become even more entrenched in their original position. This is due to the effect of social interaction on attention. * if highly attentive activists engage with Low-attentive conservatives, the increased attention induces hysteresis in low-attentive conservatives