News from Web Science 2012

1. News from Web Science 2012 • New modelsfor power law: • Hans Akkermans: usingcontinuous time • Manymodelsmayleadto power laws • Differential equations • October 18: Talk by Hans Akkermans • Hot topic: Distinguishinginfluencefromco-occurrence • Fowler • Political influence – doingexperimentsmotivatingpeopletovote • Sinan Aral • New probabilisticmodels • Tests with viral marketing on the Web

2. News from Web Science 2012 • Strengthofweakties: • Nodes with high in-betweennessaresaidtoexcel • In practicetheyoften do not • New researchformalizesmorepreciselywhenthey do excel (e.g. earnmoremoneyratherthangettingnervousbreakdowns)

3. Network Theory and Dynamic SystemsCascading Behavior in Networks Prof. Dr. Steffen Staab Dr. Christoph Ringelstein

4. Howcould • thefirsttelephone/ • firsthundredtelephones everbesold?

5. Diffusion in Networks • Information cascade, Network effects, and Rich-get-richer: • View the network as a relatively amorphous population of individuals, and look at effects in aggregate; • Global level • Diffusion in networks: • View the fine structure of the network as a graph, and look at how individuals are influenced by their particular network neighbors. • Local level

6. Diffusion of Innovations • Considering how new • behaviors, • practices, • opinions, • conventions, and • technologies spread from person to person through a social network • Success depends on: • Relative advantage • Complexity • Observability • Trialability • Compatibility

7. A Networked Coordination Game • If nodes v and w are linked by an edge, then there is an incentive for them to have their behaviors match • v and w are the players • Two possible behaviors, labeled A and B • A and B are the possible strategies • The payoffs are defined as follows: • if v and w both adopt behavior A, they each get a payoff of a > 0; • if they both adopt B, they each get a payoff of b > 0; and • if they adopt opposite behaviors, they each get a payoff of 0.

8. A Networked Coordination Game • Suppose that some of v neighbors adopt A, and some adopt B; what should v do in order to maximize its payoff? • v has d neighbors • a p fraction of them have behavior A • a1-p fraction have behavior B • If v chooses A, it gets a payoff of pda • If v chooses B, it gets a payoff of (1-p)da

9. A Networked Coordination Game • The game is played along all edges

10. A Networked Coordination Game • A is a better choice if: Or: • Threshold rule: If at least a q = b/(a+b) fraction of your neighbors follow behavior A, then you should too.

11. Cascading Behavior • Two obvious equilibria: • Everyone adopts A, and • Everyone adopts B. • How easy is it to ‘tip’ a network? • Initially everyone is using B • A few early adopters are using A

12. Tip a = 3 b = 2 q = 2/(3+2) = 2/5

13. Intermediate Equilibria a = 3 b = 2 q = 2/(3+2) = 2/5

14. Intermediate Equilibria

15. Intermediate Equilibria tightly-knit communities in the network can work to hinder the spread of an innovation

16. Cascades of adoption • The chain reaction of switches to Ais called a cascade of adoptions of A, • Two fundamental possibilities exist: (i) that the cascade runs for a while but stops while there are still nodes using B, or (ii) that there is a complete cascade, in which every node in the network switches to A. • If the threshold to switch is q, we say that the set of initial adopters causes a complete cascade at threshold q.

17. Viral Marketing • Strategy I: Making an existing innovation slightly more attractive can greatly increase its reach. • E.g. in our example: Changing a = 3 to a = 4 (new threshold q = 1/3) will results in a complete cascade.

18. Viral Marketing • Strategy II: convince a small number of key people in the part of the network using B to switch to A. • E.g. in our example: Convincing node 12 or 13 to switch to A will cause all of nodes 11–17 to switch. (Convincing 11 or 14 would not work)

19. Reflection • Population-level network effects • Decisions based on the overall fraction. • Hard to start for a new technology, even when it is better. • Network-level cascading adoption • Decisions based on immediate neighbors • A small set of initial adopters are able to start a cascade.

20. Cascades and Clusters • Definition (Densely connected community): 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. • the set of all nodes is always a cluster of density 1 • the union of two clusters with density p has also density p • Homophily can often serve as a barrier to diffusion, by making it hard for innovations to arrive from outside densely connected communities. Three four node clusters with density d = 2/3

21. Relationship between Clusters and Cascades Claim: Consider a set of initial adopters of behavior A, with a threshold of q for nodes in the remaining network to adopt behavior A. (i) 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. (ii) 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.

22. Example Clusters of density greater than 1 − 2/5 = 3/5 block the spread of A at threshold 2/5 .

23. Part (i): Clusters are Obstacles to Cascades The spread of a new behavior, when nodes have threshold q, stops when it reaches a cluster of density greater than (1 − q).

24. Part (ii): Clusters are the Only Obstacles to Cascades If the spread of A stops before filling out the whole network, the set of nodes that remain with B form a cluster of density greater than 1 − q.

25. Diffusion, Thresholds, and the Role of Weak Ties There is a crucial difference between learning about a new idea and actually deciding to adopt it. The years of first awareness and first adoption for hybrid seed corn in the Ryan-Gross study.

26. Example Steps 11-14 become aware of A but never adopt it.

27. Strength-of-weak-ties The u-w and v-w edges are more likely to act as conduits for information than for high-threshold innovations. Initial adopters: w and x Threshold: q = 1/2

28. Extensions of the Basic Cascade Model Heterogeneous Thresholds: • Suppose that each person in the social network values behaviors A and B differently => each node v, has its own payoffs avand bv • Now, A is the better choice if • Each node v has its own personal threshold qv, and it chooses A if at least a qvfraction of its neighbors have done so. = qv

29. Example & Influenceable nodes Influenceable nodes are nodes with a low threshold

30. Example & Blocking cluster Blocking cluster in the network is a set of nodes for which each node v has more than a 1−qvfraction of its friends also in the set. A set of initial adopters will cause a complete cascade (with a given set of node thresholds) if and only if the remaining network does not contain a blocking cluster.

31. Knowledge, Thresholds, and Collective Action Integrating network effects at both the population level and the local network level. • We consider situations where coordination across a large segment of the population is important, and the underlying social network is serving to transmit information about people’s willingness to participate. • Collective action problem: A positive payoff if a lot of people participate, a negative payoff if only a few participate (e.g. protest under a repressive regime). • Pluralistic ignorance: People have wildly erroneous estimates about the prevalence of certain opinions in the population at large.

32. A Model for the Effect of Knowledge on Collective Action • Suppose that each person in a social network has a personal threshold which encodes her willingness to participate. • A threshold of k means, “I will show up for the protest if I am sure that at least k people in total (including myself) will show up.” • Each node only knows its and its neighbors threshold

33. Common Knowledge and Social Institutions. • A widely-publicized speech, or an article in a high-circulation newspaper, has the effect not just of transmitting a message, but of making the listeners or readers realize that many others have gotten the message as well

34. Andthefirsttelephonescouldbesold, becausetheycouldprovidesufficientbenefit in a tightlyenoughknitnetwork! • Not due topopulationcounts!