1. New models for power law:
News from Web Science 2012
New models for power law:
Hans Akkermans: using continuous time
Many models may lead to power laws
Differential equations
October 18: Talk by Hans Akkermans
Hot topic: Distinguishing influence from co-occurrence
Fowler
Political influence – doing experiments motivating people to vote
Sinan Aral
New probabilistic models
Tests with viral marketing on the Web

2. Strength of weak ties: News from Web Science 2012
Nodes with high in-betweenness are said to excel
In practice they often do not
New research formalizes more precisely when they do excel (e.g. earn more money rather than getting nervous breakdowns)

3. Network Theory and Dynamic Systems Cascading Behavior in Networks
Prof. Dr. Steffen Staab
Dr. Christoph Ringelstein /

4. How could
the first telephone/
first hundred telephones
ever be sold?

5. Information cascade, Network effects, and Rich-get-richer:
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
a 1-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. Two obvious equilibria:
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

14. Intermediate Equilibria

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

16. Two fundamental possibilities exist:
Cascades of adoption
The chain reaction of switches to A is 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. Population-level network effects
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 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 av and 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 qv fraction 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−qv fraction 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. And the first telephones could be sold, because they could provide sufficient benefit in a tightly enough knit network!
Not due to population counts!