Steffen Staab 1WeST Web Science & Technologies University of Koblenz ▪ Landau, Germany Network Theory and Dynamic Systems Cascading Behavior in Networks Prof. Dr. Steffen Staab

Steffen Staab 2WeST How could  the first telephone/  first hundred telephones ever be sold? Why is Facebook less suitable for propagating political ideas than Twitter? (speculation!)

Steffen Staab 3WeST 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

Steffen Staab 4WeST Diffusion of Innovations  Considering how new  behaviors,  practices,  opinions,  conventions, and  technologies spread from person to person through a social network Success criteria for adoption of innovation depend on: Relative advantage Complexity For people to understand Observability For people to observe that others are using it Trialability Gradual take-up possible (counterexample: magnetic monorail) Compatibility With social system

Steffen Staab 5WeST 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

Steffen Staab 6WeST 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)db

Steffen Staab 7WeST A Networked Coordination Game  The game is played along all edges

Steffen Staab 8WeST 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.

Steffen Staab 9WeST 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

Steffen Staab 10WeST Tipping a = 3 b = 2 q = 2/(3+2) = 2/5

Steffen Staab 11WeST Tipping a = 3 b = 2 q = 2/(3+2) = 2/5

Steffen Staab 12WeST Intermediate Equilibria a = 3 b = 2 q = 2/(3+2) = 2/5

Steffen Staab 13WeST Intermediate Equilibria

Steffen Staab 14WeST Intermediate Equilibria tightly-knit communities in the network can work to hinder the spread of an innovation

Steffen Staab 15WeST 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.

Steffen Staab 16WeST 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 result in a complete cascade.

Steffen Staab 17WeST 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)

Steffen Staab 18WeST 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.

Steffen Staab 19WeST 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

Steffen Staab 20WeST 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.

Steffen Staab 21WeST Example Clusters of density greater than 1 − 2/5 = 3/5 block the spread of A at threshold 2/5.

Steffen Staab 22WeST 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).

Steffen Staab 23WeST 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.

Steffen Staab 24WeST 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.

Steffen Staab 25WeST Example Steps become aware of A but never adopt it.

Steffen Staab 26WeST 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

Steffen Staab 27WeST Heterogeneous Thresholds:  Suppose that each person in the social network values behaviors A and B differently => each node v, has its own payoffs a v and b v  Now, A is the better choice if  Each node v has its own personal threshold q v, and it chooses A if at least a q v fraction of its neighbors have done so. Extensions of the Basic Cascade Model = q v

Steffen Staab 28WeST Example & Influenceable nodes Influenceable nodes are nodes with a low threshold

Steffen Staab 29WeST Example & Blocking cluster 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. Blocking cluster in the network is a set of nodes for which each node v has more than a 1−q v fraction of its friends also in the set.

Steffen Staab 30WeST 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.

Steffen Staab 31WeST 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

Steffen Staab 32WeST  Weak ties  Better for access to new information  Strong ties  Better for collective action

Steffen Staab 33WeST 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

Steffen Staab 34WeST  And the first telephones could be sold, because they could provide sufficient benefit in a tightly enough knit network!  Not due to population counts!

Steffen Staab 35WeST