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Steffen Staab staab@uni-koblenz.de 1WeST Web Science & Technologies University of Koblenz ▪ Landau, Germany Network Theory and Dynamic Systems Cascading Behavior in Networks Prof. Dr. Steffen Staab
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Steffen Staab staab@uni-koblenz.de 2WeST How could the first telephone/ first hundred telephones ever be sold? Why is Facebook less suitable for propagating political ideas than Twitter? (speculation!)
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 7WeST A Networked Coordination Game The game is played along all edges
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Steffen Staab staab@uni-koblenz.de 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.
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 10WeST Tipping a = 3 b = 2 q = 2/(3+2) = 2/5
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Steffen Staab staab@uni-koblenz.de 11WeST Tipping a = 3 b = 2 q = 2/(3+2) = 2/5
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Steffen Staab staab@uni-koblenz.de 12WeST Intermediate Equilibria a = 3 b = 2 q = 2/(3+2) = 2/5
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Steffen Staab staab@uni-koblenz.de 13WeST Intermediate Equilibria
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Steffen Staab staab@uni-koblenz.de 14WeST Intermediate Equilibria tightly-knit communities in the network can work to hinder the spread of an innovation
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Steffen Staab staab@uni-koblenz.de 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.
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Steffen Staab staab@uni-koblenz.de 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.
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Steffen Staab staab@uni-koblenz.de 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)
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Steffen Staab staab@uni-koblenz.de 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.
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 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.
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Steffen Staab staab@uni-koblenz.de 21WeST Example Clusters of density greater than 1 − 2/5 = 3/5 block the spread of A at threshold 2/5.
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Steffen Staab staab@uni-koblenz.de 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).
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Steffen Staab staab@uni-koblenz.de 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.
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Steffen Staab staab@uni-koblenz.de 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.
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Steffen Staab staab@uni-koblenz.de 25WeST Example Steps 11-14 become aware of A but never adopt it.
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 28WeST Example & Influenceable nodes Influenceable nodes are nodes with a low threshold
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Steffen Staab staab@uni-koblenz.de 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.
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Steffen Staab staab@uni-koblenz.de 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.
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 32WeST Weak ties Better for access to new information Strong ties Better for collective action
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Steffen Staab staab@uni-koblenz.de 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
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Steffen Staab staab@uni-koblenz.de 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!
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Steffen Staab staab@uni-koblenz.de 35WeST
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