Midge CozzensRutgers University

NIMBIOS Graph WorkshopAugust 16-18, 2010

What is Viral Marketing?Start off with a small group of individuals

and spread by “word-of-mouth”, email, etc.Initial individual tells two of his friends and

they tell…Not necessarily selling a product, but

attempts to change behavior or condition, to activate!

Spread of viruses, disruptions etc. through an ecosystem.

Small-Scale Behavioral Viral SpreadingSmall-Scale Behavioral Viral Spreading

• Two people start singing

• Next their friends join in

• By the end of the song, the whole bar is singing Mariah Carey’s Fantasy…true story

Graph Theory - Viral spread

Game TheoryGame Theory

ConfessDon’t

Confess

Confess (10,10) (0,20)

Don’t Confess (20,0) (1,1)

Game Theory and Graph TheoryGraphVertices = individualsEdge (i,j) exists if i and j have a relationshipWeights on edges = strength of relationshipDirected graphDirected edge, or arc (k,g) exists if k has influence over g

Varying ApplicationsStrategies of political parties in getting

voters to vote for their candidate. REU student Jim Manning showed that the optimal strategy for the Republicans in 2008 was to go after moderates first then bring in the conservatives. Had they done that they would have taken 60% of the vote and beat Obama. Instead they brought in Sarah Palin to get the conservative base in place.

Another ApplicationChange the behavior of college students

concerning unprotected sex. REU student Jordan Mitchell wrote a paper to be published in a sociology journal showing strategies to convince students to use protection. She used real data and measured parameters governing use such as attractiveness, drinking, multiplicity of partners, etc.

Another ApplicationVolvo car dealers have decided to

concentrate on online marketing to sell cars across the country. Ajay Matapalli, REU student, developed an optimal progressive strategy for marketing where the dealers gained significantly and the incentives were far more lucrative then would have been just walking in off the street.

Incentive ModelIndividual gain is represented by v(xh) where xh

represents an individual (or in terms of a graph, a specific node).

The subsequent factor is the gain (loss) that the individual receives from his or her peers owning the product as well. The inspiration for separating the individual and collective gain came from H. Peyton Young’s work. A simple example would be if my neighbor had sneakers, then I would gain the enjoyment of being able to run with him if I purchased sneakers.

Incentive Modelu(xh , xj) is a binary function, where xj is any

node in the graph G not equal to xh, such that if there exists an edge between xh and xj , and xj owns the product, then the function equals 1; if there doesn’t exist an edge or xj doesn’t own the product, then let the function equal 0.

r = constant that represents the benefit of a node, which you are connected to having the product.

Incentive ModelEdge weights (the strength of the connection

between two people); contained in the set (0, 1]. w(xh , xj) be a function, where xj is any node in

the graph G not equal to xh, such that if there exists an edge between xh and xj , then the output value is the weight of the edge; if there doesn’t exist an edge, then the output value is 0.

This accounts for the fact that all connections which you have to other people are not equal; thus the edges with aren’t as strong don’t influence the buyer node as much.

Total Effects in the ModelThe total effects of viral marketing is the sum all the benefits

from each peer connection

∑ u(xh , xj) w(xh , xj)r

The graph theory comes into play here because when forming this function we are concerned with the converted node who is reaching out to the new potential customers nodes (its neighbors).

 

 

 

 

q = constant that represents a value that will normalize the weight of the edge to use in our final equation.

y(xh , xo) = the weight of the edge coming from the previously converted node if there is one; if there is no previously converted node, then this function equals

The final incentive equation is:I(xh) = Incentive for Individual xh

I(xh) = z(xh) – v(xh) – y(xh , xo)q - [ ∑ u(xh , xj) w(xh , xj)r ]

The incentive must at least be equal to the gap between the cost of the product to the individual and the gain and ease of the buying it.

Overall Gain Model

G(xh) Overall Gain to Individual xh

G(xh) = v(xh) + [ ∑ u(xh , xj) w(xh , xj)r ] + y(xh , xo)q

Profit ModelB = cost of marketing I = incentiveM = cost of manufacture or equivalentT = retail pricek = number expected to purchase productC(xh) Cost to Company for xh

C(xh) = I(xh) + B/k + M/k

P(xh) Profit for Company from xh

P(xh) = T – C(xh)

Payoff Matrix

Data for tableConsumer ResearchPast history dataCDC data for spread of disease,Public and World Health data on the

effectiveness of innoculations

Tipping Point and Equilibrium

Graph Theory QuestionsFind the k most influential nodes to

market to:In general, identifying the k most

influential nodes is NP-hard.A natural greedy algorithm exists which

is a 1−1/e−ε approximation for selecting a target set of size k, using probabilistically determined simulations.

Tuberculosis Network SW Oklahoma 2002

Spread of Viruses and InnoculationSocial Networks relative to virus spreading

most often look like stars linked together as in the next slide which shows a portion of the first to set to contract TB and their contacts.

Current procedures focus on inoculating the vulnerable -- often the very young and the very old. Network analysis tells us that it may be smarter, and more efficient, to focus on the spreaders -- those with many contacts to many groups.

Possible Research DirectionsFind classes of graphs where finding the k-most

influential nodes can be done in polynomial time.Consider ways of subdividing the social network

into subgraphs where the k-most influential nodes are easy to find.

What type of dominating sets in a graph should one look for?

How does the connectivity of a social network influence spread?

What other connectivity parameters make sense?

Research DirectionsWhat other connectivity parameters make sense?Once the most influential node(s) are targeted,

how do you reach other (or all) nodes in the graph?• Find a Hamiltonian path – hard!• Find an Eulerian path – easy if connected and

vertices have only 2 odd degree vertices;• Partition the graph into connected components with

one target node in each component and add or subtract edges so Eulerian path can be found in each.

Find an euler path

Remove 3 edges

Fleury’s Algorithm to get path

ReferencesInfluential Nodes in a Diffusion Model for Social Networks (David Kempe, Jon Kleinberg, Eva Tardos):http://www-rcf.usc.edu/~dkempe/publications/influential-nodes.pdf

Diffusion on Social Networks (Matthew O. Jackson):http://economiepublique.revues.org/docannexe1777.html

The Diffusion of Innovations in Social Networks (H. Peyton Young):http://www.brookings.edu/~/media/Files/rc/reports/1999/05fixtopicname_young/diffusion.pdf

Information Diffusion in Online Social Networks (Lada Adamic):http://www.eecs.harvard.edu/~parkes/nagurney/adamic.pdf

Diffusion in Complex Social Networks (Dunia Lopez-Pintado):http://www.sisl.caltech.edu/pubs/diffusion.pdf

Social Networks and the Diffusion of Economic Behavior (Matthew O. Jackson):http://www.stanford.edu/~jacksonm/yer-netbehavior.pdf

Cascading Behavior in Networks: Algorithmic and Economic Issues (Jon Kleinberg):http://www.cs.cornell.edu/home/kleinber/agtbook-ch24.pdf