Cascading Behavior in Networks Andrew Ouzts

Cascading Behavior in Networks

Cascading Behavior in NetworksAndrew OuztsBased on Cascading Behavior in Networks: Algorithmic and Economic Issues in Algorithmic Game Theory (Jon Kleinberg, 2007)andCh.16 and 19 of Networks, Crowds, and Markets: Reasoning about a Highly Connected World (David Easley, Jon Kleinberg, 2010)MotivationSimple ExampleModelsInfluence MaximizationSimilar Work

OverviewWhat is a network cascade?A series of correlated behavior changesWhy do we want to study cascading behavior?Social ContextsEpidemic DiseaseViral MarketingCovert Organization ExposureWhat are some of the interesting questions to be raised?How can we model a cascade?What can initiate or terminate a cascade?What are some properties of cascading behavior?Can we identify subsets of nodes or edges that have greater influence in a cascade than others?

MotivationWhat restaurant do you want to eat at the one thats full or empty?Could be good or bad good (spread of ideas), bad (herding)Herding vs direct benefit (choice of social networking site)Milgram stargazing experiment3A jar either contains 2 red and 1 blue marble or 2 blue and 1 red marblePeople sequentially come and remove 1 marble and verbally announce which configuration they believe to be present (there is an incentive for guessing correctly)Claim: All guesses beyond the first two are fixed if they matchSimple Example Marble Game4Marble GameCascades can result in bad outcomes: 1 in 9 chance of population-wide errorCascades are fragile: what if participants cheat?Cascades can be based on very little information5ModelsFirst, some definitionsExample2-way infinite path q = ,S = {0}Models-101-22t=0

t=1

t=2

-101-22-101-22Example2-way infinite path q = ,S = {-1,0,1}Models-101-22t=0

t=1

-101-22The contagion threshold of this graph is : any set with larger q can never extend!In fact, we can prove that the maximum contagion threshold of any graph is !

Question: what causes cascades to stop?

Definition: 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 neighbors in the setModels

Progressive vs. Non-ProgressiveOur prior model was non-progressive nodes could change back and forth between statesA progressive model is also interesting once a node switches from A to B, it remains B from then on (consider the behavior of pursuing an advanced degree)Intuition: it is easier to find contagious sets with a progressive modelActuality: for any graph G, both models have the same contagion threshold

ModelsOur model thus far is limitedThreshold is uniform for nodes everyone is just as predisposed to study algorithms as you areAll neighbors have equal weight all your facebook friends are just as important as your immediate familyUndirected graph the influence you have on your boss is the same as he has on youWe will now introduce several models to ameliorate these limitations

ModelsModelsModelsCascade Model, cont.Replace the g function from the General Threshold Model with an incremental function that returns the probability of success of activating a node v given initiator u and a set of neighbors X that already attempted and failedProvably equivalent to general threshold model in utilityIndependent Cascade ModelIncremental function is independent of X and depends only on u andvModelsDomingos and Richardson influential work that posed the question: if we can convince a subset of individuals to adopt a new product with the goal of triggering a cascade of future adoptions, who should we target?NP-hard, even for many simple special cases of the models weve discussedCan construct instances of those models for which approximation within a factor of n is NP-hard

Influence MaximizationInfluence MaximizationBy identifying instances where the influence function f is submodular and monotone, we can make use of the following theorem of Nemhauser, Wolsey, and Fisher:

Influence Maximization

Identifying instances in which we have a submodular influence functionAny instance of the Cascade Model in which the incremental functions pv exhibit diminishing returns has a submodular influence functionAny instance of the Independent Cascade Model has a submodular influence functionAny instance of the General Threshold Model in which all the threshold functions gv are submodular has a submodular influence function

Influence MaximizationThe anchored k-core problem (Bhawalker et al.)Model each user has a cost for maintaining engagement but derives benefits proportional to the number of engaged neighborsA k-core is the maximal induced subgraph with minimum degree at least k

Similar Work

The Anchored k-Core Problem

Cascade scheduling (Chierichetti et al.)Ordering nodes in a cascade to maximize a particular outcomeIdentifying failure susceptibility (Blume et al.)Notion of cascading failure-risk maximum failure probability of any node in the graphWhat about the structure of the underlying graph causes it to have high -risk?Similar WorkLawrence Blume, David Easley, Jon Kleinberg, Robert Kleinberg, and va Tardos. 2011. Which Networks are Least Susceptible to Cascading Failures?. In Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS '11). IEEE Computer Society, Washington, DC, USA, 393-402.K. Bhawalkar, J. Kleinberg, K. Lewi, T. Roughgarden, and A. Sharma. Preventing Unraveling in Social Networks: The Anchored k-Core Problem. In ICALP '12.Flavio Chierichetti, Jon Kleinberg, Alessandro Panconesi. How to Schedule a Cascade in an Arbitrary Graph. In Proceedings of EC 2012.Pedro Domingos and Matt Richardson. Mining the network value of customers. In Proc. 7th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 5766, 2001.D. Easley, J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, 2010.David Kempe, Jon Kleinberg, and Eva Tardos. Maximizing the spread of influence in a social network. In Proc. 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 137146, 2003.J. Kleinberg. Cascading Behavior in Networks: Algorithmic and Economic Issues. In Algorithmic Game Theory (N. Nisan, T. Roughgarden, E. Tardos, V. Vazirani, eds.), Cambridge University Press, 2007.

References