Based on “Cascading Behavior in Networks: Algorithmic and Economic Issues” in Algorithmic Game Theory (Jon Kleinberg, 2007) and Ch.16 and 19 of Networks, Crowds, and Markets: Reasoning about a Highly Connected World (David Easley, Jon Kleinberg, 2010)

 Motivation  Simple Example  Models  Influence Maximization  Similar Work

 What is a network cascade? ◦ A series of correlated behavior changes  Why do we want to study cascading behavior? ◦ Social Contexts ◦ Epidemic Disease ◦ Viral Marketing ◦ Covert Organization Exposure  What 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?

 A jar either contains 2 red and 1 blue marble or 2 blue and 1 red marble  People 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 match

 Example ◦ 2-way infinite path ◦ q = ½, ◦ S = {0} t=0 t=1 t=

 Example ◦ 2-way infinite path ◦ q = ½, ◦ S = {-1,0,1} t=0 t= The contagion threshold of this graph is ½: any set with larger q can never extend! In fact, we can prove that the maximum contagion threshold of any graph is ½!

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 set

 Progressive vs. Non-Progressive ◦ Our prior model was non-progressive – nodes could change back and forth between states ◦ A 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 model ◦ Actuality: for any graph G, both models have the same contagion threshold

 Our model thus far is limited ◦ Threshold is uniform for nodes – everyone is just as predisposed to study algorithms as you are ◦ All neighbors have equal weight – all your facebook friends are just as important as your immediate family ◦ Undirected graph – the influence you have on your boss is the same as he has on you  We will now introduce several models to ameliorate these limitations

 Cascade 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 failed ◦ Provably equivalent to general threshold model in utility  Independent Cascade Model ◦ Incremental function is independent of X and depends only on u andv

 Domingos 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 we’ve discussed  Can construct instances of those models for which approximation within a factor of n is NP-hard

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

 Identifying instances in which we have a submodular influence function ◦ Any instance of the Cascade Model in which the incremental functions p v exhibit diminishing returns has a submodular influence function ◦ Any instance of the Independent Cascade Model has a submodular influence function ◦ Any instance of the General Threshold Model in which all the threshold functions g v are submodular has a submodular influence function

 The 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 neighbors ◦ A k-core is the maximal induced subgraph with minimum degree at least k

 Cascade scheduling (Chierichetti et al.) ◦ Ordering nodes in a cascade to maximize a particular outcome  Identifying failure susceptibility (Blume et al.) ◦ Notion of cascading failure ◦ μ-risk – maximum failure probability of any node in the graph ◦ What about the structure of the underlying graph causes it to have high μ-risk?

1. Lawrence Blume, David Easley, Jon Kleinberg, Robert Kleinberg, and Éva Tardos 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, K. Bhawalkar, J. Kleinberg, K. Lewi, T. Roughgarden, and A. Sharma. Preventing Unraveling in Social Networks: The Anchored k-Core Problem. In ICALP ' Flavio Chierichetti, Jon Kleinberg, Alessandro Panconesi. How to Schedule a Cascade in an Arbitrary Graph. In Proceedings of EC 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 57– 66, D. Easley, J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, 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 137–146, 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.