Information Cascades

1. Information Cascades

2. Cascades • Information/behavior spreading through a network • Useful for studying • Actual viral contagion • Technology diffusion, adoption of new products • Cascading failures (e.g. power grids) • Spread of information/rumor, viral marketing

3. How to model diffusion • Initial models • Assumed that everyone has global knowledge of what fraction has adopted • First mathematical models for local information • [Schelling '70/'78, Granovetter '78] • Large body of subsequent work: • [Rogers '95, Valente '95, Wasserman/Faust '94] • Probabilistic models • with each neighbor that has the contagion, with some prob. the user could have it too • Ex: disease • Decision based models • Each neighbor typically has their own threshold. Makes decision based on how many neighbors have contagion. • Ex: adopting a product; Joining demonstrations

4. Decision based model: two states A, B • Payoff for two linked nodes (x, y) • Both nodes play A => (a, a). Both play B => (b, b). Else (0, 0) • In a large network, consider each node playing this game with each of its neighbors • Assume infinite graph • initialization is some mix of A and B • When will any node x choose B over A? • q = a/(a+b) • when fraction of neighbors playing B is > q*d(x)

5. Definitions • Starting with set S, continue the above process k times • be the current set of nodes adopting B • Non-progressive • Nodes can switch back: • Progressive • Nodes cannot switch back: • Contagion threshold of a graph • the maximum q for which there exists an infinite cascade • is a property of the graph only

6. Simple example • q = ½ (break ties by adopting B) • Case 1: S={0} adopts B • {0} -> {-1, 1} -> {0, 2, -2} -> … • Case 2: S={-1, 0, 1} starts with B • {-1, 0, 1} -> ?? • For S = {0,1}? • Contagion threshold for G is ½ • why?

7. Contagion Threshold • Do the progressive and non-progressive models have different thresholds? • Can the threshold be arbitrary in [0,1]?

8. Contagion Threshold • Do the progressive and non-progressive models have different thresholds? • Nope [Mor00] • Can the threshold be arbitrary in [0,1]? • It is always <= ½ for any graph G [Mor00]

9. Progressive vsNonprogressive:sketch • S = contagious wrt q in progressive model • Build T that is contagious; S1 = S + neighborhood of S • T is “robust” enough that the non-progressive model proceeds to infinity • through induction prove that henceforth the two processes are identical S S1

10. Contagion Threshold • For any graph G, threshold <= ½ • Suppose not, and S is contagious for q > ½ in G • For any set X, define potential(X) = outgoing-degree(X) • Claim: potential of active set decreases at every step • Only nodes that switch have majority of neighbors in active set • Can only decrease a finite amount of times. Hence finite steps! X

11. Viral marketing • Optimization formulation • Bounded marketing budget, how to spend it best • Want to utilize “network effects” • At least two different variants • Pay a small set of users to start a cascade. Maybe their friends will listen to them? • How to choose this set of users? • Offer incentives to whoever buys, if they recommend to their friends

12. Viral marketing: empirical Study I(Leskovec,Adamic, Huberman) • Recommendation incentive variant • Online store data on various categories (DVD, books, cds..) • 16 M recommendations • 4M users, 0.5M items • users who buy items can recommend to friends • both users get discount if results in buys • Some data issues regarding observed reward • sometimes inferred

13. DVD recommendation • Majority does not cause purchases (only 7% does) • Many star patterns and disconnected components • Giant component has 19% of nodes • Cascades form by chains of recommend-buy-recommend

14. Multiple recommendations • Latter recommendations matter less (on avg) • recall similar result on group affiliation in LiveJournal • We only see user receive recommendation and then purchase product • Do not know: • How long it took to act • Whether there were other effects • When did user become aware of friend’s recommendations • Is the average representative of individual users?

15. Other observations • Success depends most on the type of product • Books : rate 3%; DVDs: 7%. Anime DVDs: 29% • Sending more recommendations does result in more purchases (dvds) • Strategy of what a user should do to maximize reward incentive • However, repeated recommendations to one person causes decrease in success probability

16. Seeding variant:Finding good set of seeds • If we select a small set of nodes that are paid to spread information, how should we select them ? • Heuristics methods: • degree, random, some “centrality” notion? • Need a little more stylized influence model [KKT’03] • Suppose f(S) is the set of nodes reached when cascade starts with S

17. Linear Threshold Model • A node v has random threshold θv ~ U[0,1] • A node v is influenced by each neighbor w according to a weight bvwsuch that • A node v becomes active when at least (weighted) θv fraction of its neighbors are active

18. Independent Cascade Model • When node v becomes active, it has a single chance of activating each currently inactive neighbor w. • The activation attempt succeeds with probability pvw

19. Submodularity • fis submodular if • Example: C1, C2,…Cnare sets • is submodular • Bad news: maximizing f(S), when submodular is NP hard • Note: f(S) is actually the expected number of nodes reached

20. Good News • When monotone, we can use Greedy Algorithm! • Start with an empty set S • For k iterations: Add node v to S that maximizes f(S +v) - f(S). • How good (bad) it is? • Theorem: The greedy algorithm is a (1 – 1/e) approximation. • The resulting set S activates at least (1- 1/e) > 63% of the number of nodes that any size-k set S could activate.

21. Key 1: Prove submodularity

22. 0.6 0.2 0.2 0.3 0.1 0.4 0.5 0.3 0.5 Submodularity for Independent Cascade • Coins for edges are flipped during activation attempts.

23. Submodularity for Independent Cascade 0.6 • Coins for edges are flipped during activation attempts. • Can pre-flip all coins and reveal results immediately. 0.2 0.2 0.3 0.1 0.4 0.5 0.3 0.5 • Active nodes in the end are reachable via green paths from initially targeted nodes. • Study reachability in green graphs

24. Submodularity, Fixed Graph • Fix “green graph” G. g(S) are nodes reachable from S in G. • Submodularity: • g(T +v) - g(T) <= g(S +v) - g(S) when S T. • g(S +v) - g(S): nodes reachable from S + v, but not from S. • From the picture: g(T +v) - g(T) <= g(S +v) - g(S) when • S T

25. Submodularity of the Function Fact: A non-negative linear combination of submodular functions is submodular • gG(S): nodes reachable from S in G. • Each gG(S): is submodular (previous slide). • Probabilities are non-negative.

26. Submodularity for Linear Thresholds • Use similar “green graph” idea. • Once a graph is fixed, “reachability” argument is identical. • How do we fix a green graph now? • Each node picks at most one incoming edge, with probabilities proportional to edge weights. • Equivalent to independent cascade model (trickier proof).

27. Evaluating f(S) • How to evaluate ƒ(S)? • Still an open question of how to compute efficiently • But: very good estimates by simulation • repeating the diffusion process often enough (polynomial in n; 1/ε) • Achieve (1± ε)-approximation to f(S). • Generalization of Nemhauser/Wolsey proof shows: Greedy algorithm is now a (1-1/e- ε′)-approximation.

28. More in the paper [KKT’03] • More general model that captures both • Experimental results that show performance on greedy • For simulated cascades • Choosing for the non-progessive case • More realistic marketing scenarios • Likelihood of initial activation depends on amount spent

29. Experimental Results (KKT) • To test efficacy of greedy against other algorithms • Co-authorship data • Linear Threshold Model: multiplicity of edges as weights • weight(v→ω) = Cvw / dv, weight(ω→v) = Cwv / dw • Independent Cascade Model: • Case 1: uniform probabilities p on each edge • Case 2: edge from v to ω has probability 1/ d(w)of activating ω. • Compare with other 3 common heuristics • (in)degree centrality, distance centrality, random nodes. • Simulate the cascades a number of times…

30. Facebook study on contagions(Sun, Rosenn, Marlow, Lento) • Diffusion on FB • Pages “liked” by users • Diffusion happens through newsfeed • How do the cascades look like? • Distribution of sizes, connectedness • Small seed? • Any way to distinguish the seed nodes? • Dataset • sample set of pages and all associated fans • seeding variant

31. Cascade structure Bosnia • Large connected clusters • median page had 70% of fans in one component • second largest comp. much smaller • Multiple chains merge to form cluster Slovenia Croatia

32. Cascade starters • Large number of starters • 46% of entire set of users; 17% of users in largest component • belies the typical assumption that large cascades start from a small set of nodes; however, it does not say that it cannot • maximum chain length can be large ~80 • Tried to predict chain length by looking at different properties of the starter • age, gender, FB friends/activity/age, feed exposure, popularity • after controlling for popularity and friend-count, other variables do not have impact • Main takeaway • Contagions typically have lots of start-points • Looking at contagion w/o the effect of external sources is inadequate

33. Structure of diffusions(Goel, Watts, Goldstein’12) • Study on multiple domains • Yahoo! Kindness; Twitter; Zync; Secretary game… • Main difference with previous studies is • Multiple platforms • Takes very large number of diffusion events • Interested in studying the structure of the “typical” diffusion • Are the cascades large? • Are the trees interesting?

34. Structure of diffusions 1

35. Structure of diffusions 2

36. So then.. • Nice theory problems associated with simple models, but… • Empirical studies show simple viral model not accurate • Large cascades often need multiple starters • Even when propagation happens it is well approximated by one-step process • What lessons to take away? • Incorporate external channels? • Not trust cascade model too much? • Verdict is not clear yet: in some domains (e.g. RDS, computer virus infection) cascades do happen. Is there some missing characteristic? • (financial incentives in RDS, not voluntary in virus spread?)