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# Information Cascades - PowerPoint PPT Presentation

Information Cascades. 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. How to model diffusion.

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

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

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
• 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
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
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?
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
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
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

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

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Submodularity for Independent Cascade
• Coins for edges are flipped during activation attempts.
Submodularity for Independent Cascade

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• Coins for edges are flipped during activation attempts.
• Can pre-flip all coins and reveal results immediately.

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• Active nodes in the end are reachable via green paths from initially targeted nodes.
• Study reachability in green graphs
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
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.
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).
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.
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
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…
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

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

• 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
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?
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?)