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Guest lecture II: Amos Fiat’s Social Networks class Edith Cohen TAU, December 2014

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Today Diffusion of information/contagion in networks: Applications: Influence queries Influence maximization Influence similarity Reachability-based diffusion: Models & Scalable computation Basic reachability IC model Set of instances

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Diffusion in Networks Contagion, information, news, opinions, … spread over the network. When two nodes are connected, infection can pass from one to the other.

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Diffusion in Networks Model of how information/infection spreads

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Challenges Modeling: Formulate a model that captures what we want Scalability: Very efficient computation of many queries on very large networks

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Modeling Diffusion

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Simplest Model: Reachability “You infect everyone you can reach”

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Simplest Model: Reachability “You infect everyone you can reach” Submodular and monotone !

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Scalability: Node sketches

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Reachability diffusion model: Issues “You infect everyone you can reach” Intuition that contagion is probabilistic in nature Reachability does not capture:

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Reachability diffusion model: Issues “You infect everyone you can reach” Asymmetry: Distinguish strong or weak connections. Even if network is undirected, influence is not (may depend, say, on how many friends you have) Reachability does not capture: Strong tie weak tie

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Reachability diffusion model: Issues “You infect everyone you can reach” Asymmetry: Distinguish strong or weak connections. Even if network is undirected, influence is not (may depend, say, on how many friends you have) Reachability does not capture: More influencial Less influencial

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Reachability diffusion model: Issues “You infect everyone you can reach” Not robust: Can be very sensitive to presence or deletions of one or few (weak) edges Reachability does not capture:

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Reachability diffusion model: Issues “You infect everyone you can reach” Infection probability should decrease with path length and increase with path multiplicity. Reachability does not capture:

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Reachability diffusion model: Issues “You infect everyone you can reach” Intuition that contagion is probabilistic in nature Asymmetry: Distinguish strong or weak connections. Even if network is undirected, influence is not (may depend, say, on how many friends you have) Not robust: Can be very sensitive to presence or deletions of one or few (weak) edges Infection probability should decrease with path length and increase with path multiplicity. Reachability does not capture:

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Independent Cascade (IC) diffusion model [Kempe, Kleinberg, Tardos 2003]

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Independent Cascade (IC) Intuition that contagion is probabilistic in nature Asymmetry: Distinguish strong or weak connections. Even if network is undirected, influence is not (may depend, say, on how many friends you have) Not robust: Can be very sensitive to presence or deletions of one or few (weak) edges Infection probability should decrease with path length and increase with path multiplicity. IC model does capture:

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Independent Cascade (IC) Asymmetry: Distinguish strong or weak connections. Even if network is undirected, influence is not (may depend, say, on how many friends you have) More influencial Less influencial

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Independent Cascade (IC) Infection probability should decrease with path length and increase with path multiplicity.

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Scalability: Sketches for an IC model

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Sketches for a fixed set of instances We first consider a fixed (arbitrary) set of instances (edge sets): Influence is the average (or sum) of reachable set sizes, over instances. Motivation: When the instances come from “enough” Monte Carlo simulations of an IC model, the sketches capture the model. Capture “median” behavior of IC model Can capture relations beyond IC model (edges not independent)

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Fixed set of instances

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2 10 8 6

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Sketches for a fixed set of instances Approach I [CWY KDD 2009]

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Combined reachability sketches

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Sketches for an IC model

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IC model sketching

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Influence Maximization We can consider influence maximization: On a single instance (“static” graph) A set of instances An IC model Single instance captures the basic scalability challenges

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Influence Maximization Bad news: Problem is NP-hard even for a single instance (one “static” graph) Elements Sets Reduction to max/set cover:

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Influence Maximization Good news: Monotone and submodular

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Greedy Influence Maximization Greedy generates a sequence of nodes The approximation guarantee is for each prefix

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Greedy Sequence 1 2 3

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Scalability Greedy does not scale well even on a single “static” graph – We can not afford much more than linear time on very large networks In each step we need to determine the node with maximum marginal gain. Exact computation of the cardinality for each node is costly (search from each node)

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Scalability

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SkIM: Sketch Based Influence Maximization [CDPW CIKM 2014] We show SkIM for one instance (similar for multiple instances)

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SkIM: Sketch Based Influence Maximization [CDPW CIKM 2014]

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Sampled Sketch size Inverted sketch 1 1 1 1

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Sampled Sketch size Inverted sketch 1 1 1 1 1 1 1 1

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Sampled Sketch size Inverted sketch 1 1 1 1 1 1 1

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Sampled Sketch size Inverted sketch 1 1 2 2 1 1 1 1 1

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Sampled Sketch size Inverted sketch 1 1 2 2 1 1 1 1 1

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Sampled Sketch size Inverted sketch 1 1 2 2 1 1 1 1 1 1 1 1

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Sampled Sketch size Inverted sketch 2 2 3 3 1 1 1 1 1 1 1 1 1

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Sampled Sketch size Inverted sketch 2 2 3 3 1 1 1 1 1 1 1 1 1

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Sampled Sketch size Inverted sketch 2 2 3 3 1 1 1 1 1 1 1 1 1

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Sampled Sketch size Inverted sketch 2 2 3 3 0 1 1 1 1 1 1 1 1 1

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Sampled Sketch size Inverted sketch 3 0 Residual problem 1 1 1

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SkIM correctness

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SkIM running time (1 instance)

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SkIM: Sketch Based Influence Maximization [CDPW CIKM 2014]

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Engineering SkIM

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…Engineering SkIM

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SkIM on multiple instances

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SkIM: Sketch Based Influence Maximization [CDPW CIKM 2014]

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[CDPW 2014] data sets from SNAP

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Use of reachability sketches for influence: Chen, Wang, Young. Efficient Influence Maximization in Social Networks. KDD 2009 Combined reachability sketches and scalable influence maximization: Cohen, Delling, Pajor, Werneck. Sketch-based Influence Maximization and Computation: Scaling up with Guarantees. CIKM 2014 IC model: Kempe, Kleinberg, Tardos “Maximizing the spread of influence through a social networks” KDD 2003 Greedy algorithm for monotone submodular functions: Nemhauser, Wolsey, Fisher. “An analysis of the approximations of maximizing submodular set functions” 1978 Bibliography: Reachability-based diffusion in networks Reachability sketches: E. Cohen “Size-Estimation Framework with Applications to Transitive Closure and Reachability” JCSS 1997 KDD 2012 tutorial: Castillo, Chen, Lakshmanan “information and influence spread in social networks” http://research.microsoft.com/en- us/people/weic/kdd12tutorial_inf.aspxhttp://research.microsoft.com/en- us/people/weic/kdd12tutorial_inf.aspx There is a huge literature on scalable IM implementations (without guarantees…)

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M. Gomez-Rodriguez, D. Balduzzi, and B. Schölkopf. Uncovering the temporal dynamics of diffusion networks. In ICML, 2011. Enhanced model and scalable algorithms: Cohen, Delling, Pajor, Werneck. Timed-influence: Computation and Maximization. http://arxiv.org/abs/1410.6976http://arxiv.org/abs/1410.6976 N. Du, L. Song, M. Gomez-Rodriguez, and H. Zha. Scalable influence estimation in continuous-time diffusion networks. In NIPS. 2013 All-Distances Sketches: E. Cohen “Size-Estimation Framework with Applications to Transitive Closure and Reachability” JCSS 1997 All-Distances skethces, revisited. PODS 2014 http://arxiv.org/abs/1306.3284 Further Modelling flexibility: “timed” influence Distance-based diffusion in networks

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