Linear Threshold Model ◆ A user “activates” only when enough neighbors are active ◆ Can be seen as an extreme case of Voter Model v ▶ can solve infuence maximization ▶ limited user behavior prediction [1] Watts and Dodds. “Influentials, networks, and public opinion formation.” Journal of consumer research 34.4, 2007: 441-458 [2] Galuba et al. “Outtweeting the twitterers …” OSN, USENIX, 2010 [3] Singh et al. “Threshold-limited spreading in social networks with multiple initiators.”, Nature, Scientific reports 3, 2013 28 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195
Linear Threshold Model ◆ A user “activates” only when enough neighbors are active ◆ Can be seen as an extreme case of Voter Model v ▶ can solve infuence maximization ▶ limited user behavior prediction [1] Watts and Dodds. “Influentials, networks, and public opinion formation.” Journal of consumer research 34.4, 2007: 441-458 [2] Galuba et al. “Outtweeting the twitterers …” OSN, USENIX, 2010 [3] Singh et al. “Threshold-limited spreading in social networks with multiple initiators.”, Nature, Scientific reports 3, 2013 29 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195
Linear Threshold Model ◆ A user “activates” only when enough neighbors are active ◆ Can be seen as an extreme case of Voter Model v ▶ can solve infuence maximization ▶ limited user behavior prediction [1] Watts and Dodds. “Influentials, networks, and public opinion formation.” Journal of consumer research 34.4, 2007: 441-458 [2] Galuba et al. “Outtweeting the twitterers …” OSN, USENIX, 2010 [3] Singh et al. “Threshold-limited spreading in social networks with multiple initiators.”, Nature, Scientific reports 3, 2013 30 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195
Linear Threshold Model ◆ A user “activates” only when enough neighbors are active ◆ Can be seen as an extreme case of Voter Model v v ▶ can solve infuence maximization ▶ limited user behavior prediction [1] Watts and Dodds. “Influentials, networks, and public opinion formation.” Journal of consumer research 34.4, 2007: 441-458 [2] Galuba et al. “Outtweeting the twitterers …” OSN, USENIX, 2010 [3] Singh et al. “Threshold-limited spreading in social networks with multiple initiators.”, Nature, Scientific reports 3, 2013 31 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195
Independent Cascade Model ◆ Each user has single attempts to probabilistically infuence its neighbors ▶ can solve infuence maximization [2] ▶ works for the case of multiple types of opinions [5] [1] Goldenberg et al. “Talk of the network…” Marketing letters 12.3 (2001): 211-223. [2] Kempe et al. “Maximizing the spread of influence through a social network.” ACM SIGKDD, 2003. [3] Nemhauser et al. “An analysis of approximations for maximizing submodular set functions—I.” Math. Programming 14.1 (1978) [4] Mossel and Roch. “On the submodularity of influence in social networks.” TOC. ACM, 2007. 32 [5] Carnes et al. “Maximizing influence in a competitive social network…”, Electronic commerce. ACM, 2007
Independent Cascade Model ◆ Each user has single attempts to probabilistically infuence its neighbors ▶ can solve infuence maximization [2] ▶ works for the case of multiple types of opinions [5] [1] Goldenberg et al. “Talk of the network…” Marketing letters 12.3 (2001): 211-223. [2] Kempe et al. “Maximizing the spread of influence through a social network.” ACM SIGKDD, 2003. [3] Nemhauser et al. “An analysis of approximations for maximizing submodular set functions—I.” Math. Programming 14.1 (1978) [4] Mossel and Roch. “On the submodularity of influence in social networks.” TOC. ACM, 2007. 33 [5] Carnes et al. “Maximizing influence in a competitive social network…”, Electronic commerce. ACM, 2007
Independent Cascade Model ◆ Each user has single attempts to probabilistically infuence its neighbors ▶ can solve infuence maximization [2] ▶ works for the case of multiple types of opinions [5] [1] Goldenberg et al. “Talk of the network…” Marketing letters 12.3 (2001): 211-223. [2] Kempe et al. “Maximizing the spread of influence through a social network.” ACM SIGKDD, 2003. [3] Nemhauser et al. “An analysis of approximations for maximizing submodular set functions—I.” Math. Programming 14.1 (1978) [4] Mossel and Roch. “On the submodularity of influence in social networks.” TOC. ACM, 2007. 34 [5] Carnes et al. “Maximizing influence in a competitive social network…”, Electronic commerce. ACM, 2007
Virus Spread Models (SIR) ◆ Each user is either S(usceptible), I(nfected), or R(ecovered) ◆ S → I with rate β ◆ I → R with rate μ ◆ R do not change ◆ Modeling total shares of users of types S, I, and R ◆ Example: h t t p : / / c s . u c s b . e d u / ~ v i c t o r / p u b / u c s b / m a e / s i r - s i m u l a t i o n . m p 4 [1] Youssef and Scoglio. “An individual-based approach to SIR epidemics…” JTB 283.1, 2011 35 [*] SIR Simulation, YouTube (c) Kishoj Bajracharya
Virus Spread Models (SIR) ◆ Each user is either S(usceptible), I(nfected), or R(ecovered) ◆ S → I with rate β ◆ I → R with rate μ ◆ R do not change ▶ allow for asymptotic analysis (“what β causes epidemy?”) ▶ can predict total shares S, I, R ▶ can predict state for each user - by approximating Pr of being infected in a particular n'hood with average Pr of being infected in the network [1] Youssef and Scoglio. “An individual-based approach to SIR epidemics…” JTB 283.1, 2011 36
Bayesian Models ◆ User state – distribution over an unknown value θ (e.g., the best candidate in elections) ◆ User state is updated based upon reception of a signal s (e.g., info from neighbors) ◆ Model assumes each user has reliable model of the world ▶ can do asymptotic analysis [1] Acemoglu and Ozdaglar. “Opinion dynamics and learning in social networks.”, DGA 1.1, 2011 37
Models vs. Applications Asymptotic Influence Anomaly Model \ Problem Forecasting Analysis Maximization Detection + DeGroot-Stone + Voter + ± Linear Threshold + Independent Cascade + ± Virus Spread + ± 1 Bayesian ◆ Linear deterministic and physical models allow for asymptotic analysis ◆ Probabilistic models allow for infuence maximization ◆ Model alone is not enough for anomaly detection and forecasting 1 Any probabilistic model can be used for influence maximization via simulation 38
Agenda ◆ Dynamic Networks – focus on dynamics of node states rather than network structure ◆ Social Networks and Opinion Dynamics – target applications: forecasting, influence maximization, anomaly detection ◆ Models of Information Dynamics – models alone do not solve forecasting and anomaly detection problems ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion 39
Agenda ◆ Dynamic Networks – focus on dynamics of node states rather than network structure ◆ Social Networks and Opinion Dynamics – target applications: forecasting, influence maximization, anomaly detection ◆ Models of Information Dynamics – models alone do not solve forecasting and anomaly detection problems ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion 40
Anomaly Detection: Existing Methods ◆ Community-based methods ▶ anomalies in community structure evolution ◆ Decomposition-based methods ▶ track change of spectral properties ◆ Distance-based methods ▶ look at how new network state deviates from past states [1] Ranshous et al. “Anomaly detection in dynamic networks: A survey.” Technical Report, NCSU, 2014 [2] Pincombe “Anomaly detection in time series of graphs using ARMA processes.”, ASOR BULLETIN 24.4, 2005. 43 [3] Papadimitriou et al. “Web graph similarity for anomaly detection.” JISA 1.1 (2010)
Anomaly Detection: Existing Methods – structure-driven ◆ Community-based methods ▶ anomalies in community structure evolution – structure-driven ◆ Decomposition-based methods ▶ track change of spectral properties ◆ Distance-based methods ▶ look at how new network state deviates from past states – depends on how we define the distance [1] Ranshous et al. “Anomaly detection in dynamic networks: A survey.” Technical Report, NCSU, 2014 [2] Pincombe “Anomaly detection in time series of graphs using ARMA processes.”, ASOR BULLETIN 24.4, 2005. 44 [3] Papadimitriou et al. “Web graph similarity for anomaly detection.” JISA 1.1 (2010)
Distance-based Methods for Anomaly Detection (1) ◆ Threshold deviation from the previous state 45
Distance-based Methods for Anomaly Detection (2) ◆ Threshold deviation from “the average” Recent Network States median 46
Distance-based Methods for Anomaly Detection (3) ◆ Assess how model for recent states fts new state Time Series of Recent Network States embed Time Series in Embedding Space model Time Series Model (AR) – what is the error of the model with respect to the new state? 47 [1] Linial et al. “The geometry of graphs and some of its algorithmic applications.” Combinatorica 15.2 (1995)
Distance-based Methods for Anomaly Detection (4) ◆ Cluster network states based on their pairwise distances ◆ If have training data, classify network states 48 [1] Riesen and Bunke. “Graph classification by means of Lipschitz embedding.” SMC, Part B, IEEE Transactions on 39.6 (2009)
Distance-based Methods for Anomaly Detection (4) ◆ Cluster network states based on their pairwise distances ◆ If have training data, classify network states – using SVM require a distance measure (either for embedding or as a kernel) 49 [1] Riesen and Bunke. “Graph classification by means of Lipschitz embedding.” SMC, Part B, IEEE Transactions on 39.6 (2009)
Forecasting: Existing Methods ◆ General forecasting problem – unsolvable ◆ Need to make assumptions about time series ▶ Time series modeling - assume time series has a simple “structure” ▶ Extrapolation along similar past time series - assume network's evolution may “repeat” 50
Forecasting: Existing Methods (1) ◆ Find a model for time series of network states; use it for prediction Time Series of Recent Network States Candidate States embed Time Series in Embedding Space model Time Series Model (AR) 51 forecast
Forecasting: Existing Methods (2) ◆ Query similar time series (observed in the past); use them for prediction Time Series of Recent Network States Matched Time Series of Past Behavior DB – matching can happen in an embedding space 52
Forecasting: Existing Methods (2) ◆ Query similar time series (observed in the past); use them for prediction Time Series of Recent Network States Matched Time Series of Past Behavior DB – matching can happen in an embedding space match 53
Forecasting: Existing Methods (2) ◆ Query similar time series (observed in the past); use them for prediction Time Series of Recent Network States Candidate States Matched Time Series of Past Behavior DB – matching can happen in an embedding space match 54
Anomaly Detection and Forecasting: Summary of Methods ◆ Anomaly detection ▶ thresholding deviation from “the average” ▶ model ftting ▶ classifcation ◆ Forecasting ▶ time series modeling ▶ querying similar time series 55
Anomaly Detection and Forecasting: Summary of Methods ◆ Anomaly detection ▶ thresholding deviation from “the average” ▶ model ftting ▶ classifcation – each method requires a distance measure ◆ Forecasting ▶ time series modeling ▶ querying similar time series 56
Anomaly Detection and Forecasting: Summary of Methods ◆ Anomaly detection ▶ thresholding deviation from “the average” ▶ model ftting ▶ classifcation – each method requires a distance measure ◆ Forecasting that should measure the ▶ time series modeling distance with respect to how opinion propagates ▶ querying similar time series 57
Agenda ◆ Dynamic Networks – focus on dynamics of node states rather than network structure ◆ Social Networks and Opinion Dynamics – target applications: forecasting, influence maximization, anomaly detection ◆ Models of Information Dynamics – models alone do not solve forecasting and anomaly detection problems ◆ Forecasting and Anomaly Detection – existing methods require a distance measure ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion 58
Agenda ◆ Dynamic Networks – focus on dynamics of node states rather than network structure ◆ Social Networks and Opinion Dynamics – target applications: forecasting, influence maximization, anomaly detection ◆ Models of Information Dynamics – models alone do not solve forecasting and anomaly detection problems ◆ Forecasting and Anomaly Detection – existing methods require a distance measure ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion 59
Distance Measures from Vector Spaces ◆ ◆ Hamming ◆ Canberra ◆ Jaccard ◆ Cosine ◆ Kulback-Leibler ◆ Quadratic Form ◆ Mahalanobis 60
Distance Measures from Vector Spaces ◆ ◆ Hamming ◆ Canberra ◆ Jaccard ◆ Cosine ◆ Kulback-Leibler ◆ Quadratic Form ◆ Mahalanobis – do not work well for networks 61
Network Distance Measure Wanted ◆ Need a distance measure d(G 1 , G 2 ) specifc for social network states ◆ Should be metric ◆ Must be scalable 62
Distance Measures for Networks ◆ Isomorphism-based ◆ Graph Edit Distance ◆ Iterative ◆ Feature-based ◆ Graph Kernels 63
Isomorphism-based Distance Measures ◆ Measure how much two networks are structurally similar [1-3] ▶ Structure-driven ▶ NP-hard to compute precisely ▶ Still hard to compute approximately [4] [1] Bunke and Shearer. “A graph distance metric based on the maximal common subgraph.”, Pattern recognition letters 19.3 (1998) [2] Bunke et al. “On the minimum common supergraph of two graphs.”, Computing 65.1 (2000) [3] Fernández et al. "A graph distance metric combining [MCS] and [LCS]." Pattern Recognition Letters 22.6 (2001) 64 [4] Umeyama. “An eigendecomposition approach to weighted graph matching problems.” PAMI, IEEE Transactions on 10.5 (1988)
Graph Edit Distance (GED) ◆ GED(G 1 , G 2 ) = min cost of transforming G 1 into G 2 using node / edge insertion, deletion, and substitution [1] 2 2 3 3 4 4 1 1 5 5 2 2 3 3 5 5 1 4 5 5 ▶ Mostly, structure-driven; cannot capture opinion dynamics ▶ Expensive to compute [1] Bunke and Allermann. “Inexact graph matching for structural pattern recognition.” Pattern Recognition Letters 1.4 (1983) [2] Riesen and Bunke. “Approximate [GED] computation by means of bipartite graph matching.” IVC 27.7 (2009) 65 [3] Bunke “On a relation between [GED] and [MCS].” Pattern Recognition Letters 18.8 (1997)
Iterative Distance Measures ◆ “Nodes are similar if their neighborhoods are similar” [1-5] v 2 v 3 v 2 u 2 v 2 u 3 v 3 u 2 v 3 u 3 v 1 v 5 v 4 ... ... v 1 u 1 u 2 u 3 u 1 v 5 u 4 v 4 u 4 u 4 ▶ Hard to account for difference in node states in a meaningful way ▶ Expensive to compute [1] Blondel et al. “A measure of similarity between graph vertices…” SIAM review 46.4 (2004) [2] Heymans and Singh. “Deriving phylogenetic trees from the similarity analysis of metabolic pathways.” Bioinformatics 19 (2003) [3] Leicht et al. “Vertex similarity in networks.” Physical Review E 73.2 (2006) [4] Melnik et al. “Similarity flooding…” ICDE 2002 66 [5] Jeh and Widom. “SimRank: a measure of structural-context similarity”, SIGKDD 2002
Feature-based Distance Measures ◆ Instead of comparing networks, compare their features Feature Space Graph Space Distribution Space (possibly, dim-reduced) embed summarize ◆ Commonly used features ▶ degree, clustering coeffcient, betweenness ▶ diameter, curvature ▶ frequent substructures ▶ spectral features [1] Macindoe and Richards. “Graph comparison using fine structure analysis.” SocialCom, 2010 [2] Berlingerio et al. “NetSimile: a scalable approach to size-independent network similarity.” arXiv:1209.2684 (2012)] [3] Zhu and Wilson. “A Study of Graph Spectra for Comparing Graphs.” BMVC. 2005. 67 [?] TODO-hammond-wavelets-graphs-spectral-2011.pdf – or – ODO-shuman-spectral-signal-graph-2013
Feature-based Distance Measures ◆ Instead of comparing networks, compare their features Feature Space Graph Space Distribution Space (possibly, dim-reduced) embed summarize ◆ Commonly used features ▶ degree, clustering coeffcient, betweenness Open Problem: how to ▶ diameter, curvature extract socially relevant network features (fast)? ▶ frequent substructures ▶ spectral features [1] Macindoe and Richards. “Graph comparison using fine structure analysis.” SocialCom, 2010 [2] Berlingerio et al. “NetSimile: a scalable approach to size-independent network similarity.” arXiv:1209.2684 (2012)] [3] Zhu and Wilson. “A Study of Graph Spectra for Comparing Graphs.” BMVC. 2005. 68 [?] TODO-hammond-wavelets-graphs-spectral-2011.pdf – or – ODO-shuman-spectral-signal-graph-2013
Graph Kernels ◆ Kernels – distance measures that can serve as inner products ◆ Compare substructures – walks, paths, cycles, trees, … ◆ Main use – structure comparison of non-aligned small graphs Random Walk Kernel v 2 u 2 v 2 u 3 v 3 u 2 v 3 u 3 + + + + ... ... v 1 u 1 v 5 u 4 v 4 u 4 sim(v, u) = “number of common walks of any length in-coming to v and u ” 69 [1] Vishwanathan et al. “Graph kernels.” JMLR 11 (2010)
Graph Kernels ◆ Kernels – distance measures that can serve as inner products ◆ Compare substructures – walks, paths, cycles, trees, … ◆ Main use – structure comparison of non-aligned small graphs Random Walk Kernel Iterative Distance Measure v 2 u 2 v 2 u 2 v 2 u 3 v 3 u 2 v 3 u 3 v 2 u 3 v 3 u 2 v 3 u 3 + + + + ... ... ... ... v 1 u 1 v 1 u 1 v 5 u 4 v 4 u 4 v 5 u 4 v 4 u 4 sim(v, u) = “number of common walks sim(v, u) = “share of common 'long' walks of any length in-coming to v and u ” in-coming to and out-going from v and u ” 70 [1] Vishwanathan et al. “Graph kernels.” JMLR 11 (2010)
Graph Kernel for Polar Opinion Dynamics? ◆ Networks aligned – non need for all-to-all comparison ◆ Compare aligned neighborhoods in terms of “infuence potential” ◆ Walker should be aware of how polar opinion propagates v 's neighborhood summary: 3 × 1 × v ◆ Doable from operational point of view ◆ How to express analytically? – Open problem (future work) 71
Graph Kernel for Polar Opinion Dynamics? ◆ Networks aligned – non need for all-to-all comparison ◆ Compare aligned neighborhoods in terms of “infuence potential” ◆ Walker should be aware of how polar opinion propagates – Does not care about a particular configuration of node states in the neighborhood (only reachability is captured) v v v – Our approach using EMD will capture both node states and their locations 72
Distance Measures: Summary ◆ Existing distance measures – good for structure comparison ◆ At large scale – either too time-complex or not enough discriminating ◆ No existing distance can capture polar sentiment dynamics ◆ Perfect distance measure: 74
Distance Measures: Summary ◆ Existing distance measures – good for structure comparison ◆ At large scale – either too time-complex or not enough discriminating ◆ No existing distance can capture polar sentiment dynamics ◆ Perfect distance measure: state(v 2 ) – determined by a model of information propagation state(v 1 ) 75
Distance Measures: Summary ◆ Existing distance measures – good for structure comparison ◆ At large scale – either too time-complex or not enough discriminating ◆ No existing distance can capture polar sentiment dynamics ◆ Perfect distance measure: state(v 2 ) – determined by a model of information propagation state(v 1 ) Exponential size Hard, since models are “prescriptive” 76
Agenda ◆ Dynamic Networks – focus on dynamics of node states rather than network structure ◆ Social Networks and Opinion Dynamics – target applications: forecasting, influence maximization, anomaly detection ◆ Models of Information Dynamics – models alone do not solve forecasting and anomaly detection problems ◆ Forecasting and Anomaly Detection – existing methods require a distance measure ◆ Distance Measures – existing measures are inadequate for the analysis of opinion dynamics ◆ Earth Mover's Distance ◆ Conclusion 77
Agenda ◆ Dynamic Networks – focus on dynamics of node states rather than network structure ◆ Social Networks and Opinion Dynamics – target applications: forecasting, influence maximization, anomaly detection ◆ Models of Information Dynamics – models alone do not solve forecasting and anomaly detection problems ◆ Forecasting and Anomaly Detection – existing methods require a distance measure ◆ Distance Measures – existing measures are inadequate for the analysis of opinion dynamics ◆ Earth Mover's Distance ◆ Conclusion 78
Earth Mover's Distance (1) ◆ Comparing two aligned networks, we want to account for both ▶ differences in node states ▶ where these differences are located ◆ Natural formulation: cross-bin distance between distributions How much does it cost to transform G 1 into G 2 using opinion propagation as edit operations? 79
Earth Mover's Distance (2) ◆ Comparing two aligned networks, we want to account for both ▶ differences in node states ▶ where these differences are located ◆ Natural formulation: cross-bin distance between distributions Look at network states as distributions 80
Earth Mover's Distance (3) ◆ Comparing two aligned networks, we want to account for both ▶ differences in node states ▶ where these differences are located ◆ Natural formulation: cross-bin distance between distributions ... cross-bin ground distance D Look at network states as distributions where distance between bins is defined as the distance between corresponding nodes 81
Earth Mover's Distance (4) ◆ Comparing two aligned networks, we want to account for both ▶ differences in node states ▶ where these differences are located ◆ Natural formulation: cross-bin distance between distributions ... cross-bin ground distance D Compute optimal cost of transforming one distribution into another with respect to ground distance D – known as Earth Mover's Distance 82
Earth Mover's Distance (5) ◆ Comparing two aligned networks, we want to account for both ▶ differences in node states ▶ where these differences are located ◆ Natural formulation: cross-bin distance between distributions v ... D(u, v) u cross-bin ground distance D D(u, v) depends on Key observation: – shortest-path(u, v) – states of nodes separating u and v 83
Earth Mover's Distance: Summary ◆ Allows to capture how opinion propagates in the network ◆ Is spatially sensitive (unlike the proposed kernel) ◆ Metric ◆ Generally, expensive ▶ but can be computed fast if applied to aligned neighborhoods ◆ Original EMD does not work well with distributions having different mass ▶ our version of EMD does ◆ Can be adapted to capture structural difference [1] Rubner et al. “The Earth Mover's Distance as a metric for image retrieval.”, IJCV 40.2, 2000 [2] Ljosa et al. “Indexing spatially sensitive distance measures using multi-resolution lower bounds.” EDBT, 2006 84 [3] Pele and Werman, “A linear time histogram metric for improved SIFT matching,” ECCV, 2008
Agenda ◆ Dynamic Networks – focus on dynamics of node states rather than network structure ◆ Social Networks and Opinion Dynamics – target applications: forecasting, influence maximization, anomaly detection ◆ Models of Information Dynamics – models alone do not solve forecasting and anomaly detection problems ◆ Forecasting and Anomaly Detection – existing methods require a distance measure ◆ Distance Measures – existing measures are inadequate for the analysis of opinion dynamics ◆ Earth Mover's Distance – good distance measure for opinion distributions over networks ◆ Conclusion 85
Agenda ◆ Dynamic Networks – focus on dynamics of node states rather than network structure ◆ Social Networks and Opinion Dynamics – target applications: forecasting, influence maximization, anomaly detection ◆ Models of Information Dynamics – models alone do not solve forecasting and anomaly detection problems ◆ Forecasting and Anomaly Detection – existing methods require a distance measure ◆ Distance Measures – existing measures are inadequate for the analysis of opinion dynamics ◆ Earth Mover's Distance – good distance measure for opinion distributions over networks ◆ Conclusion 86
Open Problems: Summary BFS over custom semirings Fast Extraction of Socially Relevant Features v – want “smarter” DeGroot-Stone Model – should capture both node states and – want kernel aware of polar opinions network structure Perfect Distance Measure Earth Mover's Distance s t a t e ( v 2 ) – how to compute it fast without coarsening the input too much? state(v 1 ) – how to handle transitive learning? – how to use models to define costs? – how to reduce size of search space? – how to properly capture topological diff? 87
Future Plans ◆ “EMD-Based Distance Measure for the Analysis of Polar Sentiment Dynamics in Social Networks” ▶ fnish experiments (June) ▶ submit (Summer) ◆ 3-4 papers in 2014-16; target venues: KDD, ICDM, SIGMOD, VLDB ◆ Proposal: Fall, 2015 ◆ Defense: Fall, 2016 88
~ Thanks ~ 89
References (1) 1) Network structure Watts, Strogatz. “Collective dynamics of ‘small-world’networks.” Nature 393.6684, 1998 Barabási, Albert. “Emergence of scaling in random networks.” Science 286.5439, 1999 Leskovec et al. “Graphs over time: densifcation laws, shrinking diameters and possible explanations.", SIGKDD, 2005 Granovetter, “The strength of weak ties.” American Journal of Sociology 78.6, 1973 2) Models of Information Dynamics 2.a) DeGroot-Stone Model DeGroot, “Reaching a consensus.” Journal of the ASA 69.345, 1974 Patterson, Bamieh. “Interaction-driven opinion dynamics in online social networks.”, WSMA. ACM, 2010 Macropol et al. “I act, therefore I judge: …” ASONAM, ACM 2013 Hegselmann and Krause. “Opinion dynamics and bounded confdence models...”, ASSS 5.3, 2002 2.b) Voter Model Even-Dar and Shapira. “A note on maximizing the spread of infuence in social networks.”, INE 2007 Kimura et al. “Learning to Predict Opinion Share in Social Networks.” AAAI, 2010 Yildiz et al. “Discrete opinion dynamics with stubborn agents.” SSRN eLibrary, 2011 2.c) Linear Threshold Model Watts and Dodds. “Infuentials, networks, and public opinion formation.” JCR 34.4, 2007 Galuba et al. “Outtweeting the twitterers …” OSN, USENIX, 2010 Singh et al. “Threshold-limited spreading in social networks with multiple initiators.”, Nature,SR 3, 2013 Saito et al. “Selecting information diffusion models over social networks …”, MLKDD, 2010. 180-195 90
References (2) 2.d) Independent Cascade Model Goldenberg et al. “Talk of the network…” Marketing letters 12.3, 2001 Kempe et al. “Maximizing the spread of infuence through a social network.” ACM SIGKDD, 2003 Nemhauser et al. “Analysis of approximations for maximizing submodular set functions MP 14.1, 1978 Mossel and Roch. “On the submodularity of infuence in social networks.” TOC, ACM, 2007 Carnes et al. “Maximizing infuence in a competitive social network…”, EC ACM, 2007 2.e) Virus Spread Models Youssef and Scoglio. “An individual-based approach to SIR epidemics…” JTB 283.1, 2011 2.f) Bayesian Models Acemoglu and Ozdaglar. “Opinion dynamics and learning in social networks.”, DGA 1.1, 2011 3) Applications Ranshous et al. “Anomaly detection in dynamic networks: A survey.” Technical Report, NCSU, 2014 Pincombe “Anomaly detection in time series of graphs using ARMA processes.”, ASOR 24.4, 2005 Papadimitriou et al. “Web graph similarity for anomaly detection.” JISA 1.1, 2010 4) Graph Embedding Linial et al. “The geometry of graphs and some of its algorithmic applications.” Combinatorica 15.2, 1995 Riesen and Bunke. “Graph classifcation by means of Lipschitz embedding.” SMC, Part B, 39.6, 2009 91
References (3) 5) Distance Measures for Networks 5.a) Isomorphism-based Bunke and Shearer. “A graph distance metric based on the maximal common subgraph.”, PRL 19.3, 1998 Bunke et al. “On the minimum common supergraph of two graphs.”, Computing 65.1, 2000 Fernández et al. "A graph distance metric combining [MCS] and [LCS].", PRL 22.6, 2001 5.b) Graph Edit Distance Bunke and Allermann. “Inexact graph matching for structural pattern recognition.” PRL 1.4, 1983 Riesen and Bunke. “Approximate [GED] computation by means of bipartite graph matching.” IVC 27.7, 2009 Bunke “On a relation between [GED] and [MCS].” PRL 18.8, 1997 5.c) Iterative Blondel et al. “A measure of similarity between graph vertices…” SIAM review 46.4 (2004) Heymans and Singh. “Deriving phylogenetic trees from the similarity analysis of metabolic pathways.” BI 19, 2003 Melnik et al. “Similarity fooding …” ICDE 2002 Jeh and Widom. “SimRank: a measure of structural-context similarity”, SIGKDD 2002 5.c) Feature-based Macindoe and Richards. “Graph comparison using fne structure analysis.” SocialCom, 2010 Berlingerio et al. “NetSimile: a scalable approach to size-independent network similarity.” arXiv:1209.2684, 2012 Zhu and Wilson. “A Study of Graph Spectra for Comparing Graphs.” BMVC, 2005 5.d) Graph Kernels Vishwanathan et al. “Graph kernels.” JMLR 11, 2010 5.e) Earth Mover's Distance Rubner et al. “The Earth Mover's Distance as a metric for image retrieval.”, IJCV 40.2, 2000 Ljosa et al. “Indexing spatially sensitive distance measures using multi-resolution lower bounds,” EDBT, 2006 92 Pele and Werman, “A linear time histogram metric for improved SIFT matching,” ECCV, 2008
Backup Slides 93
DeGroot-Stone Model: Details [1] Backup Slide ◆ Each user adopts the weighted sum of the states of its in-neighbors as its new state v 2 v 3 v 1 v 4 v 4 ◆ Applications: study of asymptotic consensus (“stationary state”) ▶ For arbitrarily connected network, consensus not guaranteed ▶ Strongly connected aperiodic network: - P 's eigenspace corresponding to λ = 1 has dim=1 ◆ Extensions: accounting for user interaction frequency [2], allowing few users to act at a time [3], using dynamic P [4, 5] [1] DeGroot, “Reaching a consensus.” Journal of the ASA 69.345 (1974): 118-121 [2] Patterson, Bamieh. “Interaction-driven opinion dynamics in online social networks.”, WSMA. ACM, 2010 [3] Macropol et al. “I act, therefore I judge: …” ASONAM. ACM 2013 [4] Hegselmann and Krause. “Opinion dynamics and bounded confidence models, analysis, and simulation.”, ASSS 5.3, 200 94 [5] Mirtabatabaei and Bullo. “Opinion dynamics in heterogeneous networks …”, SIAM Control and Optimization 50.5 (2012)
DeGroot-Stone Model: Details [1] Backup Slide ◆ Each user adopts the weighted sum of the states of its in-neighbors as its new state v 2 v 3 v 1 v 4 Open Problem: can we use a “smarter” update rule (e.g., if-then) and still be able to do asymptotic analysis (matrix v 4 algebra over a custom semiring)? ◆ Applications: study of asymptotic consensus (“stationary state”) ▶ For arbitrarily connected network, consensus not guaranteed ▶ Strongly connected aperiodic network: - P 's eigenspace corresponding to λ = 1 has dim=1 ◆ Extensions: accounting for user interaction frequency [2], allowing few users to act at a time [3], using dynamic P [4, 5] [1] DeGroot, “Reaching a consensus.” Journal of the ASA 69.345 (1974): 118-121 [2] Patterson, Bamieh. “Interaction-driven opinion dynamics in online social networks.”, WSMA. ACM, 2010 [3] Macropol et al. “I act, therefore I judge: …” ASONAM. ACM 2013 [4] Hegselmann and Krause. “Opinion dynamics and bounded confidence models, analysis, and simulation.”, ASSS 5.3, 200 95 [5] Mirtabatabaei and Bullo. “Opinion dynamics in heterogeneous networks …”, SIAM Control and Optimization 50.5 (2012)
Voter Model: Details [1] Backup Slide ◆ Discrete-time process like DeGroot-Stone's ◆ of user's choosing state A is proportional to # of A 's proponents in the neighborhood ◆ Converges to consensus in with high probability if no stubborn users ◆ Applications: infuence maximization ▶ Infuence B users, so that information maximally spreads through the network ▶ Uniform infuence costs: exact solution is top-B highest degree nodes ▶ General case: exact solution is NP-hard; exists FPTAS ◆ Extensions: non-binary user states [2], stubborn users → persistent disagreement [3] [1] Even-Dar and Shapira. “A note on maximizing the spread of influence in social networks.”, INE 2007. 281-286. [2] Kimura et al. “Learning to Predict Opinion Share in Social Networks.” AAAI, 2010. 96 [3] Yildiz et al. “Discrete opinion dynamics with stubborn agents.” SSRN eLibrary, 2011.
Linear Threshold Model: Details Backup Slide ◆ A user activates only when suffcient number of neighbors get activated ◆ Applications: ▶ Detecting infuentials: experimentally study average size of triggered cascade [1] ▶ User behavior prediction: prediction of URL tweeting probabilities [2] ◆ Extensions: ▶ Allowing for multiple initiators [3] ▶ Asynchronous linear threshold (activation time delay) [4] [1] Watts and Dodds. “Influentials, networks, and public opinion formation.” Journal of consumer research 34.4, 2007: 441-458 [2] Galuba et al. “Outtweeting the twitterers …” OSN, USENIX, 2010 [3] Singh et al. “Threshold-limited spreading in social networks with multiple initiators.”, Nature, Scientific reports 3, 2013 97 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195
Independent Cascade Model: Details Backup Slide ◆ Active user undertakes single attempt to activate its out-neighbors (“edge activation”) v 2 v 3 v 2 v 3 v 2 v 3 success success v 1 v 1 v 1 success failure v 5 v 4 v 4 v 4 v 5 v 5 ◆ Applications: ▶ Experimental study: strong ties vs. weak ties vs. external infuence (ads) [1] ▶ Infuence maximization [2] is evaluated via simulation - maximization via hill-climbing with approximation guarantee [3] ◆ Extension: may depend on the nodes who have tried to activate node j [2, 4] [1] Goldenberg et al. “Talk of the network…” Marketing letters 12.3 (2001): 211-223. [2] Kempe et al. “Maximizing the spread of influence through a social network.” ACM SIGKDD, 2003. [3] Nemhauser et al. “An analysis of approximations for maximizing submodular set functions—I.” Math. Programming 14.1 (1978) 98 [4] Mossel and Roch. “On the submodularity of influence in social networks.” TOC. ACM, 2007.
Independent Cascade for Polar Opinions: Details Backup ◆ A hybrid of Independent Cascade and Voter Models [1] ▶ Two opinions , spread through the network; I A , I B – initial adopters A B ▶ Infuence maximization: given I B , choose I A (| I A | = k ) to maximize spread of A Distance-based Model Wave-propagation Model v Node v 's state determined by voting within the Each time step, set of active nodes “infects” the smallest sphere centered at v having nodes of I nodes 1-hop away, again, using voting ▶ Infuence function is submodular, as in regular IC → using hill-climbing [1] Carnes et al. “Maximizing influence in a competitive social network…”, Electronic commerce. ACM, 2007 99
Virus Spread Models (SIR): Details Backup Slide ◆ Each user is either S(usceptible),I(nfected), or R(ecovered) ◆ At each time step, each infected user infects susceptible neighbors and tries to recover ◆ SIR homogeneous mean feld (for heterogenous, shares defned per node degree k ): A type of asymptotic analysis: if infection rate is above τ , all / most users eventually get infected. ◆ Individual-based Markov chain SIR [1] user 1 S S I R user 1 I user 2 S I R S S I R user 2 [1] Youssef and Scoglio. “An individual-based approach to SIR epidemics…” JTB 283.1, 2011 100
Models of Statistical Physics Backup Slide ◆ System of stochastic oscillators [1] ◆ State of oscillator – P r distribution over statuses {+1, -1} ▶ transition rates determined based on neighborhood ◆ Example: h t t p : / / c s . u c s b . e d u / ~ v i c t o r / p u b / u c s b / m a e / o s c i l l a t o r s . g i f ◆ Can be seen as a virus spread model (SIS – no Recovery) ◆ Allows for asymptotic analysis [1] Turalska et al. “Complexity and synchronization”” Physical Review E 80.2, 2009 101
Extended Summary of Models Backup Slide ◆ DeGroot-Stone Model + deterministic, linear → analytically tractable using linear algebra – update is a rule of thumb – constant update matrix → information duplication – possible infnite fuctuations uses : asymptotic analysis of consensus ◆ Voter Model + captures user's ability to choose (in contrast to averaging over all neighbors) ± progressive (prevent fuctuation, but prohibit opinion change) – non-deterministic decision making → harder to analyze uses : infuence maximization ◆ Linear Threshold Model + captures thresholding in human behavior (study: for unanimous decision, τ ≈ 5 0 % ) + can be used the case of polar sentiment uses : detecting infuentials via simulation (applicable to any probabilistic model), user behavior prediction, infuence maximization ◆ Bayesian Model ◆ Independent Cascade Model + probabilistic + probabilistic – implies users have reliable model of the world + can be used the case of polar sentiment uses : asymptotic analysis of consensus ± progressive uses : infuence maximization ◆ Model of Statistical Physics ◆ Virus Spread Models + analytically tractable – not very closely describe social interaction + analytically tractable – unnatural oscillation in the absence of consensus – bimodal distributions of infection sizes uses : synchronization of coupled oscillators – individual-based SIR requires undirected graphs uses : analytical study of epidemic threshold, 102 prediction of state for each user
Quadratic Form Distance Backup Slide ◆ Quadratic Form ◆ Mahalanobis 103
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