Information Dynamics in Social Networks Major Area Exam Date: June - - PowerPoint PPT Presentation

Information Dynamics in Social Networks

Major Area Exam

Examinee: Victor Amelkin Committee:

• Prof. Ambuj Singh (chair)
• Prof. Xifeng Yan
• Prof. John Gilbert

Date: June 5, 2014

2

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion

3

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion

4

Networks are Everywhere

Traffc Networks Social Networks Brain Networks

5

Dynamic Networks

◆ Structure may change in time ◆ Node states may change in time

6

Dynamics: Node State vs. Structure (1)

◆ Social Network ◆ For many networks, structure evolves much slower than node states

▶ How often users message / tweet / post

vs.

▶ How often users add new connections

7

Dynamics: Node State vs. Structure (1)

◆ Social Network ◆ For many networks, structure evolves much slower than node states

▶ How often users message / tweet / post

vs.

▶ How often users add new connections

– often – not so often

8

Dynamics: Node State vs. Structure (2)

◆ Traffc Network ◆ For many networks, structure evolves much slower than node states

▶ How often speeds on a freeway change

vs.

▶ How often new freeways appear

9

Dynamics: Node State vs. Structure (2)

◆ Traffc Network ◆ For many networks, structure evolves much slower than node states

▶ How often speeds on a freeway change

vs.

▶ How often new freeways appear

– often – not so often

10

Focus: Node State Dynamics

◆Assume

▶ network structure is fxed ▶ node states may change

11

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure

12

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure

13

Online Social Networks

◆ Very large and sparse

▶ Facebook: 1.23bln active users monthly [1]; |E| / |V| ~ 22 (for a sample) [2] ▶ Twitter: 255mln active users monthly

◆ Quaint structure ◆ Host complex dynamic processes ◆ Applications in marketing, sociology, …

▶ Facebook: $2.34bln from ads for Q4-2013 [1] ▶ Twitter: 500mln tweets sent daily [3]

[1] Facebook's Q1-2014 Report: http://investor.fb.com/releasedetail.cfm?ReleaseID=821954 [2] Sparsity data for SNAP's Facebook subgraph: http://snap.stanford.edu/data/egonets-Facebook.html [3] About Twitter, Inc.: https://about.twitter.com/company

14

Social Networks: Research Directions (1)

◆ Direction #1: Studying Network Structure

▶ degree distribution, clustering coeffcient [1, 2, 3] ▶ subgraph frequency [4] ▶ densifcation and shrinking diameter over time [5] ▶ strength of dyadic ties [6]

B-A degree distribution (log scale) ArXiV, shrinking diameter over time

[1] Watts, Strogatz. “Collective dynamics of ‘small-world’networks.” Nature 393.6684 (1998): 440-442. [2] Barabási, Albert. “Emergence of scaling in random networks.” Science 286.5439 (1999): 509-512. [3] Albert et al. “Internet: Diameter of the world-wide web.” Nature 401.6749 (1999): 130-131. [4] Ugander, et al. “Subgraph frequencies: Mapping the empirical and extremal geography of large graph collections.” WWW, 2013. [5] Leskovec et al. “Graphs over time: densification laws, shrinking diameters and possible explanations.", SIGKDD, 2005 [6] Granovetter, “The strength of weak ties.” American Journal of Sociology 78.6 (1973): l.

15

Brand A , Brand B

Social Networks: Research Directions (2)

◆ Direction #2: Studying Information (Opinion, Sentiment) Dynamics

▶ Word-of-mouth information propagation ▶ How to model information propagation by users? ▶ What problems can a model solve?

16

Information Dynamics: Problems

◆ Asymptotic analysis of conditions for consensus

What the network will look like at t = ∞?

◆ Network state forecasting

What the network will look like 1 month from now?

◆ Infuence maximization

How to infuence as many users as possible?

◆ Anomaly detection

How to detect moments when network behaves anomalously?

17

Network State Forecasting

◆ Can we predict how network state will evolve? ◆ If not, can we at least analyze the asymptotic case?

18

Influence Maximization

◆ Which k users to pick so that they infuence as many users as possible?

▶ Bad choice ▶ Better choice

19

Anomaly Detection

◆ Detect when network behavior deviates from what is expected

▶ At both t1 and t2 , exactly 4 nodes change their states ▶ However

• transition t1 → t

2 is probable, but

• transition t2 → t

3 is anomalous

20

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection

21

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection

22

Models of Information Dynamics

◆ Model describes how users pass / acquire opinion to / from neighbors ◆ Popular models of information dynamics

▶ DeGroot-Stone Model ▶ Voter Model ▶ Linear Threshold Model ▶ Independent Cascade Model ▶ Virus Spread Models ▶ Bayesian Models

◆ Can models alone solve target problems?

23

Models of Information Dynamics

◆ Model describes how users pass / acquire opinion to / from neighbors ◆ Popular models of information dynamics

▶ DeGroot-Stone Model ▶ Voter Model ▶ Linear Threshold Model ▶ Independent Cascade Model ▶ Virus Spread Models ▶ Bayesian Models

◆ Can models alone solve target problems?

– some of them

24

DeGroot-Stone Model [1-5]

◆ Each time step, each node collects states of its neighbors

▶ simple linear model ▶ can do asymptotic analysis of consensus ▶ infuence maximization – problematic (at least due to determinism)

[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 [5] Mirtabatabaei and Bullo. “Opinion dynamics in heterogeneous networks …”, SIAM Control and Optimization 50.5 (2012)

3 2 1 1.5

0.2 0.1 0.15 0.55

25

DeGroot-Stone Model [1-5]

◆ Each time step, each node collects states of its neighbors

▶ simple linear model ▶ can do asymptotic analysis of consensus ▶ infuence maximization – problematic (at least due to determinism)

[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 [5] Mirtabatabaei and Bullo. “Opinion dynamics in heterogeneous networks …”, SIAM Control and Optimization 50.5 (2012)

3 2 1 1.5

0.2 0.1 0.15 0.55 Open Problem: can we use a “smarter” update rule (e.g., if-then) and still be able to do asymptotic analysis?

26

Voter Model [1]

◆ Each user updates its state probabilistically, based on the number

• f “infected” immediate neighbors

▶ more natural than DeGroot-Stone (admits persistent disagreement) ▶ can solve infuence maximization

• uniform costs: top-degree heuristic – optimal
• general case – NP-hard; FPTAS exists

[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. [3] Yildiz et al. “Discrete opinion dynamics with stubborn agents.” SSRN eLibrary, 2011.

1 1

27

v Linear Threshold Model

◆ A user “activates” only when enough neighbors are active ◆ Can be seen as an extreme case of Voter Model

▶ 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 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195

28

v Linear Threshold Model

◆ A user “activates” only when enough neighbors are active ◆ Can be seen as an extreme case of Voter Model

▶ 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 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195

29

v Linear Threshold Model

◆ A user “activates” only when enough neighbors are active ◆ Can be seen as an extreme case of Voter Model

▶ 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 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195

30

v Linear Threshold Model

◆ A user “activates” only when enough neighbors are active ◆ Can be seen as an extreme case of Voter Model

▶ 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 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195

31

v Linear Threshold Model

◆ A user “activates” only when enough neighbors are active ◆ Can be seen as an extreme case of Voter Model

▶ can solve infuence maximization ▶ limited user behavior prediction

v

[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 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195

32

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. [5] Carnes et al. “Maximizing influence in a competitive social network…”, Electronic commerce. ACM, 2007

33

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. [5] Carnes et al. “Maximizing influence in a competitive social network…”, Electronic commerce. ACM, 2007

34

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. [5] Carnes et al. “Maximizing influence in a competitive social network…”, Electronic commerce. ACM, 2007

35

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: http://cs.ucsb.edu/~victor/pub/ucsb/mae/sir-simulation.mp4

[1] Youssef and Scoglio. “An individual-based approach to SIR epidemics…” JTB 283.1, 2011 [*] SIR Simulation, YouTube (c) Kishoj Bajracharya

36

Virus Spread Models (SIR)

[1] Youssef and Scoglio. “An individual-based approach to SIR epidemics…” JTB 283.1, 2011

◆ 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

37

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

38

Models vs. Applications

Model \ Problem Asymptotic Analysis Influence Maximization Forecasting Anomaly Detection DeGroot-Stone

+

Voter

+

Linear Threshold

+ ±

Independent Cascade

+

Virus Spread

+ ±

Bayesian

+ ±1

◆ 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

39

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection – models alone do not solve forecasting and anomaly detection problems

40

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection – models alone do not solve forecasting and anomaly detection problems

43

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. [3] Papadimitriou et al. “Web graph similarity for anomaly detection.” JISA 1.1 (2010)

44

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

– structure-driven – structure-driven – 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. [3] Papadimitriou et al. “Web graph similarity for anomaly detection.” JISA 1.1 (2010)

45

Distance-based Methods for Anomaly Detection (1)

◆ Threshold deviation from the previous state

46

Distance-based Methods for Anomaly Detection (2)

◆ Threshold deviation from “the average”

Recent Network States

median

47

Distance-based Methods for Anomaly Detection (3)

◆ Assess how model for recent states fts new state

Time Series of Recent Network States Time Series Model (AR) Time Series in Embedding Space embed model – what is the error of the model with respect to the new state?

[1] Linial et al. “The geometry of graphs and some of its algorithmic applications.” Combinatorica 15.2 (1995)

48

Distance-based Methods for Anomaly Detection (4)

◆ Cluster network states based on their pairwise distances ◆ If have training data, classify network states

[1] Riesen and Bunke. “Graph classification by means of Lipschitz embedding.” SMC, Part B, IEEE Transactions on 39.6 (2009)

49

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

• r as a kernel)

[1] Riesen and Bunke. “Graph classification by means of Lipschitz embedding.” SMC, Part B, IEEE Transactions on 39.6 (2009)

50

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”

51

Forecasting: Existing Methods (1)

◆ Find a model for time series of network states; use it for prediction

Time Series of Recent Network States Time Series Model (AR) Time Series in Embedding Space embed model forecast

Candidate States

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

53

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

match

– matching can happen in an embedding space

54

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

Candidate States

DB

match

– matching can happen in an embedding space

55

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

56

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

– each method requires a distance measure

57

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

– each method requires a distance measure that should measure the distance with respect to how opinion propagates

58

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection – models alone do not solve forecasting and anomaly detection problems – existing methods require a distance measure

59

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection – models alone do not solve forecasting and anomaly detection problems – existing methods require a distance measure

60

Distance Measures from Vector Spaces

◆ ◆ Hamming ◆ Canberra ◆ Jaccard ◆ Cosine ◆ Kulback-Leibler ◆ Quadratic Form ◆ Mahalanobis

61

Distance Measures from Vector Spaces

◆ ◆ Hamming ◆ Canberra ◆ Jaccard ◆ Cosine ◆ Kulback-Leibler ◆ Quadratic Form ◆ Mahalanobis

– do not work well for networks

62

Network Distance Measure Wanted

◆ Need a distance measure d(G1, G2) specifc for social network states ◆ Should be metric ◆ Must be scalable

63

Distance Measures for Networks

◆ Isomorphism-based ◆ Graph Edit Distance ◆ Iterative ◆ Feature-based ◆ Graph Kernels

64

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) [4] Umeyama. “An eigendecomposition approach to weighted graph matching problems.” PAMI, IEEE Transactions on 10.5 (1988)

65

Graph Edit Distance (GED)

◆ GED(G

1, G2 ) = min cost of transforming G1 into G2 using node / edge

insertion, deletion, and substitution [1]

▶ 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) [3] Bunke “On a relation between [GED] and [MCS].” Pattern Recognition Letters 18.8 (1997)

3 2 5 4 1 3 2 5 4 3 2 5 4 5 1 3 2 5 1 5

66

Iterative Distance Measures

◆ “Nodes are similar if their neighborhoods are similar” [1-5]

▶ 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 [5] Jeh and Widom. “SimRank: a measure of structural-context similarity”, SIGKDD 2002

v1 v2 v3 v5 v4 u1 u2 u3 u4 v1 u1 v2 u2 v2 u3 v3 u2 v3 u3 v5 u4 v4 u4

... ...

67

Feature-based Distance Measures

◆ Instead of comparing networks, compare their features ◆ Commonly used features

▶ degree, clustering coeffcient, betweenness ▶ diameter, curvature ▶ frequent substructures ▶ spectral features

Graph Space Feature Space

(possibly, dim-reduced)

embed summarize

Distribution Space

[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. [?] TODO-hammond-wavelets-graphs-spectral-2011.pdf – or – ODO-shuman-spectral-signal-graph-2013

68

Feature-based Distance Measures

◆ Instead of comparing networks, compare their features ◆ Commonly used features

▶ degree, clustering coeffcient, betweenness ▶ diameter, curvature ▶ frequent substructures ▶ spectral features

Graph Space Feature Space

(possibly, dim-reduced)

embed summarize

Distribution Space

Open Problem: how to extract socially relevant network features (fast)?

[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. [?] TODO-hammond-wavelets-graphs-spectral-2011.pdf – or – ODO-shuman-spectral-signal-graph-2013

69

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

v1 u1 v2 u2 v2 u3 v3 u2 v3 u3 v5 u4 v4 u4

... ... Random Walk Kernel sim(v, u) = “number of common walks

• f any length in-coming to v and u”

+ + + +

[1] Vishwanathan et al. “Graph kernels.” JMLR 11 (2010)

70

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

v1 u1 v2 u2 v2 u3 v3 u2 v3 u3 v5 u4 v4 u4

... ...

v1 u1 v2 u2 v2 u3 v3 u2 v3 u3 v5 u4 v4 u4

... ... Random Walk Kernel Iterative Distance Measure sim(v, u) = “number of common walks

• f any length in-coming to v and u”

sim(v, u) = “share of common 'long' walks in-coming to and out-going from v and u” + + + +

[1] Vishwanathan et al. “Graph kernels.” JMLR 11 (2010)

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 ◆ Doable from operational point of view ◆ How to express analytically? – Open problem (future work)

v's neighborhood summary: 3 × 1 × v

72

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

– Does not care about a particular configuration

• f node states in the neighborhood (only

reachability is captured) – Our approach using EMD will capture both node states and their locations

v v

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:

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(v1) state(v2)

– determined by a model of information propagation

76

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(v1) state(v2)

– determined by a model of information propagation Hard, since models are “prescriptive” Exponential size

77

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection – models alone do not solve forecasting and anomaly detection problems – existing methods require a distance measure – existing measures are inadequate for the analysis of opinion dynamics

78

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection – models alone do not solve forecasting and anomaly detection problems – existing methods require a distance measure – existing measures are inadequate for the analysis of opinion dynamics

79

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 G1 into G2 using opinion propagation as edit operations?

80

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

81

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

Look at network states as distributions where distance between bins is defined as the distance between corresponding nodes

...

cross-bin ground distance D

82

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

83

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 u v

...

cross-bin ground distance D

Key observation:

D(u, v) depends on – shortest-path(u, v) – states of nodes separating u and v

D(u, v)

84

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 [3] Pele and Werman, “A linear time histogram metric for improved SIFT matching,” ECCV, 2008

85

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection – models alone do not solve forecasting and anomaly detection problems – existing methods require a distance measure – existing measures are inadequate for the analysis of opinion dynamics – good distance measure for opinion distributions over networks

86

Agenda

◆ Dynamic Networks ◆ Social Networks and Opinion Dynamics ◆ Models of Information Dynamics ◆ Forecasting and Anomaly Detection ◆ Distance Measures ◆ Earth Mover's Distance ◆ Conclusion – focus on dynamics of node states rather than network structure – target applications: forecasting, influence maximization, anomaly detection – models alone do not solve forecasting and anomaly detection problems – existing methods require a distance measure – existing measures are inadequate for the analysis of opinion dynamics – good distance measure for opinion distributions over networks

87

Open Problems: Summary

BFS over custom semirings – want “smarter” DeGroot-Stone Model – want kernel aware of polar opinions v Fast Extraction of Socially Relevant Features

state(v1) s t a t e ( v

2

)

Perfect Distance Measure – how to use models to define costs? – how to reduce size of search space? – should capture both node states and network structure Earth Mover's Distance – how to compute it fast without coarsening the input too much? – how to handle transitive learning? – how to properly capture topological diff?

88

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

89

~ Thanks ~

90

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

91

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

92

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 Pele and Werman, “A linear time histogram metric for improved SIFT matching,” ECCV, 2008

93

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94

DeGroot-Stone Model: Details [1]

◆ Each user adopts the weighted sum of the states of its in-neighbors as its new

state

◆ 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]

v2 v1 v4 v3 v4

[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 [5] Mirtabatabaei and Bullo. “Opinion dynamics in heterogeneous networks …”, SIAM Control and Optimization 50.5 (2012)

Backup Slide

95

DeGroot-Stone Model: Details [1]

◆ Each user adopts the weighted sum of the states of its in-neighbors as its new

state

◆ 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]

v2 v1 v4 v3 v4

[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 [5] Mirtabatabaei and Bullo. “Opinion dynamics in heterogeneous networks …”, SIAM Control and Optimization 50.5 (2012)

Open Problem: can we use a “smarter” update rule (e.g., if-then) and still be able to do asymptotic analysis (matrix algebra over a custom semiring)? Backup Slide

96

Voter Model: Details [1]

◆ 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. [3] Yildiz et al. “Discrete opinion dynamics with stubborn agents.” SSRN eLibrary, 2011.

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97

Linear Threshold Model: Details

◆ 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 [4] Saito et al. “”Selecting information diffusion models over social networks for behavioral analysis.”, MLKDD, 2010. 180-195

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98

Independent Cascade Model: Details

◆ Active user undertakes single attempt to activate its out-neighbors (“edge

activation”)

◆ Applications:

▶ Experimental study: strong ties vs. weak ties vs. external infuence (ads) [1] ▶ Infuence maximization [2]

• 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) [4] Mossel and Roch. “On the submodularity of influence in social networks.” TOC. ACM, 2007.

v2 v1 v4 v3 v5 v2 v1 v4 v3 v5 v2 v1 v4 v3 v5

success success failure success

is evaluated via simulation

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99

Independent Cascade for Polar Opinions: Details

◆ A hybrid of Independent Cascade and Voter Models [1]

▶ Two opinions , spread through the network; IA, IB – initial adopters ▶ Infuence maximization: given IB, choose IA (|IA| = k) to maximize spread of ▶ 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

v

Distance-based Model

Node v's state determined by voting within the smallest sphere centered at v having nodes of I

Wave-propagation Model

A B A

Each time step, set of active nodes “infects” the nodes 1-hop away, again, using voting Backup

100

Virus Spread Models (SIR): Details

◆ 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):

◆ Individual-based Markov chain SIR [1]

[1] Youssef and Scoglio. “An individual-based approach to SIR epidemics…” JTB 283.1, 2011 A type of asymptotic analysis: if infection rate is above τ, all / most users eventually get infected.

user2 user1 S I R S I S user2 S I R S I R user1

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101

Models of Statistical Physics

◆ System of stochastic oscillators [1] ◆ State of oscillator – Pr distribution over statuses {+1, -1}

▶ transition rates determined based on neighborhood

◆ Example: http://cs.ucsb.edu/~victor/pub/ucsb/mae/oscillators.gif

◆ 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

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102

Extended Summary of Models

◆ 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 ≈ 50% ) + can be used the case of polar sentiment uses: detecting infuentials via simulation (applicable to any probabilistic model), user behavior prediction, infuence maximization

◆ Independent Cascade Model

+ probabilistic + can be used the case of polar sentiment ± progressive uses: infuence maximization

◆ Virus Spread Models

+ analytically tractable – bimodal distributions of infection sizes – individual-based SIR requires undirected graphs uses: analytical study of epidemic threshold, prediction of state for each user

◆ Bayesian Model

+ probabilistic – implies users have reliable model of the world uses: asymptotic analysis of consensus

◆ Model of Statistical Physics

+ analytically tractable – not very closely describe social interaction – unnatural oscillation in the absence of consensus uses: synchronization of coupled oscillators Backup Slide

103

Quadratic Form Distance

◆ Quadratic Form ◆ Mahalanobis

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104

Why Metric?

◆ Non-negativity and identity axioms are natural to expect ◆ Symmetry is useful at least for determinism if input is unordered ◆ Subadditivity helps accelerate search

[1] Elkan, “Using the triangle inequality to accelerate k-means.”, ICML, Vol. 3. 2003

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105

Isomorphism-based Distance Measures: Details

◆ Quantifcation of graph isomorphism ◆ Measures how much two networks are structurally similar

▶ Maximum Common Subgraph (MCS) distance [1] ▶ Least Common Supergraph (LCS) can be used instead of MCS [2] ▶ MCS & LCS [3]

◆ Characteristics:

▶ Structure-driven ▶ NP-hard to compute precisely ▶ Still hard to compute approximately [4]

v1 v2 v4 v3 v5 u1 u3 u4 u5 u2 [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) [4] Umeyama. “An eigendecomposition approach to weighted graph matching problems.” PAMI, IEEE Transactions on 10.5 (1988)

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106

Graph Edit Distance (GED): Details

◆ GED(G1, G2 ) is the min cost of transforming G1 into G2 using node / edge

insertion, deletion, and substitution [1]

◆ Can be computed approximately as an assignment problem (or optimal bipartite fow)

[2]

◆ Characteristics: (mostly) structure-driven; expensive (MCS – special case of GED [3])

[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) [3] Bunke “On a relation between [GED] and [MCS].” Pattern Recognition Letters 18.8 (1997)

substitution insertion deletion substitution with itself

3

2 5 4 1

3

2 5 4

– matrix of node transform costs – gets extended with edge transform costs (for substitutions, requires edge alignment) – optimal assignment of rows to columns is computed using Munkres' algorithm

• r any min-cost max flow solver (push-relabel )

3

2 5 4 5 1

3

2 5 1 5

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107

Iterative Distance Measures: Details

◆ “Nodes are similar if their (immediate) neighborhoods are similar” [1-5]

[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 [5] Jeh and Widom. “SimRank: a measure of structural-context similarity”, SIGKDD 2002

v1 v2 v3 v5 v4 u1 u2 u3 u4 v1 u1 v2 u2 v2 u3 v3 u2 v3 u3 v5 u4 v4 u4

... ...

Sylvester-type equaition; use vec(*) or iterate

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EMD: Definition

[1] Rubner et al. “The Earth Mover's Distance as a metric for image retrieval.”, IJCV 40.2, 2000

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109

EMD-Banks: Definition

P1 P2 P3 γ Q1 Q2 Q3 γ γ γ γ γ

◆ Adding global “bank bin” to penalize for total mass mismatch

[1] Ljosa et al. “Indexing spatially sensitive distance measures using multi-resolution lower bounds.” EDBT, 2006

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110

EMD*: Idea

v1 v2 v3 v6 v5 v4

◆ Global bank bin does not work as we expect ◆ Use local bank bins instead

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