The document discusses predicting whether reciprocal (two-way) relationships will form on social networks based on initial one-way relationships. It proposes a Triad-based Factor Graph model (TriFG) that incorporates social theories like homophily and structural balance to model relationship formation over time. Experimental results on a large Twitter network show TriFG can accurately infer 90% of reciprocal relationships and outperforms baselines like SVM and CRF. Further analysis demonstrates the impact of factors like time span and influence of elite users on relationship formation.

1. Who Will Follow You
Back? Reciprocal
Relationship
1
1
2
Prediction* 3
John Hopcroft, Tiancheng Lou, Jie Tang
Department of Computer Science, Cornell University,
2Institutefor Interdisciplinary Information Sciences, Tsinghua
University
3Department of Computer Science, Tsinghua University

2. reciprocal
parasocial
Motivation v2
v3
v1
 Two kinds of relationships in social network, v4
v5
 one-way(called parasocial) relationship and,
 two-way(called reciprocal) relationship
v6
 Two-way(reciprocal) relationship
 usually developed from a one-way relationship
 more trustful. after 3 days prediction
 Try to understand(predict) the formation of two-
way relationships
 micro-level dynamics of the social network.
v2
v3
 underlying community structure? v1
 how users influence each other?
v4
v5
v6

3. Example : real friend relationship
On Twitter : Who Will Follow You Back?
30%
?
100%
?
60%
? JimmyQiao
Ladygaga 1%
?
Shiteng
Obama Huwei

4. Several key challenges
 How to model the formation of
two-way relationships? y2= ?
y2= ?
y2
y2
 SVM & LRC y4 = 0
4
 How to combine many social y4
4
theories into the prediction y1= 1 y1
y1= 1 y1
model? y3
y3
y3 = ?
y3 = ?
v2
v3
v1
v4
v5
(v1,, v2)
(v1 v2)
(v3, v4)
3 4
v6 (v1,, v5)
(v1 v5)
(v2,, v4)
(v2 v4)

5. Outline
 Previous works
 Our approach
 Experimental results
 Conclusion & future works

6. Link prediction
 Unsupervised link prediction
 Scores & intution, such as preferential attachment [N01].
 Supervised link prediction
 supervised random walks [BL11].
 logistic regression model to predict positive and negative links [L10].
 Main differences:
 We predict a directed link instead of only handles undirected social networks.
 Our model is dynamic and learned from the evolution of the Twitter network.

7. Social behavior analysis
 Existing works on social behavior analysis:
 The difference of the social influence on difference topics and to model the topic-
level social influence in social networks. [T09]
 How social actions evolve in a dynamic social network? [T10]
 Main differences:
 The proposed methods in previous work can be used here
 but the problem is fundamentally different.

8. Twitter study
 The twitter network.
 The topological and geographical properties. [J07]
 Twittersphere and some notable properties, such as a non-power-law follower
distribution, and low reciprocity. [K10]
 The twitter users.
 Influential users.
 Tweeting behaviors of users.
 The tweets.
 Utilize the real-time nature to detect a target event. [S10]
 TwitterMonitor, to detect emerging topics. [M10]

9. Outline
 Previous works
 Our approach
 Experimental results
 Conclusion & future works

10. Factor graph model
 Problem definition
 Given a network at time t, i.e., Gt = (Vt, Et, Xt, Yt)
 Variables y are partially labeled.
 Goal : infer unknown variables.
 Factor graph model
 P(Y | X, G) = P(X, G|Y) P(Y) / P(X, G) = C0 P(X | Y) P(Y | G)
 In P(X | Y), assuming that the generative probability is conditionally independent,
 P(Y | X, G) = C0P(Y | G)ΠP(xi|yi)
 How to deal with P(Y | G) ?

11. Random Field
Given an undirected graph G=(V,
Ya Yb E) such that Y={Yv|v∈V}, if
the probability of Yv given X and
Yc those random variables
corresponding to nodes
neighboring v in G. Then (X, Y)
Ye is a conditional random field.
Yd
undirected graphical model
Yf globally conditioned on X

12. Conditional random field(CRF)
 Model them in a Markov random field, by the Hammersley-Clifford theorem,
 P(xi|yi) = 1/Z1 * exp {Σαj fj (xij, yi)}
 P(Y|G) = 1/Z2 * exp {ΣcΣkμkhk(Yc)}
 Z1 and Z2 are normalization factors.
 So, the likelihood probability is :
 P(Y | X, G) = 1/Z0 * exp {Σαj fj (xij, yi) + ΣcΣkμkhk(Yc)}
 Z0 is normalization factors.

13. Maximize likelihood
 Objective function
 O(θ) = log Pθ(Y | X, G) = ΣiΣjαj fj (xij, yi) + ΣΣμk hk(Yc) – log Z
 Learning the model to
 estimate a parameter configuration θ= {α, μ} to maximize the objective
function :
 that is, the goal is to compute θ* = argmax O(θ)

14. Learning algorithm
 θ*
Goal : = argmax O(θ) L (k ) (k) ( Z ( x ( k ) ))'
Fj ( y ,x )
j k Z ( x(k ) )
 Newton methold.
 The gradient of each μk with Z ( x(k ) ) exp F ( y, x ( k ) )
y
regard to the objective
exp( F ( y, x ( k ) )) F j ( y, x ( k ) )
function. (Z ( x (k )
)) '
y
 dθ/ dμk= E[hk(Yc)] – EPμ (Yc|X,
k Z (x (k )
) exp F ( y, x ( k ) )
G)[hk(Yc)]
y
exp( F ( y, x ( k ) )
(k )
) F j ( y, x ( k ) )
y exp F ( y, x )
y
p ( y | x ( k ) ) F j ( y, x ( k ) )
y
E p (Y | x( k ) ) F j (Y , x ( k ) )

15. Learning algorithm
 One challenge : how to calculate the marginal distribution Pμ (yc|x, G).
k
 Intractable to directly calculate the marginal distribution.
 Use approximation algorithm : Belief propagation.

16. Belief propagation
 Sum-product
 The algorithm works by passing real values functions called “messages” along
the edges between the nodes.
 Extend edges to triangles

17. Belief propagation
 A “message” from a variable node v to a factor node u is
μv->u(xv) = Πu*∈N(v){u}μu*->v(xv)
 A “message” from a factor node u to a variable node v is
μu->v(xu) = Σx’ux’v=xvΠu*∈N(u){v}μv*->u(x’u)

18. Learning algorithm(TriFG model)
Input : network Gt, learning rateη
Output : estimated parametersθ
Initalize θ= 0;
Repeat
Perform LBP to calculate marginal distribution of unknown variables P(yi|xi, G);
Perform LBP to calculate marginal distribution of triad c, i.e. P(yc|Xc, G);
Calculate the gradient of μk according to :
dθ/ dμk= E[hk(Yc)] – EPμ (Yc|X, G)[hk(Yc)]
k
Update parameter θ with the learning rate η:
θ new = θold + ηd θ
Until Convergence;

19. Prediction features
 Geographic distance
 Global vs Local
 Homophily
 Link homophily
 Status homophily
 Implicit structure
 Retweet or reply
(A) and (B) are balanced, but (C) and (D) are not.
 Retweeting seems to be Users who share Elite users have a
more helpful Global
common links will Local
much stronger
 Structural balance have a tendency to tendency to
 Two-way relationships are follow each other. follow each other
balanced (88%),
 But, one-way relationships
are not (only 29%).

20. Our approach : TriFG
 TriFG model
 Features based on observations
 Partially labeled
 Conditional random field
 Triad correlation factors

21. Outline
 Previous works
 Our approach
 Experimental results
 Conclusion & future works

22. Data collection
 Huge sub-network of twitter
 13,442,659 users and 56,893,234 following links.
 Extracted 35,746,366 tweets.
 Dynamic networks
 With an average of 728,509 new links per day.
 Averagely 3,337 new follow-back links per day.
 13 time stamps by viewing every four days as a time stamp

23. Prediction performance
 Baseline algorithms
 SVM & LRC & CRF
 Accurately infer 90% of reciprocal relationships in twitter.
Data Algotithm Precision Recall F1Measure Accuracy
SVM 0.6908 0.6129 0.6495 0.9590
Test LRC 0.6957 0.2581 0.3765 0.9510
Case
CRF 1.0000 0.6290 0.7723 0.9770
1
TriFG 1.0000 0.8548 0.9217 0.9910
SVM 0.7323 0.6212 0.6722 0.9534
Test LRC 0.8333 0.3030 0.4444 0.9417
Case
CRF 1.0000 0.6333 0.7755 0.9717
2
TriFG 1.0000 0.8788 0.9355 0.9907

24. Convergence property
 BLP-based learning algorithm can converge in less than 10 iterations.
 After only 7 learning iterations, the prediction performance of TriFG
becomes stable.

25. Effect of Time Span
 Distribution of time span in which follow back action is performed.
 60% of follow-backs are performed in the next time stamp,
 37% of follow-backs would be still performed in the following 3 time stamps.
 How different settings of the time span.
 The prediction performance of TriFG drops sharply when setting the time span
as two or less.
 The performance is acceptable, when setting it as three time stamps.

26. Factor contribution analysis
 We consider five different factor functions
 Geographic distance(G)
 Link homophily(L)
 Status homophily(S)
 Implicit network correlation(I)
 Structural balance correlation(B)

27. Outline
 Previous works
 Our approach
 Experimental results
 Conclusion & future works

28. Conclusion
 Reciprocal relationship prediction in social network
 Incorporates social theories into prediction model.
 Several interesting phenomena.
 Elite users tend to follow each other.
 Two-way relationships on Twitter are balanced, but one-way relationships are
not.
 Social networks are going global, but also stay local.

29. Future works
 Other social theories for reciprocal relationship prediction.
 User feedback.
 Incorporating user interactions.
 Building a theory for different kinds of networks.

30.  Thanks!
 Q&A

31. Reference
 [BL11] L.Backstrom and J.Leskovec. Supervised random walks :
predicting and recommending links in social networks. In WSDM’11
 [C10] D.J.Crandall, L.Backstrom, D. Cosley, S.Suri, D.Huttenlocher, and J.
Kleinberg. Inferring social ties from geographic coincidences. PNAS, Dec.
2010
 [W10] C.Wang, J. Han, Y.Jia, J.Tang, D.Zhang, Y. Yu and J.Guo. Mining
advisor-advisee relationships from research publication networks. In
KDD’10.
 [N01]M.E.J. Newman. Clustering and preferential attachment in growing
networks. Phys. Rev. E, 2001
 [L10] J.Leskovec, D.Huttenlocher, and J.Kleinberg. Predicting positive and
negative links in online social networks. In WWW10.
 [T10] C.Tan, J. Tang, J. Sun, Q.Lin, and F.Wang. Social action tracking
via noise tolerant time-varying factor graphs. In KDD10
 [T09] J. Tang, J. Sun, C. Wang, and Z. Yang. Social influence analysis in
large-scale networks. In KDD09.

32. Reference
 [J07]A. Java, X.Song, T.Finin, and B.L. Tseng. Why we twitter : An
analysis of a microblogging community. In KDD2007.
 [K10]H. Kwak, C.Lee, H.Park, and S.B. Moon. What is twitter, a social
network or a news media? In WWW2010.
 [M10]M.Mathioudakis and N.Koudas. Twittermonitor : trend detection over
the twitter stream. In SIGMOD10.
 [S10]T. Sakaki, M. Okazaki, and Y.Matsuo. Earthquake shakes twitter
users: real-time event detection by social sensors. In WWW10.