2013 IEEE 13th International Conference on Data Mining
Learning, Analyzing and Predicting Object Roles on
Dynamic Networks
Kang Li∗ ,Suxin Guo† ,Nan Du‡ , Jing Gao§ and Aidong Zhang¶
Department of Computer Science and Engineering
The State University of New York at Buffalo
Emails: {kli22∗ ,suxinguo† , nandu‡ , jing§ and azhang¶ }@buffalo.edu
types of object roles of interest. [1] defines vulnerable nodes as
the nodes that the deletion of them could cause maximum network fragmentation. [2] assumes important objects should have
high PageRank scores. Generally, there are two drawbacks of
such methods. First, the topological properties in each task are
subjectively selected, thus critical information characterizing
the object roles in the networks may be missed. Even worse,
when object roles are complex, the aforementioned strategy
may not be able to characterize the object roles using any
existing topological property. Second, existing methods specified for static networks ignore the influence of the evolving
vertices and links in dynamic networks, thus are not applicable
for analyzing the dynamic patterns of object roles. To sum up,
these existing methods are not effective in mining the object
roles and their evolving patterns in dynamic networks.
Abstract—Dynamic networks are structures with objects and
links between the objects that vary in time. Temporal information
in dynamic networks can be used to reveal many important
phenomena such as bursts of activities in social networks and
human communication patterns in email networks. In this area,
one very important problem is to understand dynamic patterns
of object roles. For instance, will a user become a peripheral
node in a social network? Could a website become a hub on the
Internet? Will a gene be highly expressed in gene-gene interaction
networks in the later stage of a cancer? In this paper, we propose
a novel approach that identifies the role of each object, tracks
the changes of object roles over time, and predicts the evolving
patterns of the object roles in dynamic networks. In particular,
a probability model is proposed to extract latent features of
object roles from dynamic networks. The extracted latent features
are discriminative in learning object roles and are capable of
characterizing network structures. The probability model is then
extended to learn the dynamic patterns and make predictions
on object roles. We assess our method on two data sets on the
tasks of exploring how users’ importance and political interests
evolve as time progresses on dynamic networks. Overall, the
extensive experimental evaluations confirm the effectiveness of
our approach for identifying, analyzing and predicting object
roles on dynamic networks.
I.
In this paper, we propose a novel probability model to
address the problems of learning, analyzing and predicting
object roles in dynamic networks. Different from the aforementioned existing methods, the proposed model does not rely on
any subjectively selected topological properties, and is capable
of characterizing network dynamics. At the heart of our
framework is the latent feature representation of object roles,
which captures the structural information of each time point,
incorporates evolving information in dynamic networks, and
is discriminative in learning object roles. Specifically, at each
time point, objects of similar roles have close latent feature
representations, and the interactions of the latent object roles
can effectively reconstruct the observable network of the time
point. Moreover, the extracted latent feature representation can
well fit the existing supervised information in the proposed
model. Furthermore, to incorporate the evolving information in
the dynamic networks, the latent feature representation of each
time point is close to the prediction obtained from the previous
time points. This representation of object roles can be used
to build a variety of sophisticated analysis tools for dynamic
networks. We utilize the effective representation strategy for
exploring the evolution of object roles. This representation
method endues the proposed approach with simple and direct
visualizations that clearly show how the roles of individual
objects evolve as time progresses.
I NTRODUCTION
Dynamic networks exist in many different settings, such as
computer networks, social networks, biological networks and
sensor networks. During the formation and the dynamics of
these systems, objects usually play various types of changing
roles. For example, in Facebook, some users play as topic
hubs which are highly involved in various activities while some
other users play as peripheral objects who seldom participate
in discussions. As time goes by, highly involved users may
reduce their engagement with Facebook due to the increased
engagement with other social networks, and peripheral users
may become more active in the communities.
In the research of dynamic networks, a critical problem
is to understand object roles and their evolving patterns.
This problem is meaningful in many applications and has
attracted much attention. For instance, detecting users having
abnormal roles can be used to filter spammers in email
networks, analyzing the dynamics of customer engagement
in social networks can help improve the quality of service,
and predicting the informative genes in gene-gene interaction
networks is essential for preventing the onset of cancers.
Although mining dynamic networks using latent feature
representation on objects has been a hot topic recently, the proposed work in this paper significantly differs from the existing
work in both the method and the aimed task. In the existing
methods, exponential-family random graph models (ERGMs)
are the canonical way for representing observable networks
with latent variables. As discussed in [3], ERGMs usually
Nevertheless, existing methods for this problem usually
focus on static networks and make strong assumptions on the
relationships between specific topological properties and the
1550-4786/13 $31.00 © 2013 IEEE
DOI 10.1109/ICDM.2013.95
428
suffer from high computational and statistical cost thus hardly
work in practice. To avoid the drawbacks of ERGMs, several
alternative approaches [4]–[6] have been proposed in recent
years to use latent feature vectors as ”coordinates” to represent
the characteristics of each node. Compared with the proposed
probability model in this paper, these ”coordinates” methods
always assume that highly connected objects have close latent
feature representations and vice versa. Since objects of the
same roles are not necessarily highly connected or may be
even far away from each other, these existing methods can not
encode object roles as latent feature representations in many
applications and are not effective in the learning of object roles
in dynamic networks.
TABLE I: Notation
t
ni
r
c
G
Gi
Vni ×1
i
i
En
i ×ni
i
Yn ×c
i
Λni ×r
i
Hr×r
W
the Hadamard product of two matrices A and B of the same
size, and (A◦B)ij = Aij ·Bij . Similarly, (AB)ij = Aij /Bij
is the Hadamard division. Besides, a Gaussian distribution of
A with mean B and variance σ 2 is denoted as N (A|B, σ 2 ).
The proposed approach is built upon two bases: first, object
roles can be extracted from structural information; and second,
object roles should change gradually rather than abruptly. The
second base means that previous data can provide hints about
the current status of the network, and this assumption has
been widely used in dynamic network mining such as [4]–
[6]. In experiments, we investigate how people’s importance
and political interests evolve in dynamic networks. The results
well support these two bases.
We summarize the major variable matrices used in this
paper in Table I. Each dynamic network is assumed to be
periodically sampled into t snapshots, and denoted as G =
{Gi |i ∈ [1, t]}. Gi = {Vnii ×1 , Eni i ×ni } is the i-th snapshot,
where V i is the set of ni vertices, and E i is the set of links
between the vertices. In our context, each vertex is an object.
In the dynamic network G, we suppose there are c object
role classes, and the label matrix for the objects in Gi is Ynii ×c .
In the paper, we suppose labels of objects are provided in the
first snapshot, and focus on the following three tasks:
Overall, the contributions of this work include:
• In Section II-B, a Gaussian model is proposed to effectively extract latent features of object roles for both
supervised and unsupervised cases. A solution based
on variational Bayesian inference is then provided to
efficiently optimize the proposed model.
• In Section II-C, we extend the Gaussian model to incorporate dynamic information for learning and analyzing
object roles in dynamic networks.
• In Section II-D, we also provide the details of implementing the proposed model for predicting object roles
according to the learned evolving patterns.
• In Section III, we experimentally evaluate the proposed
methods on the tasks of mining people’s evolving importance and political interests in dynamic networks. The
overall performance well confirms the effectiveness of the
proposed model.
II.
number of snapshots in the dynamic network
number of objects in the i-th snapshot
number of features in latent feature representations of object roles
number of object role classes
the dynamic network
the i-th snapshot of the dynamic network
the set of vertices in the i-th snapshot
the set of links in the i-th snapshot
the label matrix for each node
the latent feature matrix of object roles in the i-th snapshot
the interaction matrix of object roles in the i-th snapshot
the coefficient matrix for learning object role classes
• Learning the role of each object at each snapshot of the
dynamic network;
• Analyzing how object roles evolve over time; and
• Predicting the object roles at the (t + 1)-th snapshot of
the dynamic network using the first to the t-th snapshots.
B. Detection on Static Networks
In this section, we propose a probability model for object
role detection on static networks, as a ground for the dynamic
object role analysis.
Different from the existing methods [4]–[6] that view
object roles as object ”coordinates”, we interpret the role of
each object as its properties that determine to what extent
and how the object impacts the other objects in the formation
and the dynamics of the network.
M ETHODOLOGY
In this section, we present the details of the proposed
LAP (Learning, Analyzing and Predicting) model for mining
object roles on dynamic networks. Specifically, we start from
detecting object roles on static networks, then extend the
developed model to dynamic cases. The prediction of object
roles is achieved through the extended model.
To better explain the intuition of this definition and its
difference from the ”coordinates” concept, we give an example
on the trade history of the ancient Tamil country [7] which is a
region in southern India. During the ancient time, people there
were frequently involved in both local and international, and
both inland and overseas trade. We conclude the rules of the
trade as follows.
A. Notation
1) The closer two objects were, the more likely they could
trade/interact. In the Ancient Tamil, most trade was by
barter, which was prevalent locally. As a result, more
trades were performed locally than internationally.
2) Objects with larger activity range had higher ability to
interact with other objects. As an evidence, the development of seamanship and the discovery of new routes significantly increased the trades between Tamil and Rome.
3) The properties of two objects determined how they interacted. For instance, on the trades between Tamil and
We first introduce the notation rule of this paper. Without
further notification, a scalar is denoted by a lower case letter,
e.g., a, b and λ; a matrix is denoted by an upper case letter such
as A, B and Λ. An×m represents that the matrix A contains
n rows and m columns. Besides, Ai,: and A:,j represent the
i-th row and the j-th column of the matrix A, respectively,
and Aij is the element at the i-th row and the j-th column.
In the formulation of our model, T r(U
s×s ) is the trace of
s
the square matrix Us×s and T r(Us×s ) = i=1 Uii . A ◦ B is
429
The posterior of Λ and H is then:
Rome, according to the different goods they could produce, Tamil exported pepper, ivory and gold, and imported
glass, coral and wine. Another example is that the changes
of the Emperor of Rome had significant impact on the
trade between Tamil and Rome.
p(Λ, H|E, σ 2 ) =
p(E|Λ, H, σ 2 )p(Λ)p(H)
.
p(E)
(1)
By the model in Eq.1, we can extract the latent feature
representation Λ of object roles as well as the role interaction
matrix H from an observable link matrix E of a static network.
In a network representation of the trades of the Ancient
Tamil, each node denotes an object that participated in the
trade, and each link represents a trade; whether there is a
link between any two objects is determined by the rule 1 and
the rule 2; and the weights of the links, which represent the
interaction types of the objects, are determined by the rule 3.
The Supervised Model on Learning Object Roles
In our experiment setting, we assume at the first time point,
there are several labeled objects to guide the learning of each
type of object roles and to maximize the margins between
different object roles.
In this process, there are two factors impacting the formation of the network: object coordinates and object roles. The
coordinates of objects, covering rule 1, determine the closeness
of each two objects. The roles of objects, covering rule 2 and
rule 3, determine to what extent and how each object impacts
the others. The objective of this paper is to explore how to
detect, analyze and predict such object roles.
For the first snapshot, the label matrix for the labeled
objects is Ŷm×c , in which m is the number of the labeled
objects. To extend the unsupervised model for supervised
cases, we introduce a feature coefficient matrix Wr×c which
measures the contribution of each feature in Λ to the object
role classes:
m c
N Yˆij |(Λ̂ · W )ij , σY2 .
(2)
p(Ŷ |Λ, W ) =
The Unsupervised Model on Learning Object Roles
i=1 j=1
Suppose object coordinates are a-dimensional and object
roles are r-dimensional. Let Ξn×a and Λn×r be the latent
coordinate matrix and latent role matrix for n objects, respectively. An observed link Eij from a source object i to a target
object j is generated as:
In Eq.2, Λ̂ denotes the related latent role features for the
labeled objects. σY2 is the noise variance in the Gaussian
distribution.
The priors for the feature
weighting
matrix−1W is:
2
c
tr(W CW W )
i
=
exp
−
.
p(W ) ∝ exp − i=1 W
2CWi
2
In the prior, W is column-wise independent and CW =
diag{CW1 , CW2 , ..., CWc } is the covariance matrix of the
prior.
Eij = Λi,: · M→ · Λj,: · M← · K(Ξi,: , Ξj,: ) + .
In the above equation, M→ measures how the role Λi,:
of the source object i impacts the link Eij ; and similarly,
M← measures how the role Λj,: of the target object j impacts
the link Eij . K(Ξi,: , Ξj,: ) is a closeness function measuring
how close two objects (object i and object j) are. The larger
K(Ξi,: , Ξj,: ) is, the more likely two objects i and j interact.
During the past few years, extensive work has been done to
estimate the latent closeness of two objects in networks. In this
paper, we use the truncated Katz kernel [8] for this purpose.
By this
rkernel, the closeness matrix K on a network E is
K = i=1 αi · E i , and K(Ξi,: , Ξj,: ) = Kij .
For the supervised cases, the posterior is:
p(Λ, H, W |E, Ŷ ) =
p(E|Λ, H)p(Ŷ |Λ, W )p(Λ)p(H)p(W )
p(E, Ŷ )
.
(3)
Given E and Ŷ , by maximizing Eq.3, we are able to obtain
the latent feature presentation Λ of object roles, the interaction
matrix H and the coefficient matrix W for learning object role
classes on Λ.
If we assume each Eij is independently generated
from a Gaussian distribution with mean Λi,: · M→ · Λj,: ·
M← · Kij and noise variance σ 2 , the conditional distribu2
tion
nthe observed link matrix E2 is: p(E|Λ, H, σ ) =
n of
i=1
j=1 N (Eij |(ΛHΛ ◦ K)ij , σ ), in which H = M→ ·
M← is the object interaction matrix measuring how each object
contributes to each link.
Solution of the Supervised Model
1) Variational Bayesian Inference: In this part, we apply
the Variational Bayesian technique [9], [10] to maximize Eq.3.
Suppose there is a trial distribution on the matrices
Λ,
H and W as Q(Λ, H, W ) which has the constraint
Q(Λ, H, W )dΛdHdW = 1. The free energy of the system
is:
Suppose the priors of the latent feature matrix Λ of the
object roles and the object interaction matrix H follow the
spherical Gaussian distributions and they
are column-wise
Λ 2
r
independent, we have: p(Λ) ∝ exp − f =1 2CfΛ
=
f
−1 2
tr(ΛCΛ Λ )
H r
, and p(H) ∝ exp − f =1 2CfH
=
exp −
2
f
−1
tr(HCH H )
.
exp −
2
F = EQ(Λ,H,W ) [log p(E, Ŷ , Λ, H, W ) − log Q(Λ, H, W )],
(4)
which can be further formulated into:
F =EQ(Λ,H,W ) [log p(Λ, H, W |E, Ŷ ) + log p(E, Ŷ )
− log Q(Λ, H, W )]
= log p(E, Ŷ ) − KL(Q(Λ, H, W )p(Λ, H, W |E, Ŷ ))
In the above priors, CΛ = diag{CΛ1 , CΛ2 , ..., CΛr } and
CH = diag{CH1 , CH2 , ..., CHr } are the prior variances of
those columns.
≤ log p(E, Ŷ ).
430
(5)
In Eq.5, Eq (f (x)) denotes the expectation of the role f (x)
q(x)
with respect to a distribution q(x). KL(qp) = q(x) p(x)
dx
represents the Kullback-Leibler (KL) divergence of a distribution p with regards to q. Since the value of a KL divergence is
always non-negative, in the end of the above formulation, we
can conclude that the lower bound for the evidence log p(E, Ŷ )
is F.
where ρ is a constant which is irrelevant to Λ. δ(Ŷ , i) is the
indicator function that equals 1 if the labeled set includes
object i and equals 0 otherwise.
Comparing with Eq.6, the derivation leads to the following
rules:
n
1
−1
exp − (Λi,: − Λi,: ) ΦΛ (Λi,: − Λi,: ) , (9)
Q(Λ) ∝
2
i=1
Through maximizing the free energy F, we can obtain the
optimum if and only if Q(Λ, H, W ) = p(Λ, H, W |E, Ŷ ). It
is usually intractable to approximate Q(Λ, H, W ) due to the
complexity caused by the high dimensionality of the interacted
members Λ, H and W . Therefore, we apply the variational
approximation Q(Λ, H, W ) = Q(Λ)Q(H)Q(W ).
where
Λi,:
By the variational Bayesian framework, the variational
posteriors of the matrices Λ, H and W can be iteratively
updated through the following rules:
1
exp EH,W log p(E, Ŷ , Λ, H, W ) ,
Q(Λ) =
zΛ
1
exp EΛ,W log p(E, Ŷ , Λ, H, W ) , (6)
Q(H) =
zH
1
exp EΛ,H log p(E, Ŷ , Λ, H, W ) ,
Q(W ) =
zW
Φ−1
Λi
The variational posteriors for the factor matrix H is computed in a similar way, by which we obtain:
r
1
exp − (H:,j − H :,j ) Φ−1
(H
−
H
)
,
Q(H) ∝
:,j
j,:
H
2
j=1
(11)
where
n
1 H :,j = ΦHj
Êij Λi,: ,
2
σE
i=1
(12)
n
1 −1
−1
ΦHj = CH + 2
(Λi,: Λi,: + ΦΛi ).
σE i=1
wherezΛ , zH and zW are constants which enforce the conditions Q(Λ)dΛ = 1, Q(H)dH = 1 and Q(W )dW = 1.
In the iterative solution, the conditional probability over the
observed link p(E|Λ, H) will introduce a term ΛHΛ ΛHΛ
which includes the quadratic multiplications of H and the
quartic integrations of Λ. This term may cause high time and
space complexities. Therefore, in each iteration, we utilize the
Λ∗ optimized in the last iteration as:
p(E|Λ, H) =
n r
2
N ((Ê)ij |Λi,: H:,j , σE
),
⎞
r
c
δ(
Ŷ
,
i)
1
=⎝ 2
Êij H :,j +
Ŷi,k W :,k ⎠ ΦΛ ,
σE j=1
σY2
k=1
⎛
r
1 = ⎝CΛ−1 + 2
(H :,j H :,j + ΦHj )
σE j=1
c
δ(Ŷ , i) +
(W :,k W :,k + ΦWk )) .
σY2
k=1
(10)
⎛
Similarly, the variational posterior of W is computed
through:
c
1
Q(W ) ∝
exp − (W:,k − W :,k ) Φ−1
(W
−
W
)
,
:,k
:,k
W
2
k=1
(13)
where
n
1 W :,k = ΦWk
δ(Ŷ , i)Ŷik Λi,: ,
σY2 i=1
(14)
n
1 −1
−1
Φ Wk = C W + 2
δ(Ŷ , i)Ŷik (Λi,: Λi,: + ΦΛi ).
σY i=1
(7)
i=1 j=1
in which Ê = E K/Λ∗ .
To verify the correctness of this simplification is to prove
that the optimal Λ and H obtained by this simplification
can also satisfy the optimization of the original conditional
probability. The proving process is straightforward. Due to the
limited pages of this paper, we skip the details here.
Plugging in the model for p(Λ, H, W |E, Ŷ ), we can compute the variational posterior Q(Λ) by:
EH,W log p(E, Ŷ , Λ, H, W )
⎡
⎛
r
n
1 ⎣
1 =−
Λi,: ⎝CΛ−1 + 2
EH (H:,j H:,j
)
2 i=1
σE j=1
c
δ(Ŷ , i) +
EW (W:,k W:,k ) Λi,:
(8)
σY2
k=1
⎡
⎛
r
n
1 1 ⎣
−2Λi,: ⎝ 2
−
Êij EH (H:,j )
2 i=1
σE j=1
c
δ(Ŷ , i) +
+ ρ,
Ŷi,k EW (W:,k )
σY2
By simply updates Λ, H and W through the above derivations iteratively until convergence, we can obtain the optimal
solution.
2
, σY2 ,
2) Parameter Setting: To set the hyper-parameters σE
CΛ , CH and CW , we take derivatives of the expectation of the
logarithm evidence EΛ,H,W log p(E, Ŷ , Λ, H, W ) with respect
to each of the hyper-parameters and set the derivatives to 0,
then we can obtain:
n
r
1 2
2
Êij − 2Êij Λi,: H :,j
=
σE
n · r i=1 j=1
(15)
+T r (ΦΛ + Λi,: Λi,: )(ΦH + H :,j H :,j ) ,
k=1
431
Ȧͳ
Ȧʹ
Ȧ–
ͳ
ʹ
–
ͳ
ʹ
ͳ
Since p(E i |Di ) ≡ p(E i |Λi , H i ), p(E i |Di ) can be well
approximated by Eq.7. The only thing remaining unsolved is
how to estimate p(Di |D1:i−1 ) ≡ p(Λi , H i |Λ1:i−1 , H 1:i−1 ).
We place Gaussian priors on Λi and H i as:
p(Λi |ΩiΛ , ΣiΛ ) =
–
j=1
p(H i |ΩiH , ΣiH ) =
–
ʹ
1
n c
δ(Ŷ , i) Ŷik2 − 2Ŷik Λi,: W :,k
Ŷ c i=1 k=1
+T r (ΦΛ + Λi,: Λi,: )(ΦW + W :,k W :,k ) ,
n
1 2
(ΦΛi )l,l + Λi,l
n i=1
r
1 2
(ΦHj )ll + H il
=
r j=1
C Wl
(16)
(19)
i
N (H:,j
|ΩiH:,j , ΣiHj I).
(20)
Let Λi for i ∈ [1, t] as a time series matrix independent of
other variables. When i = 2, since we only have Λ1 and no
change of Λ has been observed, the best prediction of Ω2Λ is
Λ1 . For i ≥ 3, we can approximate ΩiΛ through the MAR(1)
(Multivariate Autoregressive) model [11] as:
Λi = Λi−1 · AΛ + iΛ ,
(17)
(21)
where i is a Gaussian noise having zero mean and precision ΣiΛ . Suppose XΛi = [Λ1 , Λ2 , ..., Λi−2 ] and BΛi =
[Λ2 , Λ3 , ..., Λi−1 ], by maximum likelihood estimation, we
have:
(22)
AΛ = (XΛi XΛi )−1 XΛi BΛi ,
r
1 2
=
(ΦWl )ll + W hl
r
h=1
C. Detection and Analysis on Dynamic Networks
ΩiΛ = Λi−1 · AΛ ,
In this section, we extend the above probabilistic model for
detecting and analyzing object roles on dynamic networks.
ΣiΛ =
i
Let E denote the observed link matrix at the i-th sampled
time point. Suppose we have a sequence of such sampled
link matrices {E 1 , E 2 , ..., E t }, our goal of analyzing network dynamics is to obtain a sequence of low-rank matrices
{{Λ1 , H 1 }, {Λ2 , H 2 }, ..., {Λt , H t }}. The i-th pair {Λi , H i },
which approximates E i , captures the role of each object at
the i-th time point and the interaction patterns of these object
roles.
(23)
1
(B i − XΛi AΛ ) (BΛi − XΛi AΛ ).
i − 2 − r2 Λ
i
i
Similarly, XH
= [H 1 , H 2 , ..., H i−2 ] and BH
[H 2 , H 3 , ..., H i−1 ]. For H we have:
i
i −1 i
i
XH
) X H BH
,
AH = (XH
ΩiH
ΣiH =
The Object Role Model for Dynamic Networks
Let Di = {Λi , H i } denote the pair of the low-rank
matrices we wish to extract at the i-th time point. We present
the graphic model of our analyzing framework in Fig.1. The
observed network E i at the i-th snapshot is generated by
the latent parameters Di , and Di is determined by three
factors: the previous latent parameters D1:i−1 (D1 to Di−1 ),
the current object roles Λi and the current object interaction
pattern H i .
=H
i−1
· AH ,
1
i
i
i
(B i − XH
AH ) (BH
− XH
AH ).
i − 2 − r2 H
(24)
=
(25)
(26)
(27)
Since Λi and Hi are independent from each other, we
estimate p(Di |Di−1 ) as:
p(Di |Di−1 ) p(Λi |ΩiΛ , ΣiΛ )p(H i |ΩiH , ΣiH )
r
r
(28)
i
=
N (Λi:,j |ΩiΛ:,j , ΣiΛj I)
N (H:,j
|ΩiH:,j , ΣiHj I).
j=1
j=1
Thus, the evidence we wish to maximize is:
In the first snapshot, no previous latent parameters D1:i−1
exist. The object roles in E 1 can be effectively extracted
through the models proposed in Section II-B. In this paper,
we assume that labeled information exists in the first snapshot,
therefore we use the supervised model in Section II-B.
p(Λi , H i , E i , Di−1 ) p(E i |Λi , H i )p(Λi )p(H i )
p(Λi |ΩiΛ , ΣiΛ )p(H i |ΩiH , ΣiH ).
(29)
The Eq.29 is the model for learning object roles on
dynamic networks. Specifically, given previously extracted
feature D1:i−1 and the current observable network E i , we can
learn the current latent object roles Λi and their interaction
pattern H i .
i−1
has already been
At each snapshot i for i ∈ [2, t], D
calculated and E i is observable. With the assumption that
D1:i−1 and E i are independent, the posterior distribution of
on the parameter set Di is:
p(Di |D1:i−1 , E i ) = p(Di |D1:i−1 )p(Di |E i ).
N (Λi:,j |ΩiΛ:,j , ΣiΛj I),
In the above Gaussian priors, ΩiΛ and ΩiH are the
best estimations of Λi and H i based on D1:i−1 , respectively. ΣiΛ = diag{ΣiΛ1 , ΣiΛ2 , ..., ΣiΛr } and ΣiH =
diag{ΣiH1 , ΣiH2 , ..., ΣiHr } are the related variance matrices.
CΛ l =
C Hl
r
j=1
Fig. 1: The Graphical Model Representation
σY2 =
r
Solution of the Model for Dynamic Networks
(18)
432
Since E 1:t are all observable and Λt has already been extracted, by maximizing the logarithm of the above posterior
probability, we obtain Λt+1 = Λt · AΛ .
1) Variational Bayesian Inference: To optimize the objective in Eq.29, we follow an optimizing process which is almost
identical to Section II-B1, thus we skip the inference process
here. In this solution, we iteratively update:
r
r
i
1 i i ΩΛj,h
i
Λj,: =
ΦiΛ ,
Êjh H :,h +
2
σE
ΣiΛh
h=1
h=1
r
1
−1
i −1
i −1
i
(H :,h H :,h + ΦHh )
(Φ )Λj = CΛ + (ΣΛ ) + 2
σE
h=1
(30)
⎛
⎞
n
r
i
Ω
1
i
i
H
g,h
⎠,
H :,h = ΦiHh ⎝ 2
Ê i Λ +
σE j=1 jh j,: g=1 ΣiHh
(31)
n
1 i i
−1
i −1
i −1
i
(Φ )Hh = CH + (ΣH ) + 2
(Λ Λ + ΦΛj ).
σE j=1 j,: j,:
By sequential inference, we canpredict the object roles in
s
the (t + s)-th snapshot as Λt+s = j=1 Λt · AΛ .
With the predicted Λt+s , the object role probability in the
(t + s)-th snapshot is estimated as p(Fjt+s |Vit+s ) = Λt+s
i,: W:,j .
We can then obtain the role of each object following the
method at the end of Section II-C. .
III.
In this part, we experimentally evaluate the proposed LAP
algorithm on two real data sets: SocialEvolution [12] and
Robot.Net. The SocialEvolution data set is available upon
request1 and the Robot.Net data set is publicly available2 .
The experiments include three parts. On each data set, we
first evaluate the performance of LAP on the task of detecting
object role classes at each time point. Case studies are then
performed to evaluate the correctness of the extracted dynamic
patterns of object roles. To the end, we test the performance
of the proposed LAP algorithm on the task of object role
predictions. In the experiments, baselines are investigated
to quantitatively prove the superiority of the proposed LAP
algorithm.
2) Parameter Setting: By the derivatives similar to those
in Section II-B2, we can obtain:
n
r
1 i 2
i
i
2
(Êjh ) − 2Êjh Λj,: H :,h
=
σE
n · r j=1
(32)
h=1
i i
i
i
i
i
+T r (ΦΛ + Λi,: Λj,: )(ΦH + H :,h (H :,h ) ) ,
CΛi l =
i
CH
=
l
1 i
i
(ΦΛj )l,l + (Λj,l )2
n j=1
n
r 1
i
(ΦiHg )ll + (H gl )2
r g=1
A. Experiments on SocialEvolution Data Set
(33)
1) Dataset Description: We first evaluate LAP on the
SocialEvolution data set to demonstrate its capabilities in
analyzing the evolving patterns of people’s political interests
in dynamic networks. This data set was collected from October
2008 to May 2009, and it contains information of locations,
phone calls, music sharing logs, surveys on relationships,
political interests and etc.. In the experiments, we build the
dynamic networks of the SocialEvolution data set as: initially
the weights of links between the objects (people in the data set)
are set to 0 which stands for no link. If there is a phone call,
message or music share between two users at a time point, we
add the weight of the undirected link between the two users
by 1 for that time point. The constructed continuous dynamic
network is then divided into 5 snapshots which are denoted as
E 1 to E 5 , respectively.
Detection and Analysis on Object Role Classes
To make the final decision on the role label of each node,
we estimate p(Fj |Vi ), which is the conditional probability that
object Vi belongs to the role class Fj , through the estimation:
p(Fj |Vi ) = Ep(Λ),p(W ) Λi,: W:,j = Λi,: W :,j .
E XPERIMENTS AND A NALYSIS
(34)
By Eq.34, we can obtain the probability of each object
belonging to each object role class at each time point. For the
detection of object roles, each object is then assigned to the
class in which the object has the highest probability. For the
analysis of object roles, the trend of varying p(Fj |Vi ) over
different time points clearly reveals the dynamic patterns of
object roles. We explain more about how to use p(Fj |Vi ) in
the analysis of dynamic object roles in Section III-A3.
In this data set, surveys of some users’ political interests
at different time points have also been provided. According to
the surveys on whether people are interested in politics, we
divide the users into three classes: Indifferent, Moderate and
Enthusiastic. In the experiments, we suppose the surveys at the
first time point are known and use them in the training. The
surveys at the other time points are then used to numerically
evaluate the performance.
D. Prediction on Dynamic Networks
Based on the aforementioned object role detections and
analysis model, in this section, we develop the approach for
predicting object roles on dynamic networks.
2) Object Role Detection: In this part, we investigate the
performance of the proposed LAP algorithm on the task of
identifying object roles at each time point. For comparisons,
we also consider four baselines in this experiment as follows.
Suppose there are t + 1 periodically sampled snapshots
of a dynamic network, and among them, the first t snapshot
networks are observable. We predict the object roles at t +
1 based on the observable data by maximizing the posterior
probability below:
p(Λt+1 |E 1:t ) =
p(Λt+1 |Λt )p(Λt |E 1:t ).
(35)
As discussed in Section I, subjectively selected structural
properties are usually insufficient to cover the characteristics
1 http://realitycommons.media.mit.edu/socialevolution.html
2 http://www.trustlet.org/datasets/robots
Λt
433
net/
ROC curve
0.5
PageRank
Centrality
Coordinates
Prior
LAP
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.6
0.5
PageRank
Centrality
Coordinates
Prior
LAP
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.9
0.8
0.7
0.6
0.5
PageRank
Centrality
Coordinates
Prior
LAP
0.4
0.3
0.2
0.1
0
1
1
True Positive Rate
0.6
0.8
ROC curve
ROC curve
1
0.9
True Positive Rate
True Positive Rate
True Positive Rate
0.7
0
ROC curve
1
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
True Positive Rate
ROC curve
1
0.9
0.9
0.8
0.7
0.6
0.5
PageRank
Centrality
Coordinates
Prior
LAP
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.9
0.8
0.7
0.6
0.5
PageRank
Centrality
Coordinates
Prior
LAP
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
False Positive Rate
False Positive Rate
False Positive Rate
False Positive Rate
False Positive Rate
E1
E2
E3
E4
E5
(a)
(b)
(c)
(d)
(e)
0.9
1
Fig. 2: ROC curves for detecting Indifferent objects on SocialEvolution
TABLE II: Object Role Detection on SocialEvolution
of the aimed object roles. To prove it, we test two popular
topological properties PageRank [2] and Katz Centrality [13]
in the detection of object roles at each time point.
As explained in Section II-B, we assume that object roles
describe how and to what extent could the objects impact
the others. In the contrast, existing methods such as [4]–[6]
view object roles as the ”coordinates” of the objects, and they
assume objects of the same roles are highly connected. To
measure the impact of these two different assumptions, we
build a baseline named as Coordinates which is identical to
the proposed LAP model
H, σ 2 )
n that it defines p(E|Λ,
n except
2
as: p(E|Λ, H, σ ) = i=1 j=1 N (Kij |(ΛHΛ )ij , σ 2 ). By
this modification, highly connected objects tend to obtain close
latent feature representations in the results.
In the proposed LAP model, we predict the object roles at
time t base on the object roles at time 1 to time t − 1. The
major idea of the proposed model in utilizing the evolving
information is that the extracted object roles at time t should
be close to this prediction. However, in most existing methods
such as [6], the extracted object roles at time t are assumed
to be close to the object roles at time t − 1. To measure
the impact of these two different assumptions, we build a
baseline named as Prior which is identical to the proposed LAP
model except that it defines p(D i |Di−1 ) as: p(Di |Di−1 ) p(Λi |Λi−1 , CΛi−1 )p(H i |H i−1 , CH i−1 ). By this modification,
object roles at time t tend to be close to the learned object
roles at time t − 1.
E1
E2
E3
E4
E5
PageRank
0.537
0.581
0.534
0.520
0.572
E1
E2
E3
E4
E5
PageRank
0.586
0.503
0.606
0.552
0.549
E1
E2
E3
E4
E5
PageRank
0.622
0.724
0.661
0.600
0.502
Indifferent
Centrality
Coordinates
0.534
0.816
0.631
0.736
0.653
0.561
0.630
0.654
0.631
0.649
Moderate
Centrality
Coordinates
0.513
1.000
0.675
0.746
0.706
0.523
0.646
0.540
0.643
0.602
Enthusiastic
Centrality
Coordinates
0.540
1.000
0.709
0.649
0.688
0.536
0.597
0.594
0.575
0.505
Prior
1.000
0.890
0.774
0.721
0.795
LAP
1.000
0.890
0.863
0.827
0.882
Prior
1.000
0.811
0.766
0.675
0.741
LAP
1.000
0.811
0.784
0.700
0.728
Prior
1.000
0.888
0.543
0.569
0.540
LAP
1.000
0.888
0.875
0.843
0.840
performance is still good with the guidance of trained object
roles at E 1 . As time progresses, the guidance of the labeled
information at E 1 becomes weak, thus the performance of
Coordinates is very bad from E 3 to E 5 . The bad performance
on E 3 to E 5 indicates that Coordinates can not characterize the
object roles well without the guidance of labeled information.
This result supports that labeled information is critical for
extracting discriminant representations of object roles, and that
viewing object roles as ”Coordinates” does not work in these
cases.
For each investigated method, we test its performance on
the known labeled data with varying parameter values so as to
find the best parameter setting. In our model, we obtain that
the optimal number of latent features is 63.
An interesting finding in this experiment is that the fourth
baseline Prior performs significantly worse than the proposed
method LAP on detecting the Indifferent and Enthusiastic objects while achieving close performance on Moderate objects.
To seek the reason for this result, we analyze the data and
observe that the probabilities of Indifferent, Moderate and
Enthusiastic objects changing their levels of interests in politics
are on average 24.63%, 11.17% and 23.61%, respectively. This
observation indicates that Moderate objects are more consistent
in political interests while Indifferent and Enthusiastic objects
are more likely to change. Since Prior assumes the object
roles are close to previous object roles, it performs better
on Moderate objects that change less and performs worse on
Indifferent and Enthusiastic objects that change more.
Since the numbers of objects in different classes are highly
imbalanced, accuracy is not meaningful in evaluating the
performance. Therefore, we calculate the Receiver Operating
Characteristics (ROC) curve and use the Area Under the Curve
(AUC) to capture the quality of the ROC curve.
Table II summarizes the performance of all the investigated
methods on the SocialEvolution data set.
We first notice that PageRank and Centrality generally do
not perform well on detecting all the three object roles over
different time points, and in some cases the AUC scores are
even close to random guess. This bad performance indicates
that PageRank and Centrality are not able to capture sufficient
information from the structure to characterize the object roles.
Comparing with all these baseline methods, LAP achieves
the best performance on almost all the experiments. Compared
with the best baseline Prior, LAP improves the AUC scores by
up to 61.14% and on average 13.75%. To better illustrate the
advantage of the proposed LAP model, in Fig.2, we show the
ROC curves on detecting Indifferent objects (due to the space
limit, we only show this case).
For the third baseline Coordinates, the performance varies
significantly over different cases. On E 1 , since labeled information for each object role is provided, Coordinates can
reach very high AUC scores. On E 2 which is close to E 1 , the
434
Enthusiastic
Enthusiastic
Enthusiastic
Enthusiastic
(d) E 4
Moderate
TABLE III: Object Role Prediction on SocialEvolution
Indifferent
(c) E 3
Moderate
Indifferent
(b) E 2
Moderate
Indifferent
(a) E 1
Moderate
Indifferent
Indifferent
Enthusiastic
Moderate
(e) E 5
(d) E 4
Enthusiastic
Enthusiastic
Enthusiastic
Enthusiastic
E1
E2
E3
E4
E5
DyPageRank
0.533
0.606
0.552
0.549
E1
E2
E3
E4
E5
DyPageRank
0.558
0.661
0.600
0.502
Moderate
Indifferent
(c) E 3
Moderate
Indifferent
(b) E 2
Moderate
Indifferent
(a) E 1
Moderate
Indifferent
Indifferent
Enthusiastic
Fig. 3: Role Dynamics of Object 42 in SocialEvolution
Moderate
E
E2
E3
E4
E5
DyPageRank
0.518
0.534
0.520
0.572
1
(e) E 5
Fig. 4: Role Dynamics of Object 70 in SocialEvolution
3) Object Role Analysis: In this section, we do case studies
to demonstrate how we implement the proposed model to
analyze changing patterns of object roles. Without loss of
generality, we use 0, 1 and 2 to represent the classes of
Indifferent, Moderate and Enthusiastic, respectively. With the
class probabilities learned by the proposed LAP model, we
calculate the class number expectation of each object at each
time point. We view the class numbers as angles, scale them
to the range of [0, π], and demonstrate the dynamic patterns
in compasses as in Fig.3 and Fig.4.
Indifferent
DyCentrality
Coordinates
0.632
0.718
0.653
0.696
0.630
0.642
0.631
0.649
Moderate
DyCentrality
Coordinates
0.676
0.777
0.706
0.774
0.646
0.542
0.643
0.602
Enthusiastic
DyCentrality
Coordinates
0.709
0.749
0.688
0.738
0.597
0.566
0.575
0.505
Prior
0.888
0.817
0.739
0.806
LAP
0.888
0.826
0.818
0.908
Prior
0.804
0.769
0.675
0.747
LAP
0.804
0.776
0.642
0.688
Prior
0.867
0.749
0.586
0.561
LAP
0.867
0.765
0.835
0.803
performance is evaluated in AUC by comparing the prediction
with the ground truth. We summarize the results in Table III.
We first notice that the prediction performance of DyPageRank and DyCentrality is very close to the detection
performance of PageRank and Centrality in Table II. This
fact indicates that DyPageRank and DyCentrality are very
effective in learning the evolving patterns of the PageRank and
Centrality scores. Nevertheless, these two baselines perform
the worst among all the investigated methods on the task of
object role predictions. The bad performance verifies the fact
that PageRank and Centrality scores can not well capture the
characteristics of the object roles considered in this experiment.
In Fig.3, we show the changing pattern of object 42.
According to the surveys on political interests, this person
appeared to be highly interested in politics in the first three
time points which are around the date of the U.S. presidential
election at 2008. During the last two time points, this person
was less interested in politics and had doubts about the
president and the congress. In Fig.3, the learned object roles
of this person well reflect the dynamics of his/her interest in
politics.
As for Coordinates, it shows better performance in prediction than in detection. Notice that for object role detection,
Coordinates integrates two kinds of information: the prediction
based on previously learned object roles and the extraction
based on the current network. In contrast, in prediction, it
only uses the previously learned object roles. Therefore, the
extraction part, in which object roles are viewed as object
”Coordinates”, reduces the overall performance of the detection. This experiment again prove that viewing object roles
as ”Coordinates” is not effective in learning, analyzing and
predicting object roles.
In Fig.4, we show the changing pattern of object 70.
Similar to the object 42, this person appeared to be more
interested in politics during the first three time points than
the rest snapshots. This person claimed himself/herself as
slightly interested in politics around the presidential election
date and frequently switched his/her preferred party between
the Independent and the Democrat. During the last two time
points, this person showed no interest in politics at all and
expressed nothing about the president, the congress or the
economy policies. In Fig.4, the learned object roles well reflect
the dynamics of his/her interest in politics.
Similar to the results in Table II, in Table III, Prior
achieves similar performance with the proposed method LAP
on predicting Moderate objects but performs much worse
than LAP on the other object roles. This fact supports the
conclusion that Prior performs better on Moderate objects that
change less and performs worse on Indifferent and Enthusiastic
objects that change more.
Due to space limit, we only show the two cases in this
section. The excellent performance supports that the proposed
LAP model is effective in analyzing and visualizing the
dynamics of object roles.
4) Object Role Prediction: In this section, we investigate
the performance of LAP on the task of predicting object roles.
For comparison, we extend the baselines in Section III-A2
to this task. Specifically, we utilize the results of PageRank
and Centrality in autoregressive model for prediction, and
obtain two baselines DyPageRank and DyCentrality, respectively. Since the other two baselines Coordinates and Prior are
dynamic model, they can be directly used in the task of object
role prediction.
Compared to all the baselines, the proposed LAP model
achieves significantly better performance in most cases than
any baseline. The experiments confirm the effectiveness of
LAP in predicting object roles.
B. Experiments on Robot.Net Data Set
1) Dataset Description: The Robot.Net data set was
crawled daily from the website Robot.Net3 since 2008. This
In this experiment, we predict the object roles at the
second to the fifth time points using the previous data. The
3 http://robots.net/
435
ROC curve
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1
0
1
0
0.1
False Positive Rate
0.2
0.5
0.6
0.7
0.8
0.9
ROC curve
0
0.1
0.2
0.3
0.4
0.5
0.6
0.2
0.7
0.8
0.9
1
False Positive Rate
(d) R4
0.4
0.5
0.6
0.7
0.8
0.9
1
(c) R3
ROC curve
1
0.9
0.8
0.7
0.6
0.5
PageRank
Centrality
Coordinates
Prior
LAP
0.4
0.3
0.2
0.1
0
0.3
False Positive Rate
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
False Positive Rate
0.9
PageRank
0.511
0.564
0.624
0.540
0.575
0.521
R1
R2
R3
R4
R5
R6
PageRank
0.645
0.556
0.619
0.659
0.585
0.580
R1
R2
R3
R4
R5
R6
PageRank
0.718
0.564
0.654
0.630
0.623
0.585
R1
R2
R3
R4
R5
R6
PageRank
0.704
0.626
0.623
0.590
0.588
0.627
Master
(c) R3
Apprentice
Apprentice
Apprentice
0.7
0.6
0.5
PageRank
Centrality
Coordinates
Prior
LAP
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
False Positive Rate
(f) R6
Fig. 5: ROC curves for detecting Observer on Robot.Net
TABLE IV: Object Role Detection on Robot.Net
R1
R2
R3
R4
R5
R6
Master
(b) R2
0.8
(e) R5
Observer
Centrality
Coordinates
0.549
0.899
0.537
0.793
0.530
0.502
0.535
0.517
0.537
0.531
0.541
0.511
Apprentice
Centrality
Coordinates
0.677
0.778
0.662
0.775
0.657
0.705
0.658
0.684
0.654
0.671
0.654
0.670
Journeyer
Centrality
Coordinates
0.511
0.911
0.519
0.809
0.511
0.504
0.518
0.522
0.527
0.524
0.528
0.545
Master
Centrality
Coordinates
0.680
0.646
0.682
0.685
0.678
0.726
0.668
0.709
0.665
0.698
0.665
0.745
Master
(a) R1
Observer
PageRank
Centrality
Coordinates
Prior
LAP
0.1
0.1
Observer
0.6
0.2
0
Observer
True Positive Rate
0.7
0.3
0.1
ROC curve
0.8
0.4
0.2
0
1
1
0.9
0.5
0.3
(b) R2
1
True Positive Rate
0.4
PageRank
Centrality
Coordinates
Prior
LAP
0.4
False Positive Rate
(a) R1
0
0.3
0.5
Journeyer
0.2
0.6
Journeyer
0.3
True Positive Rate
0
PageRank
Centrality
Coordinates
Prior
LAP
0.4
0.7
Journeyer
0.1
0.5
Apprentice
0.8
Journeyer
0.2
0.6
Apprentice
0.9
Journeyer
0.3
True Positive Rate
True Positive Rate
True Positive Rate
PageRank
Centrality
Coordinates
Prior
LAP
0.4
0.7
Observer
0.5
0.8
Observer
0.6
1
0.9
Observer
0.7
Apprentice
ROC curve
1
0.8
Journeyer
ROC curve
1
0.9
Prior
1.000
0.998
0.853
0.832
0.806
0.810
LAP
1.000
0.998
0.998
0.964
0.960
0.958
Prior
1.000
0.988
0.891
0.878
0.792
0.783
LAP
1.000
0.988
0.988
0.844
0.841
0.844
Prior
1.000
0.982
0.867
0.851
0.720
0.706
LAP
1.000
0.982
0.983
0.927
0.921
0.983
Prior
1.000
0.990
0.794
0.769
0.785
0.784
LAP
1.000
0.990
0.973
0.969
0.969
0.969
Master
Master
Master
(d) R4
(e) R5
(f) R6
Fig. 6: Role Dynamics of Object 99 in Robot.Net
TABLE V: Object Role Prediction on Robot.Net
data set contains the interactions among the users of the
website. For the experiments, we choose the first sampled
snapshot in each year during 2007 to 2012 and denote the
obtained six snapshots as R1 to R6 . In this website, each user
is labeled by the others as Observer, Apprentice, Journeyer
or Master according to his/her importance in the website.
Based on these labels, we divide the users into four object role
classes: Observer, Apprentice, Journeyer and Master. In the
experiments, we use the labeled information at R1 for training,
and test the performance of the proposed LAP on detecting,
analyzing and predicting the above object roles. Using varying
parameter values to test the performance on R1 , we obtain that
the optimal number of latent features is 70.
R1
R2
R3
R4
R5
R6
DyPageRank
0.524
0.517
0.518
0.517
0.518
R1
R2
R3
R4
R5
R6
DyPageRank
0.643
0.644
0.645
0.645
0.645
R1
R2
R3
R4
R5
R6
DyPageRank
0.681
0.678
0.678
0.677
0.677
R1
R2
R3
R4
R5
R6
DyPageRank
0.704
0.691
0.692
0.694
0.694
Observer
DyCentrality
Coordinates
0.556
0.896
0.556
0.793
0.555
0.502
0.555
0.517
0.555
0.531
Apprentice
DyCentrality
Coordinates
0.669
0.767
0.669
0.773
0.668
0.704
0.666
0.682
0.668
0.672
Journeyer
DyCentrality
Coordinates
0.534
0.907
0.534
0.808
0.534
0.504
0.535
0.520
0.535
0.524
Master
DyCentrality
Coordinates
0.670
0.857
0.671
0.791
0.671
0.570
0.672
0.573
0.672
0.575
Prior
0.997
0.998
0.852
0.832
0.806
LAP
0.997
0.998
0.997
0.964
0.960
Prior
0.988
0.988
0.890
0.878
0.791
LAP
0.988
0.986
0.986
0.844
0.841
Prior
0.983
0.983
0.867
0.853
0.712
LAP
0.983
0.981
0.983
0.927
0.921
Prior
0.990
0.989
0.870
0.854
0.772
LAP
0.990
0.989
0.989
0.912
0.907
SocialEvolution in Table II. PageRank and Centrality can not
capture the characteristics of the four object roles thus perform
the worst. Coordinates performs well when close to R1 and
perform much worse after several snapshots. Among all the
baselines, Prior performs the best.
Compared to these baselines, the proposed LAP model
achieves the best in most cases. It outperforms the best baseline
Prior by up to 39.24% and on average 11.51%. To better
illustrate the advantage of the proposed LAP model, in Fig.5,
we show the ROC curves on detecting the Observer objects
(due to the space limit, we show this case only).
3) Object Role Analysis: To evaluate the performance on
analyzing the dynamics of object roles, we do case study on
object 99. Specifically, we use 0 to 3 to represent Observer,
Apprentice, Journeyer and Master, respectively. The class
number estimated by expectation is scaled to the range of
2) Object Role Detection: Table IV summarizes the performance of all the baselines as well as the proposed LAP model
on the Robot.Net.
Overall, the results show similar trends to the results of
436
[0, 3π
2 ]. We summarize the results of the object 99 in Fig.6.
According to the votes, in the first snapshot, four users vote
this object as Apprentice and two vote the object as Journeyer.
Starting from the second snapshot, more and more users vote
the object 99 as Master. As we can observe from Fig.6, this
trend on votes is well reflected by the results of the proposed
LAP model. The case study supports that the LAP is effective
in capturing the dynamics of object roles.
for the proposed model. In experiments, we evaluated the
proposed model through the tasks of learning the dynamics
of people’s importance and political interests in two real
world data sets. Overall, the proposed LAP model significantly
outperforms the baselines on learning and predicting seven
types of object roles. Moreover, the dynamic patterns extracted
by the proposed LAP model well reflect the real changing
states of the object roles.
4) Object Role Prediction: In this part, we predict the
object roles at the second to the sixth time points using the
previous data, and summarize the results in Table V. The
results show similar patterns to those in Table III and Table
IV. Across the four baselines, DyPageRank and DyCentrality
are still not effective in characterizing the four different object
classes; Coordinates significantly suffers from its assumption
of viewing object roles as ”Coordinates”; and Prior performs
the best in the baselines.
VI.
The materials published in this paper are partially supported by the National Science Foundation under Grants No.
1218393, No. 1016929, and No. 0101244.
R EFERENCES
[1]
C. J. Kuhlman, V. S. A. Kumar, M. V. Marathe, S. S. Ravi, and D. J.
Rosenkrantz, “Finding critical nodes for inhibiting diffusion of complex
contagions in social networks,” Proc. of ECML’10, pp. 111–127, 2010.
[2] L. Page, S. Brin, R. Motwani, and T. Winograd, “The pagerank citation
ranking: Bringing order to the web,” World Wide Web Internet And Web
Information Systems, 1998.
[3] M. Handcock, G. Robins, T. Snijders, and J. Besag, “Assessing degeneracy in statistical models of social networks,” Journal of the American
Statistical Association, 2003.
[4] J. R. Foulds, C. DuBois, A. U. Asuncion, C. T. Butts, and P. Smyth,
“A dynamic relational infinite feature model for longitudinal social
networks,” Journal of Machine Learning Research - Proceedings Track,
pp. 287–295, 2011.
[5] C. Heaukulani and Z. Ghahramani, “Dynamic probabilistic models for
latent feature propagation in social networks,” Proc. of ICML’2013, pp.
275–283, 2013.
[6] Y.-R. Lin, Y. Chi, S. Zhu, H. Sundaram, and B. L. Tseng, “Analyzing
communities and their evolutions in dynamic social networks,” ACM
Trans. Knowl. Discov. Data, pp. 8:1–8:31, 2009.
[7] “Economy of ancient tamil country,” http://en.wikipedia.org/wiki/
EconomyofancientTamilcountry, accessed: 2013-06-18.
[8] Z. Lu, B. Savas, W. Tang, and I. S. Dhillon, “Supervised link prediction
using multiple sources,” Proc. of ICDM’10, pp. 923–928, 2010.
[9] S. Nakajima, M. Sugiyama, and R. Tomioka, “Global solution of variational bayesian matrix factorization under matrix-wise independence,”
Proc. of NIPS’10, 2010.
[10] C. Fox and S. Roberts, “A tutorial on variational Bayesian inference,”
Artificial Intelligence Review, pp. 85–95, 2012.
[11] W. Penny and S. Roberts, “Bayesian multivariate autoregressive models
with structured priors,” Proc. of Vision Image and Signal Processing,
pp. 33–41, 2002.
[12] A. Madan, M. Cebrian, S. Moturu, K. Farrahi, and A. Pentland,
“Sensing the ’health state’ of a community,” Pervasive Computing, pp.
36–45, 2012.
[13] L. Katz, “A new status index derived from sociometric analysis,”
Psychometrika, 1953.
[14] D. Chakrabarti, S. Papadimitriou, D. S. Modha, and C. Faloutsos, “Fully
automatic cross-associations,” in Proc. of KDD’04, 2004, pp. 79–88.
[15] E. M. Airoldi, D. M. Blei, S. E. Fienberg, and E. P. Xing, “Mixed
membership stochastic blockmodels,” J. Mach. Learn. Res., pp. 1981–
2014, 2008.
[16] C. Kemp, J. B. Tenenbaum, T. L. Griffiths, T. Yamada, and N. Ueda,
“Learning systems of concepts with an infinite relational model,” Proc.
of AAAI’06, pp. 381–388, 2006.
[17] P. Koutsourelakis and T. Eliassi-Rad, “Finding mixed-memberships in
social networks,” Proc. of AAAI Spring Symposium’08, 2008.
[18] K. Henderson, B. Gallagher, T. Eliassi-Rad, and H. Tong,
“Roix:structural role extraction & mining in large graphs,” Proc. of
KDD’12, 2012.
[19] R. Rossi and B. Gallagher, “Role-dynamics:fast mining of large dynamic networks,” Proc. of WWW’12, 2012.
The proposed LAP model significantly outperforms all
the baselines in most cases over different object roles. LAP
improves over the best baseline Prior by up to 22.69% and on
average 5.86%.
IV.
R ELATED W ORK
As reviewed in Section I, the proposed model significantly
differs from the existing studies in the area of dynamic network
mining. Besides these, there are also several other approaches
that are related to the task in this paper, thus we explicitly
discuss them in this section.
As for object role mining in graphs, existing methods [14]–
[17] usually adopt the assumption that two objects have the
same roles if they have the same relationships to all other
objects. This assumption heavily restricts the applicability of
these existing methods. For instance, suppose there are two
spammers in an email network, and they focus on spamming
users from different areas. In this case, the two spammers have
the same roles but totally different connections to other objects
in the email network, which contradicts with the aforementioned assumption used by the above existing methods.
Besides PageRank [2] and Centrality [13], several other
existing methods such as [18], [19] also use statistics in
topology to characterize object behaviors. The major drawback
of these methods is that the object roles are estimated through
subjectively selected topological features, thus these methods
may not be discriminative in learning many types of object
roles. Moreover, these methods do not incorporate the dynamic
information at each time point into the estimation of object
roles, which differs them from the proposed method.
V.
ACKNOWLEDGMENTS
C ONCLUSIONS
In this paper, we have introduced a probability model for
learning, analyzing and predicting object roles in dynamic
networks. The proposed model effectively integrates structural
information, dynamic information and supervised information
for extracting the latent feature representation of object roles.
The extracted object role representation is then used to identify
the role of each node, track the changes of object roles over
time, and predict the object roles in dynamic networks. We
have also provided the detailed variational bayesian inference
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