Scalable

1
Scalable Learning of Collective Behavior
Lei Tang, Xufei Wang, and Huan Liu
Abstract—This study of collective behavior is to understand how individuals behave in a social networking environment. Oceans
of data generated by social media like Facebook, Twitter, Flickr, and YouTube present opportunities and challenges to study
collective behavior on a large scale. In this work, we aim to learn to predict collective behavior in social media. In particular,
given information about some individuals, how can we infer the behavior of unobserved individuals in the same network? A
social-dimension-based approach has been shown effective in addressing the heterogeneity of connections presented in social
media. However, the networks in social media are normally of colossal size, involving hundreds of thousands of actors. The
scale of these networks entails scalable learning of models for collective behavior prediction. To address the scalability issue, we
propose an edge-centric clustering scheme to extract sparse social dimensions. With sparse social dimensions, the proposed
approach can efficiently handle networks of millions of actors while demonstrating a comparable prediction performance to other
non-scalable methods.
Index Terms—Classification with Network Data, Collective Behavior, Community Detection, Social Dimensions
!
1 INTRODUCTION
THE advancement in computing and communica-
tion technologies enables people to get together
and share information in innovative ways. Social
networking sites (a recent phenomenon) empower
people of different ages and backgrounds with new
forms of collaboration, communication, and collec-
tive intelligence. Prodigious numbers of online vol-
unteers collaboratively write encyclopedia articles of
unprecedented scope and scale; online marketplaces
recommend products by investigating user shopping
behavior and interactions; and political movements
also exploit new forms of engagement and collective
action. In the same process, social media provides
ample opportunities to study human interactions and
collective behavior on an unprecedented scale. In this
work, we study how networks in social media can
help predict some human behaviors and individual
preferences. In particular, given the behavior of some
individuals in a network, how can we infer the behav-
ior of other individuals in the same social network [1]?
This study can help better understand behavioral
patterns of users in social media for applications like
social advertising and recommendation.
Typically, the connections in social media networks
are not homogeneous. Different connections are as-
sociated with distinctive relations. For example, one
user might maintain connections simultaneously to
his friends, family, college classmates, and colleagues.
This relationship information, however, is not always
fully available in reality. Mostly, we have access to the
connectivity information between users, but we have
no idea why they are connected to each other. This
Lei Tang, Yahoo! Labs, Santa Clara, CA 95054, USA, l.tang@asu.edu.
Xufei Wang and Huan Liu, Computer Science & Engineering,
Arizona State University, Tempe, AZ 85287, USA, {xufei.wang,
huan.liu}@asu.edu
heterogeneity of connections limits the effectiveness
of a commonly used technique — collective inference
for network classification. A recent framework based on
social dimensions [2] is shown to be effective in ad-
dressing this heterogeneity. The framework suggests
a novel way of network classification: first, capture
the latent affiliations of actors by extracting social
dimensions based on network connectivity, and next,
apply extant data mining techniques to classification
based on the extracted dimensions. In the initial study,
modularity maximization [3] was employed to extract
social dimensions. The superiority of this framework
over other representative relational learning methods
has been verified with social media data in [2].
The original framework, however, is not scalable
to handle networks of colossal sizes because the ex-
tracted social dimensions are rather dense. In social
media, a network of millions of actors is very com-
mon. With a huge number of actors, extracted dense
social dimensions cannot even be held in memory,
causing a serious computational problem. Sparsifying
social dimensions can be effective in eliminating the
scalability bottleneck. In this work, we propose an
effective edge-centric approach to extract sparse social
dimensions [4]. We prove that with our proposed
approach, sparsity of social dimensions is guaranteed.
Extensive experiments are then conducted with social
media data. The framework based on sparse social
dimensions, without sacrificing the prediction perfor-
mance, is capable of efficiently handling real-world
networks of millions of actors.
2 COLLECTIVE BEHAVIOR
Collective behavior refers to the behaviors of individ-
uals in a social networking environment, but it is not
simply the aggregation of individual behaviors. In a

2
connected environment, individuals’ behaviors tend
to be interdependent, influenced by the behavior of
friends. This naturally leads to behavior correlation
between connected users [5]. Take marketing as an
example: if our friends buy something, there is a
better-than-average chance that we will buy it, too.
This behavior correlation can also be explained
by homophily [6]. Homophily is a term coined in
the 1950s to explain our tendency to link with one
another in ways that confirm, rather than test, our
core beliefs. Essentially, we are more likely to connect
to others who share certain similarities with us. This
phenomenon has been observed not only in the many
processes of a physical world, but also in online
systems [7], [8]. Homophily results in behavior cor-
relations between connected friends. In other words,
friends in a social network tend to behave similarly.
The recent boom of social media enables us to study
collective behavior on a large scale. Here, behaviors
include a broad range of actions: joining a group,
connecting to a person, clicking on an ad, becoming
interested in certain topics, dating people of a certain
type, etc. In this work, we attempt to leverage the
behavior correlation presented in a social network in
order to predict collective behavior in social media.
Given a network with the behavioral information of
some actors, how can we infer the behavioral outcome
of the remaining actors within the same network?
Here, we assume the studied behavior of one actor can
be described with K class labels {c1, · · · , cK}. Each
label, ci, can be 0 or 1. For instance, one user might
join multiple groups of interest, so ci = 1 denotes that
the user subscribes to group i, and ci = 0 otherwise.
Likewise, a user can be interested in several topics
simultaneously, or click on multiple types of ads. One
special case is K = 1, indicating that the studied
behavior can be described by a single label with 1
and 0. For example, if the event is the presidential
election, 1 or 0 indicates whether or not a voter voted
for Barack Obama. The problem we study can be
described formally as follows:
Suppose there are K class labels Y =
{c1, · · · , cK}. Given network G = (V, E, Y )
where V is the vertex set, E is the edge set
and Yi ⊆ Y are the class labels of a vertex
vi ∈ V , and known values of Yi for some
subsets of vertices V L
, how can we infer
the values of Yi (or an estimated probability
over each label) for the remaining vertices
V U
= V − V L
?
It should be noted that this problem shares the same
spirit as within-network classification [9]. It can also
be considered as a special case of semi-supervised
learning [10] or relational learning [11] where ob-
jects are connected within a network. Some of these
methods, if applied directly to social media, yield
only limited success [2]. This is because connections
TABLE 1
Social Dimension Representation
Actors Affiliation-1 Affiliation-2 · · · Affiliation-k
1 0 1 · · · 0.8
2 0.5 0.3 · · · 0
...
...
...
...
...
in social media are rather noisy and heterogeneous.
In the next section, we will discuss the connection
heterogeneity in social media, review the concept
of social dimension, and anatomize the scalability
limitations of the earlier model proposed in [2], which
provides a compelling motivation for this work.
3 SOCIAL DIMENSIONS
Connections in social media are not homogeneous.
People can connect to their family, colleagues, college
classmates, or buddies met online. Some relations are
helpful in determining a targeted behavior (category)
while others are not. This relation-type information,
however, is often not readily available in social media.
A direct application of collective inference [9] or label
propagation [12] would treat connections in a social
network as if they were homogeneous. To address
the heterogeneity present in connections, a frame-
work (SocioDim) [2] has been proposed for collective
behavior learning.
The framework SocioDim is composed of two steps:
1) social dimension extraction, and 2) discriminative
learning. In the first step, latent social dimensions are
extracted based on network topology to capture the
potential affiliations of actors. These extracted social
dimensions represent how each actor is involved in
diverse affiliations. One example of the social dimen-
sion representation is shown in Table 1. The entries in
this table denote the degree of one user involving in
an affiliation. These social dimensions can be treated
as features of actors for subsequent discriminative
learning. Since a network is converted into features,
typical classifiers such as support vector machine and
logistic regression can be employed. The discrimina-
tive learning procedure will determine which social
dimension correlates with the targeted behavior and
then assign proper weights.
A key observation is that actors of the same affilia-
tion tend to connect with each other. For instance, it is
reasonable to expect people of the same department
to interact with each other more frequently. Hence,
to infer actors’ latent affiliations, we need to find
out a group of people who interact with each other
more frequently than at random. This boils down to a
classic community detection problem. Since each actor
can get involved in more than one affiliation, a soft
clustering scheme is preferred.
In the initial instantiation of the framework So-
cioDim, a spectral variant of modularity maximiza-

3
tion [3] is adopted to extract social dimensions. The
social dimensions correspond to the top eigenvectors
of a modularity matrix. It has been empirically shown
that this framework outperforms other representative
relational learning methods on social media data.
However, there are several concerns about the scal-
ability of SocioDim with modularity maximization:
• Social dimensions extracted according to soft
clustering, such as modularity maximization and
probabilistic methods, are dense. Suppose there
are 1 million actors in a network and 1, 000
dimensions are extracted. If standard double pre-
cision numbers are used, holding the full matrix
alone requires 1M × 1K × 8 = 8G memory. This
large-size dense matrix poses thorny challenges
for the extraction of social dimensions as well as
subsequent discriminative learning.
• Modularity maximization requires us to compute
the top eigenvectors of a modularity matrix,
which is the same size as a given network. In
order to extract k communities, typically k − 1
eigenvectors are computed. For a sparse or struc-
tured matrix, the eigenvector computation costs
O(h(mk + nk2
+ k3
)) time [13], where h, m,
and n are the number of iterations, the number
of edges in the network, and the number of
nodes, respectively. Though computing the top
single eigenvector (i.e., k=1), such as PageRank
scores, can be done very efficiently, computing
thousands of eigenvectors or even more for a
mega-scale network becomes a daunting task.
• Networks in social media tend to evolve, with
new members joining and new connections oc-
curring between existing members each day. This
dynamic nature of networks entails an efficient
update of the model for collective behavior pre-
diction. Efficient online updates of eigenvectors
with expanding matrices remain a challenge.
Consequently, it is imperative to develop scal-
able methods that can handle large-scale networks
efficiently without extensive memory requirements.
Next, we elucidate on an edge-centric clustering
scheme to extract sparse social dimensions. With such
a scheme, we can also update the social dimensions
efficiently when new nodes or new edges arrive.
4 SPARSE SOCIAL DIMENSIONS
In this section, we first show one toy example to
illustrate the intuition of communities in an “edge”
view and then present potential solutions to extract
sparse social dimensions.
4.1 Communities in an Edge-Centric View
Though SocioDim with soft clustering for social di-
mension extraction demonstrated promising results,
its scalability is limited. A network may be sparse
Fig. 1. A Toy Example Fig. 2. Edge Clusters
TABLE 2
Social Dimension(s) of the Toy Example
Actors Modularity Edge
Maximization Partition
1 -0.1185 1 1
2 -0.4043 1 0
3 -0.4473 1 0
4 -0.4473 1 0
5 0.3093 0 1
6 0.2628 0 1
7 0.1690 0 1
8 0.3241 0 1
9 0.3522 0 1
(i.e., the density of connectivity is very low), whereas
the extracted social dimensions are not sparse. Let’s
look at the toy network with two communities in
Figure 1. Its social dimensions following modularity
maximization are shown in Table 2. Clearly, none of
the entries is zero. When a network expands into
millions of actors, a reasonably large number of social
dimensions need to be extracted. The corresponding
memory requirement hinders both the extraction of
social dimensions and the subsequent discriminative
learning. Hence, it is imperative to develop some
other approach so that the extracted social dimensions
are sparse.
It seems reasonable to state that the number of affil-
iations one user can participate in is upper bounded
by his connections. Consider one extreme case that
an actor has only one connection. It is expected that
he is probably active in only one affiliation. It is not
necessary to assign a non-zero score for each of the
many other affiliations. Assuming each connection
represents one involved affiliation, we can expect the
number of affiliations an actor has is no more than that
of his connections. Rather than defining a community
as a set of nodes, we redefine it as a set of edges. Thus,
communities can be identified by partitioning edges
of a network into disjoint sets.
An actor is considered associated with one affiliation
if one of his connections is assigned to that affiliation.
For instance, the two communities in Figure 1 can
be represented by two edge sets in Figure 2, where
the dashed edges represent one affiliation, and the
remaining edges denote the second affiliation. The
disjoint edge clusters in Figure 2 can be converted
into the representation of social dimensions as shown
in the last two columns in Table 2, where an entry is 1
(0) if an actor is (not) involved in that corresponding
social dimension. Node 1 is affiliated with both com-
munities because it has edges in both sets. By contrast,

4
10
2
10
3
10
4
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
K
α
−3.5
−3
−2.5
−2
−1.5
Density in
log scale
(−2 denotes 10
−2
)
Fig. 3. Density Upper Bound of Social Dimensions
node 3 is assigned to only one community, as all its
connections are in the dashed edge set.
To extract sparse social dimensions, we partition
edges rather than nodes into disjoint sets. The edges of
those actors with multiple affiliations (e.g., actor 1 in
the toy network) are separated into different clusters.
Though the partition in the edge view is disjoint, the
affiliations in the node-centric view can overlap. Each
node can engage in multiple affiliations.
In addition, the extracted social dimensions follow-
ing edge partition are guaranteed to be sparse. This is
because the number of one’s affiliations is no more
than that of her connections. Given a network with
m edges and n nodes, if k social dimensions are ex-
tracted, then each node vi has no more than min(di, k)
non-zero entries in her social dimensions, where di is
the degree of node vi. We have the following theorem
about the density of extracted social dimensions.
Theorem 1: Suppose k social dimensions are ex-
tracted from a network with m edges and n nodes.
The density (proportion of nonzero entries) of the
social dimensions based on edge partition is bounded
by the following:
density ≤
n
i=1 min(di, k)
nk
=
{i|di≤k} di + {i|di>k} k
nk
(1)
Moreover, for many real-world networks whose node
degree follows a power law distribution, the upper
bound in Eq. (1) can be approximated as follows:
α − 1
α − 2
1
k
−
α − 1
α − 2
− 1 k−α+1
(2)
where α > 2 is the exponent of the power law
distribution.
The proof is given in the appendix. To give a
concrete example, we examine a YouTube network1
with 1+ million actors and verify the upper bound
of the density. The YouTube network has 1, 128, 499
nodes and 2, 990, 443 edges. Suppose we want to
extract 1, 000 dimensions from the network. Since 232
1. More details are in Section 6.
nodes in the network have a degree larger than 1000,
the density is upper bounded by (5, 472, 909 + 232 ×
1, 000)/(1, 128, 499×1, 000) = 0.51% following Eq. (1).
The node distribution in the network follows a power
law with exponent α = 2.14 based on maximum like-
lihood estimation [14]. Thus, the approximate upper
bound in Eq. (2) for this specific case is 0.54%.
Note that the upper bound in Eq. (1) is network
specific, whereas Eq. (2) gives an approximate upper
bound for a family of networks. It is observed that
most power law distributions occurring in nature
have 2 ≤ α ≤ 3 [14]. Hence, the bound in Eq. (2) is
valid most of the time. Figure 3 shows the function in
terms of α and k. Note that when k is huge (close
to 10,000), the social dimensions become extremely
sparse (< 10−3
), alleviating the memory demand and
thus enabling scalable discriminative learning.
Now, the remaining question is how to partition
edges efficiently. Below, we will discuss two different
approaches to accomplish this task: partitioning a line
graph or clustering edge instances.
4.2 Edge Partition via Line Graph Partition
In order to partition edges into disjoint sets, one way
is to look at the “dual” view of a network, i.e., the line
graph [15]. We will show that this is not a practical
solution. In a line graph L(G), each node corresponds
to an edge in the original network G, and edges in
the line graph represent the adjacency between two
edges in the original graph. The line graph of the toy
example is shown in Figure 4. For instance, e(1, 3) and
e(2, 3) are connected in the line graph as they share
one terminal node 3.
Given a network, graph partition algorithms can
be applied to its corresponding line graph. The set
of communities in the line graph corresponds to a
disjoint edge partition in the original graph. Recently,
such a scheme has been used to detect overlapping
communities [16], [17]. It is, however, prohibitive to
construct a line graph for a mega-scale network. We
notice that edges connecting to the same node in the
original network form a clique in the corresponding
line graph. For example, edges e(1, 3), e(2, 3), and
e(3, 4) are all neighboring edges of node 3 in Figure 1.
Hence, they are adjacent to each other in the line
graph in Figure 4, forming a clique. This property
leads to many more edges in a line graph than in the
original network.
Theorem 2: Let n and m denote the numbers of
nodes and connections in a network, respectively, and
N and M the numbers of nodes and connections in
its line graph. It follows that
N = m, M ≥ m
2m
n
− 1 (3)
Equality is achieved if and only if the original graph
is regular, i.e., all nodes have the same degree.

5
e(1,3)
e(1,4)
e(1,5)
e(1,6)
e(1,7)
e(2,3)
e(3,4)
e(5,6)
e(5,8)
e(5,9)
e(6,7)
e(6,9)
e(7,8)
e(2,4)
e(8,9)
Fig. 4. The line graph of the toy example in Figure 1.
Each node in the line graph corresponds to an edge in
the original graph.
Many real-world large-scale networks obey a power
law degree distribution [14]. Consequently, the lower
bound in Eq. (3) is never achievable. The number of
connections in a line graph grows without a constant
bound with respect to the size of a given network.
Theorem 3: Let α denote the exponent of a power
law degree distribution for a given network, n the size
of the network, and M the number of connections in
its corresponding line graph. It follows that
E[2M/n]
diverges if 2 < α <= 3
= α−1
(α−3)(α−2) if α > 3
(4)
The divergence of the expectation tells us that as
we go to larger and larger data sets, our estimate of
number of connections with respect to the original
network size will increase without bound. As we
have mentioned, the majority of large-scale networks
follow a power law with α lying between 2 and
3. Consequently, the number of connections in a
line graph can be extremely huge. Take the afore-
mentioned YouTube network as an example again.
It contains approximately 1 million nodes and 3
million links. Its resultant line graph will contain
4, 436, 252, 282 connections, too large to be loaded into
memory. Recall that the original motivation to use
sparse social dimensions is to address the scalability
concern. Now we end up dealing with a much larger
sized line graph. Hence, the line graph is not suitable
for practical use. Next, we will present an alternative
approach to edge partition, which is much more effi-
cient and scalable.
4.3 Edge Partition via Clustering Edge Instances
In order to partition edges into disjoint sets, we treat
edges as data instances with their terminal nodes as
TABLE 3
Edge Instances of the Toy Network in Figure 1
Edge
Features
1 2 3 4 5 6 7 8 9
e(1, 3) 1 0 1 0 0 0 0 0 0
e(1, 4) 1 0 0 1 0 0 0 0 0
e(2, 3) 0 1 1 0 0 0 0 0 0
... · · · · · · · · ·
features. For instance, we can treat each edge in the
toy network in Figure 1 as one instance, and the nodes
that define edges as features. This results in a typical
feature-based data format as in Table 3. Then, a typical
clustering algorithm like k-means clustering can be
applied to find disjoint partitions.
One concern with this scheme is that the total
number of edges might be too huge. Owing to the
power law distribution of node degrees presented in
social networks, the total number of edges is normally
linear, rather than square, with respect to the number
of nodes in the network. That is, m = O(n) as stated
in the following theorem.
Theorem 4: The total number of edges is usually lin-
ear, rather than quadratic, with respect to the number
of nodes in the network with a power law distribu-
tion. In particular, the expected number of edges is
given as
E[m] =
n
2
α − 1
α − 2
, (5)
where α is the exponent of the power law distribution.
Please see the appendix for the proof.
Still, millions of edges are the norm in a large-scale
network. Direct application of some existing k-means
implementation cannot handle the problem. E.g., the
k-means code provided in the Matlab package re-
quires the computation of the similarity matrix be-
tween all pairs of data instances, which would exhaust
the memory of normal PCs in seconds. Therefore, an
implementation with online computation is preferred.
On the other hand, the data of edge instances is
quite sparse and structured. As each edge connects
two nodes, the corresponding edge instance has ex-
actly only two non-zero features as shown in Table 3.
This sparsity can help accelerate the clustering process
if exploited wisely. We conjecture that the centroids of
k-means should also be feature-sparse. Often, only a
small portion of the data instances share features with
the centroid. Hence, we just need to compute the sim-
ilarity of the centroids with their relevant instances. In
order to efficiently identify instances relevant to one
centroid, we build a mapping from features (nodes)
to instances (edges) beforehand. Once we have the
mapping, we can easily identify the relevant instances
by checking the non-zero features of the centroid.
By taking into account the two concerns above,
we devise a k-means variant as shown in Figure 5.
Similar to k-means, this algorithm also maximizes

6
Input: data instances {xi|1 ≤ i ≤ m}
number of clusters k
Output: {idxi}
1. construct a mapping from features to instances
2. initialize the centroid of cluster {Cj|1 ≤ j ≤ k}
3. repeat
4. Reset {MaxSimi}, {idxi}
5. for j=1:k
6. identify relevant instances Sj to centroid Cj
7. for i in Sj
8. compute sim(i, Cj) of instance i and Cj
9. if sim(i, Cj) > MaxSimi
10. MaxSimi = sim(i, Cj)
11. idxi = j;
12. for i=1:m
13. update centroid Cidxi
14. until change of objective value <
Fig. 5. Algorithm of Scalable k-means Variant
within cluster similarity as shown in Eq. (6)
arg max
S
k
i=1 xj ∈Si
xj · µi
xj µi
(6)
where k is the number of clusters, S = {S1, S2, . . . , Sk}
is the set of clusters, and µi is the centroid of cluster
Si. In Figure 5, we keep only a vector of MaxSim to
represent the maximum similarity between one data
instance and a centroid. In each iteration, we first
identify the instances relevant to a centroid, and then
compute similarities of these instances with the cen-
troid. This avoids the iteration over each instance and
each centroid, which will cost O(mk) otherwise. Note
that the centroid contains one feature (node), if and
only if any edge of that node is assigned to the cluster.
In effect, most data instances (edges) are associated
with few (much less than k) centroids. By taking
advantage of the feature-instance mapping, the cluster
assignment for all instances (lines 5-11 in Figure 5)
can be fulfilled in O(m) time. Computing the new
centroid (lines 12-13) costs O(m) time as well. Hence,
each iteration costs O(m) time only. Moreover, the
algorithm requires only the feature-instance mapping
and network data to reside in main memory, which
costs O(m + n) space. Thus, as long as the network
data can be held in memory, this clustering algorithm
is able to partition its edges into disjoint sets.
As a simple k-means is adopted to extract social
dimensions, it is easy to update social dimensions
if a given network changes. If a new member joins
the network and a new connection emerges, we can
simply assign the new edge to the corresponding
clusters. The update of centroids with the new arrival
of connections is also straightforward. This k-means
scheme is especially applicable for dynamic large-
scale networks.
4.4 Regularization on Communities
The extracted social dimensions are treated as features
of nodes. Conventional supervised learning can be
Input: network data, labels of some nodes,
number of social dimensions;
Output: labels of unlabeled nodes.
1. convert network into edge-centric view.
2. perform edge clustering as in Figure 5.
3. construct social dimensions based on edge partition.
A node belongs to one community as long as any of
its neighboring edges is in that community.
4. apply regularization to social dimensions.
5. construct classifier based on social dimensions of
labeled nodes.
6. use the classifier to predict labels of unlabeled ones
based on their social dimensions.
Fig. 6. Algorithm for Learning of Collective Behavior
conducted. In order to handle large-scale data with
high dimensionality and vast numbers of instances,
we adopt a linear SVM, which can be finished in
linear time [18]. Generally, the larger a community
is, the weaker the connections within the community
are. Hence, we would like to build an SVM relying
more on communities of smaller sizes by modifying
the typical SVM objective function as follows:
min λ
n
i=1
|1 − yi(xT
i w + b)|+ +
1
2
wT
Σw (7)
where λ is a regularization parameter2
, |z|+ =
max(0, z) the hinge loss function, and Σ a diagonal
matrix to regularize the weights assigned to differ-
ent communities. In particular, Σjj = h(|Cj|) is the
penalty coefficient associated with community Cj. The
penalty function h should be monotonically increasing
with respect to the community size. In other words, a
larger weight assigned to a large community results
in a higher cost.
Interestingly, with a little manipulation, the formula
in Eq. (7) can be solved using a standard SVM with
modified input. Let X represent the input data with
each row being an instance (e.g., Table 2), and ˜w =
Σ1/2
w. Then, the formula can be rewritten as:
min λ
i
|1 − yi(xT
i w + b)|+ +
1
2
wT
Σ
1
2 Σ
1
2 w
= min λ
i
|1 − yi(xT
i Σ− 1
2 Σ
1
2 w + b)|+ +
1
2
˜w 2
2
= min λ
i
|1 − yi(˜xT
i ˜w + b)|+ +
1
2
˜w 2
2
where ˜xT
i = xT
i Σ− 1
2 . Hence, given an input data
matrix X, we only need to left multiply X by Σ− 1
2 .
It is observed that community sizes of a network
tend to follow a power law distribution. Hence, we
recommend h(|Cj|) = log |Cj| or (log |Cj|)2
.
Meanwhile, one node is likely to engage in multi-
ple communities. Intuitively, a node who is devoted
to a few select communities has more influence on
2. We use a different notation λ from standard SVM formulation
to avoid the confusion with community C.

7
TABLE 4
Statistics of Social Media Data
Data BlogCatalog Flickr YouTube
Categories 39 195 47
Nodes (n) 10, 312 80, 513 1, 138, 499
Links (m) 333, 983 5, 899, 882 2, 990, 443
Network Density 6.3 × 10−3 1.8 × 10−3 4.6 × 10−6
Maximum Degree 3, 992 5, 706 28, 754
Average Degree 65 146 5
other group members than those who participate in
many communities. For effective classification learn-
ing, we regularize node affiliations by normalizing
each node’s social dimensions to sum to one. Later,
we will study which regularization approach has a
greater effect on classification.
In summary, we conduct the following steps to
learn a model for collective behavior. Given a network
(say, Figure 1), we take the edge-centric view of the
network data (Table 3) and partition the edges into
disjoint sets (Figure 2). Based on the edge clustering,
social dimensions can be constructed (Table 2). Cer-
tain regularizations can be applied. Then, discrimina-
tive learning and prediction can be accomplished by
considering these social dimensions as features. The
detailed algorithm is summarized in Figure 6.
5 EXPERIMENT SETUP
In this section, we present the data collected from
social media for evaluation and the baseline methods
for comparison.
5.1 Social Media Data
Two data sets reported in [2] are used to examine
our proposed model for collective behavior learning.
The first data set is acquired from BlogCatalog3
, the
second from a popular photo sharing site Flickr4
.
Concerning behavior, following [2], we study whether
or not a user joins a group of interest. Since the Blog-
Catalog data does not have this group information,
we use blogger interests as the behavior labels. Both
data sets are publicly available at the first author’s
homepage. To examine scalability, we also include a
mega-scale network5
crawled from YouTube6
. We re-
move those nodes without connections and select the
interest groups with 500+ subscribers. Some statistics
of the three data sets can be found in Table 4.
5.2 Baseline Methods
As we discussed in Section 4.2, constructing a line
graph is prohibitive for large-scale networks. Hence,
3. http://www.blogcatalog.com/
4. http://www.flickr.com/
5. http://socialnetworks.mpi-sws.org/data-imc2007.html
6. http://www.youtube.com/
the line-graph approach is not included for com-
parison. Alternatively, the edge-centric clustering (or
EdgeCluster) in Section 4.2 is used to extract social
dimensions on all data sets. We adopt cosine similar-
ity while performing the clustering. Based on cross
validation, the dimensionality is set to 5000, 10000,
and 1000 for BlogCatalog, Flickr, and YouTube, respec-
tively. A linear SVM classifier [18] is then exploited for
discriminative learning.
Another related approach to finding edge partitions
is bi-connected components [19]. Bi-connected compo-
nents of a graph are the maximal subsets of vertices
such that the removal of a vertex from a particular
component will not disconnect the component. Essen-
tially, any two nodes in a bi-connected component are
connected by at least two paths. It is highly related
to cut vertices (a.k.a. articulation points) in a graph,
whose removal will result in an increase in the num-
ber of connected components. Those cut vertices are
the bridges connecting different bi-connected compo-
nents. Thus, searching for bi-connected components
boils down to searching for articulation points in the
graph, which can be solved efficiently in O(n + m)
time. Here n and m represent the number of vertices
and edges in a graph, respectively. Each bi-connected
component is considered a community, and converted
into one social dimension for learning.
We also compare our proposed sparse social dimen-
sion approach with classification performance based
on dense representations. In particular, we extract
social dimensions according to modularity maximiza-
tion (denoted as ModMax) [2]. ModMax has been
shown to outperform other representative relational
learning methods based on collective inference. We
study how the sparsity in social dimensions affects
the prediction performance as well as the scalability.
Note that social dimensions allow one actor to
be involved in multiple affiliations. As a proof of
concept, we also examine the case when each actor
is associated with only one affiliation. Essentially, we
construct social dimensions based on node partition.
A similar idea has been adopted in a latent group
model [20] for efficient inference. To be fair, we adopt
k-means clustering to partition nodes of a network
into disjoint sets, and convert the node clustering
result into a set of social dimensions. Then, SVM is
utilized for discriminative learning. For convenience,
we denote this method as NodeCluster.
Note that our prediction problem is essentially
multi-label. It is empirically shown that thresholding
can affect the final prediction performance drasti-
cally [21], [22]. For evaluation purposes, we assume
the number of labels of unobserved nodes is already
known, and check whether the top-ranking predicted
labels match with the actual labels. Such a scheme
has been adopted for other multi-label evaluation
works [23]. We randomly sample a portion of nodes
as labeled and report the average performance of 10

8
TABLE 5
Performance on BlogCatalog Network
Proportion of Labeled Nodes 10% 20% 30% 40% 50% 60% 70% 80% 90%
EdgeCluster 27.94 30.76 31.85 32.99 34.12 35.00 34.63 35.99 36.29
Micro-F1(%) BiComponents 16.54 16.59 16.67 16.83 17.21 17.26 17.04 17.76 17.61
ModMax 27.35 30.74 31.77 32.97 34.09 36.13 36.08 37.23 38.18
NodeCluster 18.29 19.14 20.01 19.80 20.81 20.86 20.53 20.74 20.78
EdgeCluster 16.16 19.16 20.48 22.00 23.00 23.64 23.82 24.61 24.92
Macro-F1(%) BiComponents 2.77 2.80 2.82 3.01 3.13 3.29 3.25 3.16 3.37
ModMax 17.36 20.00 20.80 21.85 22.65 23.41 23.89 24.20 24.97
NodeCluster 7.38 7.02 7.27 6.85 7.57 7.27 6.88 7.04 6.83
TABLE 6
Performance on Flickr Network
Proportion of Labeled Nodes 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
EdgeCluster 25.75 28.53 29.14 30.31 30.85 31.53 31.75 31.76 32.19 32.84
Micro-F1(%) BiComponents 16.45 16.46 16.45 16.49 16.49 16.49 16.49 16.48 16.55 16.55
ModMax 22.75 25.29 27.30 27.60 28.05 29.33 29.43 28.89 29.17 29.20
NodeCluster 22.94 24.09 25.42 26.43 27.53 28.18 28.32 28.58 28.70 28.93
EdgeCluster 10.52 14.10 15.91 16.72 18.01 18.54 19.54 20.18 20.78 20.85
Macro-F1(%) BiComponents 0.45 0.46 0.45 0.46 0.46 0.46 0.46 0.46 0.47 0.47
ModMax 10.21 13.37 15.24 15.11 16.14 16.64 17.02 17.10 17.14 17.12
NodeCluster 7.90 9.99 11.42 11.10 12.33 12.29 12.58 13.26 12.79 12.77
TABLE 7
Performance on YouTube Network
Proportion of Labeled Nodes 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
EdgeCluster 23.90 31.68 35.53 36.76 37.81 38.63 38.94 39.46 39.92 40.07
Micro-F1(%) BiComponents 23.90 24.51 24.80 25.39 25.20 25.42 25.24 24.44 25.62 25.53
ModMax — — — — — — — — — —
NodeCluster 20.89 24.57 26.91 28.65 29.56 30.72 31.15 31.85 32.29 32.67
EdgeCluster 19.48 25.01 28.15 29.17 29.82 30.65 30.75 31.23 31.45 31.54
Macro-F1(%) BiComponents 6.80 7.05 7.19 7.44 7.48 7.58 7.61 7.63 7.76 7.91
ModMax — — — — — — — — — —
NodeCluster 17.91 21.11 22.38 23.91 24.47 25.26 25.50 26.02 26.44 26.68
runs in terms of Micro-F1 and Macro-F1 [22].
6 EXPERIMENT RESULTS
In this section, we first examine how prediction
performances vary with social dimensions extracted
following different approaches. Then we verify the
sparsity of social dimensions and its implication for
scalability. We also study how the performance varies
with dimensionality. Finally, concrete examples of ex-
tracted social dimensions are given.
6.1 Prediction Performance
The prediction performance on all data is shown in
Tables 5-7. The entries in bold face denote the best
performance in each column. Obviously, EdgeCluster
is the winner most of the time. Edge-centric clustering
shows comparable performance to modularity maxi-
mization on BlogCatalog network, yet it outperforms
ModMax on Flickr. ModMax on YouTube is not
applicable due to the scalability constraint. Clearly,
with sparse social dimensions, we are able to achieve
comparable performance as that of dense social di-
mensions. But the benefit in terms of scalability will
be tremendous as discussed in the next subsection.
The NodeCluster scheme forces each actor to
be involved in only one affiliation, yielding in-
ferior performance compared with EdgeCluster.
BiComponents, similar to EdgeCluster, also sepa-
rates edges into disjoint sets, which in turn de-
liver a sparse representation of social dimensions.
However, BiComponents yields a poor performance.
This is because BiComponents outputs highly im-
balanced communities. For example, BiComponents
extracts 271 bi-connected components in the Blog-
Catalog network. Among these 271 components, a
dominant one contains 10, 042 nodes, while all others
are of size 2. Note that BlogCatalog contains in total
10, 312 nodes. As a network’s connectivity increases,
BiComponents performs even worse. For instance,
only 10 bi-connected components are found in the
Flickr data, and thus its Macro-F1 is close to 0. In
short, BiComponents is very efficient and scalable.
However, it fails to extract informative social dimen-
sions for classification.
We note that the prediction performance on the
studied social media data is around 20-30% for F1
measure. This is partly due to the large number of
distinctive labels in the data. Another reason is that
only the network information is exploited here. Since

9
TABLE 8
Sparsity Comparison on BlogCatalog data with 10, 312 Nodes. ModMax-500 corresponds to modularity
maximization to select 500 social dimensions and EdgeCluster-x denotes edge-centric clustering to construct x
dimensions. Time denotes the total time (seconds) to extract the social dimensions; Space represent the
memory footprint (mega-byte) of the extracted social dimensions; Density is the proportion of non-zeros entries
in the dimensions; Upper bound is the density upper bound computed following Eq. (1); Max-Aff and Ave-Aff
denote the maximum and average number of affiliations one user is involved in.
Methods Time Space Density Upper Bound Max-Aff Ave-Aff
ModMax − 500 194.4 41.2M 1 — 500 500
EdgeCluster − 100 300.8 3.8M 1.1 × 10−1 2.2 × 10−1 187 23.5
EdgeCluster − 500 357.8 4.9M 6.0 × 10−2 1.1 × 10−1 344 30.0
EdgeCluster − 1000 307.2 5.2M 3.2 × 10−2 6.0 × 10−2 408 31.8
EdgeCluster − 2000 294.6 5.3M 1.6 × 10−2 3.1 × 10−2 598 32.4
EdgeCluster − 5000 230.3 5.5M 6 × 10−3 1.3 × 10−2 682 32.4
EdgeCluster − 10000 195.6 5.6M 3 × 10−3 7 × 10−3 882 33.3
TABLE 9
Sparsity Comparison on Flickr Data with 80, 513 Nodes
ModMax − 500 2.2 × 103 322.1M 1 — 500 500.0
EdgeCluster − 200 1.2 × 104 31.0M 1.2 × 10−1 3.9 × 10−1 156 24.1
EdgeCluster − 500 1.3 × 104 44.8M 7.0 × 10−2 2.2 × 10−1 352 34.8
EdgeCluster − 1000 1.6 × 104 57.3M 4.5 × 10−2 1.3 × 10−1 619 44.5
EdgeCluster − 2000 2.2 × 104 70.1M 2.7 × 10−2 7.2 × 10−2 986 54.4
EdgeCluster − 5000 2.6 × 104 84.7M 1.3 × 10−2 2.9 × 10−2 1405 65.7
EdgeCluster − 10000 1.9 × 104 91.4M 7 × 10−3 1.5 × 10−2 1673 70.9
TABLE 10
Sparsity Comparison on YouTube Data with 1, 138, 499 Nodes
ModMax − 500 N/A 4.6G 1 — 500 500.00
EdgeCluster − 200 574.7 36.2M 9.9 × 10−3 2.3 × 10−2 121 1.99
EdgeCluster − 500 606.6 39.9M 4.4 × 10−3 9.7 × 10−3 255 2.19
EdgeCluster − 1000 779.2 42.3M 2.3 × 10−3 5.0 × 10−3 325 2.32
EdgeCluster − 2000 558.9 44.2M 1.2 × 10−3 2.6 × 10−3 375 2.43
EdgeCluster − 5000 554.9 45.6M 5.0 × 10−4 1.0 × 10−3 253 2.50
EdgeCluster − 10000 561.2 46.4M 2.5 × 10−4 5.1 × 10−4 356 2.54
EdgeCluster − 20000 507.5 47.0M 1.3 × 10−4 2.6 × 10−4 305 2.58
EdgeCluster − 50000 597.4 48.2M 5.2 × 10−5 1.1 × 10−4 297 2.62
SocioDim converts a network into attributes, other
behavioral features (if available) can be combined
with social dimensions for behavior learning.
6.2 Scalability Study
As we have introduced in Theorem 1, the social di-
mensions constructed according to edge-centric clus-
tering are guaranteed to be sparse because the density
is upper bounded by a small value. Here, we examine
how sparse the social dimensions are in practice.
We also study how the computation time (with a
Core2Duo E8400 CPU and 4GB memory) varies with
the number of edge clusters. The computation time,
the memory footprint of social dimensions, their den-
sity and other related statistics on all three data sets
are reported in Tables 8-10.
Concerning the time complexity, it is interesting
that computing the top eigenvectors of a modularity
matrix is actually quite efficient as long as there is
no memory concern. This is observed on the Flickr
data. However, when the network scales to millions of
nodes (YouTube), modularity maximization becomes
difficult (though an iterative method or distributed
computation can be used) due to its excessive mem-
ory requirement. On the contrary, the EdgeCluster
method can still work efficiently as shown in Table
10. The computation time of EdgeCluster for YouTube
is much smaller than for Flickr, because the YouTube
network is extremely sparse. The number of edges
and the average degree in YouTube are smaller than
those in Flickr, as shown in Table 4.
Another observation is that the computation time
of EdgeCluster does not change much with varying
numbers of clusters. No matter how many clusters
exist, the computation time of EdgeCluster is of the
same order. This is due to the efficacy of the proposed
k-means variant in Figure 5. In the algorithm, we
do not iterate over each cluster and each centroid
to do the cluster assignment, but exploit the sparsity
of edge-centric data to compute only the similarity

10
10
2
10
3
10
4
10
5
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
Number of Edge Clusters
Macro−F1
1%
5%
10%
Fig. 7. Sensitivity to Dimensionality
of a centroid and those relevant instances. This, in
effect, makes the computational cost independent of
the number of edge clusters.
As for the memory footprint reduction, sparse social
dimension does an excellent job. On Flickr, with only
500 dimensions, ModMax requires 322.1M, whereas
EdgeCluster requires less than 100M. This effect is
stronger on the mega-scale YouTube network, where
ModMax becomes impractical to compute directly. It
is expected that the social dimensions of ModMax
would occupy 4.6G memory. On the contrary, the
sparse social dimensions based on EdgeCluster re-
quires only 30-50M.
The steep reduction of memory footprint can be
explained by the density of the extracted dimensions.
For instance, in Table 10, when we have 50,000 dimen-
sions, the density is only 5.2 × 10−5
. Consequently,
even if the network has more than 1 million nodes,
the extracted social dimensions still occupy only a tiny
memory space. The upper bound of the density is not
tight when the number of clusters k is small. As k
increases, the bound becomes tight. In general, the
true density is roughly half of the estimated bound.
6.3 Sensitivity Study
Our proposed EdgeCluster model requires users to
specify the number of social dimensions (edge clus-
ters). One question which remains to be answered is
how sensitive the performance is with respect to the
parameter. We examine all three data sets, but find no
strong pattern to determine optimal dimensionality.
Due to the space limit, we include only one case here.
Figure 7 shows the Macro-F1 performance change on
YouTube data. The performance, unfortunately, is sen-
sitive to the number of edge clusters. It thus remains
a challenge to determine the parameter automatically.
However, a general trend across all three data sets
is observed. The optimal dimensionality increases as
the portion of labeled nodes expands. For instance,
when there is 1% of labeled nodes in the network, 500
dimensions seem optimal. But when the labeled nodes
increase to 10%, 2000-5000 dimensions becomes a
better choice. In other words, when label information
is scarce, coarse extraction of latent affiliations is better
for behavior prediction. But when the label informa-
tion multiplies, the affiliations should be zoomed to a
more granular level.
1 2 3 4 5 6 7 8 9 10
0.1
0.12
0.14
0.16
0.18
0.2
0.22
Percentage of Training set(%)
MacroF1−Measure
Normalization Effect on Flickr
None
Node Affiliation
Community Size
Both
1 2 3 4 5 6 7 8 9 10
0.22
0.24
0.26
0.28
0.3
0.32
0.34
Percentage of Training set(%)
MicroF1−Measure
Normalization Effect on Flickr
None
Node Affiliation
Community Size
Both
Fig. 8. Regularization Effect on Flickr
6.4 Regularization Effect
In this section, we study how EdgeCluster perfor-
mance varies with different regularization schemes.
Three different regularizations are implemented: reg-
ularization based on community size, node affiliation,
and both (we first apply community size regulariza-
tion, then node affiliation). The results without any
regularization are also included as a baseline for com-
parison. In our experiments, the community size reg-
ularization penalty function is set to (log(|Ci|))2
, and
node affiliation regularization normalizes the summa-
tion of each node’s community membership to 1. As
shown in Figure 8, regularization on both community
size and node affiliation consistently boosts the per-
formance on Flickr data. We also observe that node af-
filiation regularization significantly improves perfor-
mance. It seems like community-size regularization is
not as important as the node-affiliation regularization.
Similar trends are observed in the other two data sets.
We must point out that, when both community-size
regularization and node-affiliation are applied, it is
indeed quite similar to the tf-idf weighting scheme for
representation of documents [24]. Such a weighting
scheme actually can lead to a quite different perfor-
mance in dealing with social dimensions extracted
from a network.
6.5 Visualization of Extracted Social Dimensions
As shown in previous subsections, EdgeCluster
yields an outstanding performance. But are the ex-
tracted social dimensions really sensible? To under-
stand what the extracted social dimensions are, we
investigate tag clouds associated with different di-
mensions. Tag clouds, with font size denoting tags’
relative frequency, are widely used in social media
websites to summarize the most popular ongoing
topics. In particular, we aggregate tags of individuals
in one social dimension as the tags of that dimension.
However, it is impossible to showcase all the extracted
dimensions. Hence, we pick the dimension with the
maximum SVM weight given a category.
Due to the space limit, we show only two examples
from BlogCatalog. To make the figure legible, we
include only those tags whose frequency is greater
than 2. The dimension in Figure 9 is about cars, autos,
automobile, pontiac, ferrari, etc. It is highly relevant
to the category Autos. Similarly, the dimension in

11
Fig. 9. Dimension selected by Autos Fig. 10. Dimension selected by Sports
Figure 10 is about baseball, mlb, basketball, and football.
It is informative for classification of the category
Sports. There are a few less frequent tags, such as
movies and ipod, associated with selected dimensions
as well, suggesting the diverse interests of each per-
son. EdgeCluster, by analyzing connections in a net-
work, is able to extract social dimensions that are
meaningful for classification.
7 RELATED WORK
Classification with networked instances are known as
within-network classification [9], or a special case of rela-
tional learning [11]. The data instances in a network are
not independently identically distributed (i.i.d.) as in
conventional data mining. To capture the correlation
between labels of neighboring data objects, typically
a Markov dependency assumption is assumed. That
is, the label of one node depends on the labels (or
attributes) of its neighbors. Normally, a relational clas-
sifier is constructed based on the relational features of
labeled data, and then an iterative process is required
to determine the class labels for the unlabeled data.
The class label or the class membership is updated for
each node while the labels of its neighbors are fixed.
This process is repeated until the label inconsistency
between neighboring nodes is minimized. It is shown
that [9] a simple weighted vote relational neighbor-
hood classifier [25] works reasonably well on some
benchmark relational data and is recommended as a
baseline for comparison.
However, a network tends to present heterogeneous
relations, and the Markov assumption can only cap-
ture the local dependency. Hence, researchers propose
to model network connections or class labels based on
latent groups [20], [26]. A similar idea is also adopted
in [2] to differentiate heterogeneous relations in a
network by extracting social dimensions to represent
the potential affiliations of actors in a network. The
authors suggest using the community membership of
a soft clustering scheme as social dimensions. The
extracted social dimensions are treated as features,
and a support vector machine based on that can be
constructed for classification. It has been shown that
the proposed social dimension approach significantly
outperforms representative methods based on collec-
tive inference.
There are various approaches to conduct soft clus-
tering for a graph. Some are based on matrix factor-
ization, like spectral clustering [27] and modularity
maximization [3]. Probabilistic methods are also de-
veloped [28], [29]. Please refer to [30] for a compre-
hensive survey. A disadvantage with soft clustering is
that the resultant social dimensions are dense, posing
thorny computational challenges.
Another line of research closely related to the
method proposed in this work is finding overlapping
communities. Palla et al. propose a clique percola-
tion method to discover overlapping dense commu-
nities [31]. It consists of two steps: first find out
all the cliques of size k in a graph. Two k-cliques
are connected if they share k − 1 nodes. Based on
the connections between cliques, we can find the
connected components with respect to k-cliques. Each
component then corresponds to one community. Since
a node can be involved in multiple different k-cliques,
the resultant community structure allows one node
to be associated with multiple different communities.
A similar idea is presented in [32], in which the
authors propose to find out all the maximal cliques of
a network and then perform hierarchical clustering.
Gregory [33] extends the Newman-Girvan
method [34] to handle overlapping communities.
The original Newman-Girvan method recursively
removes edges with highest betweenness until a
network is separated into a pre-specified number of
disconnected components. It outputs non-overlapping
communities only. Therefore, Gregory proposes to
add one more action (node splitting) besides edge
removal. The algorithm recursively splits nodes that
are likely to reside in multiple communities into
two, or removes edges that seem to bridge two
different communities. This process is repeated until
the network is disconnected into the desired number
of communities. The aforementioned methods
enumerate all the possible cliques or shortest paths
within a network, whose computational cost is
daunting for large-scale networks.
Recently, a simple scheme proposed to detect over-
lapping communities is to construct a line graph
and then apply graph partition algorithms [16], [17].
However, the construction of the line graph alone, as
we discussed, is prohibitive for a network of a reason-
able size. In order to detect overlapping communities,
scalable approaches have to be developed.
In this work, the k-means clustering algorithm is
used to partition the edges of a network into disjoint

12
sets. We also propose a k-means variant to take advan-
tage of its special sparsity structure, which can handle
the clustering of millions of edges efficiently. More
complicated data structures such as kd-tree [35], [36]
can be exploited to accelerate the process. If a network
might be too huge to reside in memory, other k-
means variants can be considered to handle extremely
large data sets like online k-means [37], scalable k-
means [38], and distributed k-means [39].
8 CONCLUSIONS AND FUTURE WORK
It is well known that actors in a network demonstrate
correlated behaviors. In this work, we aim to predict
the outcome of collective behavior given a social net-
work and the behavioral information of some actors.
In particular, we explore scalable learning of collective
behavior when millions of actors are involved in the
network. Our approach follows a social-dimension-
based learning framework. Social dimensions are ex-
tracted to represent the potential affiliations of ac-
tors before discriminative learning occurs. As existing
approaches to extract social dimensions suffer from
scalability, it is imperative to address the scalability
issue. We propose an edge-centric clustering scheme
to extract social dimensions and a scalable k-means
variant to handle edge clustering. Essentially, each
edge is treated as one data instance, and the con-
nected nodes are the corresponding features. Then,
the proposed k-means clustering algorithm can be
applied to partition the edges into disjoint sets, with
each set representing one possible affiliation. With
this edge-centric view, we show that the extracted
social dimensions are guaranteed to be sparse. This
model, based on the sparse social dimensions, shows
comparable prediction performance with earlier social
dimension approaches. An incomparable advantage
of our model is that it easily scales to handle networks
with millions of actors while the earlier models fail.
This scalable approach offers a viable solution to
effective learning of online collective behavior on a
large scale.
In social media, multiple modes of actors can be
involved in the same network, resulting in a multi-
mode network [40]. For instance, in YouTube, users,
videos, tags, and comments are intertwined with each
other in co-existence. Extending the edge-centric clus-
tering scheme to address this object heterogeneity can
be a promising future direction. Since the proposed
EdgeCluster model is sensitive to the number of
social dimensions as shown in the experiment, further
research is needed to determine a suitable dimension-
ality automatically. It is also interesting to mine other
behavioral features (e.g., user activities and temporal-
spatial information) from social media, and integrate
them with social networking information to improve
prediction performance.
ACKNOWLEDGMENTS
This research is, in part, sponsored by the Air Force
Office of Scientific Research Grant FA95500810132.
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APPENDIX A
PROOF OF THEOREM 1
Proof: It is observed that the node degree in a
social network follows power law [14]. For simplicity,
we use a continuous power-law distribution to ap-
proximate the node degrees. Suppose
p(x) = Cx−α
, x ≥ 1
where x is a variable denoting the node degree and C
is a normalization constant. It is not difficult to verify
that
C = α − 1.
Hence,
p(x) = (α − 1)x−α
, x ≥ 1.
It follows that the probability that a node with degree
larger than k is
P(x > k) =
+∞
k
p(x)dx = k−α+1
.
Meanwhile, we have
k
1
xp(x)dx
= (α − 1)
k
1
x · x−α
dx
=
α − 1
−α + 2
x−α+2
|k
1
=
α − 1
α − 2
1 − k−α+2
Therefore,
density ≤
{i|di≤k} di + {i|di>k} k
nk
=
1
k
k
d=1
|{i|di = d}|
n
· d +
|{i|di ≥ k}|
n
≈
1
k
k
1
xp(x)dx + P(x > k)
=
1
k
α − 1
α − 2
1 − k−α+2
+ k−α+1
=
α − 1
α − 2
1
k
−
α − 1
α − 2
− 1 k−α+1
The proof is completed.
APPENDIX B
PROOF OF THEOREM 2
Proof: Let G and L(G) denote a graph and its
line graph respectively, di the degree of node i in G.
Suppose G has n nodes and m edges, L(G) has N
nodes and M edges. As each edge in G becomes a
node in L(G), it is straightforward that
N = m (8)
Because the connections of one node in G form a
clique in L(G), we have
M =
1
2 i
di(di − 1)
=
1
2 i
d2
i −
1
2 i
di
=
1
2n
(d2
1 + d2
2 + · · · + d2
n)(1 + 1 + · · · + 1
n
) − m
≥
1
2n
(d1 · 1 + d2 · 1 + · · · + dn · 1)2
− m (9)
=
1
2n
4m2
− m
= m
2m
n
− 1 (10)

14
The inequality (9) is derived following the Cauchy–
Schwarz inequality [41]. The equation is achieved if
and only if d1 = d2 = · · · = dn.
APPENDIX C
PROOF OF THEOREM 3
Proof:
E[2M/n] = E
1
n
n
i=1
d2
i −
n
i=1
di
= E[X2
] − E[X]
where X denotes the random variable (node degree).
For a power law distribution p(x) = (α − 1)x−α
, it
follows that [14],
E[X] =
α − 1
α − 2
, if α > 2;
E[X2
] =
α − 1
α − 3
, if α > 3.
Hence,
E[2M/n]
diverges if 2 < α <= 3
= α−1
(α−3)(α−2) if α > 3
(11)
The divergence of the expectation tells us that as we
go to larger and larger data sets, our estimate of
number of connections in the line graph, with respect
to the original network size, will increase without
bound.
APPENDIX D
PROOF OF THEOREM 4
Proof: Suppose a network with n nodes follows a
power law distribution as
p(x) = Cx−α
, x ≥ xmin > 0
where α is the exponent and C is a normalization
constant. Then the expected number of degree for
each node is [14]:
E[x] =
α − 1
α − 2
xmin
where xmin is the minimum nodal degree in a net-
work.
This can be verified via the properties of power law
distribution. In reality, we normally deal with nodes
with at least one connection, so xmin ≥ 1. Hence,
the expected number of connections in a network
following a power law distribution is
E[m] =
1
2
α − 1
α − 2
n.
Lei Tang is a scientist at Yahoo! Labs. He
received his Ph.D. in computer science and
engineering from Arizona State University,
and B.S. from Fudan University, China. His
research interests include social computing,
data mining and computational advertising.
His book on Community Detection and Min-
ing in Social Media is published by Morgan
and Claypool Publishers in 2010. He is a pro-
fessional member of AAAI, ACM and IEEE.
Xufei Wang is a PhD student in computer
science and engineering at Arizona State
University. He received his Masters from Ts-
inghua University, and Bachelor of science
from Zhejiang University, China. His research
interests are in social computing and data
mining. Specifically, he is interested in collec-
tive wisdom, familiar stranger, tag network,
crowdsourcing, and data crawling. He is an
IEEE student member.
Huan Liu is a professor of Computer Science
and Engineering at Arizona State University.
He received his Ph.D. from University of
Southern California and Bachelor of Engi-
neering from Shanghai Jiao Tong University.
He has worked at Telecom Research Labs in
Australia, and taught at National University of
Singapore before joining ASU in Year 2000.
Huan Liu has been recognized for excellence
in teaching and research in Computer Sci-
ence and Engineering at Arizona State Uni-
versity. His research interests are in data/web mining, machine learn-
ing, social computing, and artificial intelligence, investigating prob-
lems that arise in many real-world applications with high-dimensional
data of disparate forms such as social media, modeling group inter-
action, text categorization, bioinformatics, and text/web mining. His
professional memberships include AAAI, ACM, ASEE, SIAM, and
IEEE. He can be contacted via http://www.public.asu.edu/∼huanliu.

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