SEKE 2012 Proceedings - Knowledge Systems Institute

The rest of this paper is orga nized as follows. We
present the background of LIM model and sparse learning in
Section II. In Section III, we introduce our SLIM model for
hub nodes selection. An ef ficient and scalable optim ization
technique is given in Section IV. I n the final Section V, we
present the experimental results on real twitter data.
II.
BACKGROUND
A. Linear Influence Model (LIM)
We follow the notations in [7]. Assum e there are
nodes (corresponds to users) and contagions diffused
over the network (corresponds to topics). Assuming the
entire time intervals are norm alized into units:
. In the ori ginal paper of LIM, it uses an
indicator function to present whether the node got
infected by the contagion between the time interval
and . Therefore, the value for will be either one or
zero. In this paper, we relax this assum ption by defining
to be the num ber of ti mes that the node got
infected by contagion in . Furthermore, let
denote the total volum e of contagion between and .
Under the linear influence model:
Where is the maximum lag-length and is the nonnegative
influence factor of user at the time-lag and
is the i.i.d. zero-mean Gaussian noise. To m odel the
influence for each user, we need to obtain a ro bust estimator
of . Following LIM, we could organize ,
and in a m atrix form. In particular, we define the
volume vector
to be the concatenation of
where each
; the
user's influence vector to be the concatenation of
where each
. The matrix
, whose ele ments are , is or ganized
so that (1) can be written in a matrix form
Where is the vector of noise. For the details of
constructing , please refer to [7].
Based on (2), we can formulate the problem of predicting
by a non-negative least square problem:
B. Sparse Learning
We present the necessary background for sparse learning,
starting with the high- dimensional linear r egression model.
Let denote the input data m atrix and
denote the response vector. Under the linear regression
model,
Where is the regression coefficient to be e stimated
and the noise is distributed as , To select the
most predictive features, Lasso [10] provides a sparse
estimate of by solving the followin g optimization
problem:
Where is the -norm of which
encourages the solutions t o be sparse, and is the
regularization parameter that controls the sparsity-level (the
number of non-zero elements in ). For the sparsity-patter n
of , we could obtain a set of important features which
correspond to those non-zero elements in .
When features have a natural group structure, we could
use the group Lasso penalty [8] to shrink each group of
features to zero all-together instead of each individual
feature. In particular, let denote the set of groups a nd the
corresponding group Lasso problem can be formulated as:
Where is the sub vector of for features in gr oup ;
is the vector -norm. This group
Lasso penalty achieves the effect of jointly setting all of t he
coefficients within each group to zero or nonzero values.
III. SPARSE INFLUENCE LINEAR MODEL (SLIM)
Utilizing the group Lass o penalty introduced in the
previous section, we propose a new m odel, called SLIM
(sparse linear influence m odel), which can autom atically
select the hub nodes without the prior knowledge of the
network structure. We extend the LIM m odel by introducing
another group lasso penalty on the user influence vector. In
particular, we solve the following optimization problem:
Where is the vector -norm. From the estimated , we
obtain the pattern of the influence for each user.
2