CoRe: Exploiting the Personalized Influence of
Two-dimensional Geographic Coordinates for Location
Recommendations
Jia-Dong Zhanga , Chi-Yin Chowa,∗
a
Department of Computer Science, City University of Hong Kong, Hong Kong
Abstract
With the rapid growth of location-based social networks (LBSNs), location
recommendations play an important role in shaping the life of individuals.
Fortunately, a variety of community-contributed data, such as geographical information, social friendships and residence information, enable us to
mine users’ reality and infer their preferences on locations. In this paper, we
propose an effective and efficient location recommendation framework called
CoRe. CoRe achieves three key goals in this work. (1) We model a personalized check-in probability density over the two-dimensional geographic
coordinates for each user. (2) We propose an efficient approximation approach to predict the probability of a user visiting a new location using her
personalized check-in probability density. (3) We develop a new method to
measure the similarity between users based on their social friendship and residence information, and then devise a fusion rule to integrate the geographical
influence with the social influence so as to improve the user preference mod∗
Corresponding author. Tel.: +852 3442-8679; Fax:+852 3442-0503.
Email addresses: jzhang26-c@my.cityu.edu.hk (Jia-Dong Zhang),
chiychow@cityu.edu.hk (Chi-Yin Chow)
Preprint submitted to Information Sciences
September 6, 2014
el on location recommendations. Finally, we conduct extensive experiments
to evaluate the recommendation accuracy, recommendation efficiency and
approximation error of CoRe using two large-scale real data sets collected
from two popular LBSNs: Foursquare and Gowalla. Experimental results show that CoRe achieves significantly superior performance compared to
other state-of-the-art geo-social recommendation techniques.
Keywords: Location-based social networks, Location recommendations,
Personalization, Geographical influence, Social influence
1. Introduction
In a location-based social network (LBSN) depicted in Figure 1, such as
Foursquare and Gowalla, users can establish social links with others, indicate
their residence, check in some points-of-interest (POIs), e.g., restaurants,
stores, and museums, and share their experiences of visiting specific locations
by leaving some tips or comments [12]. These community-contributed data
enable us to mine users’ reality [40, 42, 46]. For example, in LBSNs it is
prevalent to recommend locations for users based on their preferences inferred
from the community-contributed data [3, 44], which not only helps users
explore new places and enrich their life but also enables companies to launch
advertisements to potential customers and improve their profits.
The geographical feature of POIs distinguishes them from other nonspatial items, such as books, music and movies in conventional recommender
systems [5], because physical interactions are required for users to visit POIs.
Thus, the geographical information (geographic coordinates) of locations plays a significant influence on users’ check-in behaviors,
2
Social link
User
Check-in or Visit
POI or Location
<latitude, longitude>
Residence
Figure 1: A location-based social network [62]
known as geographical influence for short, which gives new opportunities
and challenges to infer users’ preferences for making location recommendations for them.
Limitations in existing methods. There are a few studies that learn
users’ preferences through exploiting the geographical influence for location
recommendations [9, 29, 36, 56]. Unfortunately, these studies use the onedimensional geographical distance between locations for location recommendations. For example, the literatures [29, 36] simply assume that the propensity of a user for a POI is inversely proportional to the distance between
the POI and her visited locations; more sophistically, two techniques [9, 56]
model the distance between locations visited by the same user as a power-law
distribution or multi-center Gaussian distribution. In general, these studies
suffer from two major limitations. (1) One-dimensional geographic distance influence. Existing work simplifies the geographical influence as a
one-dimensional and monotonous distance distribution, but it is more reasonable and intuitive to model the geographical influence as a two-dimensional
and multimodal check-in probability density. The reason is twofold. (a) The
probability of a user visiting a location is not simply monotonous respect-
3
(a) Foursquare
(b) Gowalla
Figure 2: Personal check-in probability densities
ing their distance, because the visiting probability is not only affected by
the distance but also the location’s intrinsic characteristics. For example,
in reality the check-in locations of a user are usually distributed in several
areas. (b) It is hard to compute a visiting probability for a location based
on a distance distribution, since it needs to find a reference location to derive a reasonable distance for the location in the first place. Conversely, it
is considerably intuitive to employ a two-dimensional check-in probability
density to compute a visiting probability for any location with latitude and
4
longitude. In addition, it is rational to use the two-dimensional density on
the spherical dimensions because the altitude is a dependent variable (i.e.,
the altitude is usually determined based on a pair of latitude and longitude)
and the earth is almost a sphere (i.e., the altitude of a location is negligible).
(2) Non-personalized geographical influence. They apply the same inversely proportional relation between the propensity and the distance for all
users [29, 36] or universally estimate a common distance distribution for all
users [9, 56], but in practice the geographical influence of locations should
be personalized since the user’s check-in behavior is unique [62].
Real-world motivating examples. To observe users’ unique checkin behaviors and multimodal check-in probability densities over the twodimensional geographic coordinates, a spatial analysis is conducted on two
publicly available real data sets collected from Foursquare [15] and Gowalla [11], which are two popular LBSNs. Specifically, we focus on three users
who are randomly chosen from each data set. Figure 2 depicts their individual check-in probability density over the two-dimensional geographic
coordinates. The probability density is estimated based on the kernel density estimation (KDE) technique [49]. We have the following two findings.
(1) The geographical influence of locations on these three users’ check-in behaviors is unique since their check-in probability densities are distinct from
each other. (2) These check-in probability densities are usually multimodal
rather than unimodal or monotonous.
Our approach. In this paper, we propose a new location recommendation framework, called CoRe, that achieves three key goals. (1) We infer
users’ preferences by exploiting the personalized two-dimensional geographi-
5
cal influence. Specifically, we estimate a personalized two-dimensional checkin probability density over the latitude and longitude coordinates for each
user rather than using a common one-dimensional distance distribution for all
users. In addition, since the check-in probability densities of users are diverse,
we cannot assume their forms. We thus use a nonparametric density estimation method, i.e., the popular kernel density estimation [49]. (2) We can use a
personalized check-in probability density to predict the probabilities of each
user visiting each of all new locations (that have not been visited by the user). In particular, we contrive an efficient approximation approach to derive
these visiting probabilities based on the fast Gauss transform [20], in order
to improve system efficiency and scalability. (3) We integrate the geographical influence with the social influence in the conventional social networks
to enhance the user preference model and obtain better quality of location
recommendations. First of all, to consider the social influence, we develop
a new method to measure the similarity between users based on their social
friendships and residence information, because nearby friends share more
commonly visited locations than others [11]; the similarity measure between
users is employed to estimate the rating of a user to a new location. We then
design a new fusion rule, called the union rule, to integrate the geographical
probability and social rating of a user to a new location into a unified score,
instead of applying the simple product or sum rule. Eventually, we can make
a location recommendation for each user by top-k locations with the highest
scores to her.
This study is significant different from our previous work [62]. The main
differences and contributions of this paper can be summarized as follows:
6
• Although both studies utilize the same kernel density estimation technique to model a geographical check-in probability density for each
user, this study exploits the two-dimensional geographic coordinates
of users’ check-in locations whereas the previous work [62] only considers the distance between users’ check-in locations. Actually, this is
the first study to learn users’ preferences for location recommendations
by utilizing the personalized two-dimensional geographical influence.
Moreover, to improve the computational complexity of location recommendations, this study proposes an efficient approximation approach
to predict the probability of a user visiting any new location rather
than using the exact computation method as in the previous work [62].
(Section 2)
• Both studies employ a fusion rule to integrate the geographical influence
with the social influence in order to improve the quality of location
recommendations. However, this study develops a more sophisticated
fusion rule whereas the previous work [62] applies the simple product or
sum rule. In addition, this work devises a new method to measure the
similarity between users based on their social friendships and residence
information. (Section 3)
• We conduct extensive experiments to evaluate not only the recommendation accuracy of CoRe as in the previous work [62], but also its recommendation efficiency and approximation error using two large-scale real
data sets collected from Foursquare and Gowalla. Experimental results
show two key facts. (1) CoRe outperforms the state-of-the-art geo-
7
social recommendation techniques including the multi-center Gaussian
model (MGM) [9], power-law distribution (PD) [56] and iGSLR [62] in
terms of recommendation accuracy and efficiency. (2) CoRe is much
more efficient than the exact method, and it only leads to small approximation errors. (Sections 4 and 5)
The remainder of this paper is organized as follows. We firstly model the
personalized two-dimensional geographical influence for each user and propose an efficient approach to predict the probabilities of the user visiting new
locations in Section 2. We then integrate geographical influence with social
influence through a new user similarity measuring method and a new fusion
rule in Section 3. In Sections 4 and 5, we present our experiment settings
and analyze the performance of CoRe, respectively. Section 6 highlights related work on location recommendations. Finally, we conclude this paper in
Section 7.
2. Modeling Personalized Two-dimensional Geographical Influence
In this section, we propose a kernel density estimation (KDE) approach to
model the personalized two-dimensional geographical influence in Section 2.1
and develop an efficient approximation approach to derive users’ preferences
in Section 2.2. Table 1 summarizes the key notations used in this paper.
2.1. A KDE-Based Approach to Model the Personalized Two-dimensional
Geographical Influence
The experimental results in Section 1 inspired us to learn the users’ preferences by utilizing the personalized two-dimensional geographical influence.
8
Table 1: Key notations in this paper
Notation
Meaning
U
Set of all users in an LBSN
u
Some user and u ∈ U
L
Set of all locations (or POIs) in an LBSN
l
Some location with the latitude and longitude coordinates and
l = (lat, lon)T ∈ L
Lu
Set of locations visited by user u (i.e., u’s check-in locations) and
Lu = {l1 , l2 , . . . , ln } ⊂ L
p(l|Lu )
Predicted probability of u visiting l given Lu
Ω
A set of cells that constitute a partition of the geographical space
B
Some cell and B ∈ Ω
lB
Center of B
Fu
Set of users having social links with u and Fu ⊂ U
ru′ ,l
Actual rating of user u′ to visited location l
r̂u,l
Estimated rating of user u to unvisited location l
ŝu,l
Unified score by fusing p(l|Lu ) and r̂u,l
To this end, we estimate a personalized two-dimensional check-in probability density for each user. Specifically, we apply a general nonparametric
probability density estimation technique, known as the kernel density estimation [49] (KDE). Opposed to parametric estimation methods, a nonpara-
9
metric estimation technique does not assume a fixed distribution form in
advance, but instead learn the distribution form from data. Thus, KDE can
be used with arbitrary distributions because it does not have any assumption
on the form of the underlying distribution.
Personalized two-dimensional check-in probability density estimation. Let Lu = {l1 , l2 , . . . , ln } be the set of locations visited by the user
u that are drawn from some distribution with an unknown density f . Its
kernel density estimator fˆ using Lu is given by:
)
(
n
∑
1
x
−
l
i
fˆ(x) =
K
,
nσ 2 i=1
σ
(1)
where each location li = (lati , loni )T is a two-dimensional column vector with
the latitude (lati ) and longitude (loni ), K(·) is the kernel function and σ is
a smoothing parameter, called the bandwidth. In this paper we apply the
widely used standard two-dimensional normal kernel [49]:
K(x) =
1
1
exp(− xT x),
2π
2
and the optimal bandwidth [49]:
σ=n
√
− 16
1 T
σ̂ σ̂,
2
(2)
(3)
where σ̂ is the marginal standard deviation vector of Lu .
Thus, given u’s set of visited locations Lu = {l1 , l2 , . . . , ln }, the probability of user u visiting a new location l can be computed by:
n
1
1 ∑
exp(− 2 (l − li )T (l − li )).
p(l|Lu ) =
2
2πnσ i=1
2σ
(4)
Computational complexity. Algorithm 1 outlines the process for computing p(l|Lu ) through Equation (4). Algorithm 1 calculates a probability
10
Algorithm 1 The exact computation of p(l|Lu )
Input: u’s set of visited locations Lu = {l1 , l2 , . . . , ln }.
Output: p(l|Lu ) for each location l ∈ L − Lu .
1: Compute the bandwidth σ using Equation (3)
2: for each unvisited location l ∈ L − Lu do
3:
z ← 0 // Initializing auxiliary variable z
4:
for each li ∈ Lu do
5:
z ← z + exp(−(l − li )T (l − li )/(2σ 2 ))
6:
end for
7:
p(l|Lu ) ← z/(2πnσ 2 )
8: end for
for each unvisited location l ∈ L − Lu in order to make location recommendations for u by returning the top-k locations with the highest probability.
It is required to evaluate the sum of n Gaussian exp(−(l − li )T (l − li )/(2σ 2 ))
at each target point l ∈ L − Lu , the computational complexity of which is
O(|L − Lu |n) = O(|L|n) in which |L| ≫ |Lu | = n since users only checked in
a small fraction of locations.
2.2. An Efficient Approximation Approach to Infer Users’ Preferences
In Algorithm 1 with the computational complexity O(|L|n), when users
have checked in many POIs (i.e., with a large value of n), the calculation on
recommending top-k POIs to these users is prohibitively expensive. In this
section, we approximately compute p(l|Lu ) through the fast Gauss transform [20] to reduce its complexity to be independent of n, i.e., O(|L|). The
fast Gauss transform is an optimized variant of the fast multipole method
11
for Gaussian distributions. As for another kernel functions, we can employ
the more general fast multipole method [19].
Fast Gauss transform. The fast Gauss transform shifts a Gaussian
exp(−(l − li )T (l − li )/(2σ 2 )) centered at li = (lati , loni )T to a sum of Hermite
polynomials centered at l0 = (lat0 , lon0 )T by the Hermite expansion, given
by:
(
)q (
)
c−1
∑
l − l0
1
1
li − l0
T
√
exp(− 2 (l − li ) (l − li )) =
hq √
+ ε(c), (5)
2σ
(q!)2
2σ
2σ
q=0
where
(
)q (
)q (
)q
li − l0
lati − lat0
loni − lon0
√
√
√
=
,
2σ
2σ
2σ
)
) (
)
(
(
l − l0
lati − lat0
loni − lon0
√
√
hq √
= hq
hq
,
2σ
2σ
2σ
(6)
(7)
the Hermite functions hq (x) are defined by:
hq (x) = (−1)q
dq −x2
e ,
dxq
(8)
and ε is the error due to truncating the infinite series after c terms. When
li = (lati , loni )T closes to l0 = (lat0 , lon0 )T , a small value of c is enough
to guarantee that the error ε is negligible as these terms converge to zero
quickly.
Efficient approximation algorithm. At first, to ensure li being close
to l0 , the geographical space is subdivided into a set of cells Ω using a uniform
grid, and each cell B ∈ Ω with center lB and size σ, i.e., the optimal
bandwidth in Equation (3), as depicted in Figure 3. Accordingly, each
location li ∈ Lu is shifted into the center lB of the cell B in which li lies.
12
6σ
5σ
4σ
3σ
2σ
σ
0
σ
2σ
3σ
4σ
5σ
6σ
7σ
8σ
9σ
10σ
11σ 12σ
Figure 3: An example of the uniform grid (cell size equals bandwidth in Equation (3))
Thus, the approximation of p(l|Lu ) in Equation (4) is given by:
n
1 ∑
1
exp(−
(l − li )T (l − li ))
2πnσ 2 i=1
2σ 2
(
)q (
)
c−1
li − lB
1 ∑∑∑ 1
l − lB
√
≈
hq √
2πnσ 2 B∈Ω l ∈B q=0 (q!)2
2σ
2σ
i
(
)q (
)
c−1
l − lB
1 ∑ ∑ 1 ∑ li − lB
√
hq √
=
2πnσ 2 B∈Ω q=0 (q!)2 l ∈B
2σ
2σ
i
)
(
c−1
1 ∑∑
l − lB
=
Aq (B)hq √
2πnσ 2 B∈Ω q=0
2σ
p(l|Lu ) =
together with
(
)q
1 ∑ li − lB
√
Aq (B) =
.
(q!)2 l ∈B
2σ
(9)
(10)
i
Computational complexity. Algorithm 2 summarizes the overall process for approximating p(l|Lu ) through Equation (9). Aq (B) is independent
of any target point l ∈ L − Lu and only required to be computed once for
13
Algorithm 2 The efficient approximation of p(l|Lu )
Input: u’s set of visited locations Lu = {l1 , l2 , . . . , ln } and a constant c.
Output: p(l|Lu ) for each location l ∈ L − Lu .
1: Compute the bandwidth σ using Equation (3)
2: Subdivide the geographical space into a uniform grid
3: Compute common items Aq (B) for each combination of q and cell B ∈ Ω
in terms of Equation (10)
4: for each unvisited location l ∈ L − Lu do
5:
z ← 0 // Initializing auxiliary variable z
6:
for each cell B ∈ Ω that contains l or is immediately adjacent to l do
7:
8:
9:
for q = 0 to c − 1 do
√
z ← z + Aq (B)hq ((l − lB )/( 2σ))
end for
10:
end for
11:
p(l|Lu ) ← z/(2πnσ 2 )
12: end for
each li ∈ Lu in advance, which avoids to separately calculate the Gaussian
exp(−(l − li )T (l − li )/(2σ 2 )) for each combination of l ∈ L − Lu and li ∈ Lu
in Algorithm 1. The pre-computation step requires O(n) work. The approximation step computes p(l|Lu ) using Aq (B) (Lines 4 to 12) and it needs
O(c|L − Lu ||Ω|) = O(|L − Lu |) work, in which c and |Ω| are a small constant
relative to |L − Lu |. Therefore, the computational complexity of Algorithm 2
is O(n) + O(|L − Lu |) = O(|L|). In addition, to accelerate the evaluation
of Equation (9), we also can cut off the sum over the set of all cells Ω by
only considering the nearest cells of l (Line 6), since the cells far away from
14
l contribute a little to p(l|Lu ). For example, consider Figure 3 again. Given
l lying in the red cell, we may just consider the red cell and the nearest blue
cells around it when estimating p(l|Lu ) based on Equation (9).
3. Integrating Geographical and Social Influences
In this section, we integrate the geographical influence with the social
influence in the conventional social networks in order to gain better user
preference model and quality of location recommendations.
3.1. Using Social Influence
To use the social influence, we design a new method to measure the similarity between users based on their social friendships and residence distance,
because nearby friends share more commonly visited locations than others [11]. Specifically, we transform the residence distance of users with social
friendships into a normalized similarity through a sigmoid function that is
widely used to map an output into a probability. Formally, let F (u) be a
set of users having social friendships with u and distance(u, u′ ) be the geographical distance between the residences of users u and u′ . If u′ ∈ F (u),
the similarity between users u and u′ is defined by the sigmoid function as
follows:
sim(u, u′ ) =
2
.
1 + exp(distance(u, u′ ))
(11)
Otherwise, sim(u, u′ ) = 0. Obviously, we have sim(u, u′ ) ∈ [0, 1].
Based on the similarity, we can predict the rating r̂u,l of a user u to a new
location l via the conventional user-based collaborative filtering technique:
∑
sim(u, u′ ) · ru′ ,l
′
r̂u,l = u∑∈U
,
(12)
′
u′ ∈U sim(u, u )
15
where ru′ ,l denotes the frequency of user u′ visiting location l.
3.2. The Union Fusion Rule
Here we consider the integration of the geographical influence with the
social influence to enhance user preference model and improve the quality of
location recommendations. An easy way is to apply the simple product or
sum rule to combine the visiting probability in Equation (9) and rating in
Equation (12) of a user regarding a location into a unified score. However,
the unified score from the product rule is dominated by a smaller value of
either the visiting probability p(l|Lu ) or the estimated rating r̂u,l . As an
example, supposing p(l|Lu ) = 0.9 and r̂u,l = 0.1, the unified score given by
the product rule is 0.09 that is nearly independent of p(l|Lu ) with a large
value but strongly dominated by r̂u,l with a small value. On the other hand,
it is considerably hard to determine the relative weights of p(l|Lu ) and r̂u,l
for the sum rule to achieve the best performance. Although the weights can
be learned from data, they cost much effort to find out their optimal settings
and usually suffer from over-fitting the data.
To this end, we design a new fusion rule, called the union rule, instead of
applying the simple product and sum rules. At first, the union rule values
the geographical influence and social influence equally important under the
condition of the lack of the prior knowledge about their effect. That is,
the predicted probability in Equation (9) and rating in Equation (12) are
transformed into a normalized probability with the same scale using
p̂(l|Lu ) =
p(l|Lu )
,
maxl∈(L−Lu ) p(l|Lu )
16
(13)
and
p̂u,l =
r̂u,l
maxl∈(L−Lu ) r̂u,l
,
(14)
where maxl∈(L−Lu ) p(l|Lu ) and maxl∈(L−Lu ) r̂u,l are the normalization constants for a given user u. Further, the union rule views the difference between
the probability and rating given by Equations (13) and (14) as an uncertain
indicator and utilizes it to penalize the average of them in order to get a unified score. The underlying reason is that the difference between them usually
implies the confidence on the average of them; the larger the difference is,
the lower the confidence will be. Concretely, the union rule estimates the
mean ŝµu,l and standard deviation ŝςu,l by:
ŝµu,l =
p̂(l|Lu ) + p̂u,l
2
(15)
together with
ŝςu,l =
√
(p̂(l|Lu ) − ŝµu,l )2 + (p̂u,l − ŝµu,l )2
2
.
(16)
Then it penalizes the mean ŝµu,l with the standard deviation ŝςu,l to obtain the
final score ŝu,l of user u visiting location l, given by
ŝu,l = ŝµu,l − ŝςu,l .
(17)
It is important to note that: (1) The union rule in Equation (17) addresses
the limitations of the simple product and sum rules, since in the union rule
neither the probability nor the rating could dominate the final score and there
is no need to derive their relative weights. (2) If the difference between the
probability and rating determined by Equations (13) and (14), respectively,
is large, the final score given by the union rule will be a little far from their
17
Algorithm 3 The fusion of the geographical and social influences
Input: u’s set of visited locations Lu = {l1 , l2 , . . . , ln }.
Output: ŝu,l for each location l ∈ L − Lu .
1: for each unvisited location l ∈ L − Lu do
2:
Compute the visiting probability of u to l by Algorithm 2
3:
Compute the similarity of u with social friends by Equation (11)
4:
Compute the rating of u to l by Equation (12)
5:
Normalize the probability and rating by Equations (13) and (14)
6:
Estimate the mean and standard deviation of the normalized probability and rating by Equations (15) and (16)
7:
Compute the final score of u to l by Equation (17)
8: end for
average due to the penalty. (3) It is easy to show that the final score ŝu,l lies
in [0, 1]. We summarize the fusion process of the geographical influence with
the social influence in Algorithm 3.
4. Experimental Evaluation
In this section, we describe our experiment settings for evaluating the
performance of CoRe against the state-of-the-art geo-social recommendation
techniques. Specifically, our evaluation focuses on three aspects of CoRe:
(1) its recommendation accuracy (i.e., precision and recall); (2) its approximation error in comparison to the exact method; and (3) its recommendation
efficiency.
18
Table 2: Statistics of the two data sets
Foursquare Gowalla
Number of users
11,326
196,591
Number of locations (POIs)
182,968
1,280,969
Number of check-ins
1,385,223
6,442,890
Number of social links
47,164
950,327
User-location matrix density
2.3 × 10−4
2.9 × 10−5
Avg. No. of visited POIs per user
42.44
37.18
Avg. No. of check-ins per location 2.63
3.11
4.1. Data Sets
We use two publicly available large-scale real check-in data sets1 that were
crawled from Foursquare [15] and Gowalla [11]. The statistics of the data
sets are shown in Table 2. Figure 4 depicts the distribution of the locations
in the data sets on a world map.
4.2. Evaluated Recommendation Techniques
The recommendation techniques implemented in our experiments are classified into three categories based on the information they use in the location
recommendations.
1
The check-in data sets used for our experiments can be downloaded from http://www.
public.asu.edu/~hgao16/Publications.html and http://snap.stanford.edu/data/
loc-gowalla.html.
19
90
60
Latitude: °
30
0
−30
−60
−90
−180 −150 −120 −90
−60
−30
0
30
Longitude: °
60
90
120
150
180
(a) Foursquare
90
Latitude: °
60
30
0
−30
−60
−90
−180 −150 −120 −90
−60
−30
0
30
Longitude: °
60
90
120
150
180
(b) Gowalla
Figure 4: Distribution of locations on a world map
(1) Geographical category:
• MGM: This technique models the distance between visited locations
and centers (i.e., the most popular locations) as a multi-center Gaussian
20
distribution [9].
• PD: This technique models the distance between every pair of locations
visited by the same user as a power-law distribution [56].
• iGSLR: This technique models the distance between every pair of locations visited by the same user as a nonparametric distribution [62].
• CoRe: Our technique models a personalized two-dimensional check-in
probability density over the geographic coordinates for each user using
Algorithm 2.
• Exact: The only difference between Exact and CoRe is that Exact computes the exact probability of a user visiting a new location using Algorithm 1, while CoRe uses the approximation method to compute the
visiting probability.
(2) Social category:
• SCF: This technique measures the similarity between users by using
their social friendships and then employs the social collaborative filtering technique [5].
• CoRe: Our technique measures the similarity between users based on
their social friendships and residence distance in Equation (11) and also
utilizes the collaborative filtering technique in Equation (12).
(3) Fusion category:
• Prod: This fusion technique applies the product rule to integrate the
geographical and social influences.
21
• Sum: This technique uses the sum rule to integrate the geographical
and social influences. It is worth mentioning that before applying the
sum rule, the probability and rating have been normalized using Equations (13) and (14), respectively, which is different from our previous
work [62] and improves the performance of the sum rule.
• CoRe: Our technique applies the union rule to integrate the geographical and social influences based on Algorithm 3.
Note that we implement a different version of CoRe for each
category using the corresponding influence to achieve fair comparisons in Section 5.
4.3. Performance Metrics
Recommendation accuracy. In general, recommendation techniques
compute a score for each candidate item (i.e., a location or POI in this paper)
regarding a target user and return locations with the top-k highest scores
as a recommendation result to the target user. To evaluate the quality of
location recommendations, it is important to find out how many recommended locations are actually visited by the target user in the testing data set.
Also, it is important to know how many of the locations actually visited were
recommended by the evaluated technique. The former aspect is captured by
precision and the latter by recall. Precision and recall are standard metrics
used to evaluate the recommendation accuracy [9, 56]. We define a discovered
location as a location that is both recommended and actually visited by the
target user. Formally,
22
• Precision defines the ratio of the number of discovered locations to the
total number k of recommended locations, i.e.,
precision =
the number of discovered locations
.
k
(18)
• Recall defines the ratio of the number of discovered locations to the
total number of actually visited locations by the target user in the
testing set, i.e.,
recall =
the number of discovered locations
.
the number of visited locations
(19)
Approximation error. We evaluate the approximation error of CoRe
by comparing its recommendation accuracy with that of Exact (i.e., Algorithm 1). Note that we are more concerned about the effect of approximation
error on the recommendation accuracy than the error itself.
Recommendation efficiency. We compare the running time of CoRe
and Exact with respect to various numbers of check-in locations of users. All
algorithms were implemented in Matlab and run on a machine with 3.4GHz
Intel Core i7 Processor and 16GB RAM.
4.4. Experiment Settings
We split each data set into the training set and the testing set in terms
of the check-in time rather than using a random partition method, because
in practice we can only utilize the past check-in data to predict the future
check-in events. A half of check-in data with earlier timestamps are used as
the training set and the other half of check-in data are used as the testing
set. In CoRe, unless otherwise specified, in Algorithm 2 the cell size is set to
23
the optimal bandwidth in Equation (3) and the number of truncated terms
in Equation (5) is set to a small value: c = 6.
5. Experimental Results
This section analyzes our extensive experimental results. We first compare our CoRe against the state-of-the-art geo-social recommendation techniques in terms of the recommendation accuracy (Section 5.1). We then
study the approximation error of CoRe (Section 5.2). Finally, we evaluate
the recommendation efficiency of CoRe (Section 5.3).
5.1. Recommendation Accuracy
Here we separately investigate the performance of three new techniques
proposed in this paper, including the efficient approximation approach to
exploit the personalized two-dimensional geographical influence in Algorithm 2 (Section 5.1.1), the method to measure the similarity of users in Equation (11) (Section 5.1.2), and the union rule to integrate the geographical
and social influences in Algorithm 3 (Section 5.1.3). Moreover, we discuss
the effect of the number k of recommended locations for users (Section 5.1.4),
the number n of check-in locations of users (Section 5.1.5), and the cell size
on the recommendation accuracy (Section 5.1.6).
5.1.1. Geographical Recommendation Techniques
Figures 5 and 6 compare the recommendation accuracy of CoRe (i.e., Algorithm 2), iGSLR [62], PD [56] and MGM [9] with the effect of the number
of recommended locations for users (top-k) and the number of check-in locations of users (given-n) using two large-scale real data sets collected from
24
0.3
0.3
MGM
PD
iGSLR
CoRe
0.25
0.25
0.2
Recall
Precision
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
5 10 15 20 25 30 35 40 45 50
top−k
(a) Foursquare (Precision)
5 10 15 20 25 30 35 40 45 50
top−k
(b) Foursquare (Recall)
0.3
0.3
MGM
PD
iGSLR
CoRe
0.25
0.25
0.15
0.15
0.1
0.1
0.05
0.05
0
MGM
PD
iGSLR
CoRe
0.2
Recall
0.2
Precision
MGM
PD
iGSLR
CoRe
5 10 15 20 25 30 35 40 45 50
top−k
(c) Gowalla (Precision)
0
5 10 15 20 25 30 35 40 45 50
top−k
(d) Gowalla (Recall)
Figure 5: Comparison of geo-techniques on the number of recommended locations k (CoRe
is Algorithm 2)
25
0.3
0.3
MGM
PD
iGSLR
CoRe
0.25
0.25
0.2
Recall
Precision
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
10 20 30 40 50 60 70 80 90 100
given−n
0
10 20 30 40 50 60 70 80 90 100
given−n
(a) Foursquare (Precision)
(b) Foursquare (Recall)
0.3
0.3
MGM
PD
iGSLR
CoRe
0.25
0.25
MGM
PD
iGSLR
CoRe
0.2
Recall
0.2
Precision
MGM
PD
iGSLR
CoRe
0.15
0.15
0.1
0.1
0.05
0.05
0
10 20 30 40 50 60 70 80 90 100
given−n
(c) Gowalla (Precision)
0
10 20 30 40 50 60 70 80 90 100
given−n
(d) Gowalla (Recall)
Figure 6: Comparison of geo-techniques on the number of check-in locations n (CoRe is
Algorithm 2)
26
Foursquare and Gowalla, respectively. Note that k is set to a smaller number
than n because a large number of recommended locations may not be helpful
for users. As a whole, our proposed method CoRe always exhibits the
best precision and recall for all the values of k and n. The details are
demonstrated as follows.
MGM models the distance between a location and a center as a universal multi-center Gaussian distribution for all users. Actually, its estimated
probability of p(l|Lu ) is independent of user u or Lu , that is, it recommends
the same set of locations for all users. As a result, it performs the worst.
PD models the distance between every pair of locations visited by the
same user as a power-law distribution for all users. Although it enhances
the performance of recommending locations in comparison to MGM, it still
inherits the limitation of the universal distance distribution for all users.
iGSLR estimates an individual distance distribution for each user and
then improves the recommendation accuracy compared to PD, but the improvement is very limited since it still simplifies the geographical influence
as one-dimensional distance distributions just as in MGM and PD.
CoRe estimates a personalized two-dimensional check-in probability density over the geographic coordinates for each user and always shows the best
recommendation quality in terms of precision and recall. These results verify
the superiority of exploiting the personalized two-dimensional geographical
influence for location recommendations.
5.1.2. Social Recommendation Techniques
Figures 7 and 8 contrast the recommendation accuracy of CoRe (i.e.,
Equation (12)) with that of SCF [5] on the Foursquare and Gowalla data
27
0.3
0.3
SCF
CoRe
0.25
0.2
0.2
Recall
Precision
0.25
0.15
0.15
0.1
0.1
0.05
0.05
0
0
5 10 15 20 25 30 35 40 45 50
top−k
(a) Foursquare (Precision)
5 10 15 20 25 30 35 40 45 50
top−k
(b) Foursquare (Recall)
0.3
0.3
SCF
CoRe
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
SCF
CoRe
0.25
Recall
Precision
SCF
CoRe
5 10 15 20 25 30 35 40 45 50
top−k
(c) Gowalla (Precision)
0
5 10 15 20 25 30 35 40 45 50
top−k
(d) Gowalla (Recall)
Figure 7: Comparison of social techniques on the number of recommended locations k
(CoRe is Equation (12))
sets. By transforming the residence distance and social friendships of users
into a normalized similarity through a sigmoid function in Equation (11),
CoRe is able to reflect the fact that nearby friends share more commonly
visited locations than others. As a result, CoRe peforms better than SCF in
terms of both precision and recall.
On the other hand, both CoRe and SCF suffer from the effect of data
28
0.3
0.3
SCF
CoRe
0.25
0.2
0.2
Recall
Precision
0.25
0.15
0.15
0.1
0.1
0.05
0.05
0
10 20 30 40 50 60 70 80 90 100
given−n
0
10 20 30 40 50 60 70 80 90 100
given−n
(a) Foursquare (Precision)
(b) Foursquare (Recall)
0.3
0.3
SCF
CoRe
0.25
SCF
CoRe
0.25
0.2
0.2
Recall
Precision
SCF
CoRe
0.15
0.15
0.1
0.1
0.05
0.05
0
10 20 30 40 50 60 70 80 90 100
given−n
(c) Gowalla (Precision)
0
10 20 30 40 50 60 70 80 90 100
given−n
(d) Gowalla (Recall)
Figure 8: Comparison of social techniques on the number of check-in locations n (CoRe is
Equation (12))
sparsity. For instance, their precision and recall in the Gowalla data set are
obviously lower than those in the Foursquare data set, because the density
of the Gowalla data set is one order-of-magnitude lower than that of the
Foursquare data set as shown in Table 2.
29
0.5
0.5
Prod
Sum
CoRe
0.4
Recall
Precision
0.4
0.3
0.2
0.1
Prod
Sum
CoRe
0.3
0.2
5
0.1
10 15 20 25 30 35 40 45 50
top−k
5
(a) Foursquare (Precision)
(b) Foursquare (Recall)
0.5
0.5
Prod
Sum
CoRe
Prod
Sum
CoRe
0.4
Recall
Precision
0.4
0.3
0.2
0.1
10 15 20 25 30 35 40 45 50
top−k
0.3
0.2
5
10 15 20 25 30 35 40 45 50
top−k
(c) Gowalla (Precision)
0.1
5
10 15 20 25 30 35 40 45 50
top−k
(d) Gowalla (Recall)
Figure 9: Comparison of fusion rules on the number of recommended locations k (CoRe is
Algorithm 3)
5.1.3. Fusion Rules
Figures 9 and 10 depict the performance of three fusion rules for integrating geographical and social influences, including the product rule (Prod), the
sum rule (Sum), and the union rule (CoRe, i.e., Algorithm 3).
Prod and Sum are comparable to each other on both Foursquare and
Gowalla data sets and they improve the performance of the geo-social rec-
30
0.5
0.5
Prod
Sum
CoRe
0.4
Recall
Precision
0.4
0.3
0.3
0.2
0.1
10 20 30 40 50 60 70 80 90 100
given−n
0.1
10 20 30 40 50 60 70 80 90 100
given−n
(a) Foursquare (Precision)
(b) Foursquare (Recall)
0.5
0.5
Prod
Sum
CoRe
0.4
Recall
0.4
Precision
Prod
Sum
CoRe
0.2
0.3
0.3
Prod
Sum
CoRe
0.2
0.2
0.1
10 20 30 40 50 60 70 80 90 100
given−n
0.1
10 20 30 40 50 60 70 80 90 100
given−n
(c) Gowalla (Precision)
(d) Gowalla (Recall)
Figure 10: Comparison of fusion rules on the number of check-in locations n (CoRe is
Algorithm 3)
ommendation techniques only using either geographical information or social
information in Sections 5.1.1 or 5.1.2, respectively. Promisingly, using the
union rule that considers the difference of two scores in Equations (13) and
(14) given by the geographical and social recommendation techniques respectively as an uncertain indicator to penalize the average of them, CoRe can
further raise the precision and recall in comparison to Prod and Sum.
31
5.1.4. Effect of Number of Recommended Locations
Figures 5, 7 and 9 describe the recommendation accuracy of a variety
of recommendation techniques with respect to the change of the number of
recommended locations, i.e., k from 5 to 50. As k increases, the precision
gradually gets lower but the recall steadily becomes higher on the two data
sets. Our explanation is that, in general, by returning more locations for
users, it is able to discover more locations that users would like to visit.
However, the extra recommended locations are less possible to be liked by
users due to the lower visiting probabilities of these locations.
5.1.5. Effect of Number of Check-in Locations
Figures 6, 8 and 10 depict the recommendation accuracy of a variety
of recommendation techniques with regard to various numbers of check-in
locations of users, i.e., n from 10 to 100, on the two data sets. As users visit
more locations, more check-in data are available for these recommendation
techniques; hence, they can more accurately estimate the scores of these users
for new locations. Accordingly, the precision inclines. Nonetheless, users who
have checked in many locations in the training data set usually have visited
many locations in the testing data set, so the recall fluctuates.
5.1.6. Effect of Cell Sizes
Heretofore, the cell size in CoRe is set to the optimal bandwidth σ in
Equation (3). Figure 11 depicts the effect of cell sizes on the recommendation
accuracy of CoRe in the two data sets. (1) In general, CoRe achieves the
best precision and recall when the cell size equals the optimal bandwidth
σ. (2) With the decrease of the cell size, the recommendation accuracy
32
0.25
Foursquare
Gowalla
Precision
0.2
0.15
0.1
0.05
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Cell size (unit: σ)
(a) Precision
0.25
Foursquare
Gowalla
Recall
0.2
0.15
0.1
0.05
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Cell size (unit: σ)
(b) Recall
Figure 11: Effect of cell sizes on recommendation accuracy of CoRe (Algorithm 2)
of CoRe gradually declines. The reason is that: For efficiency CoRe only
takes into account the nearest cells of a location when computing its visiting
probability according to Algorithm 2 and a smaller cell size leads to an end
that more visited locations out of the nearest cells are ignored. (3) As the cell
size increases, the recommendation accuracy of CoRe decreases as well. Our
explanation is that: A larger cell size accordingly requires a larger number of
truncated terms in Equation (5) in order to complete the same approximation
33
0.22
0.2
0.2
0.18
0.18
Recall
Precision
0.22
0.16
0.14
Exact
CoRe
0.12
0.16
0.14
0.1
0.08
0.1
1
0.08
2 3 4 5 6 7 8 9 10
No. of the truncated terms: c
(a) Foursquare (Precision)
1
0.22
0.22
0.2
0.2
0.18
0.18
0.16
0.14
Exact
CoRe
0.12
0.16
0.14
Exact
CoRe
0.12
0.1
0.08
2 3 4 5 6 7 8 9 10
No. of the truncated terms: c
(b) Foursquare (Recall)
Recall
Precision
Exact
CoRe
0.12
0.1
1
2 3 4 5 6 7 8 9 10
No. of the truncated terms: c
(c) Gowalla (Precision)
0.08
1
2 3 4 5 6 7 8 9 10
No. of the truncated terms: c
(d) Gowalla (Recall)
Figure 12: Approximation error of CoRe (Algorithm 2) in comparison to Exact (Algorithm 1)
accuracy.
5.2. Approximation Error
Figure 12 compares the approximation error of CoRe (Algorithm 2) with
that of Exact (Algorithm 1) regarding the change of the number of truncated
terms in Equation (5) in terms of the recommendation accuracy rather than
the error itself, since the error itself is not meaningful for location recom-
34
mendations and the effect of approximation error on the recommendation
accuracy is more significant.
As shown in Figure 12, with the increase of the number of truncated
terms, the precision and recall of CoRe quickly rise and approach to the performance of Exact in both the Foursquare and Gowalla data sets. The reason
is that the approximation error caused by truncating the infinite series with
the first c terms descends exponentially. For instance, in previous experiments, the default value of c is set to 6, in which the caused approximation
error is negligible according to the experimental results depicted in Figure 12.
5.3. Recommendation Efficiency
Figure 13 depicts the running time of CoRe (Algorithm 2), Exact (Algorithm 1), MGM, PD and iGSLR with respect to various numbers of check-in
locations of users. (1) With the increase of the number of check-in locations
of users, CoRe maintains a low consistent running time while Exact takes
a linearly increasing running time, since CoRe has the computational complexity of O(|L|) and Exact with O(|L|n). (2) Although the running time
of MGM also remains constant because of its computational complexity of
O(M |L|) in which M is the number of centers (i.e., the most popular POIs),
MGM requires more running time than CoRe. (3) As the number of check-in
locations of users gets larger, the running time of both PD and iGSLR with
O(|L|n) also increases linearly, but increasing faster than that of Exact, not
to mention CoRe.
Therefore, CoRe is more scalable to the web-scale calculation in the process of location recommendations.
In addition, all evaluated techniques
take more time to recommend locations in the Gowalla data set than the
35
3.5
MGM
PD
iGSLR
Exact
CoRe
Running time (seconds)
3
2.5
2
1.5
1
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
No. of check−in locations of users: n
(a) Foursquare
20
Running time (seconds)
18
16
MGM
PD
iGSLR
Exact
CoRe
14
12
10
8
6
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
No. of check−in locations of users: n
(b) Gowalla
Figure 13: Recommendation efficiency of CoRe (Algorithm 2) in comparison to Exact
(Algorithm 1), MGM, PD and iGSLR
36
Foursquare data set, because the former contains more location candidates
for recommendations than the latter as shown in Table 2.
6. Related Work
In this section, we highlight related work about location recommendations
in location-based social networks [12].
Mobile context-aware recommender systems. Location recommendations have been widely studied in mobile context-aware recommender systems for tourism [18]. In addition to user locations, potentially useful context
information contains mobility trajectory, distance from POIs, budget, time,
weak day, season, time available for sightseeing, transportation mode, weather condition, etc. For example, to make location recommendations, Barranco
et al. [4] employed users’ mobility trajectories and speeds, Savage et al. [47]
utilized users’ transportation modes and moods, and the works [1, 17] focus
on using time of a day and weather conditions. However, most these studies
do not take into account the social interactions between users.
Social recommendations. With the rapid growth of social networks,
like Facebook and Twitter, social network information, e.g., personal profile
content, tags and friendships, has been utilized to improve the quality of
recommender systems. Such information can be considered as implicit user
feedbacks to alleviate the data sparsity problem of explicit user feedbacks (i.e.,
the rating matrix) in traditional recommender systems [35, 50]. In particular,
recommendation techniques integrating social friendships have been widely
studied [5], including model-based methods [13, 25, 26, 27, 33, 35, 38, 41, 43,
48, 53, 54] and memory-based methods [24, 28]. The rationale behind these
37
methods is that friends are more likely to share common interests and thus
the social influence should be considered when making recommendations.
Location recommendations in LBSNs. Some recent studies provide POI recommendations by using the conventional collaborative filtering
techniques on users’ check-in data [14, 21, 45, 64, 65, 70], GPS trajectory
data [8, 23, 30, 40, 58, 66, 67, 68, 69], or text data [22, 37, 51, 52]. However,
these studies have not leveraged any geographical influence when generating
recommendations. In reality, the geographical information of users and locations plays a significant influence on users’ check-in behaviors [9, 56], since
physical interactions are required for users to visit locations that are totally
different from other non-spatial items, e.g., books, music and movies [56].
Location recommendations using geographical influence. To exploit geographical influence for improving the quality of location recommendations, some techniques [15, 16, 31, 55, 59, 61] employ the geographical
influence of users to derive their similarity weights as an input of the conventional collaborative filtering techniques [6, 7]. However, the performance
is considerably limited due to no consideration for the geographical influence
of locations. In contrast, other techniques explore the geographical influence
of locations. For example, the studies [2, 10, 44, 57] view locations as ordinary non-spatial items and consider the geographical influence of locations
by predefining a range; locations only within this range will be possibly recommended to users. The literatures [29, 32, 34] present a geo-topic model
respectively by assuming that if a location is closer to the locations visited
by a user or the current location of a user, it is more likely to be visited by
the same user. More sophistically, some works model the distance between
38
two locations visited by the same user as a common distribution for all users, e.g., a power-law distribution [36, 39, 56, 60] or a multi-center Gaussian
distribution [9]. In particular, our previous papers [62, 63] personalize the
geographical influence by modeling the distance between locations visited by
the same user as a personalized distance distribution for each user.
7. Conclusion
In this paper, we have proposed an effective and efficient location recommendation framework called CoRe. In CoRe, we have modeled the personalized two-dimensional geographical influence and employed the kernel density estimation technique to derive the personalized two-dimensional check-in
probability density for a user visiting a new location. To improve the system
efficiency and scalability, we have used the fast Gauss transform to develop
the approximation method to compute visiting probabilities. In addition, we
have also designed a new method to measure the similarity between users
based on their social friendships and residence information and a new fusion
rule to combine the geographical and social influences into a unified score.
Finally, we have conducted extensive experiments to evaluate the recommendation accuracy, recommendation efficiency, and approximation errors
of CoRe using two data sets from Foursquare and Gowalla. Experimental results show that CoRe outperforms all other evaluated recommendation
techniques.
39
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