# Hete-CF: Social-Based Collaborative Filtering Recommendation using
Heterogeneous Relations

Chen Luo  College of Computer Science
and Technology
Jilin University, China
Email<EMAIL_ADDRESS>Wei Pang  School of Natural and
Computing Sciences
University of Aberdeen, UK
Email<EMAIL_ADDRESS>Zhe Wang  College of Computer Science
and Technology
Jilin University, China
Email<EMAIL_ADDRESS>
###### Abstract

Collaborative filtering algorithms haven been widely used in recommender
systems. However, they often suffer from the data sparsity and cold start
problems. With the increasing popularity of social media, these problems may
be solved by using social-based recommendation. Social-based recommendation,
as an emerging research area, uses social information to help mitigate the
data sparsity and cold start problems, and it has been demonstrated that the
social-based recommendation algorithms can efficiently improve the
recommendation performance. However, few of the existing algorithms have
considered using multiple types of relations within one social network. In
this paper, we investigate the social-based recommendation algorithms on
heterogeneous social networks and proposed Hete-CF, a Social Collaborative
Filtering algorithm using heterogeneous relations. Distinct from the exiting
methods, Hete-CF can effectively utilize multiple types of relations in a
heterogeneous social network. In addition, Hete-CF is a general approach and
can be used in arbitrary social networks, including event based social
networks, location based social networks, and any other types of heterogeneous
information networks associated with social information. The experimental
results on two real-world data sets, DBLP (a typical heterogeneous information
network) and Meetup (a typical event based social network) show the
effectiveness and efficiency of our algorithm.

##  I Introduction

With the advent of Internet era and the emergence of Big Data, users
constantly suffered from information overload. Therefore, recommendation
systems, as effective methods to deal with information overload, become a very
popular research topic in recent years [1, 2]. On the other hand,
recommendation systems also help e-commerce companies provide personalised
services [3]. Many companies, including YouTube and Amazon, have launched
their own personalised recommender systems so that they can provide better
services.

Among many recommendation algorithms, Collaborative Filtering [2] has been
widely used in both social networks and online stores. Most of the
collaborative filtering methods aim to provide recommendations or rating
predictions based on historical user-item preference records [1]. However, in
the real world, users only rate a limited number of items. For example, in
Amazon, a user always buys a small fraction of all available commodities,
which makes the corresponding user-item information matrix very sparse.
Consequently, collaborative filtering recommendation algorithms severely
suffer from the cold start and data sparsity problems [4].

In order to deal with the data sparsity problem, many algorithms have been
proposed. Social-based recommendation, as one of the efficient and emerging
methods, has attracted much attention in recent years [5, 6, 7, 2, 8, 1, 9].
Social-based recommendation utilises social information to help improve the
recommendation performance. For example, Zhang _et al._ [5] consider the
recommendation system on EBSN (Event Based Social Network) [6], which contains
both online and offline networks. The algorithm presented in [5] uses the
social information extracted from offline networks to help make
recommendations in online networks. Another recommendation system research on
EBSN also demonstrates the effectiveness of this method [7]. In [7], the
author proposed LCARS, a location-content-aware recommender system, and LCARS
considers both personal interests and local preferences to make
recommendations. In IJCAI-13, Yang _et al._ proposed the TrustMF
recommendation algorithm [2]. TrustMF considers the trust and trustee
information between users from the social network. Similar to TrustMF, Xiao Yu
_et al._ proposed several recommendation algorithms [8, 9] based on
heterogeneous information networks by introducing the relationships between
items. In [1], the cross-domain knowledge was used to improve the
recommendation performance. In this paper, we will introduce the above-related
research in detail in Section V.

Previous research has demonstrated that more effective information could lead
to better recommendation results [6, 2]. However, most of the above algorithms
only use part of the information in social networks (either user-user or item-
item information). In order to make better use of the social information, in
this work we study the collaborative filtering method on heterogeneous social
networks. Different from previous research, in this paper we will utilize all
three types of relations, that is, not only the user-user and item-item
relations, but also the user-item relations.

As in [9, 10], a heterogeneous social network contains more information than
homogeneous ones and may have more semantic meaning. Fig.1 shows a simple
heterogeneous network (Bibliographic Network), and from this figure we can see
that there are three types of objects in this network: Author, Paper, and
Conference. It is also noted that there is more than one relation in the
network. In order to utilize these relations in a heterogeneous network, we
can use meta-path [11] to present each type of relations. Meta-path is an
effective way to represent relationships in heterogeneous information networks
[11]. For example, the co-author relationship can be represented as a meta-
path “Author$-$Paper$-$Author”; the co-conference relationship can be
represented as a meta-path “Author$-$Paper$-$Conf.$-$Paper$-$Author”. It is
pointed out that, the heterogeneous social networks in our research denotes
all the HIN (heterogeneous information networks) [12] with social information,
for example, EBSNs (Event-based Social Networks)[6] and LBSNs (Location-based
Social Networks) [11]. We will discuss the relations between these networks in
Section II-A.

Figure 1: A Simple Heterogeneous Network

In this research, we model the three types of relations (user-user, user-item,
and item-item) respectively and propose a unified model. However, not all the
social information is of benefit to the recommendation system. If the
historical user-item rating is not very sparse, the social information may
bias the recommendation result [1]. Therefore, we also proposed a leveraging
method to evaluate the weight of the introduced social information.

Our contributions in this research are listed as follows:

1\. We proposed a general recommendation algorithm (Hete-CF) on heterogeneous
social networks. The proposed algorithm can be used in many types of
heterogeneous social networks, including LBSN and EBSN, etc.

2\. Three types of relations have been integrated to form a unified model, and
a leveraging method has been proposed to leverage the impact of user-item
relation in our model.

3\. Our approach has been tested on on two real-world data sets: DBLP, a
typical heterogeneous information network with social information, and Meetup,
a typical event-based social network. Experimental results demonstrate the
effectiveness and efficiency of our algorithm.

The rest of the paper is organised as follows: relevant background knowledge
is introduced in Section II; the proposed recommendation model is described in
detail in Section III. In Section IV, we present our experimental results.
This is followed by the introduction of some work related to our research in
section V. Finally, we conclude our research and explore future work in
Section VI.

##  II Preliminary

In this section, we first introduce some background knowledge for our
research, including heterogeneous social networks [12] and meta-path [11].
Then we briefly introduce the collaborative filtering algorithm.

###  II-A Heterogeneous Social Network

The Heterogeneous Social Networks (HSNs) in our work can be regarded as HINs
[12] containing social information, for instance, bibliographic networks, or
Facebook relationship network. The relations between HSN, HIN, EBSN and LBSN
are illustrated in Fig. 2.

Figure 2: Relationships between various types of Heterogeneous Networks.

From this figure, we can see that LBSN and EBSN are special cases of HSN, and
HSN is a special case of HIN. In this research, we consider all kinds of HSNs,
including LBSN [13], EBSN [6] and any other types of HIN [12] associated with
social information.

As HSN is a special case of HIN, we define HSN by following the concepts of
HIN. Refer to the definition of heterogeneous information networks [12], the
heterogeneous social network is defined as follows:

######  Definition 1 (Heterogeneous Social Network)

Suppose we have $m$ types of data objects, denoted by
$X_{1}=\\{x_{11},...,x_{1n_{1}}\\},...,X_{m}=\\{x_{m1},...,x_{mn_{m}}\\}$, a
heterogeneous social network is in the form of a graph $G=\langle
O,E,W\rangle$, where $O=\bigcup_{i=1}^{m}X_{i}$ ($m\geq 2$), and in $O$ there
exists at least one $X_{i}$ which is the type of social objects (e.g., Person,
Author, or Actor). $E$ is the set of links between any two data objects in
$O$. $W$ is the set of weight values on the links.

Obviously, $G$ will reduce to a homogeneous network when $m=1$.

In [11], the concept of meta-path is used to denote different relations in
heterogeneous information networks. In the same way as in [11], in this
research, meta-path is used to represent the relations in heterogeneous social
networks. Following [11], the meta-path is defined as follows:

######  Definition 2 (Meta-path [11])

Given a heterogeneous social network $G=\langle O,E,W\rangle$, A meta path $P$
is in the form of
$A_{1}\overset{R_{1}}{\rightarrow}A_{2}\overset{R_{2}}{\rightarrow}...A_{l-1}\overset{R_{l-l}}{\rightarrow}A_{l}$,
which defines a composite relation $P=R_{1}\circ R_{2}\circ...\circ R_{l-1}$
between two objects, and $\circ$ is the composition operator on relations.

In addition, the reverse meta-path is defined as follow:

######  Definition 3 (Reverse Meta-path [11])

Given a meta-path $P$:
$A_{l}\overset{R_{1}}{\rightarrow}A_{2}\overset{R_{2}}{\rightarrow}...A_{l-1}\overset{R_{l-l}}{\rightarrow}A_{l}$,
${P}^{-1}$ is the reverse meta-path of $P$ if ${P}^{-1}$ is
$A_{l}\overset{{R_{l-1}}^{-1}}{\rightarrow}A_{l-1}\overset{{R_{l-2}}^{-1}}{\rightarrow}...A_{2}\overset{R_{l}^{-1}}{\rightarrow}A_{1}$,
where $R^{-1}$ is the reverse relation of $R$.

After describing the concept of meta-path, we need to know how to measure it.
Here, we use a topological measure, PathSim, which is proposed in [11]:

Given a meta-path, denoted as $P$, the PathSim between two objects $s$ and $t$
can be calculated as follows:

$\small
S_{P}^{PS}(s,t)=\frac{2*S_{P}^{PC}(s,t)}{S_{P}^{PC}(s,:)+S_{P^{-1}}^{PC}(:,t)}$
(1)

In the above, $S_{P}^{PC}(s,t)$ is a _Path Count_ measure [11] and can be
calculated as the number of path instances between $s$ and $t$. $P^{-1}$
denotes the reverse meta-path of $P$. $S_{P}^{PC}(s,:)$ denotes the path count
value following $P$ and starting with $s$; and $S_{P}^{PC}(s,:)$ denotes the
path count value following $P^{-1}$ and ending with $t$.

###  II-B Collaborative Filtering Model

Collaborative filtering uses historical user-item ratings to make
recommendations. Suppose $\mathbb{R}$ is an $n-$by$-m$ matrix, which denotes
the user-item rating matrix ($n$ is the number of users and $m$ the number of
items). $\mathbb{R}_{i,j}$ denotes the rating that User $i$ gives to Item $j$.
One can map users and items to a lower dimension vector, denoted as $U_{i}$
and $V_{j}$, respectively. The lower dimension vectors (also called feature
vectors), $U_{i}$ and $V_{j}$, can be learned by minimizing the following loss
function [2]:

$\small\underset{U,V}{\min}\sum_{i=0}^{m}\sum_{j=0}^{n}(U_{i}^{T}V_{j}-\mathbb{R}_{i,j})^{2}+\lambda(\left\|U\right\|_{F}^{2}+\left\|V\right\|_{F}^{2}),$
(2)

where $\left\|*\right\|_{F}^{2}$ is the Frobenius norm [14].
$\lambda\in{(0,1)}$ captures the importance of each term in the loss function.
$U=[U_{1},U_{2},...,U_{n}]$ and $V=[V_{1},V_{2},...,V_{m}]$ are the low rank
representation of users and items [2]. Then the rating can be calculated as
$\hat{R_{ij}}=U_{i}^{T}V_{j}$, where $U_{i}^{T}$ denotes the transpose of
$U_{i}$. Several algorithms have been proposed to improve the performance of
learning $U$ and $V$ [15, 16]. Theses algorithms will be introduced in Section
V.

##  III The Hete-CF Model

In this section, we first propose to combine the three types of relations
(user-user, user-item, and item-item) in heterogeneous social networks into a
unified model. Then we introduce the leaning method of our model. At last, we
introduce the complete algorithm and analyse its parameter and time
complexity.

###  III-A The Recommendation Model

####  III-A1 Modelling the relations between users

as in [2], when we recommend items to users, the relationships between users
can be used to improve the recommendation performance. This is because it is
common sense that similar users may have similar orientations on a certain
range of items. Let us consider an example of recommending conferences to
authors for submitting their papers. Given two authors Tom and Peter, who
focus on similar research topics and are both interested in artificial
intelligence, if Tom frequently publishes his papers in the conference AAAI,
Peter may have a high possibility to publish his paper in AAAI as well.

We model the above common sense by using graph regularization [17], and the
object function is shown as follows:

$\small\underset{U,A}{\min}\sum_{k=0}^{N_{A}}\alpha_{k}\sum_{i=0}^{n}\sum_{j=0}^{n}S_{A}^{k}(i,j)\left\|U_{i}-U_{j}\right\|_{F}^{2}$
(3)

In the above, $\alpha_{k}$ denotes the importance of the $k-$th meta-path
between users and $A=[\alpha_{1},\alpha_{2},...,\alpha_{N_{A}}]$; $N_{A}$ is
the number of meta-paths between users. $U_{i}$ has the same meaning as
mentioned in Section II-B. $S_{A}^{k}$ is the similarity matrix for users
under the $k-$th meta-path relation, and it is calculated as follows (recall
_PathSim_ described in Section II-A):

$\small S_{A}^{k}(i,j)=S_{P_{k}^{A}}^{PS}(i,j),$ (4)

where $P_{k}^{A}$ is the $k-$th meta-path between User $i$ and User $j$.

####  III-A2 Modelling the relations between items

as in [15], we can see that by introducing the relations between items the
recommendation performance can be improved. It is common sense that a user may
be interested in similar items. For example, if an author is interested in
publishing his/her papers in ICML, this author may also be interested in
publishing his/her papers in similar conferences, for instance, NIPS (NIPS and
ICML are both top conferences in the field of machine learning). Here we also
employ graph regularization [17] to model this common sense as follows:

$\small\underset{V,B}{\min}\sum_{k=0}^{N_{B}}\beta_{k}\sum_{i=0}^{m}\sum_{j=0}^{m}S_{B}^{k}(i,j)\left\|V_{i}-V_{j}\right\|_{F}^{2}$
(5)

In the above $\beta_{k}$ denotes the importance of the $k-th$ meta-path
between items and $B=[\beta_{1},\beta_{2},...,\beta_{N_{B}}]$; $N_{B}$ is the
number of meta-paths between items; $V_{i}$ has the same meaning as mentioned
in Section II-B. $S_{B}^{k}$ is the similarity matrix for items;
$S_{B}^{k}(i,j)$ is calculated as follows:

$\small S_{B}^{k}(i,j)=S_{P_{k}^{B}}^{PS}(i,j),$ (6)

where $P_{k}^{B}$ is the $k-$th meta-path between Item $i$ and Item $j$.

####  III-A3 Modelling the relations between users and items

as mentioned in Section II-B, the collaborative filtering algorithm uses
historical user-item ratings to make recommendations. In a heterogeneous
social network, there are also many other types of relations between users and
items, and these relations may be used to further improve the recommendation
performance.

For instance, if an author often cites papers from a particular conference, he
may also be very likely to submit his papers to that conference. The common
sense here is that users may be highly interested in the items which are
“close” to them. The term “close” here in the context of HSNs can be
represented as the larger similarity of the meta-path between a user and an
item, and this distance can be calculated by using PathSim [11], the concept
of which has been introduced in Section II-A. Same as the historical user-item
ratings, these relationships are also between users and items. So, we can use
the collaborating filtering method to model such relationships, and the model
is shown below:

$\small\underset{U,V,W}{\min}\sum_{k=0}^{N_{W}}w_{k}\sum_{i=0}^{n}\sum_{j=0}^{m}(U_{i}^{T}V_{j}-\mathbb{R}_{i,j}^{k})^{2}$
(7)

Here, $m$ and $n$ are the numbers of items and users, respectively; $w_{k}$
denotes the importance of the $k-$th meta-path between a user and an item, and
$W=[w_{1},w_{2},...,w_{N_{W}}]$; $N_{W}$ is the number of meta-paths between
users and items. $\mathbb{R}^{k}$ is the relation graph for the $k-$th meta-
path between users and items, and it can be calculated as below:

$\small\mathbb{R}^{k}(i,j)=S_{P_{k}^{W}}^{PS}(i,j),$ (8)

where $P_{k}^{W}$ is the $k-$th meta-path between User $i$ and Item $j$.

####  III-A4 A Unified Model

TABLE I:  The three Sub-models Factor | Sub-model
---|---
User$-$User | $\underset{U,A}{\min}\sum\limits_{k=0}^{N_{A}}\alpha_{k}\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{n}S_{A}^{k}(i,j)\left\|U_{i}-U_{j}\right\|_{F}^{2}$
Item$-$Item | $\underset{V,B}{\min}\sum\limits_{k=0}^{N_{B}}\beta_{k}\sum\limits_{i=0}^{m}\sum\limits_{j=0}^{m}S_{B}^{k}(i,j)\left\|V_{i}-V_{j}\right\|_{F}^{2}$
User$-$Item | $\underset{U,V,W}{\min}\sum\limits_{k=0}^{N_{W}}w_{k}\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{m}(U_{i}^{T}V_{j}-\mathbb{R}_{i,j}^{k})^{2}$

Having proposed the modelling approaches in Sections III-A1 $\sim$ III-A3, we
intend to put all the three factors together. The three corresponding sub-
models are illustrated in Table I. And the unified model for recommendation in
a heterogeneous social network is proposed as follows:

$\displaystyle\underset{U,V,A,B,W}{\min}$
$\displaystyle\sum_{i=0}^{m}\sum_{j=0}^{n}(U_{i}^{T}V_{j}-\mathbb{R}_{i,j})^{2}$
(9) $\displaystyle+$
$\displaystyle\sum_{k=0}^{N_{A}}\alpha_{k}\sum_{i=0}^{n}\sum_{j=0}^{n}S_{A}^{k}(i,j)\left\|U_{i}-U_{j}\right\|_{F}^{2}$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{N_{B}}\beta_{k}\sum_{i=0}^{m}\sum_{j=0}^{m}S_{B}^{k}(i,j)\left\|V_{i}-V_{j}\right\|_{F}^{2}$
$\displaystyle+$
$\displaystyle\mu\sum_{k=0}^{N_{W}}w_{k}\sum_{i=0}^{n}\sum_{j=0}^{m}(U_{i}^{T}V_{j}-\mathbb{R}_{i,j}^{k})^{2}$
$\displaystyle+$
$\displaystyle\lambda(\left\|U\right\|_{F}^{2}+\left\|V\right\|_{F}^{2}+\left\|A\right\|_{F}^{2}+\left\|B\right\|_{F}^{2}+\left\|W\right\|_{F}^{2})$

In the above, the symbols have the same meanings as introduced in previous
sections. $\mu$ and $\lambda$ are important parameters which captures the
importance of each term, and we will discuss these parameters in section
III-C1.

The first term of the model incorporates the collaborating filtering
component, which keeps the $U^{T}V$ closer to the user-item rating matrix
$\mathbb{R}$. The second and third terms consider the user-user and item-item
relations, respectively. The fourth term of the model is the user-item
relationship component. The last term of the model is the smoothing term.
After minimising this model, we can obtain $U$ and $V$, and then the ratings
can be obtained as $\hat{R_{ij}}=U_{i}^{T}V_{j}$.

In order to avoid over-fitting during the learning process, we introduce
weighted-$\lambda$-regularization [15] in our algorithm. This method penalizes
the feature vectors which involves more ratings. Thus the last term of our
model becomes:

$\small\lambda(\sum_{i}n_{user_{i}}\left\|U\right\|_{F}^{2}+\sum_{j}n_{item_{j}}\left\|V\right\|_{F}^{2}+\left\|A\right\|_{F}^{2}+\left\|B\right\|_{F}^{2}+\left\|W\right\|_{F}^{2}),$

where $n_{user_{i}}$ and $n_{item_{i}}$ denote the number of ratings given by
User $i$ and the number of ratings given to Item $j$, respectively.

On the other hand, in a heterogeneous social network, the similarity values
calculated by _PathSim_ are between $0$ and $1$ (recall _PathSim_ described in
section II-A). So, as suggested in [16], in order to fit the data more
conveniently, we adopt a logistic function to bound the inner product of the
latent feature vectors into the interval $[0,1]$. As in [16], we use the
logistic function $f(x)=1/{(1+{\exp}(-x))}$ in our model. Then the model to be
optimized is shown below:

$\displaystyle\underset{U,V,A,B,W}{\min}$
$\displaystyle\sum_{i=0}^{m}\sum_{j=0}^{n}(f(U_{i}^{T}V_{j})-\mathbb{R}_{i,j})^{2}$
(10) $\displaystyle+$
$\displaystyle\sum_{k=0}^{N_{A}}\alpha_{k}\sum_{i=0}^{n}\sum_{j=0}^{n}S_{A}^{k}(i,j)\left\|U_{i}-U_{j}\right\|_{F}^{2}$
$\displaystyle+$
$\displaystyle\sum_{k=0}^{N_{B}}\beta_{k}\sum_{i=0}^{m}\sum_{j=0}^{m}S_{B}^{k}(i,j)\left\|V_{i}-V_{j}\right\|_{F}^{2}$
$\displaystyle+$
$\displaystyle\mu\sum_{k=0}^{N_{W}}w_{k}\sum_{i=0}^{n}\sum_{j=0}^{m}(f(U_{i}^{T}V_{j})-\mathbb{R}_{i,j}^{k})^{2}$
$\displaystyle+$
$\displaystyle\lambda(\sum_{i}n_{user_{i}}\left\|U\right\|_{F}^{2}+\sum_{i}n_{item_{i}}\left\|V\right\|_{F}^{2}$
$\displaystyle+$
$\displaystyle\left\|A\right\|_{F}^{2}+\left\|B\right\|_{F}^{2}+\left\|W\right\|_{F}^{2})$

Then the predicted rating becomes $\hat{R_{ij}}=f(U_{i}^{T}V_{j})$. However,
this model cannot be directly optimised. So, in order to optimize it, we
rewrite the graph regularizing terms into their trace forms as in [18]. Then
the graph regularizing terms are derived as follows:

$\displaystyle\sum_{k=0}^{N_{A}}\alpha_{k}\sum_{i=0}^{n}\sum_{j=0}^{n}S_{A}^{k}(i,j)\left\|U_{i}-U_{j}\right\|_{F}^{2}$
(11) $\displaystyle=Tr(U^{T}(\sum_{k=0}^{N_{A}}\alpha_{k}L_{A}^{k})U),$

and

$\displaystyle\sum_{k=0}^{N_{B}}\beta_{k}\sum_{i=0}^{m}\sum_{j=0}^{m}S_{B}^{k}(i,j)\left\|V_{i}-V_{j}\right\|_{F}^{2}$
(12) $\displaystyle=Tr(V^{T}(\sum_{k=0}^{N_{V}}\beta_{k}~{}L_{B}^{k})V).$

In the above, $L_{A}^{k}=D_{A}^{k}-S_{A}^{k}$, where $L_{A}^{k}$ is a diagonal
matrix and $D_{A}^{k}(i,i)=\sum_{j=0}^{n}S_{A}^{k}(i,j)$. Similarly,
$L_{B}^{k}=D_{B}^{k}-S_{B}^{k}$, where $L_{B}^{k}$ is a diagonal matrix and
$D_{B}^{k}(i,i)=\sum_{j=0}^{m}S_{B}^{k}(i,j)$. Then, based on Equations (10),
(11), and (12), the unified model, denoted as $J$, can be rewritten as:

$\displaystyle J=$
$\displaystyle\sum_{i=0}^{m}\sum_{j=0}^{n}(f(U_{i}^{T}V_{j})-\mathbb{R}_{i,j})^{2}$
(13) $\displaystyle+$ $\displaystyle
Tr(U^{T}(\sum_{k=0}^{N_{A}}\alpha_{k}L_{A}^{k})U)+Tr(V^{T}(\sum_{k=0}^{N_{V}}\beta_{k}~{}L_{B}^{k})V)$
$\displaystyle+$
$\displaystyle\mu\sum_{k=0}^{N_{W}}w_{k}\sum_{i=0}^{n}\sum_{j=0}^{m}(f(U_{i}^{T}V_{j})-\mathbb{R}_{i,j}^{k})^{2}$
$\displaystyle+$
$\displaystyle\lambda(\sum_{i}n_{user_{i}}\left\|U\right\|_{F}^{2}+\sum_{i}n_{item_{i}}\left\|V\right\|_{F}^{2}$
$\displaystyle+$
$\displaystyle\left\|A\right\|_{F}^{2}+\left\|B\right\|_{F}^{2}+\left\|W\right\|_{F}^{2})$

###  III-B The Learning Algorithm

In this subsection, we introduce the learning algorithm of our model presented
in Equation. (13). The learning method of our model is a two-step iteration
method, where the predicted rating vector $U,V$ and the weight for each meta-
path $A,B,W$ mutually enhance each other. In the first step, we fix the weight
vectors $A,B,W$ and learn the best predicted rating vector $U,V$. In the
second step, we fix the predicted rating vector $U,V$ and learn the best
weight vectors $A,B,W$.

####  III-B1 Optimize $U,V$ Given $A,B,W$

When $A,B,W$ are fixed, the model becomes to a traditional Collaborative
Filtering model. Therefore, as in [2], we can use SGD (Stochastic Gradient
Descent) to solve such problem. The gradient descents on $U$ are shown as
follows:

$\displaystyle\frac{1}{2}\frac{\partial J}{\partial U_{i}}=$
$\displaystyle\sum_{i=0}^{n}f(U_{i}^{T}V_{j})^{{}^{\prime}}U_{i}^{T}(f(U_{i}^{T}V_{j})-\mathbb{R}_{i,j})+U_{i}\sum_{k=0}^{N_{A}}\alpha_{k}L_{A}^{k}$
(14) $\displaystyle+$
$\displaystyle\mu\sum_{k=0}^{N_{W}}w_{k}\sum_{i=0}^{n}f(U_{i}^{T}V_{j})^{{}^{\prime}}U_{i}^{T}(f(U_{i}^{T}V_{j})-\mathbb{R}_{i,j}^{k})+\lambda
U_{i},$

and the gradient descents on $V$ are shown as follows:

$\displaystyle\frac{1}{2}\frac{\partial J}{\partial V_{j}}=$
$\displaystyle\sum_{j=0}^{m}f(U_{i}^{T}V_{j})^{{}^{\prime}}V_{i}(f(U_{i}^{T}V_{j})-\mathbb{R}_{i,j})+V_{i}\sum_{k=0}^{N_{B}}\beta_{k}L_{B}^{k}$
(15) $\displaystyle+$
$\displaystyle\mu\sum_{k=0}^{N_{W}}w_{k}\sum_{j=0}^{m}f(U_{i}^{T}V_{j})^{{}^{\prime}}V_{i}(f(U_{i}^{T}V_{j})-\mathbb{R}_{i,j}^{k})+\lambda
V_{i}$

Then in each iteration, $U,V$ can be updated as:

$\displaystyle U_{i}\leftarrow U_{i}-\alpha^{s}\frac{\partial L}{\partial
U_{i}}~{}~{}(i=0,\cdots,n-1),$ (16)

and

$\displaystyle V_{i}\leftarrow V_{i}-\alpha^{s}\frac{\partial L}{\partial
V_{j}}~{}~{}(i=0,\cdots,m-1)$ (17)

where $\alpha^{s}\in(0,1)$ is used to control the convergence speed.

####  III-B2 Optimize $A,B,W$ Given $U,V$

When $U,W$ is fixed, the terms only involving $U,V$ can be discarded, then the
objective function can be reduced to:

$\displaystyle J_{1}=$ $\displaystyle
Tr(U^{T}(\sum_{k=0}^{N_{A}}\alpha_{k}L_{A}^{k})U)+Tr(V^{T}(\sum_{k=0}^{N_{V}}\beta_{k}~{}L_{B}^{k})V)$
(18) $\displaystyle+$
$\displaystyle\mu\sum_{k=0}^{N_{W}}w_{k}\sum_{i=0}^{n}\sum_{j=0}^{m}(f(U_{i}^{T}V_{j})-\mathbb{R}_{i,j}^{k})^{2}$
$\displaystyle+$
$\displaystyle\lambda(\left\|A\right\|_{F}^{2}+\left\|B\right\|_{F}^{2}+\left\|W\right\|_{F}^{2})$

We can see that $J_{1}$ becomes to a linear model for each $A,B,$ and $W$.
Therefore, we also use SGD (Stochastic Gradient Descent) to obtain $A,B,W$.
The gradient descents on $A,B,W$ are shown as follows:

$\displaystyle\frac{1}{2}\frac{\partial J_{1}}{\partial\alpha_{k}}=$
$\displaystyle\lambda\alpha_{k}+\sum_{i=0}^{n}\sum_{j=0}^{m}(f(U_{i}^{T}~{}V_{j})-\mathbb{R}_{i,j})^{2}$
(19) $\displaystyle+$ $\displaystyle
Tr(U^{T}(\sum_{k=0}^{N_{A}}\alpha_{k}~{}L_{A}^{k})U)$
$\displaystyle\frac{1}{2}\frac{\partial J_{1}}{\partial\beta_{k}}=$
$\displaystyle\lambda\beta_{k}+\sum_{i=0}^{n}\sum_{j=0}^{m}(f(U_{i}^{T}~{}V_{j})-\mathbb{R}_{i,j})^{2}$
(20) $\displaystyle+$ $\displaystyle
Tr(V^{T}(\sum_{k=0}^{N_{B}}\beta_{k}~{}L_{B}^{k})V)$
$\displaystyle\frac{1}{2}\frac{\partial J_{1}}{\partial w_{k}}=\lambda
w_{k}+\mu\sum_{i=0}^{n}\sum_{j=0}^{m}(f(U_{i}^{T}~{}V_{j})-\mathbb{(}R)_{i,j})^{2}$
(21)

Then in each iteration, $A,B,W$ can be updated as:

$\displaystyle\alpha_{i}\leftarrow\alpha_{i}-\alpha^{s}\frac{\partial
J_{1}}{\partial\alpha_{i}}~{}~{}(i=0,\cdots,N_{A}-1),$ (22)

and

$\displaystyle\beta_{i}\leftarrow\beta_{i}-\alpha^{s}\frac{\partial
J_{1}}{\partial\beta_{j}}~{}~{}(i=0,\cdots,N_{B}-1),$ (23)

and

$\displaystyle w_{i}\leftarrow w_{i}-\alpha^{s}\frac{\partial J_{1}}{\partial
w_{j}}~{}~{}(i=0,\cdots,N_{W}-1)$ (24)

###  III-C The Complete Algorithm

After presenting the calculation method for each relevant variable in Section
III-B1 and Section III-B2, the detailed steps of our recommendation algorithm
is given in Algorithm 1.

Input: A heterogeneous information network $G=\langle O,E,W\rangle$. Three
sets of meta-paths between user and item, users, and items. The user-item
rating matrix $\mathbb{R}$. Parameter $\lambda$, $\alpha^{s}$.

Output: The rating Matrix $\hat{R}$;

1

2Initialize $U,V,A,B,W$ randomly;

3 while  _not reaching the inner $U,V,A,B,C$ difference threshold_ do

4 while  _not reaching the inner $U,V$ difference threshold_ do

5 Update $U,V$ using Eqs.(16) and (17);

6

7 end while

8

9 while  _not reaching the inner $A,B,C$ difference threshold_ do

10 Update $A,B,W$ using Eqs. (22), (23) and (24);

11

12 end while

13

14 end while

15The prediction rating is $\hat{R_{ij}}=f(U_{i}^{T}V_{j})$.;

16 return $\hat{R_{ij}}$;

Algorithm 1 Hete-CF

####  III-C1 Parameter Settings

There are two parameters, $\lambda$ and $\mu$, in our model. As in [2, 15,
16], $\lambda$ is always assigned manually based on the experiments and
experience. Therefore, we only discuss the assignment of parameter $\mu$ in
our model.

$\mu$ is an important parameter, and it can directly affect the performance of
our algorithm. When $\mathbb{R}$ is sparse, a larger $\mu$ can improve the
recommendation results, because more information can be added to the training
process. On the other hand, when $\mathbb{R}$ is not sparse, a larger $\mu$
will bias the recommendation results.

Therefore, the value of $\mu$ depends on how sparse $\mathbb{R}$ is. In this
sense, we can use the proportion of non-zero elements in matrix $\mathbb{R}$
to calculate $\mu$ as follows:

$\small\mu=\frac{\sum_{i=1}^{n}\sum_{j=1}^{m}I_{i,j}}{m\times n},$ (25)

where $I_{i,j}$ is calculated as:

$\small I_{i,j}=\begin{cases}0&\text{ if }\mathbb{R}_{i,j}=0\\\ 1&\text{ if
}\mathbb{R}_{i,j}\neq 0\end{cases}$ (26)

####  III-C2 Time Complexity Analysis

In Algorithm 1, the time for learning our model is mainly taken by computing
the objective function (13) and its corresponding gradients against feature
vectors. In the inner iteration, the time complexity of computing the
gradients of $J$ against $U,V$ is $O(td(n+N_{W}n+N_{A}))$ and
$O(td(m+N_{W}m+N_{B}))$, respectively. The time complexity of computing the
gradients of $J$ against $W$, $A$, and $B$ is $O(td(mn))$, $O(td(mn+N_{A})$,
and $O(td(mn+N_{B})$, respectively. In the above, $t$ is the number of
iterations; $d$ is the dimensionality of feature vector. $N_{W}$, $N_{B}$, and
$N_{A}$ have the same meaning as mentioned in Section III-A, and $N_{W}$,
$N_{B}$, and $N_{A}$ are far less than $mn$. Therefore, the upper bound of the
complexity of inner iteration is $O(td(mn))$. Suppose the number of outer
iteration is $T$, the overall complexity of our method is $O(Ttd(mn))$. We can
see the complexity is linearly scaling to the dimension of the user-item
numbers. The efficiency test will be introduced in Section IV-D.

##  IV Experimental Results

In this section, we first report the experimental evaluation of our algorithm
by performing a set of experiments on two real datasets. Then we test the
efficiency of the algorithm on synthetic datasets. Finally, we study the
parameter of our algorithm and analyse the learned weight for each selected
meta-path.

###  IV-A Datasets

In this research, we use two real heterogeneous social network datasets for
our experiments: DBLP 111http://www.informatik.uni-
trier.de/$\scriptsize{\sim}$ley/db/ and MeetUp 222http://www.meetup.com/. As
the links in DBLP and MeetUp are always sparse, we can test the ability of our
algorithm on mitigating the cold start problem and data sparsity problem. On
the other hand, our algorithm is proposed for the HSNs, and these two datasets
are typical HSN data.

DBLP dataset is widely used for heterogeneous network analysis [8, 12, 11]. In
this research, we extract a sub dataset from DBLP. The sub dataset contains
the papers published in 261 computer journals and 313 computer conferences.
The schema [19] of DBLP is shown in Fig.3(a), in which Term is extracted from
the titles of the papers. For the DBLP data, our recommendation problem
becomes to recommend conferences to authors. Therefore, we model the
historical user-item relationship as the “Author$-$Paper$-$Conference” meta-
path, and the rate that an author gives to a conference is calculated as the
PathSim (see Section II-A) of the meta-path “Author$-$Paper$-$Conference”.

(a) DBLP

(b) MeetUp

Figure 3: Network Schema

As in previous research on DBLP as a heterogeneous social network [12], the
meta-paths chosen for the experiments are shown in Table II. In this table, A
stands for “Author”; P stands for “Paper”, C means “Conference or Journal”,
and T is for “Term”.

The second dataset is taken from Meetup, an online social website for offline
meetings. The dataset comes from the one published in [6]. We extract a sub-
network from the total dataset as in [5, 20, 21]. The sub-network contains the
events happened in New York City and Los Angeles and involved the
corresponding users and groups. This sub dataset is the same as the one used
in [5]. The schema of Meetup network is shown in Fig. 3(b). For the Meetup
dataset, we recommend groups to users. The meta-paths chosen for the
experiment are shown in Table II. We have two principles for choosing the
meta-paths: (1) all the types of objects need to be included into the selected
meta-path; (2) choose as few meta-paths as possible. In this table, U stands
for ”User”, L stands for ”Location”, G stands for “Group”, and E stands for
“Event”.

TABLE II:  Meta paths selected in DBLP and MeetUp Dataset | Relations | Meta-paths
---|---|---
DBLP Dataset |  Authors$-$Author | $A-P-A,~{}A-P-C-P-A,~{}A-P-T-P-A$
Conf. $-$ Conf. | $C-P-A-P-C,~{}C-P-P-C,~{}C-P-T-P-C$
Author $-$ Conf. | $A-P-T-P-C,~{}A-P-P-C$
MeetUp Dataset |  Users $-$ User |  $U-L-U$, $U-G-U$, $U-E-U$, $U-U$
Group $-$ Group | $G-U-G,~{}G-E-L-E-G,~{}G-U-U-G$
Users $-$ Group | $U-U-G,~{}U-E-G,~{}U-L-E-G$

###  IV-B Experimental Setup

####  IV-B1 Cross-validation

As in [2], we use 5-fold cross-validation for learning and testing. We
randomly select $40\%$ $(60\%)$ of the data as the training set and the rest
$60\%$ $(40\%)$ as the testing set. Each result discussed below is averaged
over ten trials.

####  IV-B2 Comparison Methods

There are three baselines and two state-of-the-art methods used in our methods
for comparison. The three baselines are listed as follows:

  * •

UserMean: prediction rate equals the mean value of the users.

  * •

ItemMean: prediction rate equals the mean value of the items.

  * •

NMF: non-negative matrix factorization function [2], with $d=5$, and $d=10$,
where $d$ denotes the dimension of the feature vector.

The three baselines cannot make use of the heterogeneous relations in the
heterogeneous social network, and they only consider the target recommendation
relation (The $A-P-C$ relation in DBLP, and the $U-G$ relation in MeetUp). So
in this experiment, we only consider the target recommendation relations in
each dataset for these three baseline algorithms.

The two state-of-the-art methods selected in our experiment are described
below:

  * •

Trust-MF [2]: this is a recommendation approach proposed by Yang _et al._ in
IJCAI-13. Trust-MF introduced the relations between users to help improve the
recommendation results. In the DBLP dataset used in our experiment, in terms
of the relations between users, we use the co-author relationship. For the
MeetUp dataset, we use the friendship relation to denote the relation between
users in our experiment. As in [2], the parameters of this algorithm is set as
follows: $\lambda=0.001$ and $\lambda_{t}=1$. For more details of parameter
assignment of Trust-MF please refer to [2].

  * •

Hete-MF [18]: this is an approach proposed by Yu _et al_. in _IJCAI-HINA’13_.
Hete-MF considers the relations between ’items’. In this paper, for the
relations between ’items’, we add all the selected relations: $C-P-P-C,~{}C-P-
A-P-C,~{}C-P-T-P-C$ in DBLP, and $G-U-G,~{}G-E-L-E-G,~{}G-U-U-G$ in Meetup.

####  IV-B3 Evaluation Methods

As in the following previous recommendation systems research [2, 18], we use
two evaluation methods in our experiment:

  * •

MAE: Mean Absolute Error, which is denoted as follows:

$\small MAE=\frac{1}{T}\sum_{i=0}^{n}\sum_{j=0}^{m}|R_{i,j}-\hat{R_{i,j}}|,$

where $R_{i,j}$ denotes the ground truth rating that User $i$ gives to Item
$j$. $\hat{R_{ij}}$ denotes the rating calculated by the selected algorithm
and $T=mn$ is the number of ratings.

  * •

RMSE: Root Mean Square Error, which is denoted as follows:

$\small
RMSE=(\frac{1}{T}\sum_{i=0}^{n}\sum_{j=0}^{m}|R_{i,j}-\hat{R_{i,j}}|^{2})^{\frac{1}{2}}$

The notations in RMSE have the same meaning as in MAE.

###  IV-C Result and Analysis

The experimental results are reported in Table III. From these results, we can
see that for our Hete-CF algorithm, with the increase of the training data,
its performance becomes better. This is because more training data can provide
more information and more importantly, our model can avoid over-fitting. On
the other hand, In the DBLP dataset, we can see that when the dimension of
feature vector $d=5$, the performance is always better. This is because our
datasets extract nearly five areas of data from the DBLP dataset (see Section
IV-A). Compared with the other algorithms, Hete-CF always performs better.
From this aspect, our algorithm has significant advantages when performing the
recommendation task on heterogeneous social networks.

TABLE III: Algorithm Performance Comparison in DBLP and MeetUp. %Training | Feature | Evaluation | UserMean | ItemMean | NMF | Trust-MF | Hete-MF | Hete-CF
---|---|---|---|---|---|---|---|---
DBLP Dataset |  |  |  |  |  |  |  |
$40\%$ | $d=5$ | MAE | $0.942\pm 0.02$ | $1.065\pm 0.02$ | $2.156\pm 0.02$ | $0.831\pm 0.01$ | $0.931\pm 0.02$ | $\textbf{0.831}\pm\textbf{0.02}$
RMSE | $1.216\pm 0.01$ | $1.123\pm 0.02$ | $2.394\pm 0.01$ | $1.013\pm 0.02$ | $1.105\pm 0.01$ | $\textbf{1.002}\pm\textbf{0.03}$
$d=10$ | MAE | $0.943\pm 0.03$ | $0.948\pm 0.01$ | $2.194\pm 0.03$ | $0.887\pm 0.01$ | $0.901\pm 0.01$ | $\textbf{0.859}\pm\textbf{0.01}$
RMSE | $1.138\pm 0.02$ | $1.256\pm 0.04$ | $2.292\pm 0.02$ | $1.083\pm 0.03$ | $1.114\pm 0.03$ | $\textbf{1.056}\pm\textbf{0.02}$
$60\%$ | $d=5$ | MAE | $0.948\pm 0.02$ | $0.919\pm 0.01$ | $2.131\pm 0.04$ | $\textbf{0.812}\pm\textbf{0.02}$ | $0.891\pm 0.02$ | $0.831\pm 0.02$
RMSE | $1.132\pm 0.02$ | $1.157\pm 0.01$ | $2.385\pm 0.01$ | $\textbf{0.907}\pm\textbf{0.02}$ | $1.010\pm 0.03$ | $0.938\pm 0.02$
$d=10$ | MAE | $0.932\pm 0.03$ | $0.978\pm 0.03$ | $2.184\pm 0.02$ | $0.873\pm 0.03$ | $0.881\pm 0.01$ | $\textbf{0.856}\pm\textbf{0.02}$
RMSE | $1.154\pm 0.02$ | $1.143\pm 0.02$ | $2.275\pm 0.01$ | $1.051\pm 0.01$ | $1.013\pm 0.02$ | $\textbf{0.994}\pm\textbf{0.03}$
MeetUp Dataset |  |  |  |  |  |  |  |
$40\%$ | $d=5$ | MAE | $1.076\pm 0.01$ | $0.954\pm 0.02$ | $1.154\pm 0.03$ | $0.887\pm 0.02$ | $1.035\pm 0.02$ | $\textbf{0.854}\pm\textbf{0.01}$
RMSE | $1.154\pm 0.02$ | $1.143\pm 0.03$ | $1.354\pm 0.03$ | $1.046\pm 0.02$ | $1.154\pm 0.02$ | $\textbf{0.986}\pm\textbf{0.02}$
$d=10$ | MAE | $0.943\pm 0.03$ | $1.065\pm 0.01$ | $1.154\pm 0.03$ | $0.876\pm 0.03$ | $1.013\pm 0.03$ | $\textbf{0.854}\pm\textbf{0.01}$
RMSE | $1.121\pm 0.04$ | $1.265\pm 0.03$ | $1.264\pm 0.02$ | $1.076\pm 0.01$ | $1.235\pm 0.01$ | $\textbf{0.967}\pm\textbf{0.02}$
$60\%$ | $d=5$ | MAE | $0.976\pm 0.02$ | $1.054\pm 0.02$ | $1.154\pm 0.01$ | $0.854\pm 0.01$ | $0.964\pm 0.02$ | $\textbf{0.837}\pm\textbf{0.03}$
RMSE | $1.167\pm 0.01$ | $1.136\pm 0.01$ | $1.343\pm 0.02$ | $1.054\pm 0.04$ | $1.132\pm 0.01$ | $\textbf{0.966}\pm\textbf{0.02}$
$d=10$ | MAE | $0.965\pm 0.02$ | $0.943\pm 0.02$ | $1.112\pm 0.02$ | $0.887\pm 0.02$ | $0.953\pm 0.03$ | $\textbf{0.876}\pm\textbf{0.01}$
RMSE | $1.254\pm 0.01$ | $1.254\pm 0.02$ | $1.254\pm 0.02$ | $1.052\pm 0.01$ | $1.012\pm 0.02$ | $\textbf{0.978}\pm\textbf{0.02}$

###  IV-D Efficiency Study

In this section we study the efficiency of our algorithm. As in [10], we use
the synthetic datasets considering the fact that we can manipulate the size of
the dataset flexibly.

(a) Varying Feature Dimension

(b) Varying Data Size

Figure 4: Efficient Study, in (a) we fix the size of dataset, and vary the
number of feature dimensions; in (b) we fix the feature dimension $(d=10)$ and
vary the datasets.

In Fig.4(a), we fix the size of data, and vary the number of the feature
dimensions. From the results we can see that the running time of Hete-CF is
almost linear to the feature dimension. In Fig. 4(b), we keep the number of
feature dimensions $(d=10)$ fixed and randomly link 2$0\%$ of the objects in
the datasets. Then we vary the data size. We can see that the CPU execution
time increases with the increase of the data size, but the increase of CPU
time is linear to the data size.

###  IV-E Parameter Study

In this section, we study the impact of parameter $\mu$ in Equation (13),
which is used for leveraging the importance of the introduced meta-paths
between users and items in our model. The result is shown in Fig. 5.

In Fig. 5, we can see that when the value of $\mu$ is too small or too large,
the result will not be good enough. Because $\mu$ is used for leveraging the
_user-item_ meta-path term (the fourth term) in our model (Equation (13)), the
extreme value of $\mu$ (too large or too small) will bias the result.

On the other hand, the $\mu$ value calculated by Equation (25) is $\mu=0.7$ in
DBLP and $\mu=0.4$ in MeetUp. And the best result just appeared when $\mu=0.7$
and $\mu=0.4$, as shown in Fig. 5. This demonstrates that our proposed method
of evaluating the $\mu$ value is effective, and it can deal with the parameter
pre-assignment issue.

(a) DBLP Dataset

(b) MeetUp Dataset

Figure 5: Parameter investigation on the DBLP and MeetUp datasets, $\mu$
ranges from $0.1$ to $1.0$. The value of $\mu$ calculated by our method is:
$\mu=0.7$ in DBLP and $\mu=0.4$ in MeetUp

###  IV-F Case Study for Meta-path Weight

The output of our algorithm not only contains the predicted ratings but also
the weights for each selected meta-path. In this section, we study the learned
weight for each meta path. The weight for each selected meta-path is shown in
Table IV and Table V. For the ease of illustration, we mapped all the weights
for each meta-path to the range $[0,1]$.

From the result in Table IV, we can see that the meta-path “$A-P-A$” has a
bigger weight, which means the co-author relationship can make an important
impact on conference recommendation. The meta-path “$C-P-A-P-C$” also has a
big weight, that is because the meta-path “$C-P-A-P-C$” contains the target
recommendation meta-path ($A-P-C$) and may have a high impact on the
conference recommendation. The meta-path “$A-P-P-C$” has a high weight too,
which is consistent with the ground truth that authors often submit their
papers to the conference which they often cites.

From the result in Table V, we can see that the meta-path “$U-U$” has a bigger
weight. This means the friendship between users has a high impact on Group
recommendation, which is consistent with human intuition. The meta-path
“$G-U-G$” also has a big weight, that is because $G-U-G$ contains the target
recommendation relation “$U-G$”. The Meta-Path “$U-E-G$” has high weight,
which means if an user has participated an Event of a Group, the user may have
high probability to be a member of that Group.

TABLE IV: _Meta-path_ Weight of DBLP Meta-path Type | Meta-path | Weight
---|---|---
Author$-$Author | $A-P-A$ | $0.4\sim 0.5$
$A-P-C-P-A$ | $0.3\sim 0.4$
$A-P-T-P-A$ | $0.2\sim 0.3$
Conf.$-$ Conf. | $C-P-P-C$ | $0.2\sim 0.3$
$C-P-A-P-C$ | $0.5\sim 0.6$
$C-P-T-P-A$ | $0.2\sim 0.3$
Author$-$ Conf. | $A-P-P-C$ | $0.7\sim 0.8$
$A-P-T-P-C$ | $0.2\sim 0.3$
TABLE V: _Meta-path_ Weight of MeetUp Meta-path Type | Meta-path | Weight
---|---|---
User$-$User | $U-U$ | $0.4\sim 0.5$
$U-L-U$ | $0.1\sim 0.2$
$U-G-U$ | $0.3\sim 0.4$
$U-E-U$ | $0.1\sim 0.2$
Group$-$Group | $G-U-G$ | $0.4\sim 0.5$
$G-E-L-E-G$ | $0.2\sim 0.3$
$G-U-U-G$ | $0.3\sim 0.4$
User$-$Group | $U-U-G$ | $0.3\sim 0.4$
$U-E-G$ | $0.4\sim 0.5$
$U-L-E-G$ | $0.2\sim 0.3$

##  V Related Work

In this section, we introduce some related works to our research.

###  V-A Collaborative Filtering and Social-based Recommendation

Collaborative Filtering Recommendation is a popular method, which has been
researched intensively in recent years. In [2], the authors introduced the
social trust information and proposed a novel collaborative filtering model.
In [15], the authors proposed a parallel collaborating algorithm for the
Netflix Prize, Alternating-Least-Squares with Weighted- -Regularization (ALS-
WR). In [16], the authors proposed a probabilistic matrix factorization model,
which can deal with problems with large-scale datasets and sparse ratings. In
this research, we have incorporated the ideas presented in [15] and [16] to
improve our model.

Social based recommendation is an emerging research topic which combines the
recommendation algorithms and social media mining algorithms. In [22], the
authors proposed a general social based recommendation framework using social
regularization. This work demonstrates that the social network information can
benefit recommender systems. In [5], the author proposed a group
recommendation method on EBSN. This method considers location features, social
features, and implicit patterns in a unified model. Another recommendation
system proposed on EBSN is LCARS [7]. LCARS consists of an off-line model,
which considers the off-line part of EBSN, and an on-line recommendation
model. By considering both the on-line and off-line parts, the recommendation
performance was significantly improved [7]. In [20], the authors proposed to
realise location recommendation services. The recommendation method in [20]
considered both the friend relationship and the geographic information. The
algorithm proposed in [1] utilised the knowledge from other domains to improve
the recommendation performance. In [8], the authors proposed a recommendation
algorithm on HIN and utilise part of the heterogeneous relationships.

In this research, we aim to bridge the gap between CF-based recommendation and
social-based recommendation. Different from the above algorithms, our proposed
method considers the recommendation in HSNs (e.g., LBSN, EBSN and other HINs
with social information). In addition, our method utilises all types of
relations in HSNs while the above algorithms utilise only part of the
relations.

###  V-B Mining Heterogeneous Social Network

Heterogeneous Social Network is defined as the heterogeneous information
network with social information in our paper. As introduced before,
Heterogeneous Social Network is a special case of HIN. HIN (Heterogeneous
Information Network) has multi-typed objects and relations, and it may contain
more meaningful information. The concept of heterogeneous information network
is first proposed by Sun _et. al_ in [12]. The works proposed by them [7, 8,
12, 11, 9, 18, 10] also demonstrated that by mining heterogeneous information
network, one can obtain more meaningful results. It also attracted other
researchers to investigate on mining the heterogeneous information network
[19, 23].

There are also some other kinds of heterogeneous social networks as previously
introduced in Section II-A. Location based networks [13] are heterogenous
social networks with location information, and they have attracted many
researchers in recent years [24, 25]. In [24], the authors proposed a model to
mine interesting locations using GPS data; in [25], the authors designed a
model to explain the human movements using some location-based network data.
Event based social networks (EBSNs) are those heterogeneous social networks
which contain not only social information but also the off-line event
information. Recently EBSNs also receive more and more attention [6, 5]. In
[6], the authors formally proposed the definition of EBSNs, and some initial
research was carried out in [6] on event based social networks, such as
community detection and link prediction.

In this research, we consider all types of heterogeneous social network
together, and our recommendation method can be used in all the networks
introduced above.

##  VI Conclusion and Future Work

Collaborative filtering recommendation are frequently suffered from the data
sparsity and cold start problems. Social based recommendation, as one of the
efficient and emerging methods to deal with these problems, has attracted much
attention in recent years. In this paper, we focus on the recommendation
problem using social heterogeneous relations, and we proposed Hete-CF, a
Collaborative Filtering recommendation method on Heterogeneous Social
Networks. Different from the previous social based recommendation methods, we
propose to effectively incorporate all the social relations, including the
relations between users, items and user-item. In addition, since Hete-CF is a
network structure based model, it can be used in many types of social networks
(e.g. , LBSN, EBSN, and other HINs with social information). For instance,
Hete-CF can be used in recommending off-line events in EBSN or recommending
locations (or hotels) in LBSN. The experiments on two real-world heterogeneous
social networks (DBLP and Meetup) demonstrate the effectiveness and efficiency
of Hete-CF.

In the future, we intend to apply Hete-CF to more real-world recommendation
problems. In addition, another direction of our future research is to explore
the potential of Hete-CF on big data problems, such as problems involving
massive amounts social media data.

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# The Pomeron — A Bootstrap Story

Chung-I Tan  Brown University, Providence, RI 02912

###### Abstract

In a contribution to the volume A Passion for Physics, a collection of essays
in honor of Geoffrey Chew’s sixtieth birthday, I wrote, together with A.
Capella, Uday Sukhatme, and Tran Thanh Van The Pomeron Story. This is a
follow-up to that contribution. This sequel 111This is a contribution to the
book “Geoff Chew: Architect of the Bootstrap”, to be published by World
Sceintific. also serves as an opportunity to acknowledge my gratitude to Geoff
as a PhD student under his tutelage.

##  1 My Years at Berkeley

In our 1984 essay, The Pomeron Story, we wrote “In their seminal paper [1]
Chew and Frautschi asserted that all hadrons are composite; they lie on Regge
trajectories. $\ldots\ldots$” To account for the near constancy of hadronic
total cross sections, it was necessary for them to postulate the existence of
a Regge trajectory with vacuum quantum numbers and zero-energy intercept at
$j=1$, even though no physical particles were known at the time to lie on the
trajectory. This Regge singularity, which was also proposed independently by
Gribov, is known as the Pomeron.

Our understanding of the Pomeron has evolved since The Pomeron Story was
written. The fundamental theory for the strong interactions, Quantum
Chromodynamics (QCD), has been established. However, much of its tests
involved perturbative aspects of QCD. Great progress has also been made via
lattice studies. Yet, near-forward high energy scattering remains a challenge.
What is the Pomeron, really?

As a student at Berkeley in the late sixties, one faced constant turmoil on
social issues. We were all caught up with the Free Speech Movement,
subscribing to Ramparts, etc. As a student of Geoff, we also found relief in
moving daily to the Rad Lab on the hill. Geoff had a large group of graduate
students, sharing a common interest in various aspects of strong interactions,
ranging from purely phenomenological to foundational aspects of S-matrix
theory. This group included Ling-Lie Chau, Jerry Finkelstein, Jiunn-Ming Wang,
Farzam Arbab, Jan Dash, Huan Lee, Richard Brower, Michael Misheloff, Carleton
DeTar, Dennis Sivers, Don Tow, and others.222Students just finished included
David Gross, John Schwarz, Shu-Yuan Chu, John Stack, etc. Other contemporary
students include Dick Haymaker, Lay-Nam Chang, and others. Together with other
students, e.g., Dick Slansky and Joe Weis, and also a large group of post-
docs, a common theme shared by most was the question of consistency and
uniqueness of the S-matrix, as exemplified by the bootstrap program[2, 3]. Can
all Regge parameters [4] be fixed within an analytic S-matrix? One could not
avoid discussing the Pomeron and Bootstrap nearly on a daily basis.

Geoff allowed us sufficient flexibility in focussing on areas of particular
interest of our own. I began my work when Geoff suggested that I take a look
at various questions of bounds, and, in particular, the question of the
Cerulus–Martin lower bound [5], which had just gained certain interest from an
experimental perspective. Asymptotic behavior of scattering amplitudes has
already been recognized as an integral part of S-matrix consistency. The Regge
hypothesis, which had become popular due to its application to experiment, was
introduced as an integral component of bootstrap consistency[2].

I was fortunate that Prof. Richard Eden of Cambridge came to Berkeley for a
year’s sabbatical, and I worked with him closely, which formed the main part
of my thesis. Although my thesis work seemed formal, it connected well with
the common interest of all my fellow students, as well as a large group of
post-docs. Although we each had our own research focus, we all shared the
excitement of the bootstrap philosophy, with regular weekly informal seminars.

The group grew steadily in size and one often could not find seats in the
Building-50 seminar/coffee room. When Geoff and Ruth moved back to the
Berkeley hills, Geoff invited his group of students for an evening get-
together at their beautiful new home. As a house-warming gift, I organized and
brought Chinese take-outs. It was a wonderful evening, the discussion
invariably turning to physics, while we enjoyed the view of the Campanile and
Golden Gate in the distance. Whether by design or not, this informal gathering
turned into a regular weekly meeting in the Berkeley hills for his
students.333In addition to discussing physics, other stories were also
exchanged, e.g., Geoff’s recollections of his student days at Chicago with
Fermi, Goldberger, Yang, Lee, etc. I also remember his story about meeting
Landau, who reminded Geoff of the significance of his being the first to
emphasize the importance of the particle–pole correspondence. I understand
this practice continued after those first timers had left Berkeley after
receiving their PhDs.

In looking back one cannot avoid concluding that the Pomeron remained the
central topic of interest to most of us at that time. In this remembrance of
Geoff, I will discuss how the issue of the Pomeron evolved over the past 50
years.

##  2 Rencontres de Moriond and the Soft Pomeron

The “Rencontres de Moriond” is a series of annual high energy physics meetings
starting in 1966. The initial impetus for this series is “to promote fruitful
collaboration by bringing together a small number of scientists in inspiring
surroundings, to discuss recent findings and new ideas in physics.” This
series was organized by Prof. Tran Thanh Van of the University of Paris,
Orsay. It has continued to thrive over more than half a century. As I will
explain below for the period of 70s and 80s, Pomeron Physics suffered a period
of “crisis.” However, during this period, these annual meetings served as a
playground where the physics of the Pomeron flourished, from the experimental
perspective. In particular, it provided support for the notion that the
Pomeron is non-perturbative and exhibits aspects of string structure with a
cylindrical topology, i.e., the exchange of a closed string.

### Crisis for the Pomeron:

There are several developments in the late 60s and early 70s which dealt
unkindly to the study of the Pomeron. (a) Since total cross sections can be
interpreted geometrically, where does the scale come from for the Pomeron
coupling? (b) Total cross sections continued to increase with energy, albeit
slowly. This is in conflict with the initial impetus of having the Pomeron
with an exact intercept at $j=1$. (c) As a collection of closed string
excitations[6, 7], flat-space string theories led to a Pomeron intercept at
$j=2$. What are the low-lying particles on this trajectory? (d) Lastly, what
is the Pomeron in QCD?

Let me comment on these aspects briefly, in reverse. QCD is a theory of quarks
and gluons. Although confinement in the infrared (IR) remains to be
demonstrated explicitly, most theorists believe this can eventually be
accomplished. This belief is further supported by lattice studies, where the
low-lying particle spectrum has been calculated. Presumably, particles on the
Pomeron trajectory should be identified with a tensor glueball and its
recurrences. As a non-perturbative phenomenon, the Pomeron is not considered a
subject of immediate concern.

Having an intercept $j=2$ for closed string excitations has led to the
possibility of a theory for quantum gravity. This, of course, is based on
string theory in flat space–time, albeit at a higher dimension. More
seriously, string theories also suffer difficulty in introducing local
currents. As a consequence, it is difficult to see how the experimental
observation of power-behaved amplitudes at large momentum transfer can be
accommodated.

### The Topological Pomeron and Bootstrap:

It has become increasingly clear that the increase in the total cross sections
is driven by an increasing rate of inelastic particle production. Since
particle production can be successfully explained by a Regge mechanism, it
remains consistent to assume that the Pomeron, through unitarity, should
emerge as a sum of ladder-like diagram, e.g., so-called multiperipheral
mechanism [8]. Indeed, this picture is consistent with string theories, with
the Pomeron entering at a higher genus, e.g., exchanging a Pomeron has the
topology of a cylinder. This picture was advocated by Veneziano, leading to
the program of “dual topological unitarization”[9] (DTU). (See also the
contribution by Veneziano in this volume.)

From a QCD perspective, one optimistically awaits for a future QCD string
theory, propagating over a curved background. In the meantime, it is important
to continue to search for evidence for such a topological structure via
particle production. Notably, in a topological expansion, the Froissart bound
no longer applies at the cylinder-level, which removes the proposition that
the Pomeron has a unit intercept. In fact once this restriction is removed,
the value of the Pomeron intercept, being greater than unity, becomes a
challenging dynamical question. Experimentally it seems to be greater by an
amount of the order of $0.1\sim 0.2$. Can one provide an upper bound? Finally,
the Froissart bound should ultimately be restored, e.g., via eikonalization.
At what level should a geometrical scale enter in the discussion?

### DTU:

Much of particle production data during this period came from experiments at
CERN. It was natural that new data was presented and discussed at Moriond,
which was then held in the French alps. Due to its close proximity to Geneva,
many theorists at CERN would regularly participate in these meetings, e.g., G.
Veneziano, A. White, D. Amati, Chan Hong-Mo, etc. Theoretical interpretation
for particle production was a regular topic of intense discussion.

During my first sabbatical at Orsay, working with A. Capella, U. Sukhatme, and
Tran Thanh Van, we began an exploration of the consequences of the Pomeron
having the desired topological structure, as suggested by string theory in
general and DTU in particular. These studies were vigorously discussed and
confronted with experiments. In particular, these were carried out regularly
at annual Moriond meetings. Based on the initial success, a systematic
calculus was formulated, which has subsequently dubbed the “Dual Parton Model”
(DPM)[10]. This is partly based on the general approach to the S-matrix
bootstrap advocated by Veneziano [9] and also adopted by Geoff [11]. The
details of this model were discussed in the contribution The Pomeron Story,
and will not be repeated here. (See Ref. [10] and also the contribution of G.
Veneziano in this volume.) Here we illustrate the key features by two
schematics in Fig. 2.1.

Figure 2.1: A cut-topological Pomeron for particle production and the
resulting inclusive cross section.

Through these efforts, the notion of a topological Pomeron was firmly
established phenomenologically. Since this identification is established via
the production of particles, which tend to have limited transverse momenta, it
has been dubbed a “soft Pomeron.” This usage also indicates that further
complications are lurking around the corner and would soon appear in the
coming years.

##  3 A Perturbative vs a Non-Perturbative Pomeron

As hinted above, it soon became clear that jet production would become an
important component of particle production. The question can then be posed,
how to account for the total cross section resulting from such processes.
However, it is unclear if this separation can be made meaningfully before a
fully comprehensive confinement picture emerges. In any event, both during
Moriond and also at other meetings, this issue became increasingly the center
of discussion. One began to talk about “hard-Pomeron” versus “soft-Pomeron.”
Loosely speaking, hard-Pomeron should be calculable based on a perturbative
approach, while soft-Pomeron remains non-perturbative. Should the bootstrap be
restricted to the soft component only?

### BFKL:

Beginning in the mid-1970s, Lev Lipatov [12] and his collaborators (the
Leningrad school), carried out a series of studies of the high-energy behavior
of scattering amplitudes in QCD. These were serious and difficult analyses,
which ultimately led to the emergence of a Pomeron-like structure. However,
this study was carried out mostly by assuming conformal invariance, leading to
a Pomeron being a branch-point, instead of a pole. Nevertheless, the location
of this singularity was calculated. To first order in the ’t Hooft coupling,
$\lambda$, it is located above unity,

$\alpha_{P}=1+(\ln 2/\pi^{2})\lambda\,.$

It should be emphasized that this result follows from summing gluon ladder
graphs, which clearly ignores confinement: thus it is perturbative.
Nevertheless, it has resulted in a robust research program, including an
application to particle production. Furthermore, the BFKL program was able to
lead to interesting predictions for deep-inelastic scattering (DIS)
experiments at HERA. However, this picture clearly runs counter to that for
DTU, responsible for a soft Pomeron. The corresponding BFKL Pomeron, is now
identified with the hard Pomeron.

### Donnachie–Landshoff:

While the work of BFKL received increasing attention from both theoretical and
experimental communities, a surprisingly simple parametrization was proposed
by Donnachie and Landshoff for a soft-Pomeron [13]. They were able to explain
experimental data for total cross sections for $\bar{p}p$, $pp$, $\pi^{-}p$,
$\pi^{+}p$, $K^{-}p$, $K^{-}p$, and $\gamma p$ over a large energy range. It
was a truly impressive fit to the data, with an effective Pomeron intercept of

$\alpha_{P}\simeq 1.08\,.$

This value is significantly lower than that expected for the BFKL-Pomeron.
This analysis is purely phenomenological, (For more discussion of this
development, see the contribution by Donnachie and Landshoff in this volume.)
No suggestion on how this trajectory can be generated dynamically was
provided, e.g., what states lie on this trajectory?

### Russians are Coming — Moriond 1992:

While the late 1960s was the period of social unrest and change in the US, the
same can be said for the Soviet Union in the 1980s. Moriond had always been
welcoming to physicists from eastern block countries. Beginning in late 1980s,
increasingly more participants were able to come from the former Soviet Union.
In particular, many from the Leningrad school began to come regularly. In
1992, while again on sabbatical at Orsay, I was fortunate to meet Genya Levin
at Moriond. Genya is a contemporary of Lev Lipatov; both were students
together working with Gribov. While I tried to convince him of the success of
DTU, leading to a string-like topological-Pomeron, Genya would argue
forcefully that the correct approach should be BFKL-Pomeron [14].

After several days of argument, we both began to see the values of the
opposite view. We clarified for each other how each respective Pomeron can be
understood in terms of the picture of random walks. The difference between the
two lies in the respective space in which this takes place. In the case of the
Soft-Pomeron, the random walk occurs in the transverse impact space. In
contrast, the BFKL Pomeron is generated primarily by random walks in
“virtuality.”

We kept up our communication after Moriond, and soon became good friends. We
also came to the conclusion that a proper treatment of high energy scattering
should incorporate both soft- and hard-features. We soon came up with a toy-
model which would do precisely that. We coined the phrase “Heterotic Pomeron,”
and I presented a talk on this proposal at the July 1992 meeting for
International Symposium on Multiparticle Dynamics (ISMD) at Santiago, Spain,
with the intention of writing up a formal paper shortly afterwards. For
various reasons, the project was delayed. (We finally were able to finish this
project nearly ten years later [15].)

##  4 The Pomeron and Gauge–String Duality

A major development in our understanding of non-pertubative gauge theory
occurred in 1997, the AdS/CFT conjecture of Maldacena. In particular,
motivated by a suggestion of Witten, Jevicki et al. and independently Csaki et
al. showed how scalar glueball masses can be calculated via AdS/CFT, once
confinement is implemented. Working with Richard Brower, together with a then
young colleague at MIT, Samir Mathur, we were able to carry out an analogous
analysis generalized to tensor glueballs [16]. This is by itself an
interesting exercise, which I learned greatly from Rich and Samir. Equally
important is a paper by Polchinski and Strassler where they explained how
point-like power behavior for scattering at large angle can emerge when
AdS/CFT is implemented. Secretly, I was extremely excited; at long last, a
promising theoretical framework lay ahead as the basis for the
phenomenologically motivated heterotic Pomeron.

In 2007, Brower and I, together with J. Polchinski and M. Strassler[17],
provided a coherent first principles treatment of the Pomeron. In the large-N
QCD-like theories, we used curved-space string-theory to describe
simultaneously both the BFKL regime and the classical Regge regime. On the
basis of gauge/string duality, this allows us to deal with high-energy small-
angle scattering in QCD. The problem reduces to finding the spectrum of a
single j-plane Schrödinger operator. Our results agree with expectations for
the BFKL Pomeron at negative t, and with the expected glueball spectrum at
positive t, but provide a framework in which they are unified.

In short, a dual-topological Pomeron is nothing but a reggeized graviton over
a curved background, with the Pomeron intercept depending on the curvature. To
first order in $1/\sqrt{\lambda}$, we find

$\alpha_{P}\simeq 2-2/\sqrt{\lambda}\,.$

This gauge-string motivated Pomeron has been referred to by others as the
“BPST Pomeron”. It is heterotic, as previously proposed. It is based on firm
principles, and should be contrasted with weak-coupling BFKL Pomeron.

Further applications of this dual-topological Pomeron has been carried out in
recent years [18]. Effects beyond the single Pomeron exchange have also been
discussed[19]. Future developments and challenges undoubtedly await, e.g., the
recent reported observation of the Odderon [20] requires further clarification
from a string-dual perspective [21]. Similarly, the influence of the pion mass
on the total and elastic cross sections has also been raised in the context of
the topological Pomeron [22]. More intriguingly, can the value of the Pomeron
intercept be fixed by the bootstrap?

##  5 Moriond 1984

After Berkeley, I moved to Princeton in 1968 as a post-doc, working primarily
with Murph Goldberger, in particular on understanding the Pomeron from the
perspective of the multiperipheral picture. After completing my post-doc, I
moved to Brown as an Assistant Professor in 1970 and stayed since. Geoff and I
kept up with our interactions during this period — in particular in 1970 when
Geoff spent his sabbatical at Princeton. Equally rewarding is the fact that I
was able to work with Carleton DeTar and Joe Weis [23], my fellow graduate
student colleagues at Berkeley, who had just moved to MIT. We had a fruitful
collaboration, making use of Veneziano models for inclusive production via the
multi-Regge picture. This is an extension of the work of Mueller, which
strengthened the usefulness of a string-like picture for particle production.

I spent a fruitful sabbatical at Orsay in 1976–77, working with Capella,
Sukhatme and Tran on DTU. After my return to Brown, my interests gradually
shifted to the study of QCD non-perturbatively via the large-N expansion. I
was greatly helped by my colleagues at Brown, in particular A. Jevicki, who
introduced me to the collective-field approach. I was also fortunate in the
following year to team up with my long-time graduate student friend, Rich
Brower, who was spending a year at Harvard, in the study of matrix models.
Together with a young post-doc at MIT, P. Rossi, we worked out and extended
several nontrivial examples of coupled matrix models [24]. Rich also
introduced me to the detailed working of lattice gauge theory, for which I am
truly grateful, and he helped direct my graduate student, Kostas Orginos, who
is by now an accomplished lattice gauge theorist at William and Mary. During
this period, I did not have much direct contact with Geoff, although I have
tried to keep up with his work when preprints arrived.

I had the fortune to overlap with Geoff again in 1984 when I spent my second
sabbatical at Orsay. However, Geoff and I were located at different research
institutes, and we ate at different canteens, preventing us from having
regular get-togethers. At the same time my research interests had also
diverged from his. I began to engage increasingly more with model studies in
QCD at large N, while maintaining my continuing interests in phenomenological
applications to multiparticle production based on the topological Pomeron. At
the same time, Geoff focused on developing a topological approach to the
S-matrix bootstrap [25].

While he would occasionally listen to our explanations about how the Pomeron
would fit in a topological setting, clearly he was less and less interested in
phenomenological applications. Although we did not meet on a daily basis at
Orsay, we tended to often catch the same commuter train back to Paris. During
the short 30-minute ride, he would, in his usual enthusiastic fashion, expound
on his latest thinking on physics. While I don’t have a specific recollection
of these conversations, I do remember one incident of interest: During one of
the return trips, while we continued to chat at the transfer point at Denfert-
Rochereau, Geoff realized that his commuter ticket was missing. After I had
passed through the gate, Geoff, while looking briefly left and right for
security guards, simply jumped over the barrier with a quick athletic move. We
each moved quickly to the next quay to catch our respective transfers, Geoff
with a smug smile on his face.

My last extended interaction with Geoff took place at the 1984 Moriond
conference at les Arcs 1600, France. I recall Geoff expounding on his
optimistic view of how unification of the standard model would emerge from a
topological bootstrap approach [25]. He was at his usual best, trying to share
his excitement on how new physics can be examined from a bootstrap
perspective. After the 1984 Moriond meeting, my interactions with Geoff came
less often.

##  6 Last Visits

I visited Geoff in 2010 while attending a conference in Berkeley. I had lunch
with Geoff at the usual Rad Lab cafeteria, together with Geoff’s longtime
collaborators, Henry Stapp and Jerry Finkelstein. After lunch, we returned to
their office, and Geoff tried to explain to me their latest research efforts.
Although the visit was short it brought back memories of the wonderful
graduate student days.

I had another chance to visit Geoff in September 2013. We had lunch together
at his home in the Berkeley hills. How it brought back memory of fifty years
ago when we went there for the first time after the house was built! Instead
of Chinese take-out Geoff had ordered pizzas for lunch. He was pleased when I
shared with him the resurgence of the Bootstrap program in current CFT studies
as well as the heavy reliance on S-matrix principles for amplitude studies. I
also updated him on the status of our topological Pomeron program. I learned
from Geoff about his current goal in understanding cosmology from a bootstrap
perspective [26]. He also talked more about his graduate student years with
Fermi. I was also fascinated later by the story of how he had interacted with
Von Neumann as an undergraduate at George Washington University and how G.
Gamov likely helped start his physics career by sending him to join the
Manhattan Project. When it was time to leave I had no idea that would be the
last time I would see him.

Figure 6.1: Photo taken during my 2013 visit.

In March 2018, we had a chance to visit DAMTP where my former student, Kostas
Orginos, was on sabbatical there. His wife, Lily, a classicist, was also on
sabbatical at Cambridge. Through her connection they were staying at Clare
Hall, where Richard Eden is a founding Fellow. The highlight of our visit was
to have lunch with Richard. It was a pleasure to listen to him re-telling the
story of his graduate days working with Dirac.

I started my research career more than 50 years ago by trying to understand
the Pomeron and the Bootstrap. It remains a personal challenge and The Pomeron
Story continues. It has been a wonderful and continuing journey. Thanks to
Geoff, my fellow graduate students and Eden 444 Richard Eden passed away in
2021, shortly before the completion of this book., for shaping my research
interest and trajectory.

## References

  * [1] G. F. Chew and S. C. Frautschi, “Regge Trajectories and the Principle of Maximum Strength for Strong Interactions”, Phys. Rev. Lett. 8, 41 (1962).
  * [2] G. F. Chew, “Hadron Bootstrap Hypothesis,” Phys. Rev. D 4, 2330 (1971); “Nuclear Democracy and Bootstrap Dynamics”, UCRL-11163,12 (1963).
  * [3] R. Dolen, D. Horn and C. Schmid, Phys. Rev. 166, 1768 (1968).
  * [4] S. Mandelstam, Phys. Rev. 166, 1539 (1968).
  * [5] F. A. Cerulus and A. Martin, Phys. Lett. 8, 80 (1964); C. Chiu and C.-I. Tan, “Power Behavior of the Scattering Amplitude at Fixed Momentum Transfer and a Reconsideration of the Cerulus-Martin Lower Bound,” Phys. Rev. 162, 1701 (1967). This was my first serious paper. I was thrilled when it was cited in the seminar paper by Veneziano, (Footnote # 9, in Ref. [6]).
  * [6] G. Veneziano, “Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories,” Nuovo Cim. A 57, 190 (1968).
  * [7] J. Scherk and J. H. Schwarz, Nucl. Phys. B 81, 118 (1974).
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  * [9] G. Veneziano, “Duality and the Bootstrap,” Phys. Lett. B 34, 59–62 (1971).
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