1 Generalized Reciprocal Perspective Kevin Dick Graduate Student Member IEEE Daniel G. Kyrollos Graduate Student Member IEEE

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Generalized Reciprocal Perspective
Kevin Dick, Graduate Student Member, IEEE, Daniel G. Kyrollos, Graduate Student Member, IEEE
and James R. Green, Senior Member, IEEE
Abstract—Across many application domains, real-world problems can be represented as a network, with nodes representing
domain-specific elements and the edges (or lack thereof) capturing the relationship between elements. Two notable example domains
that leverage link prediction algorithms are the tasks of protein-protein interaction (PPI) prediction, in the bioinformatic domain, and the
prediction of user-item five-point scale rating, in the eCommerce domain. Leveraging high-performance computing and optimized link
prediction algorithms, it is increasingly possible to evaluate every possible combination of nodal pairs enabling the generation of a
comprehensive prediction matrix (CPM) that places an individual link prediction score in the context of all possible links involving either
node (providing data-driven context). Historically, this contextual information has been ignored given exponentially growing problem
sizes resulting in computational intractability; however, we demonstrate that expending high-performance compute resources to
generate CPMs is a worthwhile investment given the improvement in predictive performance. In this work, we generalize for all pairwise
link-prediction tasks our novel semi-supervised machine learning method, denoted Reciprocal Perspective (RP). We demonstrate that
RP significantly improves link prediction accuracy by leveraging the wealth of information in a CPM. Context-based features are
extracted from the CPM for use in a stacked classifier that refines link prediction scores. Herein, we leverage three independent data
sources (MovieLens, GoodBooks, Amazon Product Categories) containing 5-point ratings data across 24 different item categories and
differing user-bases and 13 recommendation system (RS) algorithms (including an equi-weighted ensemble of all 12 component RS
algorithms). We demonstrate that the application of RP in a cascade almost always results in significantly (p < 0.05) improved
predictions. These results on RS-type problems, combined with previously published performance on both homogeneous and
heterogeneous link prediction, in both classification- and regression-task formulations, suggest that RP is applicable to a broad range
of link prediction problems and we recommend its use in any pairwise link prediction pipeline for which the CPM may be generated.
The RP framework is available for use from the following GitHub repository: https://github.com/GreenCUBIC/RP.
Index Terms—Pairwise Prediction, Link prediction, Semi-Supervised Learning, Recommendation Systems, Stacked Generalization
F
1 INTRODUCTION
INNUMERABLE systems across diverse fields of science and
engineering can be abstracted into networks of intercon-
nected elements. These networks conventionally comprise a
set of nodes representing individual and distinct elements,
and a set of edges1which occur between pairs of nodes.
Networks are flexible and abstract models that can represent
diverse concepts. Popularized examples of element-element
relationships in networks include social networks (nodes:
individuals; edges: social tie), telecommunication systems
(nodes: terminals; edges: transmission links), and biological
systems (nodes: proteins; edges: molecular interactions).
Indeed, advances within the domain of Network Science
reverberate through many disparate domains having wide
bearing on the creation and understanding of those complex
systems.
1.1 Pairwise Link Prediction in Network Science
Network Science as an academic field is methodologically
diverse and ever-evolving [1]. Notably, the field draws
•K. Dick, D. G. Kyrollos, and J. R. Green are with the Department of
Systems & Computer Engineering, Carleton University, Ottawa, Ontario,
Canada.
E-mail: {kevin.dick,daniel.kyrollos,james.green}@carleton.ca
•The authors are also members of the Institute of Data Science, Carleton
University, Ottawa, Canada.
1. In this work we use the terms edge and link interchangeably as well
as the terms network and graph.
from graph theory in mathematics [2], social structure from
sociology [3], inferential modelling from statistics [4], and
statistical mechanics from physics [5]. This work and the
methods described herein are contributions to link predic-
tion and data mining from computer science [6].
Link prediction is a fundamental problem in modern
information science. Certain traditional approaches lever-
aged methods such as Markov models and/or statistical
modeling to infer new edges [7], [8]; however, these fail
to capture structural characteristics of networks such as
communities, hubs, and hierarchical organization [9]. It is
the nascence of the Netflix competition circa. 2006-2009 that
spurred research engagement and the popularization of
similarity-based and matrix-factorization type link predic-
tors, notably for the task of predicting user-movie ratings
on the five-point star scale [10]. Through the following
decade and with the emergence of deep learning and the
growing availability of high-performance computing in-
frastructure, new classes of network representations have
emerged including network embedding that seeks to map
higher-dimensional nodes to a lower dimensional latent
space while conserving neighbourhood structure [11]. Two
notable examples include DeepWalk [12] and node2vec
[13], each with demonstrated improvement of link predic-
tion performance for diverse domain applications. Beyond
graph representation learning, an emergent class of end-to-
end graph neural networks (GNN) have been introduced
[14] enabling improved graph feature learning [15]. (Deep)
GNNs are reported to be highly competitive with state-of-
arXiv:2210.11616v1 [cs.LG] 20 Oct 2022
2
the-art (SOTA) graph kernel methods and outperform deep
learning methods for graph classification tasks [14].
1.2 Pairwise Relationships for Scientific Discovery
Accurate link prediction for applied real-world problems
have important consequence to the understanding and in-
terpretation of the complex systems that they represent.
Algorithms capable of accurately predicting missing links
enable data mining applications, accelerate network data
collection, and improve network model validation. Recent
trends have seen improved performance in link predictions
through multi-sided recommendation, model calibration,
model stacking, and/or large-scale ensembles.
In the work of Berlusconi et al., link prediction was lever-
aged to identify possible missing links in a criminal network
by considering multi-sided similarity measures of pairs of
nodes in the network and inferring them a contrario with the
assumption that putative social ties will be characterized by
opposite features [16].
Model “stacking” is an ensemble approach that learns
a meta-model that learns how to leverage the predictions
from individual component predictors [17] that differs from
conventional bagging and boosting. Unlike bagging, in
stacking, the contributing component models are typically
diverse (e.g. a variety of algorithms or (deep) learning mod-
els) and fit on the same dataset (as opposed to a sampling of
the training dataset). Unlike boosting, in stacking, a single
model is used to learn how to best combine the predictions
from the contributing models as opposed to a sequence of
models that correct the predictions of prior models.
Ghasemian et al. reported a systematic evaluation of
203 individual link predictor algorithms, representing three
popular families of methods, applied to a large corpus of 550
structurally diverse networks from six scientific domains
[18]. Excitingly, the stacked models achieved (near) optimal
levels of accuracy over synthetically generated datasets for
which the maximally achievable level of performance was
known and the stacked meta-classifiers classifiers, when
trained on real-world datasets, were consistently superior
to component models [18]. These findings demonstrate the
broad utility of stacked meta-classification methods on di-
verse problem sets.
Finally, model calibration is crucial in high-stakes scenar-
ios such as drug-target interaction (DTI) prediction where
end-users need trustworthy and interpretable decisions. In
a binary classification formulation (e.g. positive prediction
indicates a putative interaction and a negative prediction
indicates non-interaction) probability calibration is impor-
tant when the confidence in a given prediction must make
probabilistic sense. For example, if a given model predicts
a fact is true with 80% confidence, the model should be
correct 80% of the time. Adherence to this property is
evaluated by means of calibration/reliability curves which
plot the model’s mean predicted value along the x-axis and
the fraction of positives along the y-axis with the identity
function representing perfect calibration. In the work of
Tabacof and Costabello, the application of Platt scaling [19]
and isotonic regression [20] was used to calibrate knowledge
graph embedding models [21] and Wang et al. proposed
methods broadly applicable to graph neural networks [22].
Calibration methods are a form of post-processing/model
stacking for refining prediction scores, which is conceptually
similar to RP.
1.3 Reciprocal Perspective for Biomedical Discovery
The fields of bioinformatics and computational biology have
been central application domains for the study and ex-
emplification of network-based methods; these approaches
have been widely used to investigate biological systems at
various scales, whether in macroscopic ecological dynam-
ics down to microscopic and molecular interaction studies
[23]. The Reciprocal Perspective (RP) methodology was
first discovered and investigated within these domains by
reframing several applications as pairwise link prediction
problems.
Briefly, the RP framework is a cascaded classifier that
refines raw pairwise link prediction scores by considering
the context of all possible link scores involving either el-
ement of the pair. To provide a concrete example, con-
sider the task of predicting all protein-protein interactions
(PPI) among an organism’s nproteins. From all the n(n+1)
/2
possible pairs of proteins, a subset are known to interact
(positive) or not interact (negative) through experimental
validation studies. These known links are useful for training
and evaluating a PPI prediction algorithm, denoted fΘ(x).
RP begins by applying this initial predictor,fΘ(x), to infer
prediction scores between all n(n+1)
/2possible pairs (includ-
ing those that are known via a cross-validation schema).
This results in a complete Knweighted-edged graph of
predicted scores from fΘ(x)with a corresponding complete
adjacency matrix of predicted scores that we denote the
Comprehensive Prediction Matrix (CPM). A given row i
sliced from the matrix is a n×1vector of all scores between
protein iand every other protein in the proteome. Similarly,
a given column jsliced from the matrix is a 1×nvector
of all the scores between protein jand every other protein
in the proteome (including protein iin cell i, j). These
vectors, when each sorted in rank-order by monotonically
decreasing score, represent a pair of One-to-All (O2A) score
curves. The pair of O2A curves share only a single common
value representing the query pair i, j (identical in value,
but usually differing in sorted rank). These O2A curves
(which we also refer to as protein i’s perspective, and protein
j’s perspective) typically exhibit a characteristic ”S”- or ”L”-
shaped distribution with a baseline that we can attribute to
non-interacting pairs. The singular common point between
the two reciprocal perspectives enable a number of numeric
features to be computed characterizing the location of that
point in the broader context of all the inferred scores within
the two distributions and with respect to their baselines.
Intuitively, we wish to identify a pair for which their shared
score would be relatively high-scoring with respect to each
perspective’s baseline. Thus, the RP framework extracts, for
any pair of proteins i, j, a new numerical vector of O2A-
derived features that can be leveraged to subsequently train
and evaluate a cascaded predictor that refines the original
predictions. Additionally, the cascaded predictor can also
function as a means of combining multiple experts (CME)
by fusing the RP feature vectors obtained from the CPMs
generated by numerous initial predictors to function in a
stacked generalization schema.
3
The original RP formulation was proposed for intra-
species PPI prediction tasks [24] and then later for improv-
ing inter- and cross-species SOTA predictors [25]. In each
case, the link prediction problems were formulated as a
binary classification task seeking to maximize F1 score, pre-
cision, and recall. Thereafter, RP demonstrated significant
improvement for the task of micro-RNA (mRNA) target
prediction in [26]. This work formulated the link prediction
task as a classification-type problem seeking to maximize
the Area under the Reciever Operating Characteristic curve
(AUC ROC) and the Area under the Precision-Recall curve
(AUC PR). Most recently, the RP framework was lever-
aged for drug-target interaction prediction [27]; RP was
here generalized to regression-type problems to minimize
the Root Mean Squared Error (RMSE) and maximize the
Concordance Index (CI). Finally, the original PPI problem
formulation was leverage for the study of SARS-CoV-2 [28]
and adapted to use the cascaded RP model for the CME in a
n= 2 stacking of two SOTA PPI classifiers. In each problem
domain, the biological elements being operated upon were
arranged into a network-like representation. This suggests
that the RP framework may be effectively generalized to
operate on abstracted element-element link prediction in the
broader domain of Network Science.
1.4 Generalizing Reciprocal Perspective for (m)any
Pairwise Applications
Following from the brief RP methodology description
above, a complete technical description of the RP framework
is later described in section3.2, however, to understand the
generalization of the RP framework to a broader class of
problems, we first review how RP was leveraged for each
independent bioinformatic application and how these relate
to generalized link prediction. A conceptual overview of the
RP framework is depicted in Fig. 1A.
In each bioinformatic application domain, SOTA pre-
dictors (referred to as the initial predictors) were leveraged
to generate pairwise predictions for all possible combina-
tions between elements. As previously described, the infer-
ence of all pairwise prediction scores between all elements
forms the CPM, comprising a complete adjacency matrix
representing a Kncomplete graph with weighted edges.
The predictor-specific CPMs are then leveraged to extract
pairwise RP features for use in the training the cascaded
predictor. In the intra-species PPI tasks, we generated a
square CPM, Rn×n, for all possible pairs between nproteins
within an organism (e.g. n=∼21,000 for Homo sapiens).
In this work, we denote this RP formulation as Homo-RP,
indicating that CPM row and column indices represent the
same node type (element sets A=B), the CPM diagonal
represents self-predictions, and the complete graph can be
expressed as Kn.
For other prediction problem formulations between two
differing sets of elements (e.g. inter- and cross-species PPI
prediction, mRNA-target interaction prediction, or drug-
target interaction prediction), with element set sizes of n
and m, respectively, the resultant CPM, Rn×m, is typically
non-square (n6=m). In this work, we denote this RP
formulation as Hetero-RP, indicating that CPM row and
column indices are different and distinct sets, the CPM is
a complete bipartite adjacency matrix, and the complete
graph can be expressed as Kn,m.
The ability to computationally generate CPMs has
only recently become possible with the advent of high-
performance computing infrastructure and algorithmic op-
timizations. While domain-specific SOTA methods are
methodologically diverse, fortunately the CPM generation
process is embarrassingly parallel and computationally effi-
cient methods can be scaled to generate such large matrices.
Evidently, these matrices grow with the square of the num-
ber of elements (i.e. n2or nm). In generalizing RP to the
broader class of Network Science methods, it is therefore
important to select a class of link predictors applicable to
diverse datasets and problem formulations. To that end,
we opted to demonstrate the efficacy of RP on predicting
the five-point star user-item rating with diverse Recom-
mendation System (RS) algorithms leveraged as the initial
predictor. Given the previous success in leveraging RP in
diverse biomedical domains with a wide variety of problem
formulations and its demonstrated utility in combination
with RS algorithms in this work, we seek to demonstrate the
generalized utility of the RP framework for many pairwise
applications.
1.5 Reciprocal Perspective for Recommendation Sys-
tems
The prediction of user-item relationships has considerable
(eCommerce) financial consequence within diverse indus-
tries. Most notably, the study and development of RS al-
gorithms has been greatly popularized and adopted by
industry in the last 15 years. RS algorithms were widely
adopted in the technology industry and active applied re-
search reported their utility to an increasingly broad array
of prediction tasks. Fortunately, RS algorithms are usually
computationally efficient and scalable with sufficient sup-
porting compute infrastructure. This class of link prediction
algorithms is well-suited to evaluating the impact of apply-
ing RP to diverse application domains. To demonstrate the
universal application of RP to network-based abstractions
of real-world problems, we consider three different data
sources and 13 different RS algorithms. We further demon-
strate how RP can be leveraged as a combination of multiple
experts ensemble technique for n2component models
(Fig. 1B).
2 RELATED WORK
The RP framework is a cascaded semi-supervised machine
learning layer that leverages features derived from the CPM
in a space unique from the original application domain
[24]. There exist many semi-supervised approaches for link
prediction tasks [29], [30], [31], [32] including link propaga-
tion methods, label propagation, active learning, and multi-
view learning; however, each of these approaches operate
on the graph characteristics and/or feature spaces of the
original domain to discover new links among disjoint nodes.
The RP framework differs in that it leverages context-based
features derived from the predicted scores (produced from
the application of an initial predictor) of the CPM within
a cascaded machine learning method. The RP framework
4
A: Conceptual Overview of the Reciprocal Perspective Method to Improve Recommendation System Predictions
Initial
Predictor
Data
Source
Initial CPM
Predictions
Final
Predictions
B: Experimental Framework for using Reciprocal Perspective on Component Models and in an Ensemble
...
b: Recommendation
Ensembled
RS Model
...
System Algorithms
d: Extract RP
Feature Vectors
RS-RP
e: Cascaded Meta Model
Ensembled with
RP Feature Vectors
a: Benchmark
Ratings
[Train Dataset]
c: Generated Initial
Predictions in
[Prediction Model] [Trained Model]
...
f: Final Predictions &
Performance Evaluation
[Test Dataset]
RMSE
Eval.
Metric
Dataset(s) Cross-Validation
3x Benchmark
Ratings Datasets
24x Amazon Product
Ratings Datasets
+
...
...
...
Fusion of
Multiple Exp.
{
Stacked Model
users
items
users
items
...
...
Baseline
Baseline
One-to-All
User, u
Perspective
Item, i
Perspective
Feat. Extraction
Cascaded/Stacked
Learning Model
Reciprocal Perspective Framework
...
Fig. 1. Conceptual Overview of the Generalized Reciprocal Perspective Framework.
is, therefore, analogous to the use of stacked classifiers
in pattern classification/regression, however goes beyond
these approaches to consider all predictions involving either
element in the pair. These derived features reside in a space
(i.e. domain agnostic) that differs from those leveraged by
the initial predictor thereby differing conceptually from the
other forms of semi-supervised machine learning.
RP is conceptually most similar to transductive learning
in that the few labelled training samples are situated in
the context of the unlabelled data. Transductive learning
algorithms learn the latent structure of the data by clustering
all (un)labeled samples and then labeling a given cluster
using those few samples for which labels are available,
while additionally accounting for the data distribution [33],
[34]. These transduced labels are then used for prediction
decisions, as exemplified in [35]. The context-based fea-
tures leveraged by RP are, however, derived from predicted
scores which reside in a feature-space that differs from the
original feature space where transductive learning would
perform its clustering.
Another fundamental facet of RP (as emphasized by its
nomenclature) is reciprocity. The concept of reciprocity is
common to Network Science applications and integrated
within many methods given the intrinsic duality of pairwise
relationships. Ali et al. predicted reciprocal links within
directed citation networks, where author aicreates a direct
link (through citation) to disconnected author aj, creating
a one-way (parasocial) relationship with ajthat forms a
two-way (reciprocal) relationship when author ajcreates a
link (through citation) to ai[36]. Reciprocal links in directed
graphs have also been leveraged to provide additional in-
formation on directed closure triads in the work of Li et al.
[37]. RS algorithms themselves will incorporate modeling
terms integrating factors from both set contributions as seen
in matrix factorization methods such as Singular Value De-
composition (SVD) [38]. While the reciprocity underpinning
the RP framework similarly seeks to leverage information
contributed from two differing views, the RP features are
derived from the pair of O2A curves sliced from the CPM.
This enables the contextualization of a given pair’s inferred
score in the global context of (un)labelled score distributions
from these reciprocal views. This differs importantly from
the locally propagated factors described in the cited work
that will not make use of a complete graph of inferred
scores.
The final fundamental facet of the RP method is its ap-
plication in a cascade for refinement of initial predictions.
The class of calibration techniques is one that seeks to map
raw prediction scores to posterior probabilities based on
observed incidence rates at different score thresholds. As
reported in the work of [39], the two methods for calibrating
these initial model predictions are Platt Calibration and
Isotonic Regression. The former transforms initial model
predicitons into a posterior probability using a sigmoid
transform [19], whereas the latter learns an isotonic function
to adjust the calibrate (i.e. refines) the model predictions
[20]. In both cases, a held-out calibration set is required
to successfully generate these prosterior probabilited [39].
The RP framework differs from these methods in that the
model refinement is learned in a cascaded model leveraging
context-base features derived from the O2A curves.
Contributions. In this work, we generalize the RP frame-
5
A: Movielens 100k Dataset B: Movielens 1m Dataset C: Goodbooks 6m Dataset
35K
30K
25K
20K
15K
10K
5K
1.0 2.0 3.0 4.0 5.0
Frequency
0
Five-Point Rating
350K
300K
250K
200K
150K
100K
50K
0
1.0 2.0 3.0 4.0 5.0
Five-Point Rating
0.25M
0
0.50M
0.75M
1.00M
1.25M
1.50M
1.75M
2.00M
1.0 2.0 3.0 4.0 5.0
Five-Point Rating
Fig. 2. Distribution of Five-Point Ratings across Benchmark Datasets.
work outside of the bioinformatics domain to demonstrate
its applicability to link prediction and machine learning
problems capable of being abstracted to a network-based
representation. In this work we consider the class of RS
algorithms to demonstrate RP’s applicability to regression-
type tasks. We leverage three disparate benchmark datasets
with user-item ratings on a five-point scale and apply the RP
framework in a cascade to over 12 unique and scalable RS
algorithms integrating various information to inform their
predictions. The demonstration of RP’s applicability to a
regression problem here, taken together with our previous
demonstration that RP is applicable to homo- and hetero-
classification tasks in diverse applications within bioin-
foramtics, collectively support the assertion that RP is near-
universally applicable to link prediction problems.
3 DATA & EXPERIMENTAL METHODOLOGY
The following section describes the datasets considered
within this study (section 3.1), then introduces the gen-
eralized RP framework for Network Science applications
(section 3.2), describes the RS algorithms that were used to
produce CPMs (section 3.3), and finally outlines all of the
experimental design elements to report on the RP contribu-
tion to each of these prediction tasks.
3.1 Rating Benchmark Datasets
The datasets considered in this work each represent user-
item ratings on a five point star rating scale and are eval-
uated as a regression-type prediction task. We considered
the MovieLens benchmark dataset with user-movie review
ratings datasets, the GoodBooks user-book review ratings
dataset, and finally the Amazon Product Review datasets
with a broad diversity of user-product review ratings over
numerous product categories. Dataset specifics are dis-
cussed below and the distribution of five-point ratings is
illustrated in Fig. 2.
3.1.1 MovieLens Benchmark Dataset
The MovieLens datasets are a movie RS benchmark dataset
prepared by the GroupLens research group at the Uni-
versity of Minnesota [40]. By collecting user-ratings from
movie viewings, a user-item ratings matrix was created for
the purpose of recommending movies for users based on
shared interest. The MovieLens benchmarks are organized
into various-sized variants according to total rating number
among different movie and user-bases. In this work, we
considered the MovieLens 100K benchmark (with 100K rat-
ings distributed among 943 users and 1,682 movies) as well
as the superset MovieLens 1M benchmark (with 1,000,209
ratings distributed among 6,040 users and 3,900 movies). A
tabulation of the benchmark statistics including density and
sparcity metrics are listed in Table 1.
3.1.2 Goodbooks Benchmark Dataset
The GoodBooks datasets is a book RS benchmark dataset
organized from the Goodreads user-book ratings online
forum. The dataset has various version of differing sizes
and complexities that have been used to benchmark RS
algorithms as well as form the basis of numerous Kaggle
competitions [41], [42], [43]. In this work, we leveraged
one of the largest versions with six million ratings between
53,000 users and 10,000 books. A tabulation of the bench-
mark statistics including density and sparcity metrics are
listed in Table 1.
3.1.3 Amazon Review Dataset
The Amazon Review Datasets (2018 update) was originally
assembled at the University of California San Diego [44],
[45]. The updated dataset contains a total of 233.1 million
reviews spanning May 1996 - Oct 2018 and 24 different
categories of items. When subdivided according to product
category, each dataset represents a gargantuan CPM size
that is incredibly sparse. We tabulate the CPM size (in
number of unique predictions, and requisite RAM/storage
space to manipulate the object) as well as the matrix sparcity
and density in the Supplementary Materials. With a single
prediction represented by a single precision float in 4 bytes,
we computed CPM storage as nm
/2×4 bytes, revealing the
range of CPM storage infrastructure to be between 678 GB
for the smallest and 722.6 TB for the largest.
To account for these overly sparse and intractably large
CPMs, we considered the k-core of each product category
where each user and item must provide or receive at least k
ratings. This space reduction approach leverages a constant
thresholding across all product categories, however when
ranked by number of CPM elements, there are two orders
of magnitude between the smallest and largest datasets
suggestive that a constant kthresholding may be too ex-
treme for the smaller datasets (too many users and items are
truncated) and perhaps insufficient for the largest datasets
(too few user and items are truncated). In this work, we
摘要：

1GeneralizedReciprocalPerspectiveKevinDick,GraduateStudentMember,IEEE,DanielG.Kyrollos,GraduateStudentMember,IEEEandJamesR.Green,SeniorMember,IEEEAbstract—Acrossmanyapplicationdomains,real-worldproblemscanberepresentedasanetwork,withnodesrepresentingdomain-specicelementsandtheedges(orlackthereof)ca...

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