Non-Euclidean Mixture Model for Social Network Embedding
Authors: Roshni Iyer
YEWEN WANG
Wei Wang
Yizhou Sun
Venue: NeurIPS 2024 poster (2024)
Keywords: network science
social influence
representation learning
Primary area: machine_learning_for_social_sciences

Abstract: It is largely agreed that social network links are formed due to either homophily or social influence. Inspired by this, we aim at understanding the generation of links via providing a novel embedding-based graph formation model. Different from existing graph representation learning, where link generation probabilities are defined as a simple function of the corresponding node embeddings, we model the link generation as a mixture model of the two factors. In addition, we model the homophily factor in spherical space and the influence factor in hyperbolic space to accommodate the fact that (1) homophily results in cycles and (2) influence results in hierarchies in networks. We also design a special projection to align these two spaces. We call this model Non-Euclidean Mixture Model, i.e., NMM. We further integrate NMM with our non-Euclidean graph variational autoencoder (VAE) framework, NMM-GNN. NMM-GNN learns embeddings through a unified framework which uses non-Euclidean GNN encoders, non-Euclidean Gaussian priors, a non-Euclidean decoder, and a novel space unification loss component to unify distinct non-Euclidean geometric spaces. Experiments on public datasets show NMM-GNN significantly outperforms state-of-the-art baselines on social network generation and classification tasks, demonstrating its ability to better explain how the social network is formed.

Non-Euclidean Mixture Model for Social Network Embedding
Roshni G. Iyer
Yewen Wang
Computer Science Department University of California, Los Angeles Los Angeles, California, USA roshni g gerercss wwla.ellao..oooo... edu
Computer Science Department University of California, Los Angeles
Los Angeles, California, USA wyw 080 40044 gma i com
Wei Wang
Yizhou Sun
Computer Science Department University of California, Los Angeles Los Angeles, California, USA weiwang@cs wwla.ullu.u.
Computer Science Department University of California, Los Angeles Los Angeles, California, USA yzsun0cs. wwteeess..cllee...
Abstract
is largely agreed that social network links are formed due to either homophily or social influence. Inspired by this, we aim at understanding the generation of links via providing novel embedding-baid from existing graph representation learning, where link generation probabilities are model the link generation as a mixture model of the two factors. addition, we model the homophily factor in spherical space and the influence factor in hyperbolie space accommodate the fact that (1) homophily results cycles and (2) influence results in hierarchies in networks. We also design special projection to align these spaces. We call this model Non-Euclidean Mixture Model, i.e., NMM. We further integrate NMM with our non-Euclidean graph variational autoencoder (VAE) framework, NMM-GNN. NMM-GNN learns embeddings through a unified framework which uses non-Euclidean GNN encoders, non-Euclidean Gaussian priors, a non-Euclidean decoder, and a novel space unification loss component to unify distinct non-Fuclidean geometric spaces. Experiments public datasets show NMM-GNN significantly outperforms statc-of-the-a baselines on social network generation and classification tasks, demonstrating its ability to better explain how the social network is formed.
graph formation model. Different
defined simple function of the corresponding node embeddings,
Introduction
Social networks omnipresent because they used for modeling interactions among users social platforms. Social network analysis plays key role in several applications, including detecting underlying communities among users (1, classifying people into meaningful social classes (2]. and predicting user connectivity 3 Most existing embedding models are designed based on the homophily aspect of social networks 4
5]. They utilize the intuition that associated nodes in social network imply feature similarity. and an edge is usually generated between similar nodes. Prior works have used shallow embedding models to represent homophily. like matrix factorization and random-wall (Section 2), which are parameter intensive and do not employ message passing. As an improvement, graph neural network (GNN) models (Section 2 have been proposed more effectively capture homophily by representing node through its local neighborhood context.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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However, research of RaRE [6] and work of [7] show homophily is insufficient, and social influence also critical in forming connections. This is due to popular nodes having direct influence in forming links [8] For example, in Twitter network, users tend to follow celebrities in addition to users who share similar interests [9]. Though RaRE jointly models both factors, has limitations in modeling capabilittis. Specifically, it is parameter intensive as each node embedding is fully parame- terized through a Bayesian framework. Further, RaRE assumes graphs are transductive, limiting its performance in the practical inductive setting where new links not seen during training must be predicted. Moreover, nearly all works embedding social networks utilize a single zero-curvature Euclidean space, when in reality, network factors may create different topologies. Specifically, edges generated by homophily tend to form cycles [10]. while edges generated by social influence tend to form tree structures [ [2. From Riemannian geometry, people have found that networks with cycles are best represented by spherical space embeddings [13], while tree structured networks are best represented by hyperbolic space embeddings [1]. Thus, end-to-end model to bridge social network embeddings distinct non-Fuclidean geometric spaces is a promising direction.
Our motivation is two-fold: (I) We aim to understand how the social network is generated e.g., which factors affect node connectivity and what topological patterns emerge in the network as result. (2) Using our learning from (I), we aim to design more realistic deep learning model to explain how the network is generated (inferring new connections). We summarize our contributions as follows:
We propose Graph-based Non- Eucllieen Mixture Model (NMM) to explain social network generation. NMM represents nodes via joint influence by homophily (modeled in spherical space) and social influence (modeled in hyperbolic space), while seamlessly unifying embeddings via our space unification loss.
To our knowledge, we are also the first to couple NMM with a graph-based VAE learning framework, NMM-GNN. Specifical, we introduce a novel non- Euclidean VAE framework where node embeddings are learned with a powerful encoder of GNNs using spherical and hyperbolic spaces, non- Enclliide Gaussian priors, and unified hon-Euclidean optimization.
Extensive experiments on several real-world datasets on large-scale social networks, Wikipedia networks, and attributed graphs demonstrate effectiveness of NMM-GNN in social network generation and classification, which outperforms state-of-the-art (SOTA) network embedding models.
2
Preliminary and Related Work
We provide an overview of social network embedding models and discuss advancements in non- Euclidean graph learning.
2.1 Social Network Embedding
Several works that embed social networks merely model homophily 15
capturing node-node
similarity, without also considering node's social influence node popularity e.g., celebrity. Homophily-based models include shallow embedding models and GNN embedding models. Most shallow embedding models are either based on matrix factorization [16] random-walk (17). Though
GNN models [8] effectively learn on large networks and in inductive settings, they still fail to model the social influence factor in the network. Further, even the model that captures both homophily and social influence, RaRE (6, has limitations in that its fully-parameterize node embeddings require large parameter size be learned, and it models all nodes in Euclidean space, an approach not effective in capturing different topologies (e.g., cycles and hierarchy) in the social network.
2.2 Non-Euclidean Geometry for Graphs
Non-Euclidean geometric spaces, commonly used to model surfaces in mathematics and physics, are curved geometries that include spherical spaces with positive curvature, and hyperbolic spaces with negative curvature [19]. Works have recently found Euclidean space modeling be insufficient for -un-luclde graph-structured data (20]. Namely, spherical spaces have been shown to effectively embed graphs with eyclic structure due to their positive curvature 21 13. while hyperbolic spaces
have been shown to effectively embed graphs with hierarchical structure due to their negative
curvature and exponential or "tree-like" growth of the space (2 23). HGCN (4 and 25) are critical works exploring GNNs in non-Euclidean spaces. Both these hyperbolic GNN methods have shown

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significant improvement on benchmark datasets by preserving hierarchical structure of graphs. In knowledge graphs, [26] has achieved notable performance by modeling graphs in non- non-Euclidean spaces of various curvatures, using both hyperholic and spherical space.
3 Methodology
Hyperbalic Geussia Prior
recovered graph
KL Divergence
MCA
min (xf x
Hyperbolic GIN
Poincard ball (3D)
p Ley 1 (x" 2)"))
pe(eu 1)
plc diist, x7))
Misture Model
Spherical
Encoder
Gaussian Prior
Spherical
Decoder
Surface of Spherical (2D)
terms
(a)
(b)
Figure 1: Model Architecture. (a) Architecture overview NMM-GNN, a Non-Euclidean Mixture Model with non-Euclidean VAE framework. (b) Illustration of space unification loss of NMM-GNN. = from the
hyperbolic space projected its corresponding point in the spherical space, 2²² such that its geodesic distance ensured to close to v; existing spherical space representation. x!
social network G (V, A) consists of set of vertices V ] and associated adjacency matrix Gij E A, where exj is edge from to ui. We aim to design a model to jointly learn both node homophily and social influence representation, denoted 2 and z respectively, that can best explain the social network in terms of link reconstruction. In this section, we describe our architecture. First, we introduce a novel non-Euclidean mixture model, called NMM, to model the probability of a new link. Second, we add a non-Euclideea GNN encoder to enrich NMM, called NMM-GNN, which is connected to the variational autoencoder framework. Source code is in the Appendix. For details on future directions, the reader is also referred to the Limitations section in the Appendix.
3.1 Overview
To model both factors of homophily and social influence that affect graph connectivity, we model each node U; with homophily regulated representation, 2/. and social influence regulated representation, #! A unified framework is designed such that 2; and = influence each other bi-directionally and are seamlessly merged through NMM. NMM utilizes non-Euclidean geometric spaces to better represent homophily and social influence components which produce curved structures like cycles and trees. To improve NMM, we enrich its encoder by GNNs, (2 |G) and e (z,"s( G). To ensure generated embeddings are in spherical and hyperbolic spaces respectively, Eunclliddean GNNs are adopted. The enhanced model is NMM-GNN, which has a clear connection to the VAE framework.
Figure Ta aa) illustrates the architecture overview of NMM-GNN. Specifically, the encoder component maps nodes into homophily based embedding, z5, in spherical space (for homophily generated cycles) and social influence based embedding. H in hyperbolic space (for social influence generated trees), which follow non non-Euclideat prior distributions. The embeddings are passed into our mixture model decoder, which models the probability of link as mixture of a homophily based distribution component and a social influence based distribution component. The objective is to maximize the likelihood observe the links, equivalently minimize the link reconstruction loss. In addition, the two geometric spaces are ensured to be aligned together via space unification regularization term, to make sure the two embeddings of the same node are corresponding to each other.
3.2 Modeling via Non-Euclidean Mixture Model (NMM)
Geometrically, the space of NMM is a spherical surface inside a unit Poincaré ball. Each node 222 corresponds to (1) point in the Poincaré ball, z and (2) point the spherical surface,
'The geometric space community conventionally uses S to represent spherical space and H to represent hyperbolic space.
3

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zi' where both spaces are embedded inside Euclidean space. These two points are aligned by enforcing projection of " onto the spherical surface to be close to 2; as projecting a star onto carth's atmosphere, shown in Fig. [b). We note that the space alignment does not mean embeddings are enforced to be the same in two spacef Rather, we require the projection of 2." is close to i's embedding in spherical space z In this case, the two distances of spherical and hyperbolic spaces are also different from each other (norm space difference vs. spherical geodesic distance). Without the space alignment, =! has too much degree of freedom, which can move freely as long its norm is kept the same. Lastly, the probability of generating link is a mixture of the probability of generating the link in each non-Fuclidean space.
3.2.1 Modeling for homophily
Representation of homophily regulated nodes. We embed homophily based representation of node UA or 2 on the surface of the spherical ball in spherical space, S², Formally, S {2 E R=112 5 is the d-dimensional w -nrm ball, where ||-I| is the Euclidean norm and u E [0. 1) is a constant to ensure the spherical surface is inside the unit Poincaré ball. To better capture homophily distribution, p.={). we use spherical Gaussian distribution Gs(+) as spherical prior, described in Table I B is the axis or direction of the lobe controlling where the lobe is located on the sphere, and points towards the center of the lobe; À is the sharpness of the lobe such that as this value
increases, the lobe will become narrower in width; and is the amplitude or intensity of the lobe, corresponding to the height of the lobe at its peak.
Link prediction using homophily based distribution. The probability of link eiy = 1 between nodes U; and Vg. also described in Table I is determined by the geodesic distance between o. and 25.
(x) xx)) arccos( (z, >))) is the geodesic distance between 2,S and z5. J 0 and B 2 0 model parameters; and (.. denotes the inner product. The probabilistic model is designed so nodes greatly dissimilar (exhibiting low homophily), or dist, =)5) + +00, have low probability of a link being generated, or Phom(e;j = + 0. In the case of nodes exhibiting high homophily, they either (1) may be connected due to highly similar characteristics, simply because the individuals do not know each other. Our distribution models both scenarios. Note
(2) may not be connected
when dist, (2 > - O, Phom (Cij 1) Itea. which can be interpreted factor to control sparsity of the network.
Table I: Homophily regulated nodes: Distribution Prior and Link Generation Probability (a) Spherical Distribution Prior
(b) Link Generation Probability
p(x;)) Gs(=f;B,A,0) ac MB.af
Phom = 1) p(e., 1/dist (z
1 + (xd 255)+
3.2.2 Modeling for social influence
Representation of social influence regulated nodes. We embed social influence based representa- tions of node U; or z!! on the Poincaré ball from hyperbolic space, Hd+1 to better capture resulting hierarchical structures. Social influence regulated nodes are represented as points, =H belonging in- side the Poincaré (open) ball in Hdd+1, Formally, Hd+1 = {" E [R++ ||=="| = w }, w E [0. 1). is (d+ (---1)-ll -dddmennsoonal w -norm ball, where ||. is Euclidean norm. We assume center of the Poincaré ball is aligned with center of the sphere, and one dimension larger than the sphere to ensure the spherical surface is inside the Poincaré ball. To better capture a social influence regulated distribution, p(=!") ). we use Hyperbolic Gaussian distribution (- as -uc-lieli Gaussian prior, described
in Table 2 tf is the origin of (r, w) for radius r and angle w in polar coordinates. z." is the center of mass and Ç > 0 the dispersion parameter. where the dispersion dependent normalization constant Z(C) accounts for the underlying non- -uuc-ielld geometry. Z(C) is partitioned into angular w and radial r components. I'(-) is Euler's 's gamma function, and distu(.) is the hyperbolic distance between two hyperbolic space node embeddings, æ and y, where ||-|| denotes Euclidean norm.
Note that it is impossible to equate these two embeddings directly as they are in different geometric spaces.

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Link prediction using social inftuence based distribution. We model existence of an edge, Eaj = 1 between nodes v, and Uj. also described in T as function of norm space difference, dist, (-). between two hyperbolic space node embeddings, z and 2 We utilize norm space difference as opposed to hyperbolic distance, since nodes of similar social influence status may have large distance in the Poincaré (open) ball due nodes possibly being placed towards the ball's boundary. This is because, at the boundary of the ball, nodes become infinitely distanced apart. Thus, to allow for numerical stability and to capture social influence difference (in which the higher the social influence of nodes indicated by large in-degree and smaller out-degree, the closer they are embedded towards the center of the ball), we utilize norm space as indicator. Consistent with the notation of (6), node with higher social influence is associated with smaller norm value. C. and D are learned model parameters is the norm difference between z and x , C > 0 and D 2 O are model parameters; and norm (-) denotes the LI normalization function. The probabilistic model is designed such that nodes largely different in popularity (exhibiting high social influence). or dist, .> - +oc, have high probability of a link being generated, or Prank(C; = 1) - 1. In the case both nodes exhibit low social influence (such as having similar social rank), they either (1) may be connected due to highly similar characteristics, or (2) may not be connected simply because individuals do not know each
and norm space of a vector is norm x) = ||0||- dista (z. xx3" = |_ormiz?" (1) - norm (x #1 )
other. Our distribution models both scenarios. When dist, (z x + O, Prank = 1) 14.0 which can be interpreted as another factor to control sparsity of the network.
Table 2: Social influence regulated nodes: Distribution Prior and Link Generation Probability (b) Link Generation Probability
(a) Hyperbolic Distribution Prior
p(z.) = GH (z," =
i x.H5
(1) Prank (cy- = 1)
= pless = 1|dista, ,x5²) e (xdimnnee )4
di f ,7
= 2(0)
2(c) = ()) (c)
(2)
1 e (xf (f)
(2) = Vol(TH
(3)
#
T(2 1)
Z-(C)= / +o0
x< sinh
sinh"(() (r) dr
(4)
d'
22
(2k 103-d1) d)G
1)*
2
ce
2
erfe
2
V2
2|
2
y
(a, .) = arrcosh (1
(5)
(1 |
[2)
10/ 2)
3.2.3 Non-Euclidean Mixture Model
Link Prediction Using Mixed Space Distribution. Since both homophily and social influence affect the connectivity structure of social networks, we model existence of a new link between nodes, po(eyy = 1). as a weighted combination distribution of these factors. Specifically, our non-Euclidcan mixture model is weighted combination of homophily based distribution, (eam = 1), and social influence based distribution, Prank (eij = 1), with learned weight y:
pa (el, = 1) = pe (el, = 1 1|2. x5 x ,
(6)
Phom (e., = 1) + (1 7) Prank =1)
where Phom (eij = 1) is modeled in positively curved spherical space since homophily based links may form cycles due to similarity connections between node clusters. Prank = 1) is modeled in negatively curved hyperbolic space since social influence based links may form tree-like structures due to popularity-basee social hierarchy between node clusters. We would like to highlight that the link between nodes i and is a mixture model because each link a weighted combination of influence from both spherical and hyperbolic spaces (not one or the other) as evidenced in Equation 6. As shown in Figure 1b, the same node has two representations - one the spherical space and one in the hyperbolic space, and because they represent the same underlying node, they need to be aligned. Therefore, these two network factors do not contradict each other, but rather work together to explain how links are formed between users.
5

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3.3 Modeling via Non-Euclidean VAE on Graphs
We enrich the encoder of NMM to generate better embeddings. To do so, we explore GNN methods which have been shown to be more effeective than shallow embedding methods (see Experiments and Ablation Studies). We refer to this enriched NMM model as NMM-GNN. NMM-GNN uses non-Fuclidean VAE its learning framework. The framework integrates mixture of different non- Euclidean geometric spaces e.g., hyperbolic and spherical spaces, for learning of encoder, decoder, and node prior distributions, and ensures geometric spaces unified during training.
Encoder model. The encoder learns two corresponding embedding representations per node o to produce spherical embedding =f and hyperbolic embedding 2 For homophily regulated nodes in spherical space, S any spherical space GNN (SGNN) can be applied, and for social influence regulated nodes hyperbolic space, H any hyperbolic space GNN (HGNN) can be applied. The general framework for SGNN and HGNN are shown in Tables and 4
z E I and =/!(0) E Id+11)) are spherical and hyperbolic feature representations of node vs layer with dimensionality d and (d + 1) respectively. f is a message-specific neural network function of incoming messages U; from neighborhood context N,, and activation function a-), typically ReLU(() for all layers but the last one being softmax(-).
Table 3: General Framework and Example Model for Spherical Graph Neural Network. (a) General Framework: Spherical GNN
(b) Example: Spherical GCN
g(++1)
qSGNN(=) (= u4)
s(++1)
(2
zi
= =u(2i(AA,) vs)
expê (o(T((
Σ
C3 Vmym
logo (2) 8(0)
1))
=o Σ s(2$0))
JENUU((
JEN,
Table 4: General Framework and Example Model for Hyperbolic Graph Neural Network. (a) General Framework: Hyperbolic GNN
(b) Example: Hyperbolic GCN
#(++1) = QUGNN m)
H(+1) =
[= |A, v:)
= expo (o(WT(
Σ SEN,U(()
Cj Viygm
Mf logō (z, (2)(())
= ( Σ f(z!! 21
JEN;
Example Non-Euclidean Encoder models. We illustrate Non-Euclidean Graph Convolutional Neural Network (Non-Eaclidean GCN) as an example GNN. Tables 3 and describe the model architectures for both Spherical GCN and Hyperbolic GCN respectively. Node features are initialized by sampling each embedding dimension from a distribution uniformly random for all nodes where
2. 20$(0) E ( + ([0, 1))d and x fT Rd+1 + (ni 1)) respectively. The retraction operator, R(-), involves mapping between spaces. For non -uclidedd spaces, retraction is performed between non-Euclidean space and approximate tangent Euclidean space using logarithmic and exponential map functions. Specifically, log!" (z tanh (i ||> i' a logarithmic map
at center from hyperbolic space to Euclidean tangent space, and expo (z.") = tanh || ()) F|pT
x!
is an exponential map at center O from Euclidean tangent space to hyperbolic space. logø (2$) tanh -1 [[1=
x is a logarithmic map at center O from spherical space to Euclidean tangent space ||
and expo $ (2²5) = tanb([1=;11) |=PT is an exponential map at center O from Fuclidean tangent space
to spherical space. where 2:5() = (())) are embeddings of node v, at layer / [0, L), L 2: W is a layer-specific learnable weight matrix; N₂ is the set of nodes in the neighborhood context of v;; C, the edge-weight between nodes U/ -> Us, with default edge weight being 1.0 an edge exists.
m.mmy are entries of the degree matrix, with m, = 1 ΣjEN, Gj.-
Decoder model. NMM can be considered as a probabilistic decoder for link generation, which maps embeddings of two nodes into the probablity to generate a link between them.

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Joint loss function. The training loss involves components of reconstruction loss (to ensure the generated graph is consistent with the original graph), KL divergence loss (to ensure predicted em- beddings of and z closely match their non- onnFuuclidean Gaussian distributions), and space unification loss (to ensure 2. and z map to the same node et).
Reconstruction Loss. Table shows reconstruction loss, which minimizes the upper bound on the negative log- ligllikelihood. X is hyperparameter; A' = X XAXT, given X E 0, 1 fed where Xa,i 1 only if node a G is assigned to E G and Xa,: = 0 otherwise, where the predicted graph.
Table 5: Description for Reconstruction Loss.
p(A'l 1==) kkkk Σ
Aa, + (1 -a, Ao,)) Aa,b)
(7)
logpo (G 21 zj 2,
Aalogp (A 12 :2) 21" 2
(8)
Liecon (0, 0: G) = E 1, [G) Logan G z; 13)
Hi
(9)
L (2) 0 G) = E (x |o) [G)
(10)
KI. Divergence Loss. The KI. divergence loss is formed by minimizing the equations described in T Minimizing the KI. divergence loss ensures that the homophily regulated nodes and social influence regulated nodes closely align to their underlying non- non-Euclidean distribution priors. As described in Section 3 these distributions are designed to appropriately capture the distinct topologies that emerge as result of the respective social network factors.
The space unification loss is formed by minimizing the equations described in T Minimizing the space unification loss ensures that the hyperbolic space representation of node v/ in spherical space, (ro ), is close to the corresponding learned representation of node 2 in the spherical space, zi¹ Note that using normalized hyperbolic disk is not substitute for the projection operator from the social influence hyperbolic space onto the homophily spherical space. The projection operation solely projects out-of-sphere nodes onto the w norm space, or the norm the surface of the spherical ball. normalization operator would instead change the embedding values of all nodes. More importantly, the space unification loss component ensures minimal spherical geodesic distance between H 's representation on the spherical ball and 2,. illustrated in Figure (b).
Space Unification Loss.
Table 6: Description of KL Divergence Loss and Space Unification Loss. (a) KL Divergence Loss
(b) Space Unification Loss
(e G) = ( G) (p()))
(11) (12)
projs (ax) w if |||| * w
T. G) = KT./ga G)))((
T
(13)
otherwise
Tuuityy (G) = (projs x
(14)
Total Loss. The overall loss function for homophily regulated and social influence regulated nodes respectively is then a summation of the above loss components given by:
1 I (0, 0; G) + (g; G. + Trunity (G)
(15)
" = s (bb G) + L&L (w: G. + Lumitt (G)
(16)
3.4 Training
This section details NMM-GNN's training framework, using non-Fuclidean VAF, for representing social networks. We describe the optimization method for variables from Sections 3.2 and 3.3
Embedding Initialization. We randomly initialize all embeddings of 2 and x," For homophily regulated nodes, we choose a value for norm wS,, that is sampled from uniform distribution: w w E [0, 1) + Unif [0, 1)) for all nodes, and for social influence regulated nodes, we choose value for norm w assigned uniformly at random per node. We set curvature values of spherical respectively, We leave the non-trivial problem of
and hyperbolic spaces as Ks and K learning optimal curvatures as future work.
Training procedure for homophily regulated nodes. Parameter optimization for learning node embeddings is performed using Riemannian stochastic gradient descent (RSGD) for the spherical space as shown Table To ensure the updated node embeddings remain norm-w- space, we perform rescaling operation, projs, to project oft-oo-byundary embeddings back to the surface of
7

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Table 7: Training Procedure for homophily regulated nodes and social influence regulated nodes.
(a) Homophily Regulated Nodes
(b) Social Influence Regulated Nodes
2 (x9,
#1,+1 #h, nel |= V (x
1.) (1+ (
(17)
(20)
fia prois r(
(
SCD (x) m211 2i n VV (xc)
(21)
SGD (z)) 1i41 + as VL (2))
(19)
Table 8: Dataset statistics for evaluation datasets.
Datasct
# Vertices
Edges
Type
# Classes
BlogCatalog
I0.3K
334,0K
undirected
39
LiveJournal
4.8M
69.0M
directed
I0
Friendster
65.6M
1.8B
undirected
the wS--ball. We further update scalar parameters J and B (from the homophily regulated distribution) and B8 A, (from the spherical gaussian prior) through stochastic gradient descent (SGD) as defined below via SGD SGD (B),SGD) SGD" (3), SGD* SG), (a).
Training procedure for social influence regulated nodes. Parameter optimization for learning node embeddings is performed using RSGD for the hyperbolic space as shown Table 7 The
corresponding norm space, w is also learned through RSGD by updating embeddings of z We further update scalar parameters and D (from the social influence regulated distribution) and ( (from the hyperbolic gaussian prior) through SGD as defined below via SGD (C),SGD" (D), SGD"
Training procedure for NMM-GNN weights. where Liotal = L + =
Parameter optimization for y uses SGD follows,
741 f- "t- T/ Lrotal(71))
(22)
4 Experiments
We comprehensively evaluate NMM-GNN on social network generation for popular large-scale social networks through multi-label classification and link prediction tasks against competitive SOTA baselines in various categories. The Appendix section further details Ablation Studies where test quality of using (1) mixture model, (2) distinct non-Euclidean geometric spaces, (3) non-Euclidean GNN-based encoders and Euclidean Graph VAE framework (through the inductive setting), and (4) space unification loss component. All experiments and each of the ablation studies consistently show NMM-GNN outperforms bascline network embedding models on all metries for all datasets.
4.1 Datasets
For comprehensive evaluation, assess models real-world datasets from well-known social media venues: BlogCatalog (BC) (27). LiveJournal (LJ) (28), and Friendster (F) [29] which are friendship networks among bloggers. Table R provides statistics of the datasets. In the Appendix, we also include experiments for Wikipedia datasets, to show that our model can also benefit other networks. Our research goal is to design an embedding model to better explain how the social network formed. Thus, to separate model contribution from the learned representation, we focus on the setting of featureless graphs since quality of node features can be confounding factor in determining quality of our model. However, since our model can handle feature graphs, we also provide these experiments in the Appendix, with our model outperforming all baselines attributed networks.
4.2 Models
Baselines. We compare NMM-GNN to SOTA network embedding models, in Table 9 and report results in Table TO We omit comparison to prior models e.g., LINE (30] due to lower performance. We also highlight that unlike prior works like RCGCN, our model overcomes limitations of the product space, e.g., where the entire model belongs to Cartesian product non- non-Euclidean geometric spaces by default. Our work is in category called mixed space model that uses multi-geometric space framework where different portions of the graph may possibly belong to different spaces (based on the amount of impact each of homophily and social influence has for that personalized pair of
8

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Table 9: Category and description of baseline models.
Catepory
Description
Structural
GraRep
RJ shallow embeedding integrating global structural information
Embedding
RolX unsupervised learning approuch using structural role based similarity
Models
GraphWave ave [33] shallow cmbeeddnn model using spectral graph wavelet diffusion partems
GruphSAGE [4 inductive framework using node features and neighhor aggregation
GNN Emnbedding
GCN
35
supervised leurning model via gruph convolation on local neighborhoools
Models
GAT
16
graph attention model using mask self attention layers on local neighhorhoodds
(Euclidean space)
GIN 37 graph embeddiin model based on the Weisfeiler-Lennee Lehman (WL) graph isomorphism test
GRAPHCL [88, graph contrastive learning framewark for unsupervised graph data
Homophily-
GELTOR 17
emhedding method using leamning-o- to-rank with AdaSim similarity metric
Eibbedding Models
NRP 22 embedding model using pairwise personalized PageRank on the global graph
GNN Embedding Models
HGCN [4]
hyperbolic GCN model utilizing Ricmannian goometry hyperboloid model
(non-Fuclidean space)
N-GCN [39] GCN model using product space egg. product of consrant curvature spaces
Mixture Models
RaKE [d]. Bayesian probabilistic modolfor node prosimity populllllytlltttttyyy..oo via postcrior cstimation
(bomophily
NMM. our nea-Euctidean
model (see Eqn. without NMM-GNN, our mon-Fuclidean mixture model with non- non-Euclidean GraphVAE VAE framework
Graph VAE framework
social influence)
nodes). In the extreme case (Case 1) where only social influence is at play, e.g., weight of homophily representation is learned close to O, the hyperbolic space will be used. On the other hand if only homophily is at play (Case 2), e.g., weight of homophily representation is learned close to O, the spherical space will be used. In the normal case of both factors at play (Case 3), then both spaces will be used and can be jointly aligned with our space alignment mechanism. When using product space, Cases 1, 2, and 3 will all not be distinguished from each other as all cases will be modeled by one complex Euclidean geometric space as a Cartesian product of spherical and hyperbolic spaces.
Time Complexity of NMM-GNN. Regarding our model, NMM is highly efficient, with time complexity Oled + nd), where is number of nodes, e is number of edges, and d is dimension size. In comparison, the time complexity analyses for the remaining baseline models are as follows: the mixture model of RaRE is O(ed + nd) which is comparable to the NMM mixture model, and the GNN embedding models of GCN, GAT (with one-head attention), and K-GCN (for λ. 0) are O + nd²). K-GCN (for n o 0) and HGCN's time complexities are O(ed + a nd?), where a is the filter length, and NMM-GNN is O + nd?) which is comparable GNN embedding models. As our work focuses on improving accuracy of learned embeddings, we further use Graph VAE training with NMM to achieve SOTA performance. We would like to point out that Graph VAE (of AMM-GNN training is also designed to be highly parallelizable, which allows for scalability. Moreover, our model is capable of learning on world, highly large-scale graphs on the order of millions of nodes and billion of edges, c.g., Friendster, while achieving the best performance, which attests to its practical value to the network science community.
4.3 Evaluation
We detail our evaluation procedure for multi-label classification and link prediction. For all experi- ments, for faimess of comparison to baselines, we utilize the experiment procedure of (6. Specifically, 90% of links are randomly sampled as training data. We do not perform cross-validation, since it may cause overfitting occur our framework uses learnable parameters e.g., z z J, B, C, D. 7, B, X, a, W. and ( which is function of z equivalently interpreted as mean square error. Per dataset, we choose hyperparameter values for X in reconstruction loss: {0, 1, 2, 4. 8, 16, 32, 64), step sizes me: (0.005, 0.001, 0.01, 0.05, 0.1). and experiments are performed on AWS cluster (8 Nvidia GPUs).
4.3.1 Classification
Evaluation results are in Table TO We observe that mixture models (homophily and social influence), achieve better performance on all datasets for all metrics. Specifically, it improves over structural em- bedding models (Graph Wave), GNNs (GAT, HGCN), and homophily-based models (GELTOR, NRP). We also see learning embeddings in non-Euclidean geometric spaces helps better represent structures in social networks (HGCN GAT, RaRE vs. NMM). We further observe using GNN- based encoders with Graph VAE learning yields additional improvement (NMM vs. NMM-GNN).
4.3.2 Link Prediction
As from 66. we measure quality of link prediction by sorting probability scores of every pair of nodes per model and evaluating them using area under the ROC curve (AUC) score. Specifically, 10% of existing edges and non-existing edges are hidden from training set, and probabilities examined by

---

Table 10: Results of social network classification and link prediction for Jaccard Index (JT), Hamming Loss (HL), F1 Score (F1), and AUC in % using embedding dimension 64, Our NMM and its variants gray shading. For each group of models, the best results are bold face-f The overall results on each dataset are underscored. 'Ablation study variant models using distinct non- Son-uuclidean geometric spaces for NMM (homophily/sooial influence) where IE, S. and Hl denote Euclidean, Spherical, and Hyperbolic spaces.
Datasets
BlogCatalog
LiveJournll
Friendster
Metrics
J1
HI
F1
AUC
J1
HL
AUC
J1
HL
F1
AUC
GraRep
36.0
28.2
45.6
87.9
40.1
41.1
35.2
56.7
53.6
34.2
40.6
89.8
RolX
37.2
25.4
48.7
90.4
40.9
38.0
35.6
60.1
58.8
31.9
40.9
90.3
GraphWave
39.5
22.8
48.9
92.3
42.2
37.6
35,9
60.1
59.0
31.5
41.1
90.5
GraphSAGE
45.4
20.1
49.3
92.0
45.5
34.7
34.1
59.0
64.1
28.7
43,4
90.5
GCN
47.3
19.5
55.1
91.6
46.7
31.2
47.8
62.6
66.5
28.0
47.2
91.9
GAT
47,9
19.3
54.5
91.4
47.4
28.5
49.0
65.3
66.3
28.0
46.8
92.0
GIN
47.1
19.7
56.2
91.5
48.6
28.3
48.
67.2
66.0
27.7
48.1
92.3
GRAPHCL
47.5
19.4
55.8
91.3
49.7
27.9
69.4
68.1
25.5
49,9
92.8
GELTOR
47.4
19.3
54.9
92.0
51.0
28.9
48.6
65.3
66.7
27.9
47.3
91.7
NRP
61.6
20.4
65.2
95.5
69.7
24.5
64.0
78.7
72.2
22.6
52.8
92.2
HGCN
56.7
19.2
60.9
92.7
58.8
27.1
57.7
68.5
69.9
24.3
49.9
93.3
K-GCN
61.6
20.7
65.4
95.3
63.6
27.3
57.2
69.1
69.4
24.1
50.3
93.1
RaKE
61.4
20.6
95.
74.2
23.8
65.1
75.7
94.4
NM yt
56.6
74.0
74.6
(S ST
74.7
75.3
N St
57,9
19.5
66.3
95.4
75.1
25.0
58.4
71.2
77.0
24.7
52.8
94.5
NMM(8" 13"11
59,2
19.2
67.1
95.5
75.3
24.4
59.3
74.5
77.5
23.3
54.3
94.5
NM /E
58.4
19.0
66.7
95.3
75.6
24.6
61.9
76.0
78.8
23.3
55.0
94.7
NMM )
60.3
19.1
67.8
95.7
76.2
23.2
64.4
79.2
79.
22.6
55.4
94.5
NMM (ours)
62.7
19.0
76.5
22.7
84.2
79.8
94.8
NMM-GNN (ours)
62.6
17.3
78.8
96.9
78.6
20.4
67.3
86.8
83.3
21.8
57.7
94,9
the model. Further, 10% of non-training edges are used for validation. For fairness against baselines on undirected networks, we treat all directed networks as undirected. Table][[jhoow evaluation results for AUC score. Comparing relative score differences between best performing homophily embedding model (NRP) to RaRE, we observe that LiveJournal and Friendster datasets contain relatively more node social influence than BlogCatalog, which homophily- basdd models do not capture. Due to the above observation, it is likely that LiveJournal and Friendster (which are also larger datasets) show more realistic heterogeneity in network structure compared to Blog BlogCatalog dataset c.g., cyclic structures produced by homophily based nodes and tree-like structures produced by social influence based nodes. Thus, modeling these structures in non-Euclidean spaces (RaRE vs. NMM) also shows more improvement. Moreover, in NMM variant models, every node is influenced by both factors of homophily and social influence through our non-Euclidean mixture model. It is weighted combination personalized per node of these factors that influence the links formed, rather than being generated solely through homophily vs. social influence.
5 Conclusions
We are among the first to explore a Graph-based non-Euclidean mixture model for social networks. As social networks are influenced by homophily and social influence, we design model to represent both factors jointly for nodes. Further, we model resulting unique network topologies (cycles and trees) using distinct non-Euclidean geometric spaces and introduce a GNN-based non-Euclidean variational autoencoder framework for our model, to effectively learn embeddings. The resulting model, NMM-GNN, significantly outperforms various state-of-the-arr models for social networks. As future work, we hope to explore alternatives to graph-based VAE methods for improved learning.
10

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6 Acknowledgments and Disclosure of Funding
This work was partially supported by DARPA HR001 1-24-9-0370; NSF 1937599, 2106859, 2119643, 2200274, 221 2211557, 2303037, 23 12501;; NIH U54HG012517, U24DK097771; Optum AI; NASA; SRC JUMP 2.0 Center; Amazon Research Awards; and Snapchat Gifts.
References
[1] Huiwen Xiang, Mingjing Xie, and Yiyi Fang. Study on the architecture space-social network characteristics based on social network analysis: A case study of anshun tunpu settlement. Elsevier, 2024.
[2] Farzana Afridi, Amrita Dhillon, and Swati Sharma. The ties that bind us: Social networks and productivity in the factory. Elsevier, 2024.
[3] Austin P. Logan, Phillip M. LaCasse, and Brian J. Lunday, Social network analysis of twitter interactions: a directed multilayer network approach. Springer, 2023.
[4] Wei Jiang, Xinyi Gao, Guandong Xu, Tong Chen, and Hongzhi Yin. Challenging low homophily in social recommendation. WWW, pages 3476-3484, 2024.
[5] Kazi Khanam, Gautam Srivastava, and Vijay Mago. The homophily principle in social network analysis: A survey. Springer, 2023.
[6] Yupeng Gu, Yizhou Sun, Yanen Li, and Yang Yang. Rare: Social rank regulated large-scale network embedding. In WWW, pages 359-368, 2018.
[7] Liye Ma, Ramayya Krishnan, and Alan L. Montgomery. Latent homphily social influence? an empirical analysis of purchase within social network. In Management Science INFORMS, pages 454-473, 2015.
[8] A.L. Barabasi, H. Jeong, Neda, E. Ravasz, A. Schubert, and T. Vicsek. Evolution of the social network of scientific collaborations. Elsevier Science, 2002.
[9] Julian McAuley and Jure Leskovee. Learning to discover social circles in ego networks. NeurlPS, 2012.
[10] Li Sun, Mengjie Li, Yong Yang. Xiao Li, Lin Liu, Pengfei Zhang, and Haohua Du Rcoco: contrastive collective link prediction across multiplex network in riemannian space. Springer, 2024.
[11] Xuelian Ni, Fei Xiong, Yu Zheng, and Liang Wang. Graph contrastive learning with kernel dependence maximization for social recommendation. WWW, pages 481-492, 2024.
[12] Zitai Qiu, Congbo Ma, Jia Wu, and Jian Yang. An efficient automatic meta-path selection for social event detection via hyperbolic space. WWW. pages 2519 22292 2024.
[13] Cosimo Gregucci, Mojtaba Nayyeri, Daniel Hernández, and Steffen Staab. Link prediction with attention applied multiple knowledge graph embedding models. WWW, pages 2600-2610, 2023.
[14] Min Zhou, Menglin Yang, Bo Xiong, Hui Xiong, and Irwin King. Hyperbolic graph neural networks: A tutorial on methods and applications. KDD, pages 3843-5844, 2023.
[15] Azad Noori, Mohammad Ali Balafar, Asgarali Bouyer, and Khosro Salmani. Review of heterogeneous graph embedding methods based on deep learning techniques and comparing their efficiency in node classification. Springer, 2024.
[16] Xin Liu, Tsuyoshi Murata, Kyoung-Sook Kim, Chatchawan Kotarasu, and Chenyi Zhuang. A general view for network embedding as matrix factorization. In WSDM, pages 375-383, 2019.
[17] Masoud R. Hamedani, Jin-Su Ryu, and Sang- Wook Kim. Geltor: graph embedding method based on listwise learning to rank. WWW, pages 6-16, 2023.
11

---

[18] Guohao Li, Chenxin Xiong, Ali Thabet, and Bernard Ghanem. train deeper gens. IEEE TPAMI, 2023.
Deepergen: All you need to
[19] Peter Petersen, Riemannian geometry, volume 171. Springer, 2006.
[20] Gabriel Moreira, Manuel Marques, João P. Costeira, and Alexander Hauptmann. Hyperbolic vs euclidean embeddings few-shot learning: Two sides of the same coin. IEEE WACV, pages 2082-2090, 2024.
[21] Yansong Ning, Hao Liu, Hao Wang, Zhenyu Zeng, and Hui Xiong. Uukg: Unified urban knowledge graph dataset for urban spatiotemporal prediction. NeurIPS, 2024.
[22] Jongmin Park, Seunghoon Han, Soohwan Jeong, and Sungsu Lim. Hyperbolic heterogeneous graph attention networks. wwW, pages 561-564, 2024.
[23] Fragkiskos Papadopoulos, Maksim Kitsak, M. Ángeles Serrano, Marián Boguñá, and Dmitri Krioukov. Popularity versus similarity in growing networks. Nature, pages 537-540, 2012.
[24] Ines Chami, Zhitao Ying, Christopher Ré, and Jure Leskovec. Hyperbolic graph convolutional neural networks. NeurIPS, 32, 2019.
[25] Qi Liu, Maximilian Nickel, and Douwe Kiela. Hyperbolic graph neural networks. NewrIPS, 32, 2019.
[26] Roshni G. Iyer, Yunsheng Bai, Wei Wang, and Yizhou Sun. Dual- -eomeeeo space embedding model for two-view knowledge graphs. In KDD, pages 376-686, 2022.
[27] Renchi Yang, Jieming Shi, Xiaokui Xiao, Yin Yang, and Sourav S. Bhowmick. Homogeneous network embedding for massive graphs via reweighted personalized pagerank. VLDB, pages 2020.
[28] Feng Xia, Lei Wang, Tao Tang, Xin Chen, Xiangjie Kong, Giles Oatley, and Irwin King. Cengen: Centralized convolutional networks with vertex imbalance for scale-free graphs. IEEE TKDE, pages 4555 4569,, 2022.
[29] Danah M. Boyd. Friendster and publicly articulated social networking. ACM CHI, 2004.
[30] Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. Line: Large- scale information network embedding. Www, pages 1067-1077, 2015.
[31] Shaosheng Cao, Wei Lu, and Qiongkai Xu. Grarep: Learning graph representations with global structural information. CIKM, pages 891-900, 2015.
[32] Keith Henderson, Brian Gallagher, Tina Eliassi-Rat Rad, Hanghan Tong, Sugato Basu, Leman Akoglu, Danai Koutra, Christos Faloutsos, and Lei Li. Rolx: structural role extraction & mining large graphs. KDD, pages 1231-1239, 2012.
[33] Claire Donnat, Marinka Zitnik, David Hallac, and Jure Leskovec. Learning structural node embeddings via diffusion wavelets. KDD, pages 1320-1329, 2018.
[34] William L. Hamilton, Rex Ying, and Jure Lescovec. Inductive representation learning on large graphs. NeurIPS, 30, 2017.
[35] Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. ICLR, 2017.
[36] Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. ICLR, 2018
[37] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? ICLR, 2019.
[38] Yuning You, Tianlong Chen, Yongduo Sui, Ting Chen, Zhangyang Wang, and Yang Shen. Graph contrastive learning with augmentations. NeurIPS, 2020.
12

---

[39] Gregor Bachmann, Gary Bécigneul, and Octavian Ganea. Constant curvature graph convolu- tional networks. ICML. pages 486-496. PMLR, 2020.
[40] Jiaxuan You, Rex Ying, Xiang Ren, William L. Hamilton, and Jure Lescovek. Graphrnn: Generating realistic graphs with deep auto-regressive models. ICML, 2018.
[41] Renchi Yang, Jieming Shi, Xiaokui Xiao, Yin Yang, Sourav S. Bhowmick, and Juncheng Liu. Scaling attributed network embedding to massive graphs. VLDBJ, pages 37-49, 2023.
[42] Qiaoyu Tan, Xin Zhang, Xiao Huang, Hao Chen, Jundong Li, and Xia Hu. Collaborative graph neural networks for attributed network embedding. IEEE TKDE, 2023.
[43] Jure Leskovec, Kevin J. Lang, Anirban Dasgupta, and Michael W. Mahoney. Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics, 2009.
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Appendix
A Limitations
NMM is highly efficient, with time complexity Oled + nd), where n is number of nodes, is number of edges, and d is dimension size. As our work focuses on improving accuracy of learned embeddings, we further use GraphVAE training with NMM to achieve SOTA performance. called NMM-GNN. Using the Graph VAB training pipeline may be seen as a limitation as it is less efficient compared to GNNs for training time. However, in the worst case scenario, NMM-GNN still achieves efficiency comparable to popular heterogeneous graph models including GraphRNN 40., and training designed to be highly parallelizable, which allows for scalability. Moreover, our model is capable of learning on real-world, highly large-scale graphs on the order of millions of nodes and billion of edges, e.g., Friendster, while achieving the SOTA performance, which attests to its practical value to the network science community.
Our model also makes the assumption the homophily regulated nodes lie on the surface of the spherical ball and the social influence regulated nodes lic on the open Poincare ball, which are both natural representations for modeling nodes. People have observed that more popular celebrity nodes tend to have smaller norm space and are embedded towards the center of the ball (similar to higher order concepts in the knowledge graph space), while less popular nodes (containing less social influence) are embedded towards the boundary of the ball (similar to entities in the knowledge graph space). To ensure that we can compute an intersection space (for the space unification component to unify the homophily and social influence representation for the same node), we enforce that the spherical surface norm is in the Poincare ball e.g., any learned soft value norm space between and 1. In compute, while we evaluate on a cluster of 8 GPUs, at least one GPU is necessary for running our experiments (which may been seen as compute limitation).
Regarding ethical considerations and fairess, privacy and fairness are often topics of focus regarding social networks. Our model uses information like node popularity based on graph structure (in degree vs. out degree ratio), as well as attributed features if available. Further it also infers links through neighborhood context by looking at connected nodes. In general, a network embedding model uses personal information from the user which may impinge on their privacy. However, we make every attempt in our model to allow the user to select their granular choice of model personalization (e.g.. we provide the option to choose from whether or not users want to enable their attributed information like profile interest and history being learned).
B A Note on Graph-Level Learning
Our work can be generalized to the graph-level because our method learns to represent social science network factors based on topologies in the graph on clusters of nodes and edges. Thus, the cluster of nodes and edges comprised of the entire graph and we subsequently applied graph pooling per node embedding (that we currently learn), we can reduce the embedding from node level to graph level. In this way, homophily and social influence can be modeled at the graph level. That said, it is unclear whether graph-level modeling would be specifically useful or interpretable for social network embedding models as compared to node-level modeling. This is due to the social network setting requiring links to be generated that are per node and not at the graph level, because a user (modeled as node) is recommended to another specific user in the practical social network setting. This is different from other network domains like molccular classification where the entire graph represents one molecule c.g., atoms form individual nodes and chemical bonds form the edges. For this reason, node-level learning is in fact consistent with the recent state-of-the-ar NN methods in the network science community though RMM-GNN can still be generalized to learning at the graph-level.
C Additional Experiments
In this section, we provide experiment results on Wikipedia networks and on attributed graphs. As in the case for social networks, we also evaluate on multi-label classification and link prediction for Wikipedia networks and attributed graphs. For all experiments, for fairness of comparison to baselines, we utilize experiment procedure of (6). Specifically, 90% of links are randomly sampled
14

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as training data. We do not perform cross- cross-validation, since it may cause overfitting to occur as our framework uses learnable parameters e.g., z², z", J, B, C. D. y. 3, X, a, W,, and Ç which is a function of z equivalently interpreted as mean square error. Per dataset, we choose hyperparameter values for XA in reconstruction loss: [0, 1, 2, 4, 8, 16, 32, 64), and step sizes n/: (0.005, 0.001, 0.01, 0.05, 0.1).
Evaluation on Wikipedia networks. In Table 12 we additionally present multi-label classification and link prediction experiments on Wikipedia datasets to show the benefits of our model on other networks. These include Wikipedia Clickstream [6], which contains counts of (referrer, resource) pairs extracted from the request logs of Wikipedia, and Wikipedia Hyperlink [6] which contains edges as hyperlinks from one page to another. Dataset statistics for the Wikipedia datasets are summarized in Table [1 Our NMM model variants consistently achieves the best performance over all the competitive baseline models belonging to categories of (1) structural embedding models, (2) GNN embedding models (Euclidean space), (3) homophily-basee baaed embedding models, (4) GNN embedding models (non-Euclidean space), and (5) mixture models. This shows that NMM's model is generalizable and widely applicable because it can learn effective representations on online information networks that go beyond social networks.
Table 11: Dataset statistics for evaluation datasets.
Dataset
# Vertices
# Edges
Type
# Classes
Wikipedia Clickstream
2.4M
15.0M
directed
6
Wikipedia Hyperlink
488K
5.5M
directed
6
Table 12: Results of social network classification and link prediction for Jaccard Index (JI), Hamming Loss (HL), F1 Score (F1), and AUC % using embedding dimension 64. Our NMM and variants are in gray shading. For each group of models, best results are bold-faced. The overall best results on each dataset are underscored. Ablation study variant models using distinct non-Fuclidean geometric spaces for NMM (homophily/social influence) where IR, S. and HI denote Fuclidean, Spherical, and Hyperbolic spaces.
Datasets
Wikipodin Clicksrramm
Wakapodia Hypertink
Metrics
HL
AUC
GraRep
31,9
28.3
44.4
79.6
48.3
22.7
50.2
76.8
RoIX
32.2
28.1
57.1
17.6
GraphWave
32.8
27.0
44.6
$6.1
55.3
18.1
54.8
81.8
GraphSAGE
33.1
26.6
45.1
89.3
62.8
8.3
68.7
89.4
GCN
33.1
24.1
63.7
76.4
92.0
GAT
33.4
24.2
51.2
89.0
64.2
6.5
76.2
92.1
GIN
33.7
65.1
92.3
GRAPHCL
33,3
24.1
51.0
90.9
64.5
6.6
76.4
92.7
GELTOR
33.2
24.
50.9
91.4
64.0
6.8
76.7
92.3
46.5
75.5
HCCN
38.1
20.6
34.3
89.9
69.8
6.0
78.4
94.1
R-GCN
37.6
20.9
55.0
90.2
75.0
7..1
81.6
96.9
RaRE
47.8
20.7
58.0
93.0
75.7
82.3
97.3
NM (2
44.8
27.8
44.2
86.5
76.0
8.2
76.4
92.1
(g yT
45.1
22.9
55.7
90.3
78.4
7.1
79.7
97.0
NMM 7
46.2
21.7
56.0
90.6
79.2
6.8
82.0
97.5
NMM(S
47.7
20.9
56.2
91.0
79.9
6.6
82.3
97.6
(M
47.4
20.9
56.8
91.2
81.0
6.6
81.6
97.3
NMM(E (E" )
48.1
20.7
57.9
91.7
80.5
82.4
97.8
NMM (OurS)
49,2
18.5
59.0
92.8
80.3
5.8
82.5
98.0
NMM-GNN (ours)
49.7
16.5
60.8
95.6
81.7
5.5
83.0
97.3
Evaluation on attributed graphs. As our model can also handle attributed graphs, we provide experiments against SOTA attributed network embedding models that are summarized below. Since these models require the presence of network attributes, we do not include them in our experiments on featureless graphs for fairness of comparison. We conduct evaluation on well-known large- scale attributed graph datasets which include Facebook [9] and Google+ [9] social networks where consistent with (27]. we treat each ego-network as a label and extract attributes from their user profiles. Dataset statistics are in Table [3] and results are reported in Table [4] It can be seen that our NMM-GNN model consistently achieves the best performance in both the large-scale graph datasets, indicating that our model's inherent learning ability is effective to learn graph structure and topology, which goes beyond simply exploiting information from node and edge attributes. Below summary of the baseline attributed network embedding models:
15

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100
100
100
MMP delino Su
90
AUC 90
AUCC (192
AUC (%%
40
%
20
40
60
80
100
BC
% training nodes
DC
V11
(a)
(b)
Ablation studies. (a) Quality of mixture model, where NMM and NMMrank are homophily- only and social influence-only deconstructed NMM components. (b) Inductive reasoning for NMM-GNN and RaRE on LiveJoumal, % nodes {10, 30, 50, 70, 90} are sampled ensuring overlap with test. (c) Ablation study on quality of using space unification loss (SUL) component.
Figure 2:
PANE
41,
random-wal based attirubted network embedding (ANE) model
NRP
27
described in Table of the main paper
CONN
42
GNN for attributed networks via collaborative aggregation bipartite graphs
Table 13:
Dataset statistics for evaluation datasets.
Dataset
Vertices
Edges
#Attributes
Type
Classes
Facebook
4.0K
88.2K
1.3K
undirected
193
Google+
107.6K
13.7M
15.9K
directed
468
Table 14: Results of social network classification and link prediction for Jaccard Index (JJ), Hamming Loss (HL). F1 Score (F1), and AUC in % using embedding dimension 64. Our NMM and variants are in gray shading. The overall best results on each dataset are bold-faced.
Datasets
Facehook
Google-
Metrics
F1
HL
AUC
NRP
48.0
19.7
33.8
96.1
40.7
26.5
51.1
81.3
PANE
CONN
51.2
16.4
61.2
96.8
44.9
23.3
61.3
90.8
(ours)
The source
code
and
datasets
for
our
work
can
be
found
at:
sstps://githhe con/ http://oonngiier//mmmm
D
Ablation Studies
This section details ablation studies where test quality of using (1) mixture model, (2) distinct non- Euclidean geometric spaces, (3) non-Euclidean GNN-based encoders and non-Euclidean Graph VAE framework (through the inductive setting), and (4) space unification loss component. Each of the ablation studies provides insight to motivate our architecture design choices, as well as consistently shows NMM-GNN outperforms baseline network embedding models.
Quality of using a mixture model architecture. We deconstruct our mixture model NMM, to observe the effect different network factors have on learning embeddings, where and / are learnalle. We report results on embedding dimension 64, evaluated on AUC score, with the models summarized below:
NMM deconstructed homophily component: po(e; = 1) = a Phom (eis = 1)
NMMrank deconstructed social influence component: pe (eig 1) B Prank (eik
NMM, which is our mixture model defined by Eqn.
As shown in Figure a), the mixture model of NMM outperforms that of its subcomponents NMM and NMM on all datasets for link prediction. This validates effectiveness of using mixture model architecture for modeling both homophily and social influence factors jointly.
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Quality of using distinct non-Euclidean geometric spaces. We study combinations of geomet- ric spaces to model NMM (homophily/social influence) to observe the effect it has on learning topological structure, denoted with NMM in Table (main paper). As shown, the choice of modeling homophily based nodes spherical space and modeling social influence based nodes in hyperbolic space leads to the best performance. Further, NMM outperforms its Euclidean space counterpart showing that the social network exhibits structures (cycles and hierarchy) that need to be appropriately represented in curved geometric spaces. There is also evidence of non- non-Euclidean topologies in the datasets, For example, the average node on LiveJournal has in-degree of 17 but outdegree of 25, showing several hierarchical structures present, and 17.7% of LiveJournal data contains cycles [43]
Link prediction unseen nodes (inductive task). We study ability of NMM-GNN to learn the inductive setting (in addition to the standard transductive setting of Table TO of the main paper). To do so, we randomly sample % of nodes being {10, 30, 50, 70, 90) and their corresponding links as our training set. Test nodes are ensured to have no overlap with training nodes, to allow for link prediction unseen graphs. Figure reports results NMM-GNN and RaRE for LiveJournal on embedding dimension of 64 for AUC score. NMM-GNN outperforms RaRE on all settings of training nodes. This shows the effectiveness of NMM-GNN in using non-Euclidean GNN- based encoders and a non-Fuclidean GraphVAF training framework during the learning process, two components that RaRE lacks. Further, as less training nodes are observed, NMM-GNN outperforms RaRE by larger margins (c.g.. 10% vs. 70% training nodes), showing NMM-GNN better generalizes to unseen graphs.
Quality of using space unification loss. We test quality of using our proposed space unification loss (SUL) our NMM-GNS by conducting experiments for with the loss component (NMM-GNN) and without it SMMM-GNN-n--SULE
We report results on embedding dimension 64, evaluated on AUC score in Figure Cc), for datasets Wikipedia Clickstream (WC). BlogCatalog (BC), LiveJournal (LJ), Wikipedia Hyperlink (WH), and Friendster (F). As shown, using SUL improves performance on all datasets, indicating importance of bridging together representations of distinct non- oonEuulidean spaces (spherical and hyperbolic space) at node level.
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The abstract summarizes the main paper claims and we also provide con- tribution summary with bullet points in the Introduction. Further, all these main claims of the paper are supported through Section 3 (detailing our model architecture), Section 4 on Experiments, as well as our (four) ablation studies in the Appendix.
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Our model novelly represents both homophily and social influence factors when modeling the social network, an advancement over the recent network embedding methods. Further, the novel utilization of non- -on-Euclidean geometric spaces to model the resulting topologies due to network factors through appropriate positive and negative curvatures naturually exibiting properties of cycles and hierarchy (the observed topological heterogeneity). tremendously improves the representation capability of network embedding modes. This is evidenced by the empirical results as ablation studies for our work. Moreover, NMM-GNN not only improves against SOTA models in the performance metrics (Jaccard Index, Hamming Loss,, Fl score, AUC Score), but is also applicable to the Inductive Setting, other large-scale infor- mation networks like Wikipedia networks, and attributed networks (Facebook and Google+), while consistently achieving the best performance indicating model generalizability.
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Justification: We have an entire section on Limitations in the Appendix (Section A) that addresses all the guidelines below. Our model also makes the assumption the homophily regulated nodes lie on the surface of the spherical ball and the social influence regulated nodes lie on the open Poincare ball, which are both natural representations for modeling nodes. People have observed that more popular celebrity nodes tend to have smaller norm space and are embedded towards the center of the ball (similar higher order concepts in the knowledge graph space), while less popular nodes (containing less social influence) are embedded towards the boundary of the ball (similar to entities in the knowledge graph intersection space (for the space unification
space). To ensure that we can compute component to unify the homophily and social influence representation for the same node), enforce that the spherical surface norm is in the Poincare ball e.g., any learned soft value norm space between 0 and 1. In compute, while we evaluate on a cluster 8 GPUs, at least one GPU is necessary for running our experiments (which may been seen compute limitation).
Regarding ethical considerations and fairness, privacy and fairness are often topics of focus regarding social networks. Our model uses information like node popularity based on graph structure (in degree vs. out degree ratio), well as attributed features if available. Further it also infers links through neighborhood context by looking at connected nodes. In general, a network embedding model uses personal information from the user which may impinge on their privacy. However, we make every attempt in our model to allow the user to select their granular choice of model personalization (c.g., provide the option to choose from
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Justification: The paper provides clear motivation of the importance of the research problem of investigation and further identifies key challenges in the network science community in sate-of-the-att embedding models, which thus motivates our model architecture. Further, our paper provides clear intuition and justification for each design choice decision in our model components. Additionally, impact of our model is discussed with respect to the social network community and more broadly for other online information networks e.g., Wikipedia datasets and attributed networks (in the Appendix section on Ablation studeies). Lastly, we also discuss Limitations of our model Appendix Section A.
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Reviews:
[{"review_id": "gyjz1JmLG8", "reviewer": "Reviewer_iEGq", "summary": "This work addresses the embedding of social networks with downstream tasks such as link prediction in mind. In this manuscript the authors propose to model the link as the mixture of two factors of node embedding, Spherical and Hyperbolic. Concretely, two kinds of node embedding are unified into a single loss by projecting from hyperbolic space into spherical space, and jointly trained in the framework of VAE on Graaph. The authors also conduct experiments against a wide range of baseline methods to highlight the capacity of the proposed method.", "questions": "N/A", "limitations": "Well addressed.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The idea is novel. Overall things are good. The idea of bridging both is novel, and according to experiments performance better than purely hyperbolic embedding. Writing is overall clear and, although there are many details, it’s generally feasible to follow. Extensive comparison with a wide range of baselines from different categories. Good discussion of limitations.", "weaknesses": "The major issue is on limited datasets in experiments. I would suggest the authors consider other datasets used in previous works, such as synthetic datasets, citation networks (PubMed, wikipedia citation,  DBLP, Microsoft Academic Graph ) and other social networks (e.g. Twitter). \n\nAlso another factor limiting the impact of the proposed method is that the proposed method does not work on the whole-graph level, and thus only applies to node classification and link prediction tasks. If by any extension the embeddings could be aggregated to the whole-graph leve, there could be more significance (e.g. tasks like molecular classification, protein-to-protein interactions)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks a lot for your insightful comments and feedback. Please find our response below to your questions.\n\nQ1: The major issue is on limited datasets in experiments. I would suggest the authors consider other datasets used in previous works, such as synthetic datasets, citation networks (PubMed, wikipedia citation, DBLP, Microsoft Academic Graph) and other social networks (e.g., Twitter)\n\nA1: We would like to clarify that the primary goal of our work is to explain how links are generated in social networks e.g., user to other users. As such, for datasets, in the main paper we specifically showcase results on social network datasets as that is the domain focus of this research. However, in the Appendix (Tables 12 and 14), we also include experiments for the citation networks of Wikipedia datasets, to show that our model can also benefit other networks, as well as attributed network datasets.  In total, we evaluate on seven datasets. The social network datasets include BlogCatalog, LiveJournal, Friendster which are the most prominent and recent large-scale networks from the latest research in the literature, and which are a balanced mix of small-scale and the largest scale size (node counts: of 10k, 5M, 66M respectively) and both directed and undirected. For citation networks, we evaluate on Wikipedia Clickstream and Wikipedia Hyperlink. For attributed networks, we evaluate Facebook and Google+ (containing ~16K attributes), and these numerous datasets show the generalizability performance and scalability of our approach. \n\nQ2: Also another factor limiting the impact of the proposed method is that the proposed method does not work on the whole-graph level, and thus only applies to node classification and link prediction tasks. If by any extension the embeddings could be aggregated to the whole-graph level, there could be more significance (e.g., tasks like molecular classification, protein-protein interactions)\n\nA2: \nOur work can definitely be generalized to the graph-level because our method learns to represent the social science network factors based on topologies in the graph on clusters of nodes and edges. Thus, if the cluster of nodes and edges comprised of the entire graph and we subsequently applied graph pooling per node embedding (that we currently learn), we can reduce the embedding from node level to graph level. In this way, homophily/social influence can be modeled at the graph level. That said, it is unclear whether graph-level modeling would be specifically useful/interpretable for social network embedding models (as compared to node-level modeling). This is due to the social network setting requiring links to be generated that are per node (not at the graph level), since a user is recommended to another specific user in the practical social network setting. This is different from other network domains like molecular classification where the entire graph represents one molecule e.g., atoms form individual nodes and chemical bonds form the edges. On the other hand, one node represents one user (unlike molecular graphs). We’d also like to clarify that for this reason, node-level learning is in fact consistent with the recent state-of-the-art NN methods in the network science community though our model can still be generalized to learning at the graph-level.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "e40PeHg6h7", "reviewer": "Reviewer_3fkq", "summary": "This paper proposes a new Graph-based non-Euclidean mixture model for social networks.\nUnder the assumptions that social network links are formed due to either homophily or social influence, \nthe homophily factor is modeled in spherical space and the social influence factor is in hyperbolic space.\nThe homophily regulated nodes lie on the surface of the spherical ball and the social influence-regulated\nnodes lie on the open Poincare ball. \nThe projection to align these two spaces is also proposed.\nThe non-Euclidean GraphVAE is also integrated into the model.\nExperiments show the proposed model outperforms other baselines.", "questions": "- Why is the space unification necessary? Each node has two coordinates, in the spherical and the hyperbolic space. The link between nodes i and j is generated either from the spherical or hyperbolic proximity (it is the \"explaining away\" situation as explained in the RaRE paper [6] ). The homophily and social influence relationships may be unrelated or may even contradict to each other (you may hate your boss). Therefore the two coordinates may not necessarily be aligned. It would be appreciated if the authors would elaborate on this. \n\n- The $\\kappa$-GCN proposed in [39] also deals with spherical and hyperbolic spaces. I would like to know the similarities and/or differences between the two approaches as the two coordinates in the proposed model can also be regarded as a coordinate in the product space. The authors do not discuss them. They just compare the experimental performance.\n\n- What is $i$ in the definition of $\\log_0^H$ and $\\exp_o^H$ ?", "limitations": "As for the computational efficiencies, more formal and quantitative comparisons would be necessary in the main texts.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed framework models both the homophily and social influence factors for social network generation. \n- It is integrated into the non-Euclidean graph-based VAE to further improve performance.\n- Experiments show the proposed model outperforms other SOTA baselines.", "weaknesses": "- The motivation of the space unification is unclear (See the questions below).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks a lot for your insightful comments and feedback. Please find our response below to your questions.\n\nQ1: The motivation of the space unification is unclear: Why is the space unification necessary? Each node has two coordinates, in the spherical and the hyperbolic space. The link between nodes i and j is generated either from the spherical or hyperbolic proximity (it is the “explaining way” situation as explained in the RaRE paper [6]). The homophily and social influence relationships may be unrelated or may even contradict each other (you may hate your boss). Therefore, the two coordinates may not necessarily be aligned. It would be appreciated if the authors would elaborate on this. \n\nA1: We would like to clarify and highlight that the link between nodes i and j is a mixture model because each link is a weighted combination of influence from both spherical and hyperbolic spaces (not one or the other) as evidenced in Equation 6 of the paper. Hence, as shown in Figure 1b, the same node has 2 representations – one in the spherical space and one in the hyperbolic space, and because they represent the same underlying node, they need to be aligned. In the case of “you may hate your boss” if both you and your boss are not highly popular nodes e.g., celebrity nodes, then likely the social rank may be similar due to low social influence. At the same time, regardless of liking each other, you and your boss may share many similar characteristics like working at the same company, studied the same field like “computer science”, live in the same location/country, working on the same project problems etc. the homophily is still high by its very definition. Therefore, these two network factors do not contradict each other, but rather work together to explain how links are formed between users. We address this as well as scenarios of noise such as where nodes may still not be connected to each other even after exhibiting high homophily, as well as in the case they are not connected when having low social influence (or similar social rank). Please refer to sections 3.2.1 “Link prediction using homophily based distribution” and 3.2.2 “Link prediction using social influence based distribution” where we explain this and how our distribution models also represent factors to control sparsity of the network.\n\nQ2: The K-GCN proposed in [39] also deals with spherical and hyperbolic spaces. I would like to know the similarities and/or differences between the two approaches as the two coordinates in the proposed model can also be regarded as a coordinate in the product space. The authors do not discuss them. They just compare the experimental performance. \n\nA2: First, our model is not in the product space (e.g., where the entire model belongs to a cartesian product of non-Euclidean geometric spaces by default). Rather, our work is in a category called mixed space model that uses a multi-geometric space framework where different portions of the graph may possibly belong to different spaces (based on the amount of impact each of homophily and social influence has for that personalized pair of nodes). In the extreme case (Case 1) where only social influence is at play (e.g., weight of homophily representation is learned close to 0), the hyperbolic space will be used. On the other hand if only homophily is at play (Case 2)  e.g., weight of homophily representation is learned close to 0, the spherical space will be used. In the normal case of both factors at play (Case 3), then both spaces will be used and can be jointly aligned with our space alignment mechanism. When using product space, Cases 1/2/3 will all not be distinguished from each other (though that would be a better representation) as all cases will be modeled by one complex non-Euclidean geometric space as a cartesian product of spherical and hyperbolic spaces.  \n\nQ3: What is ‘i’ in the definition of log_0^H and exp_0^H? \n\nA3: The ‘i’ is the mathematical symbol - sqrt(-1) (the formal term: imaginary number). i (which is in italics) is notation e.g., z_i, which is referring to the i-th node.\n\nQ4: As for the computational efficiencies, more formal quantitative comparisons would be necessary in the main texts. \n\nA4: Thank you for your comment. In the main paper, we will be sure to include formal quantitative comparisons for computational efficiency. Regarding our model, NMM is highly efficient, with time complexity O(n * d), where n is number of nodes and d is dimension size. In comparison, here are the time complexity analyses for the remaining baseline models, mixture model of RaRE is O(n * d) which is comparable to the NMM mixture model, the GNN embedding models of GCN is O(n^2 * d + n * d^2), and GAT is O(n * d^2), the non-Euclidean GNN embedding models of k-GCN is O(k * [n^2 * d + n * d^2]) and HGCN is O(2d^2 + a * n * d^2) = O(a * n * d^2) where ‘a’ is the filter length, and NMM-GNN is O(n^2 * d + n * d^2) which is comparable to GNN embedding models. Moreover, we would like to point out that GraphVAE (of NMM-GNN) training is designed to be highly parallelizable, which allows for scalability. Moreover, our model is capable of learning on real-world, highly large-scale graphs on the order of millions of nodes and billion of edges, e.g., Friendster, while achieving the SOTA performance, which attests to its practical value to the network science community.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fMmiXegooV", "reviewer": "Reviewer_BmsX", "summary": "The authors understand why links are generated through node-embedded representations of social networks. Specifically, spherical space is utilized to represent the homogeneity of nodes and hyperbolic space is utilized to represent the hierarchy and influence of nodes. By mixing these two spaces together, the corresponding link representation is finally obtained.", "questions": "1.How to analyze the reasonable causes of social network link generation from complex feature representations?\n2.Can this model explain small-world networks in complex networks?", "limitations": "1.The model does not really give an adequate explanation of links in complex networks, but only from the perspectives of homogeneity and hierarchy.", "rating": 5, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1.The writing is clear and easy to understand.\n2.The experiments were adequate and demonstrated the power of the models", "weaknesses": "1.The idea of the article is not novel. Both hyperbolic and spherical spaces are geometric models that are very commonly used in the field of social networking.[1]\n2.Although the authors are trying to explain linking relationships in terms of homogenization and hierarchy, however the learning of the model embedding is still unknown, which does not explain the emergence of social networks.\n3.There has been much better work on embedding methods in hybrid spaces. [2]The authors don't contribute as much to this as their article suggests.\n[1]Network geometry. Nature physics\n[2]Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learning AAAI 2024\n[3]Product Manifold Learning. AISTATS 2021", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q1: Idea is not novel. Hyperbolic/spherical spaces are are commonly used in social networking. [1] Network geometry. Nature physics\n\nA1: We would like to clarify and highlight that the novelty of our work lies in representing the factors in network science that explain how links are generated in the social network: homophily and social influence. We are among the first works to jointly model both factors in the network when comprehensively surveying the latest literature of work of 15 baseline models (up until the year of 2024) in all categories of learning models including (1) structural embedding models, (2) GNN embedding models, (3) homophily-based embedding models, and (4) mixture models. As we observe through resulting topological patterns formed by homophily (cycles) and social influence (hierarchy), we have motivation to utilize the non-Euclidean geometric spaces of spherical/hyperbolic to model the resulting topologies. Specifically, we are among the first to not only jointly model jointly for homophily/social influence, but also doing so with the novel integration of multiple non-Euclidean geometric spaces used together (with novel space unification architecture) through an efficient Non-Euclidean Mixture Model (NMM).\n\nQ2: Although authors are trying to explain linking relationships in terms of homogenization and hierarchy, however learning of the model embedding is still unknown, which does not explain the emergence of social networks. \n\nA2: We utilize feature learning through latent embeddings to represent nodes in terms of homophily and social influence factors. Homophily is based on feature similarity which can be captured through cosine similarity of embeddings, and social rank can be captured via norm space. Furthermore, these learned embeddings are highly interpretable. Specifically, in our visualizations of the network embeddings, we see that celebrity nodes are embedded towards the center of the Poincare disk, while nodes with lower social rank are embedded towards the boundary. Nodes of high homophily can be embedded closer to each other on the spherical space compared to what the hyperbolic or Euclidean spaces even allows for (better capturing cyclic influence). We will add this interpretable visualization to the paper.\n\nQ3: There has been much better work on embedding methods in hybrid spaces. [2] Motif-aware Riemannian Graph Neural Network with Generative-Contrastive Learning AAAI 2024. The authors don’t contribute as much to this as their article suggests. \n\nA3: The novelty of our work lies in representing the factors in network science that explain how links are generated in the social network: homophily and social influence which form resulting topological patterns formed by homophily (cycles) and social influence (hierarchy) motivating our use of non-Euclidean geometric spaces of spherical/hyperbolic. Specifically, we are among the first to not only jointly model jointly for homophily/social influence, but also doing so with the novel integration of multiple non-Euclidean geometric spaces used together (with novel space unification architecture) through an efficient Non-Euclidean Mixture Model (NMM). \n\nThe paper you reference is not modeling social networks (critical for our problem of investigation) but rather some general purpose graphs Cora etc. (1) this model does capture laws of how links are formed in social networks (homophily/social influence), and (2) do not exhibit the types of resulting topologies from those factors. Also distinct/individual non-Euclidean geometric spaces are considered but rather does not explore how to jointly integrate different curvature non-Euclidean geometric spaces (to have joint influence from social network factors).\n\nQ4: How to analyze the reasonable causes of social network link generation from complex feature representations?\n\nA4: Please refer to sections 3.2.1 “Link prediction using homophily based distribution” and 3.2.2 “Link prediction using social influence based distribution” where we explain the probability distributions for our mixture model NMM by considering the impact of high/low homophily (+ noise/sparsity factor) in addition to high/low social influence (+ noise/sparsity factor). To empirically validate our claims and quality of our NMM mixture model’s learned embeddings, in our experiment results on Table 10, we rigorously evaluate the results of social network classification and link prediction against 15 other SOTA baseline models.\n\nQ5: Can model explain small-world networks in complex networks? \n\nA5: Definitely, which is substantiated through rigorous empirical evidence. We evaluate on numerous datasets including: social network datasets (domain focus of this research), citation networks (Appendix), and attributed networks (Appendix). For example, the social network datasets include BlogCatalog, LiveJournal, Friendster which are a balanced mix of small-scale and the largest scale size (node counts: of 10k, 5M, 66M respectively) and both directed and undirected. These numerous datasets show the generalizability performance and scalability of our approach as they are in all sizes for vertices (4K to 66M), edges, edge types, attributes, classes (6 to 500) etc. Please see Tables 8, 11, 13 of main paper for details.\n\nQ6: The model doesn't really give an adequate explanation of links in complex networks, but only from perspectives of homogeneity and hierarchy.\n\nA6: It is largely agreed that social network links are formed due to either homophily or social influence from the network science community through numerous years of research development. In fact, most existing embedding models are designed based on the homophily aspect of social networks [4, 5]. However, research of RaRE [6] and work of [7] show homophily is insufficient, and social influence is also critical in forming connections. This is due to popular nodes having direct influence in forming links [8]. Refer to our paper for [4],[5],[6],[7].", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KOo68kyoVn", "reviewer": "Reviewer_BsLx", "summary": "This paper proposes NMM-GNN, a non-Euclidean mixture model that captures both homophily and hierarchies in social networks for embedding.", "questions": "1. My main concern is that social network embedding algorithms considering social impact are not novel. For example, when performing edge prediction, one can directly assume the probability of the edge is proportional to the degree of both nodes. I hope the author could further explain why the proposed method is novel enough in terms of encoding both similarity and impact.\n\n2. Despite the experimental results and ablation study in the main article showing the good performance of the proposed method, I want to know how the speed of the method compares with existing baselines.", "limitations": "The author discussed the potential limitations of the paper in detail in the appendix.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.The paper is well-structured, clearly written, and easy to follow.\n\n2.In the experiments section, the author compared baselines from different categories on multiple tasks (classification and link prediction). The results show that the NMM-GNN proposed in the paper consistently achieves the best performance.", "weaknesses": "1. The related work section does not cover most existing social embedding works. For example, many works, in addition to RaRE, also consider both the similarities and the social impact of nodes.\n\n2. There is no clear proof or at least empirical experiments to demonstrate that it is more reasonable to embed both node similarity and social impact in spherical space instead of Euclidean space.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks a lot for your insightful comments and feedback. Please find our response below to your questions.\n\nQ1: The related work section does not cover most existing social embedding works. For example, many works, in addition to RaRE, also consider both the similarities and the social impact of nodes. \n\nA1: Please refer to Table 9 in our main paper of “Category and Description of Baseline models” where we discuss and empirically evaluate against all the latest state-of-the-art models for social network embedding models. We cover 15 baseline models (up until the year of 2024) in all categories of learning models including (1) structural embedding models, (2) GNN embedding models, (3) homophily-based embedding models, and (4) mixture models. Our model consistently outperforms all the models for both classification and link prediction on four comprehensive metrics of Jaccard Index (JI), Hamming Loss (HL), F1 Score (F1), and AUC. Please refer to Tables 10, 12 (Appendix), and 14 (Appendix).\n\nQ2: There is no clear proof or at least empirical experiments to demonstrate that it is more reasonable to embed both node similarity and social impact in the spherical space instead of Euclidean space.\n\nA2: We have conducted extensive ablation studies, as shown in Table 10, which could be used as empirical experiments to justify our approach. In these studies, we evaluated several variations of the NMM model, using different non-Euclidean geometric spaces for homophily and social influence. Specifically, we compared the performance of models where E, S, and H denote Euclidean, Spherical, and Hyperbolic spaces, respectively. Our results demonstrate that modeling homophily in spherical space and social rank in hyperbolic space significantly outperforms other configurations, including those using Euclidean space. \n\nQ3: My main concern is that social network embedding algorithms considering social impact are not novel. For example, when performing edge prediction, one can directly assume the probability of the edge is proportional to the degree of both nodes. I hope the author could further explain why the proposed method is novel enough in terms of encoding both similarity and impact. \n\nA3: We would like to clarify and highlight that the novelty of our work lies in representing the factors in network science that explain how links are generated in the social network: homophily and social influence. As we observe through resulting topological patterns formed by homophily (cycles) and social influence (hierarchy), we have motivation to utilize the non-Euclidean geometric spaces of spherical/hyperbolic to model the resulting topologies. Specifically, we are among the first to not only jointly model for homophily/social influence, but also do so with the novel integration of multiple non-Euclidean geometric spaces used together (with novel space unification architecture) through an efficient Non-Euclidean Mixture Model (NMM). This addresses the dual aspects of homophily and social influence which we model in personalized way for each link of the social network in a unified framework, which, to the best of our knowledge, has not been previously explored.\n\nQ4: Despite the experimental results and ablation study in the main article showing the good performance of the proposed method, I want to know how the speed of the method compares with existing baselines. \n\nA4: In the main paper, we will be sure to include formal quantitative comparisons for computational efficiency. Regarding our model, NMM is highly efficient, with time complexity O(n * d), where n is number of nodes and d is dimension size. In comparison, here are the time complexity analyses for the remaining baseline models, mixture model of RaRE is O(n * d) which is comparable to the NMM mixture model, the GNN embedding models of GCN is O(n^2 * d + n * d^2), and GAT is O(n * d^2), the non-Euclidean GNN embedding models of k-GCN is O(k * [n^2 * d + n * d^2]) and HGCN is O(2d^2 + a * n * d^2) = O(a * n * d^2) where ‘a’ is the filter length, and NMM-GNN is O(n^2 * d + n * d^2) which is comparable to GNN embedding models. Moreover, we would like to point out that GraphVAE (of NMM-GNN) training is designed to be highly parallelizable, which allows for scalability. Moreover, our model is capable of learning on real-world, highly large-scale graphs on the order of millions of nodes and billion of edges, e.g., Friendster, while achieving the SOTA performance, which attests to its practical value to the network science community.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Dear Reviewers: Thank you for your time in reading our paper and for your useful comments/questions/suggestions on our paper. We have responded individually to each reviewer, however, we are also including a general summary answer to some common questions:

Datasets: In the main paper we specifically showcase results on social network datasets as that is the domain focus of this research. However, in the Appendix (Tables 12 and 14), we also include experiments for the citation networks of Wikipedia datasets, to show that our model can also benefit other networks, as well as attributed network datasets.  In total, we evaluate on seven datasets ranging from small-scale to large scale for nodes, edges, types of edges, attributes, classes etc. on a very diverse range of datasets. These numerous datasets show the generalizability performance and scalability of our approach (which we rigorously evaluate on 15 SOTA baseline models of four different learning method categories on both social network classification and link prediction on 4 metrics).

Computational complexity: In the main paper, we will be sure to include formal quantitative comparisons for computational efficiency. Regarding our model, NMM is highly efficient, with time complexity O(n * d), where n is number of nodes and d is dimension size. In comparison, here are the time complexity analyses for the remaining baseline models, mixture model of RaRE is O(n * d) which is comparable to the NMM mixture model, the GNN embedding models of GCN is O(n^2 * d + n * d^2), and GAT is O(n * d^2), the non-Euclidean GNN embedding models of k-GCN is O(k * [n^2 * d + n * d^2]) and HGCN is O(2d^2 + a * n * d^2) = O(a * n * d^2) where ‘a’ is the filter length, and NMM-GNN is O(n^2d + nd^2) which is comparable to GNN embedding models. Moreover, we would like to point out that GraphVAE (of NMM-GNN) training is designed to be highly parallelizable, which allows for scalability. Moreover, our model is capable of learning on real-world, highly large-scale graphs on the order of millions of nodes and billion of edges, e.g., Friendster, while achieving the SOTA performance, which attests to its practical value to the network science community.

Novelty: We would like to clarify and highlight that the novelty of our work lies in representing the factors in network science that explain how links are generated in the social network: homophily and social influence. We are in fact one of the first works to jointly model both factors in the network when comprehensively surveying the latest literature of work of 15 baseline models (up until the year of 2024) in all categories of learning models including (1) structural embedding models, (2) GNN embedding models, (3) homophily-based embedding models, and (4) mixture models. As we observe through resulting topological patterns formed by homophily (cycles) and social influence (hierarchy), we have motivation to utilize the non-Euclidean geometric spaces of spherical/hyperbolic to model the resulting topologies. Specifically, we are among the first to not only jointly model jointly for homophily/social influence, but also doing so with the novel integration of multiple non-Euclidean geometric spaces used together (with novel space unification architecture) through an efficient Non-Euclidean Mixture Model (NMM). This addresses the dual aspects of homophily and social influence which we model in a personalized way for each link of the social network in a unified framework, which, to the best of our knowledge, has not been previously explored.

Analyzing Causes of Social Network Link Generation: As mentioned in detail in the paper, it is widely agreed from the network science community that two factors (homophily and social influence) affect how links are generated in the social network. Please refer to sections 3.2.1 “Link prediction using homophily based distribution” and 3.2.2 “Link prediction using social influence based distribution” where we explain the probability distributions for our mixture model NMM by considering the impact of high/low homophily (+ noise/sparsity factor) in addition to high/low social influence (+ noise/sparsity factor). To empirically validate our claims and quality of our NMM mixture model’s learned embeddings, in our experiment results on Table 10, we evaluate the results of social network classification and link prediction for Jaccard Index (JI), Hamming Loss (HL), F1 Score (F1), and AUC. Our model is comprehensively evaluated against 15 other state-of-the-art baseline models belonging to four different categories of learning representations.

Meta-review: The paper proposes a graph embedding methodology in which link generation is modeled via a mixture model that leverages non-Euclidean geometric spaces. The proposed approach has merits that the reviewers have highlighted. These mostly include novel aspects of the methodology, such as modeling both homophily and social influence, as well as well-executed experiments.  The response of the authors to the different questions raised further clarified several aspects of the paper. Therefore, I recommend accepting the paper.
Decision: Accept (poster)