N2GON: Neural Networks for Graph-of-Net with Position Awareness
Authors: Yejiang Wang
Yuhai Zhao
Zhengkui Wang
Wen Shan
Ling Li
Qian Li
Miaomiao Huang
Meixia Wang
Shirui Pan
Xingwei Wang
Venue: ICML 2025 poster (2025)
Keywords: Graph-of-Net
Neural Network
Representation Learning
Supervised Learning
Primary area: deep_learning->graph_neural_networks

Abstract: Graphs, fundamental in modeling various research subjects such as computing networks, consist of nodes linked by edges. However, they typically function as components within larger structures in real-world scenarios, such as in protein-protein interactions where each protein is a graph in a larger network. This study delves into the Graph-of-Net (GON), a structure that extends the concept of traditional graphs by representing each node as a graph itself. It provides a multi-level perspective on the relationships between objects, encapsulating both the detailed structure of individual nodes and the broader network of dependencies. To learn node representations within the GON, we propose a position-aware neural network for Graph-of-Net which processes both intra-graph and inter-graph connections and incorporates additional data like node labels. Our model employs dual encoders and graph constructors to build and refine a constraint network, where nodes are adaptively arranged based on their positions, as determined by the network's constraint system. Our model demonstrates significant improvements over baselines in empirical evaluations on various datasets.

N2GON: Neural Networks for Graph-of-Net with Position Awareness
Yejiang Wang" 12
Yuhai Zhao
Zhengkui Wang* Wen Shan Ling Li Qian Li Miaomiao Huang 12 Meixia Wang 12 Shirui Pan? Xingwei Wang'
Abstract
Graphs, fundamental in modeling various re- search subjects such as computing networks, con- sist of nodes linked by edges. However, they typi- cally function components within larger struc- tures in real-world scenarios, such as in protein- protein interactions where each protein is a graph in a larger network. This study delves into the Graph-of-Nae (GON), a structure that extends the concept of traditional graphs by representing each node graph itself. It provides multi-level perspective the relationships between objects, encapsulating both the detailed structure of in- dividual nodes and the broader network of de- pendencies. To learn node representations within the GON, we propose a position-awes aware neural net- work for Graph-of-Ner which processes both intra- graph and inter- graph connections and incorpo- rates additional data like node labels. Our model employs dual encoders and graph constructors to build and refine a constraint network, where nodes are adaptively arranged based on their posi- tions, as determined by the network's 's constraint system. Our model demonstrates significant im- provements over baselines in empirical evalua- tions on various datasets.
Figure 1:
An example of a Graph-of-Net, where each node is graph in itself, interconnected to form a net.
comprising a set of nodes and a set of edges, with each edge connecting two nodes. For example, in computing networks, nodes can represent computational units (such servers, CPUs), while edges denote communication links between them. To address the intricacies of real-world applications, more intricate graph structures such as hypergraphs and het- erogeneous graphs (Zhou et al., 2006; Zhang et al., 2019) have been employed. Hypergraphs extend the notion of standard graphs by allowing edges to connect any number of nodes, while heterogeneous graphs encompass various types of nodes and/or edges. Recent years have witnessed burgeoning interest in the utilization of graph-structured data, including graph kernels (Tang & Yan, 2022), graph embeddings (Zhou et al., 2023), and graph neural networks (Liu et al., 2023; Wang et al., 2024a).
1. Introduction
Graphs, mathematical structures that represent relation- ships between entities, pervade diverse domains. A con- ventional graph is a fundamental construct in graph theory,
Schoool of Computer Science and Engineering, Northeastern University, China Key Laboratory Intelligent Computing in Medical Image of Ministry of Education, Northeastern Univer- sity, China IInfoComm Technology Cluster, Singapore Institute of Technology. SIT X NVIDIA AI Centre. Singapore 'Singapore Uni- versity of Social Sciences, Singapore Shanxi University, China 'Shandong University, China Griffith University, Australia. Cor- respondence to: Yuhai Zhao <zhaoyuhai @ mww..neu.edu.cco
Despite substantial exploration in mathematics and various scientific fields, it becomes apparent that existing graph structures may inadequately capture the nuances of modern, complex real-world data, thereby hindering effective graph- based learning. Real-world scenarios exhibit a spectrum of graph structures that encompass diverse scales, types, and hierarchical levels. A myopic focus on individual graphs may overlook the intricate multi-level structures and inter-
Proceedings of the nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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N2GON: Neural Networks for Graph-of-Net with Position Awareness
dependencies inherent in these datasets.
nodes based on their labels and contextual relationships with positions. This network is conceived from two critical in- sights: firstly, nodes with similar labels should have closely aligned embeddings, whereas those with dissimilar labels should diverge; secondly, the interactive proximity between node- sade-graphs diminishes as the separation between their rela- tive positions grows. By employing this constraint network, we refine the similarities among the node embeddings. This layered, network- wide approach, which accounts for both untra-graph and inter-graph dynamics provides framework for understanding the complex structures. Our contributions are summarized as follows:
To address this gap and effectively model majority of real- world systems, this paper investigates new graph structure, named Graph-of-Net (GON). Within a Graph-of-No of-Net, each node itself is a graph. This construct incorporates two types of links: 1) intra-graph connections within each node-graph, and 2) inter-graph dependency links between mode-graphe Such framework facilitates in-depth analysis and permits the exploration of relationships spanning multiple layers. An example, illustrated in 1 arises in biological net- works, where individual proteins are represented as small- scale graphs, while protein-protein interaction networks constitute large-scale graphs, often referred to as networks.
We explore a novel Graph-of-Ne structure that lends itself to easy generalization across various fields, e.g.. citation networks, and biomedical networks.
Existing graph learning methodologies typically either uti- lize multiple small-scale graphs to independently represent proteins, overlooking critical interactions, or amalgamate all proteins into single large-scale graph, representing nodes as vectors, thus neglecting the intricate internal structure of proteins. GON shares conceptual similarities with pre- viously used data structures in specific domains, such as the exploration of theoretical properties in physical sciences (Gao et al., 2011; Ni et al., 2014) the prediction of in- teractions among drug molecules in the field of biology (Wang et al., 2021). However, it exhibits versatility across diverse domains. For instance, an Internet can be modeled as a GON, where local area networks interconnect as nodes within a vast web of data exchange. Similarly, a citation network can be captured as a GON, wherein individual pa- pers, conceptualized as text graphs, interconnect through citations.
We propose a position-aware neural network designed specifically for the GON structure. Our model innova- tively manages both intra- and inter-graph interactions, significantly enhancing our ability to decipher and in- terpret intricate data.
We conduct extensive experiments on various types of data, including 9 benchmark network datasets and 7 biomedical datasets. The results show that our model significantly outperforms SOTA baselines.
2. Related Work
2.1. Graph and Graph-of-Net
This paper delves into the intricacies of GON structures, their representation, and their applicability in modeling complex real-world systems. By embracing the inherent graph-of-net paradigm, we aim to provide a comprehensive framework that addresses the multifaceted nature of modern data structures and their interdependencies.
Graphs, with vertices and edges, serve well for depicting bi- nary relationships (West et al., 2001; Wang et al., 2024b;c). The discipline has evolved, incorporating complex struc- tures such as hypergraphs (Zhou et al., 2006) and heteroge- neous graphs (Liu et al., 2024a; Mo et al., 2024). Hyper- graphs permit edges to connect multiple vertices, enabling the representation of complex many-to-many relationships. Heterogeneous graphs, characterized by their diverse types of vertices and edges. In this evolving landscape, the con- cept of the GON introduces a distinct conceptual framework. In contrast to these traditional graphs, the GON incorpo- rates complete graphs within the nodes of larger graph. This approach offers a unique perspective for representing hierarchical and multi-layered relationships. However, exist- ing research on Graph-of-Net is primarily domain-specifer For instance, in the physical sciences, (Gao et al., 201 1) has been dedicated to establishing precise percolation law applicable to network comprising interdependent net- works. Similarly, in the pharmaceutical sector, research (Wang et al., 2021) has predominantly focused on the pre- diction of drug-drug/ weaedduu/chhmiic--chhmeeee contrast to these specialized domains, our study adopts a broader approach. We concentrate on universal structure
The Graph-of-Net (GON) presents paradigm divergent from conventional graphs. As such, traditional graph leam- ing methodologies, such as graph neural networks (GNNs), are not inherently suited for direct application to GONs. To efficaciously learn the representations within GONs, we introduce an position-aware neural network specifically tai- lored for Graph-of-Net (N2GON). This model meticulously incorporates the dual link types inherent in GONs while synergistically integrating supplementary data, such as node labels, to discern and encapsulate more nuanced patterns and structural complexities. Our approach involves the develop- ment of dual graph encoders: one dedicated to embedding the constituent node-graphs and the other to embedding the overarching network, utilizing the representations de- rived from the first encoder. Furthermore, we construct implicit constraint network that interconnects these graph
interactions. In
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N2GON: Neural Networks for Graph-of-Net with Position Awareness
each node as graph, highlighting data's s hierarchical and interconnected aspects.
Encoder 11
Constrainer
3. Methodology
Grapher II
In this section, we will introduce a structure, named Graph- of-Net (GON), and propose novel neural network model for learning from Graph-of-Net data. Actually, the Graph- of-Net structure is a generalization of the traditional graph structure, where each node is graph. Given input Graph- of-Net, we aim to learn the representations of nodes in it. To effectivvly achieve this goal, we propose an position- aware neural network model, which considers both intra- intra-graph connections within node-graphs and inter-graph dependen- cies among them and integrates additional data like node labels capture complex patterns.
Figure 2: The framework of the N2GON model.
of GON, aiming for applicability in wide range of fields. This includes, but is not limited to, social networks, citation networks, and biomedical networks.
2.2. Graph Representation Learning
Graph representation learning (GRL) is experiencing signifi- cant development, focusing on transforming input nodes into low-dimensio- nal vectors. These vectors are primarily used for tasks at both the graph and node levels. The initial phase of GRL centered around shallow networks, with notable examples including DeepWalk (Perozzi et al., 2014), which employed random walks to generate node embeddings, and Node2Vec (Grover& & Leskovec, 2016), which introduced a more flexible approach for neighborhood sampling. In recent years, the rise of graph neural networks (GNNs) has marked significant advancement in graph-based learn- ing. GNNs operate by aggregating the representations of both a node and its neighbors in recursive manner. They achieved ofate-of-the-art performance on various tasks (Liu et al., 2024b), such as node classification (Zhang et al., 2024; Wang et al., 2023), node clustering (Tsitsulin et al., 2023), etc. Nevertheless, despite these significant strides, applying these advanced techniques to GON structures, known for their hierarchical and multi-relational complexity, remains challenging and active area of research. This highlights the need for exploration in the domain of GON learning.
The framework of the proposed model is illustrated in Fig.2. We employ two encoders, Encoder I and Encoder II, to en- code the Graph-of-Net, with the former targeting the node- graphs and the latter focusing on the Net formed between these mode-graphs. Two graph constructors, Grapher and Grapher II, are utilized to build a node-graph constraint network. Grapher I, based on the labels of the node-graphs, connects two nodes with the same label with edge. Gra- pher II, leveraging this preliminary network in conjunction with the Graph-of Saaph-of-Neee of-Net's inherent structure, computes the final constraint network, which features edge weights indi- cating the similarity between nodes. Guided by this network, the Constrainer strategically arranges the node-graph repre- sentations based on their relative positions, aligning those with greater similarities in closer proximity each other, while distancing those is dissimilar (i.e., with different la- bels).
3.1. Definition of Graph-of-Net
A Graph-of-Net is mathematically characterized as a so- phisticated structure GN = Eg), wherein the node set Vi comprises a collection of N individual graphs, each paired with its corresponding one-hot label, repre- sented as {(G,, Y;)l E {I...., N].. The edge set &g C {G,, G, Gy E Vg) delineates the inter-graph relation- ships within this structure. The adjacency matrix Ag for GN is defined such that each element Aij signifies the con- nectivity between graphs G, and Gj. encapsulating the com- prehensive network of interactions. Each graph node G, is defined as G. = (V;, E,, Xx), where V, = .vi...., vi,) and E, C ((v) vi) v vf E V,} represent the vertices and edges of G,, respectively. It's important to note that E, symbolizes the intra-grapi grra- connections. The feature matrix X, Rexd encapsulates the attributes of vertices in G,, with being the number of vertices and d the dimensionality of these features. The adjacency matrix A; of G, assigns a value of I to Aij if (vf, vt) E E, and 0 otherwise.
2.3. Graph Coasrening Learning
Graph coarsening, a technique devised reduce the com- plexity of large graphs, involves the consolidation of mul- tiple nodes into a single node. This process traditionally relies on mathematical methods, which is exemplified in (Dhillon et al., 2007) through graph cuts. Recent advance- ments, like those by (Cai et al., 2021), have incorporated graph neural networks refine edge weights, enhancing coarsening efficiency. Additionally, (Jin et al., 2022) have contributed by focusing on preprocessing and graph conden- sation, crucial for the scalability and acceleration of graph meural networks. Graph coarsening and GON differ sig- nificantly in their functions. Graph coarsening simplifies a single graph for better processing efficiency, while GON, a complex structure, represents multi-layered systems with
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N2GON: Neural Networks for Graph-of-Net with Position Awareness
In this work, given Graph-of-Ner GN, our primary objec- tive is to engage in the task of Graph-of-Net supervised learning, This involves learning the representative vectors vG for the graph nodes, which are instrumental in predicting their labels.
technique further update these representations within the GON network. Analogously, establish the f-th message passing layer f Fe + F with a consideration that fo is network-dependerr Subsequently, by orchestrating a se- ries of operations through fg, we update the representation of each node
3.2. Encoders of Graph-of-Net
VG = fg .... (}
(3)
Given the intricate composition of a Graph-of-Net, where each node themselves possesses distinct structures and the inter-node relationships coalesce into a network, we adopt a dual-encoder strategy. This approach is tailored to harness the multifaceted structural nuances of the Graph-of-Ner 1) Encoder tasked with transforming individual node- graphs into low-dimensional representations; 2) Encoder II, which, building upon the foundations laid by the Encoder I, further refines these representations in alignment with the GON's inherent network architecture. This sequential, dual- encoder setup is meticulously designed to integrate the two- tiered structural essence of the Graph-of-Net, thus ensuring that the resultant node representations are comprehensive blend of both the nodes' intrinsic structural details and the overarching inter-node relational dynamics.
in the GON network, where Ly is the number of layers used in the net encoder.
3.3. Construction of Constraint Network
Upon finalizing the representations of node-graphs within the context of supervised learning-where each node- graph's 's label is accessible during training-we focus on constraining these representations using their associated la- bels. Given the node label matrix Y R**2, where row Y; denotes the one-hot label of the i-th node-graph, we define the label identity matrix C as
YxYXY"
(4)
to construct a constraint network (Grapher I). This matrix's s elements, Ciy. serve to denote whether the labels of the i-th and j-th node-graphs match, assigning a value of 1 for identical labels and 0 for differing ones.
In the context of a specific Graph-of-Net gN, we employs a message- message-passing framework to encode the node-graph G, with a focus on retaining the vital adjacency information intrinsic to the node-graph as follows. Let bg Fe de- notes the feature at layer E associated with node s, where F denotes arbitrary finte-dimeesional space of the form Ra (for various values of q) typically representing the feature space, the updated feature h,41 is obtained as: h,t1 = fu s (h, It E B,).. where E N₁ means that nodes and s are neighbors in the graph G, i.. (s,t) E E, and the function fupd : 2Fc - Fe+1 is a learnable fune- tion taking as input the feature vector of the center vertex hy and the multiset of features of the neighboring vertices (hū It E N... Indeed, for any such function fupd can be approximated by layer of the form
A direct approach might suggest that pairs of nodes with Is in matrix C are similar, whereas those corresponding to Os are dissimilar. However, this method neglects the nuanced interrelations among node-graphs, treating nodes as if they were independently and identically distributed. simplistically presumes uniform similarity for nodes sharing labels, assigning a consistent weight of 1.
Yet, in practical scenarios, two nodes sharing a label may exhibit varying levels of similarity. For example, in cita- tion network, paper P1 focused on graph neural networks (GNNs) and two other papers P2 and P3. although sharing a common thematic label (c.g.. neural networks), may differ in their degree of relevance to P1. If Ps discusses GNNs while P3 explores convolutional neural networks (CNNs), then P2 inherently shares a closer similarity with P1 than P3. To tackle this, we recognize that the GON's 's inherent network structure provides critical insights to differentiate such varying degrees of similarity. Nodes sharing a label and exhibiting closer connections should logically possess higher similarity, and the converse for less connected nodes. In our example, paper P1 is more likely to cite paper Pa (directly interconnected) than Ps. suggesting tighter con- nection within the GON's citation network.
hetl = (w"
-o("". (w. (
ff (hh, Whille N.B)))))) (1)
where fe 2Fc Fe41 is injective set functions in the l-th layer, & denotes vector concatenation, W is learnable weight matrix and a is an element-wiss activation function. We get the ê-th message passing layer f Fe 7 F e (note that fo depends implicitly on the node-graph). Then, by the composition of I and pool function fCool : L, - FLI+1 we obtain the representation of each graph node
Pool o ft. o fe fd(X) hg = f
(2)
Building upon these insights, we address the issue of uni- form similarity weights by evaluating node significance and adjacency within the GON. To this end, we construct a re- fined constraint network (Grapher If), which recalibrates
where L, denotes the number of layers used in the node- graph encoder. Leveraging the derived representations of node-graphs, the Encoder Il adopts message propagation
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N2GON: Neural Networks for Graph-of-Net with Position Awareness
Algorithm 1 N2GON
segregating the representations of node-graphs that are dis- similar (namely, those with differing labels). To achieve this goal, for each node-graph, we define a loss function to constrain it in relation to other nodes (Constrainer):
Input: : GN, F2: the backbone encoders.
while not converge do
Sample full-batch of node-graphs [G from GN¹
Encode IN
{- by Fi (2) to get {hG. IN
L. Update {hg, i= by F2 (3) to get {vG; J
exp(sim(vG..) VG, )/T
Lcon Σ
III C)es log
Generate constraint net C (4. and II (6);
Ligt exp(sim(yG, VG, (/)r))
Calculate constrain loss Loon (7) and NLL loss, opti-
mize the encoders F2;
where denotes the Hadamard product, VG is the represen- tation of node-graphs G. sim(a, b) denotes the similarity between vectors a and b, which is usually defined as the cosine similarity without normalization. T is tempera- ture parameter. For label-based supervision, we employ the Negative Log-LLikelihood (NI..) as the loss function. Consequently, the overall loss is formulated as the sum of the constraint loss and the NLL loss. The algorithm of the proposed model N2GON is summarized in Algorithm 1.
end
Output: The well trained model F1, F2
the similarity weights among node-graphs. This recalibra- tion is inspired by the Personalized PageRank (PPR) algo- rithm. The PPR algorithm calculates the probability of an a-random walk, which originates from source node and concludes at target node. In our model, the source node corresponds to the node-graph G, and the target node is Gj. Here, an a-random walk represents a path taken by a random surfer on the network, who either halts at the cur- rent node with a probability of a or proceeds to randomly selected outgoing neighbor with probability of 1 a. The pairwise score value x (G., G,) can be computed by solving the following equation:
4. Computational Complexity
In this section, we provide a comprehensive discussion of the computational complexity of the proposed N2GON ap- proach. Let the number of node-graphs in the Graph-of-Net be VI, and the number of edges between the node-graphs be |g]. Additionally, the number of nodes and edges within node-graph are denoted as [V] and [E]. respectively. 1) Encoder: The time complexity of the graph neural net- work backbone for Encoder and II are O(VV + (ET) and O(IV + |ε)), respectively. 2) Grapher Since the node label matrix Y is a sparse one-hot label matrix, the com- putation of the label identity matrix C is mainly related to the number of non-zero elements in Y. Therefore, the time complexity is ((6)))q), where ö <<< 1 represents the ratio of non-zero elements in Y, and is the number of classes. 3) Grapher II: In constructing the refined constraint network, we use the PageRank local partitioning algorithm, 2 log" [8) where
(G, GR) d(Gx))
TG,, G,) - aG (1-
Σ GRENS,
(5)
wherein NG signifies the set of neighboring node-graphs of G, 1(.,-) :) is an indicator, and d(G) denotes the degree of node-graph G. Intuitively, as per this definition, the value (G,, G2) assesses the importance and similarity of node Gj in relation the source G, Let II represent the score matrix, where (IT), corresponds to the importance value (G, Gj) relative to the source i. We have
II a - (1 -1
(6)
with a logarithmic running time of O is the conductivity and b E [1, log m] is constant. 4) Loss: Since the constraint loss requires the computation of the dot product between pairs of embeddings, the time complexity is O (IV)2d), where d is the dimensionality of the embed- dings. Therefore, the total computational time complexity is O 1100a+1122 2" log [8] + M IVI+EE)
where I, is the identity matrix. Hence, our model considers not only the label information of graph nodes but also the positional information between nodes, which computes the topological influence of node-graphs on each other, implic- itly revealing their relative positions in the network.
3.4. Loss
5. Experiments
After establishing the constraint network, we integrate it with the representations of our node-graphs. The goal here is to meticulously align these representations based on the pairwise importance scores defined within the constraint network. This strategic alignment is crafted to bring rep- resentations of node-graphs with higher similarity scores closer together, while placing those with lower similarity scores at proportionately greater distance, and explicitly
In this section, we conduct an extensive experimental eval- uation of N2GON, examining its performance across wide range of datasets from various domains, including social networks, citation networks, web page networks, and biomedical data. The first three categories are common in graph learning research, allowing us to compare N2GON with tate-of-thhear graph learning algorithms. However,
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N2GON: Neural Networks for Graph-of-Net with Position Awareness
Table I: Mean test classification accuracy (%) + stdev on 6 heterophily and 3 homophily datasets. The highest performance is highlighted, and the second best performance is underlined.
Texas
Wisconsin
Actor
Squirel
Chameleon
Comell
Citeseer
Pubmed
Cora
Hom. level h
0.11
0.21
0.22
0.22
0.23
0.3
0.74
0.8
0.81
#Nodes |
183
251
7,600
5,201
2,277
183
3,327
19,717
2.708
#Edges |E]
295
466
26,752 5
198,493
31,421
280
4,676 7
44,327
5,278
#Classes
MLP
81.89=4.75
85.29+3.00
35.76=0.99
29.6841.
46.36+2..00
81.08=6635
72.4142..11 18
86, 66.65-t0 65.30.35
74.75+
GCN
55.14-6.16
51.7643.00
27.32$1.10
53, 53.43=2.01
64.82 24
60.54-6.380
76.50+1.38
88, 88.42+0.50
36.90+1.00
GAT
52.14-5.16
49.41+4.09
27.44+0.89
40.72+1.55
60.26 2.50
61.89+5.00 0.05
76.55+1.22
86.33+0.44
87.30+1.10
GraphSAGE
82.43=6.14
18.
34.23=0.09
41.61=0.74
68.73+1.68
75.96=5.00 01
76.0441.33
88.4520.50
36.90+1.04
GCNII
77.57=3.88
80.39+3.40
17.4441.30
38.47=1.68
63.86=3.04
77.86=3.79
77.33+1.48
90.15::0.43
88.37
H2GCN-1
64.86=6.77
86.6714.69
35.86=1.03
36.42=1.80
57.11=1.58
82.1644.80
77.07
89.40:0.34
86.92+1.37
H2GCN-2
82.16=b.28
4..22
35.62=1.3U
37.90=2.02
59.39
1.99
82.16=6.00
76.88+1.777
89.59+0.33
87.81+1.36
ACM-GCN
87.84=4.. 4.40
88.4343.22
36 1.00
51.40=11.88
66.93
65.1446607
1.
90.00:0..2
87.91
WRGAT
83.62-5.00
86,98 43.78
36.53+0.77
48.85
63.2440.87
61.62+3.90
76.81441.89
88.52 88.52+0.92
87.95+1.11
GGCN
86.86 84.86=4.55
1..
17.18
7..1421.84
85.68:1
17.1411.45
89.15:0.37
8..95=1.05
S2GC
71.57 68.65=6.05
34.17=0.92
01.63=0.98
75.25=7.68 55.55
75.
76.0810.45 45
88. 88.31+0.35
87.73 27.99
SIGN
75.
30.5943.75
36. 1.01
40. 26
78.11 20.4
76.534 16
89.5810.45
86.7211.00
APPNP
78.37=6.01
34.64#1.61 81.4244.34
33.51+2.02
47.50=1.76
77.02=7.01
77.0641.78
87.9440.56
87.71+1.34
GPRGNN
82.12=7.72 81.16+3.17
33.29#1.39
43.29=1.66
2..89
81.08=6.59
75.56+1.62 6.
86.85+0.46
86.98+1.33
GCN+JK
66.49-6.64
64.31
34.18
40.4521.61
63.4222..0
64.39-8.68
74.51
88.41
86.7910.92
GCN-Cheby
77.30=4.07
49.41=4.46
34.11#1.09
13.86=1.64
56.2442.76
74.32=7.46
75.82+1.53
88.7210..5
86.76=0.95
MixHop
77.84=7.73
75.8824.90
32.22=2.34
43.80=1.48
60.50
73.51
76.2611.33
80.31
87.61
FAGCN
78.11=0.00
81.56+4.64
35.41#1.18
42. 42.43-2.11
56.31+3.22
76.12=7.60
74.86+2.42
85.74+0.36
83.21+2.04
DAGNN
70.27 4.93
71.7645.25
35.51#1.10
30.29:2.23
15.92+2.30
73.31=7.18
76.44+1.97
89.37:00.2
86.82+1.67
HopGNN
82.97
85.6945.43
37.09+0.97
64.23
71.21+1.45
84.05+44.48
76.69+1.56
90.28+0.42
87.57+1.38
N2GON
16.1664.11 4..2
42.57
60.98=2.12
23
92. 11 11.1123.60 6666
81.2711 30
90.6410.37
89. 14.24.2..00 01
biomedical data presents distinct challenges due its di- verse representation methods. This data is often represented in multiple complex vector formats, resulting from various processing techniques. Consequently, our evaluation also encompasses these alternative representation methods.
versity of Texas, University of Wisconsin, and Cornell Uni- versity, as well as Squirrel and Chameleon datasets based on Wikipedia topics, and the Actor network (Pei et al., 2020). These datas build networks through nodes (web pages or actors) and edges (hyperlinks or collaborations). The node labels are based on metrics such as webpage traffic or the categorization of actors in their context.
5.1. Network Benchmark Datasets
Datasets. To evaluate the effectiveness of N2GON across a spectrum of datasets, we conducted a comprehensive analy- sis using 9 benchmark network datasets. These datasets are diverse, varying in domain, size, and the degree of data smoothness. Our selection encompasses three stan- dard homogeneous citation datasets, mentioned in (Kipf & Welling, 2017), well as six well-known heterogeneous datasets, referenced in (Pei et al., 2020). The statistical information of these datasets are systematically outlined in Table 1. The network benchmark datasets focus on graph node classification, measuring the accuracy of node classi- fication. Details of datasets are as follows. For homophily datasets, the benchmarks include CiteSeer, PubMed, and Cora (Kipf& & Welling, 2017), which are prominent in the domain of citation network analysis. In these datasets, nodes are utilized to represent academic papers, while edges cor- respond to the citations amongst these papers. The labels assigned to each node categorize the paper according to its research topic. As for heterophily datasets, this group en- compasses datasets from the WebKB collection for the Uni-
Given that these benchmark datasets predominantly offer processed graph data, we were tasked with adapting this data into a GON format suitable for our analysis. To achieve this, we implemented strategy wherein the k-hop neighbor nodes of each node were sampled to induce a correspond- ing subgraph, effectively representing that particular node. This transformation method is justified by considering, for instance, a citation network where a paper's composition is not only its content but also the referenced works it cites; similarly, in social networks, user's (node's) social con- nections are an integral part of their identity.
Baselines. We compare the proposed model N2GON with various baselines, including (1) MLP: (2) standard node- interaction GNN methods: GCN (Kipf & Welling, 2017). GAT (Veličković et al., 2018), GraphSAGE (Hamilton et al., 2017) and GCNII (Chen et al., 2020), MixHop (Abu-El El- Haija et al., 2019), GCN-Cheby (Defferrard et al., 2016), GCN+JK (Xu et al., 2018); (3) heterophilic GNNs with adaptive node interaction: H2GCN (Zhu et al., 2020), WRGAT (Suresh et al., 2021), ACM-GCN (Luan et al.,
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N2GON: Neural Networks for Graph-of-Net with Position Awareness
Table 2: Predictive performance results, ROC-AUC and PR-AUC (%), on 4 relationship classification tasks PPI, PEPMHC, TCR, and MTI. The highest performance is highlighted, and the second best performance is underlined.
PPI:HuRI
TCR:Weher
MTEmiRTarBBae PR-AUC
Method
ROC-AUC
PR-AUC
ROC-AUC
PR-AUC
ROC-AUC
PR-AUC
AUC
AAC
88.81 4.14
90.62 3.09
90.57
9.01
83.18 4.39
86.51
6.91
83.97 + 4.20
86.48 + 7.36
81.24
5.74
Conjoin(Triad
87.43 = 5.29
91.01
7.18
93.19
5.29
87.21 = 3.52
85,96
= 6.77
83.98 + 8.34
81,40 + 5.35
82.82
+ 4.56
Quasi Seq
79.28 + 4.72
82.62
4: 5.23
90.75
+ 3.19
85.08 + 2.99
76.65
4.90
73.15
5.21
71.44 + 2.66
73.16
3.29
ESPF
82.94 6:20
86.38
4 4.85
93.31
+: 5.12
87.46
3.9G 85.51
83.32
+: 5.69
89.53 + 6.71
85.01
4.55
CNN
87.19 = 3.38
90.45
0 6.17
91.36
+ 1.27
81.99
= 3.86
86.21
83.72
3.19
87.81 - 4.28
81.39
6.50
CNN RNN
86.61 + 3.77
89.81
+ 5.10
86.25
+ 7.23
81.50 + 4.38
6.96 86.30
71.38
71 + 3.32
82.11
+ 3.45
80.41
+ 3.67
Transformer
88.20
90.35
4.16
91.78
4: 4.39
85.28
4.76
84.35
+: 3.80
88.45
+ 6.01
82.22
4.79
N2GON
89.83
3.52
91.37
93.77
+: 6.23
86.94 4.65
89.78 7.38
89.75 +: 2:35
91.55 + 7.49
84.66 3.95
Table 3: Predictive performance results, ROC-AUC and PR-AUC (%), on DTI datasets. The best performance is highlighted, and the second best performance is underlined.
Table 4: Predictive performance results, ROC-AUC and PR-AUC (%), on DDI datasets. The best performance is highlighted, and the second best performance is underlined.
Method
ROC-AUC
DAVIS
PR-AUC
ROC-AUC
KIBA
PR-AUC
Method
ROC-AUC
TWOSIDES
PR AUC
MogganAACC
85.19
76.77 1.63
90 + 5.53
83.18 + 6 70
Morgan
82.68 = 5.17
79.30 + 8.76
MogaanESPP
85.96 = 3.07
76.93 + 7.58
87.
13 + 6.88
85.56 + 3.59
Pubchem
81.69
6.40
78.21
8.03
Morga--CNN
82.01 = 1.78
73.96 = 1.09
90.36
2 8.61
83.43 + 5.89
Daylight
81.97 4.11
78.76
1 5.59
PubChem-AAC
86.87 + 29
77.75 = 7,99
85.51 + 8.14
81.74 1.98
RDKIT
82.29
78.75
6.74
PuhChem-FSPP
80.13 +: 3.87
78.40 + 4.91
85.72
8.38
82.23 + 3.52
CNN
62:29 5.24
79.35
+ 7.16
PuhChem-CNN
88.23 L: 5.22
81.18 1 4.16
92.61
1 7.36
00.72 1: 6.27
CNN RNN
62.58
5.77
78.61
+ 5.17
AAC
ESPP
85.54 = 3.78
84.13 +: 4.50
81.79
76.85
+: 4.34
6.18
83,96
85.84 + 8.52
03.96-40
82. 02.00411 + 6.66
76, + 5.22
MPNN
Transformer
70.30
80.27 3.15
5.18
78.26
77.34
+ 5.35
5.70
CNyligh+-CNN
84.43 3.32
75.78
1. 7.08
6.92 91.33
92.30
1 5.99
GoGNN
83.03 = 4.72
79.19
+ 5.85
RDKIT+AAC
81.71 = 6.94
80.80 =: 7.32
87.02
+: 6.38
8-1 86
+ 4-45
N2GON
85.53 5.67
80.96
6.35
RDKIT-ESPP
85.90
4.47
75.19
6.52
5.33 85.92
82.26 + 4.07
RDKIT-CNN
84.52 1. 5.33
78.56 1. 5.71
5.92 88.68
85.45 1. 6.47
ESPP+AAC
85.38
6.52
77.68
7.45
87.54
7.97
841-44 + 4.94
ESPP+ESPP
85.14 +: 3.04
76.83
+: 7.29
85.80 +: 5.43
60.81
+ 4.22
ESPF+CNN
84,83
3.31
77.65
6.69
8.60 88.45
5.72 85,
CNN+AAC
83.73 = 5.42
80.21
+: 4.57
86.87 +: 6.88
85. 15 +: 6.55
from the set {1, The selection of the probability pa- rameter a ranged from {0.1...., 0.6}. while the parameter temperature T was chosen from a range of {0.1..... 0.5]. For the training phase, the Adam optimizer (Kingma Ba, 2014) was utilized. Regarding the prediction tasks, our benchmarks included network dataset aimed at predicting graph node labels and a biomedical dataset focused on iden- tifying relationships between graph nodes. Accordingly, our model was adapted to each dataset's unique requirements: the former directly leveraged the labels of nodes, while in the latter, treated adjacent nodes as sharing the same labels.
CNN+ESPF
82.96 4.70
74.81
7.91
85.55
5.34
83..43
1: 6.31
CNN+CNN
84.39 1: 3.06
76.39
1: 7.34
46 1. 8.06
84.58
1: 6.30
MPNN+AAC
86.27 + 6.87
78.96
+ 6.56
81.41 6.97
83.41 + 6.76
MPNN+ESPE
85.90 1: 3.75
80.93 à 4.26
81.09 1: 7.54
78.51
1: 6.85
MPNN+CNN
82.78 3.86
73.59
7.10
89.41
5.47
86.76
1: 5.97
N2GON
91.53= = 4.26
83.68 +: 5.65
90.27 4: 3.15
95.08
+ 3.11
2022), GGCN (Yan et al., 2022), FAGCN (Bo et al., 2021),
Geom-GCN (Pei et al., 2020) (4) sampling GNNs: Fast- GCN (Chen et al., 2018), AS-GCN (Huang et al., 2018), ClusterGNN (Chiang et al., 2019), GraphSAINT (Zeng al., 2020); and (5) decoupled GNNs: S2GC (Zhu & Ko- niusz, 2020), SIGN (Frasca al., 2020), APPNP (Klicpera et al., 2019), GPRGNN (Chien et al., 2021), DAGNN (Liu et al., 2020), HopGNN (Chen et al., 2023). We report results from previous studies using the same experimental setup when available. For unreported results with available codes, we implement them using official codes.
Results. Table 1 displays the node classification results comparing our N2GON algorithm with various established graph representation learning algorithms. The data in Table indicates that N2GON outperforms the SOTA algorithms across a range of datasets. Notably, on the homogeneous dataset CiteSeer, N2GON shows a 3.94% improvement in performance over GCNII, and on the heterogeneous dataset ACTOR, it exceeds the performance of GGCN by 5.03%. These results suggest the potential effectiveness of N2GON in various dataset contexts. The results collectively suggest that the GON data structure, and our algorithm's 's approach to leveraging this structure, could offer certain advantages in capturing and representing the complexities of different types of datasets.
Implementation Details. In this study, we utilized PyTorch to implement our methodology. Our experimental setup con- sisted of server equipped with two NVIDIA A6000 GPUs running Ubuntu 20.04. We configured the hidden dimension of N2GON as 32 for the nine datasets. We tune the hops k from {1,2.... 63. We determined the layer counts for En- coder and Encoder II, denoted as La and Teg, by selecting
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N2GON: Neural Networks for Graph-of-Net with Position Awareness
Table 5: The statistics of the biomedical datasets. # Enl and # En2 represent entity counts in the first and second sets, respectively, and # Link signifies the number of interactions.
chem, Daylight, RDKIT, CNN, CNN_RNN, Transformer, and MPNN, while protein encoding options are AAC, Con- join(Triad, Quasi_seq, ESPF, CNN, CNN_RNN and Trans- former. Details of these encoders can be found in (Huang et al., 2020). Therefore, our study conducts comparisons of N2GON against encoder-based classification. Addition- ally, since the GoGNN (Wang et al., 2021) specifically designed for drug-drug interaction data, we also compare N2GON with GoGNN on the DDI dataset Twosides.
Data
DAVIS
KIBA
TWOSIDES
HuRI
MHC-1
miRTarBase
Weber
Em1
2,068
645
8,248
43,018
3,465
192
8,248
150
21,242
23.139
# Link
25,772
117.657
63,462
51.813
185,985
400,082
47,182
Implementation Details. For the baselines, we followed the default settings from the benchmark library (Huang et al., 2020), and the original work (Wang et al., 2021) with its specified hyperparameters. Our N2GON model, being well- suited for affinity network data, utilizes the adjacency matrix to construct the label identity matrix, assuming neighboring nodes share the same label. In these tasks, N2GON aims to predict connections between two nodes, and the aforemen- tioned assumptions align seamlessly with the requirements of the tasks. We maintained consistent hyperparameter set- tings for N2GON as outlined previously. It should be noted that within these datasets, PPI and DDI are homogeneous networks, whereas other datasets, such as DTI and MTI, are heterogeneous networks, meaning the nodes have differ- ing properties. For heterogeneous networks, we utilize the method described in the paper (Schlichtkrull et al., 2018) to convert a homogeneous GNN model into its heterogeneous equivalent. The edge splits for training, validation, and test- ing datasets were uniformly distributed across all methods using an 80/10/10 ratio. Note that during test phase, the encoder utilizes edges from both the training and validation sets for encoding and subsequently predicts edges from the testing set. We conduct 10 runs and report the average and standard deviation of ROC-AUC and PR-AUC scores for each method.
5.2. Biomedical Datasets
Datasets. We conducted evaluation of N2GON biomedical datasets, which includes 7 datasets from di- verse domains:
DrugeTarget Interaction (DTI) with datasets DAVIS and KIBA (Davis et al., 2011; Tang et al., 2014), Drug-Drug Interaction (DDI) with Twosides (Tatonetti et al., 2012), Protein- Interaction (PPI) with HuRT (Luck et al., 2020), Pcptide-MHCC Binding Prediction (PEPMHC) with MHC-I (Nielsen & Andreatta, 2016), MicroRNA- Target Interaction (MTT) with miRTarBase (Chou et al., 2018), and TCR-Epitope Binding Affinity (TCR) with We- ber (Weber et al., 2021). The biomedical datasets focus on biological entity interaction prediction. The statistics of the biomedical datasets are shown in Table 5. These datasets were assembled by experts in the biomedical field, focusing on the prediction of affinity relationships between various entities (Koh et al., 2024). For example, the DTI dataset DAVIS includes data on drug-target affinities, while the DDI dataset TWOSIDES contains information on drug- drug affinities. This type of data is naturally suited for representation in the GON format. In this structure, individ- ual nodes (e.g., drug molecules) are graph structures, and the inter-node relationships (e.g., interactions between drug molecules and targets) constitute a network of graphs. In our study, drug molecules are represented as graphs with nodes symbolizing atoms and edges denoting chemical bonds. For the node features of drug molecule construction, we can use the transformation methods provided by the library Therapeutics Data Commons to extract node features from SMILES sequences. Similarly, proteins (or peptides) are represented as graphs with nodes depicting amino acids and edges illustrating the chemical bonds between these amino acids.
Results. Tables 2, 3, and 4 present the comparative per- formance results of our N2GON algorithm against corre- sponding baselines. Analysis of these tables reveals that N2GON consistently outperforms other methods across all datasets. For example, on the KIBA dataset within the DTI domain, N2GON shows a performance improvement of 2.66% with regard to ROC-AUC over the top encoder- based classification method, Pubchem+CNN. RNNN This suggests that N2GON may offer advantages over traditional encoder-based method and the GoGNN, in the context of biomedical data.
Baselines. In the analysis of biomedical datasets, it's com- mon to use one or two encoders for encoding entities like drugs or proteins. The resulting encoded vectors are then fed into a decoder to predict the relationship between two enti- ties. Typically, this prediction task falls under binary classi- fication, where an output of 1 signifies a relationship, and O denotes its absence. To establish a comprehensive baseline for comparative analysis, various encoders for drugs and pro- teins, specified in a domain domain-specific specific benchmark library, were employed. The drug encoding options include Morgan, Pub-
5.3. Analysis of N2GON
Ablation Study. In this analysis, we evaluate the impact of the constraint network components, Grapher I and Gra- pher I1, on the N2GON model's performance. We compare N2GON's effectiveness both with and without these com- ponents, designating the models without Grapher I and
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N2GON: Neural Networks for Graph-of-Net with Position Awareness
and differentiating between node classes, highlighting its potential utility in such applications.
CURA
6. Conclusion
In this paper, we investigate a structure, graph-of-net (GON), to model real-world systems. GON provides a multi-level perspective on the coupled dependency relations of objects, it contains: 1) node as graph, where each such graph can model entities like computing networks; 2) edge connect graph. We propose a position-aware neural network to learn the representations of GON, integrating both the structural information of nodes and their interdependencies, while also considering auxiliary information between nodes (i.e., label), To demonstrate the superiority of the GON representation and the effectiveness of our proposed model, experiments were conducted across various types of datasets. The results show that our model significantly outperforms state-of-the- art algorithms in these domains.
CHAMELEON
CHRAMIE
(a) GCN
HopGNN
N2GON
Figure 3: t-SNE embeddings of nodes in the network data.
Grapher II as w/o Grapher I and w/o Grapher II, respec- tively. The comparative performance data for these models is delineated in Table 6. An examination of the results shows that the versions of the model lacking Grapher and Gra- pher generally exhibit lower performance compared to the full implementation of N2GON. It is noteworthy that the performance of w/o Grapher II is superior to that of w/o Grapher I, indicating a greater impact of Grapher on the overall effectiveness of N2GON. This finding suggests that Grapher I is a fundamental component for Grapher II's functionality, as the construction of an importance-baste -base constraint network relies on the presence of similar or iden- tical labels between nodes. Thus, these results collectively indicate that the incorporation of importance constraint network contributes positively to the performance of the proposed N2GON model.
7. Limitations and Future Work
In GON, noise in any part of the data can adversely af- feet the learned representations. For instance, inaccuracies in node attributes or connectivity can propagate through successive encoding layers, potentially degrading overall performance. To mitigate this issue, future work may ex- plore noise-robust strategies, including more sophisticated regularization techniques applied across multiple layers.
The process of constructing both intra-graph and inter-graph connections can influence the performance of the model. Different graph construction strategies-sutt as diverse node/edge selection criteria or varying ways of creating inter-graph links-could alter the quality of the resulting representations. A comprehensive comparative study on multiple construction methods would therefore be valuable. Such an investigation would offer insights into the most effective techniques for building multi-layer graphs in dif- ferent applications.
Table 6: Ablation study on the key components of N2GON.
Variants
Squirrel
Cora
Citeseer
Pubmed
N2GON
60.98-2.12
89.1442.1
81.27=1.0
90.64+a.a7
w/o Grapherff
59.10
87.22:183
78.76 88.60a036
w/o Grapherl
58.34-3.16 85.8141.3 77 88.5520.12
In addition, several integration approaches can be explored with GON in future work. For example, integrating rein- forcement learning could allow for dynamic graph adapta- tion, where RL agents iteratively rewire grter- connec- tivity in applications like drug discovery and evolving rec- ommender systems. Additionally, meta-learning approaches can be used to pre-train GON encoders on diverse tasks, es- pecially when encountering limited labeled data.
Visualization Analysis. To demonstrate the strengths of our model in an objective manner, we utilized 2D projections for visual analysis, as shown in Fig.3. These projections de- pict the representations of the Cora and Chameleon datasets as processed by GCN, HopGNN, and N2GON, using the t-SNE algorithm (Van der Maaten & Hinton, 2008). The vi- sualizations employ different colors distinguish between various classes of nodes. In these representations, N2GON notably achieves a more pronounced separation, implying an enhanced ability to maintain class structures when compared to the other methods. This distinct separation achieved by N2GON suggests its effectiveness in accurately capturing
A Acknowledgments
This project was in part supported by the following projects: the National Natural Science Foundation of China (No.62432003, No.92267206).
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N2GON: Neural Networks for Graph-of-Net with Position Awareness
Impact Statement
Davis, M. I,, Hunt, J. P., Herrgard, S., Ciceri, P., Wod- icka, L. M., Pallares, G., Hocker, M., Treiber, D. K., and Zarrinkar, P. P. Comprehensive analysis of kinase inhibitor selectivity. Nature biotechnology. (9(1))1104 1051, 2011.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Defferrard, M., Bresson, X., and Vandergheynst, Con- volutional neural networks on graphs with fast localized spectral filtering. Advances Neural Information Pro- cessing Systems, 29, 2016.
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12


Reviews:
[{"review_id": "vbtVQrA4JA", "reviewer": "Reviewer_afvE", "summary": "This paper introduces Graph-of-Net (GON), a novel graph structure where each node itself is a graph, enabling multi-level representation and analysis of complex real-world systems. To effectively learn representations within GONs, the authors propose N2GON, a position-aware neural network that captures both intra-graph and inter-graph interactions using dual graph encoders and an implicit constraint network. Extensive experiments demonstrate that N2GON outperforms state-of-the-art models in graph learning tasks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1. The partitioning of GON —particularly in datasets like CiteSeer—appears questionable, where the partitioning is performed directly using KNN, resulting in computed outcomes that merely replicate the information propagation process of GNN.**\n\n>R1: Thank you for your feedback. Below, we provide clarification as follows:\n\n>1. **Rationale for $k$-hop Sampling Strategy**. We would like to clarify that we adopt a $k$-hop sampling (instead of KNN, which only involves 1-hop) to construct a graph for each node. This method is motivated by the observation that in many real-world scenarios, a node’s full identity is not encapsulated solely by its features but also by the structural context provided by its neighboring nodes.\n\n>For example, 1) in **Citation Networks** such as CiteSeer, a paper’s identity is defined not only by its content but also by its *references* and *cited-by* relationships. By sampling $k$-hop neighbors, we explicitly model a node’s \"extended identity,\" which includes its local academic context (e.g., related works). This aligns with the GON philosophy of representing nodes as hierarchical structures. 2) in **Social Networks**, a user’s social graph (friends, followers) is a natural extension of their identity, and $k$-hop sampling preserves this contextual information. Therefore, sampling the $k$-hop neighborhood effectively captures the local context. \n   \n>The reviewer noted that, in the case of CiteSeer, direct partitioning by $k$-hop might induce outcomes very similar to standard GNN information propagation. We believe this observation is, in fact, **supportive** of our methodology. The successful aggregation of neighbor information by GNNs substantiates that a node’s neighborhood plays a crucial role in the learning process. Our method does not merely replicate the GNN propagation process but rather formalizes the node’s expanded representation through explicit graph construction. This ensures that our Graph-of-Net structure inherently reflects the multi-level composition of nodes—each node graph is a manifestation of both its own features and the collective representation of its neighbors.\n   \n>2. **Alternative Graph Construction Method**. Beyond $k$-hop sampling, we explored constructing graphs from raw node data. We appreciate the reviewer's note on the challenge of accessing original data. In response, we made significant efforts and successfully obtained the raw textual data for the CiteSeer data through the GitHub project (https://github.com/sivaramanl/Information-Retrieval/tree/master/Text%20Processing/citeseer). This repository provided us with nearly all the title and abstract information for each paper. For this alternative graph construction, we processed each paper’s text as follows: 1) **Text Preprocessing:** We used NLTK to split the text of each paper into sentences, treating each sentence as a node in the paper’s graph.  2) **Feature Extraction:** We generated embeddings for each sentence using the popular sentence transformer model `all-MiniLM-L6-v2`, which provided the node attributes. For papers where the original information is missing, we default to representing the paper as a single node with an all-zero attribute vector. 3) **Edge Construction:** Edges were formed by computing the cosine similarity between sentence embeddings and applying a threshold of 0.7 to retain only strong connections.\n\n>We then conducted experiments on this newly constructed data. The generated data statistics and performance comparisons, as presented in the tables below. The comparable performance of $k$-hop and text-based GONs (**81.27% vs. 81.82%**) demonstrates that both methods validly capture hierarchical semantics. The $k$-hop approach is a **pragmatic and effective** proxy when raw data is unavailable, while text-based construction confirms GON($k$-hop)’s flexibility for domains with explicit substructures.\n\n*Table I: Resulting GON Statistics on Citeseer*  \n\n| Metric                      | Value       |\n|-----------------------------|-------------|\n| Avg. Nodes per Graph         | 7.72        |\n| Max Nodes                    | 336         |\n| Min Nodes                    | 1           |\n| Avg. Edges per Graph         | 156.36      |\n| Max Edges                    | 106,286     |\n| Min Edges                    | 0           |\n| Node Feature Dimension       | 384         |\n\n*Table II: Accuracy on data CiteSeer*\n\n| Algorithm              | Acc (%)     |\n| ---------------------- | ---------------- |\n| APPNP                  | 77.06 ± 1.73     |\n| GPRGNN                 | 75.56 ± 1.62     |\n| MixHop                 | 76.26 ± 1.33     |\n| FAGCN                  | 74.86 ± 2.42     |\n| DAGNN                  | 76.44 ± 1.97     |\n| HopGNN                 | 76.69 ± 1.56     |\n| **N2GON($k$-hop)**     | **81.27 ± 1.30** |\n| **N2GON (Text-Based)** | **81.82 ± 1.46** |\n\n**Q2. The full name of GON is mentioned twice.**\n\n>R2: Thank you for pointing that out. We will remove the extra name.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes. However, the proposed method is designed for GON, but the way this structure is partitioned in the paper does not seem entirely reasonable. This issue is particularly evident in datasets such as CiteSeer, where the partitioning is performed directly using KNN, resulting in computed outcomes that merely replicate the information propagation process of GNN.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "In the first part of the experiments, the partitioning of GON does not seem entirely reasonable, whereas in the later part, the partitioning in the chemical datasets appears to be more appropriate.", "relation_to_broader_scientific_literature": "Yes", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "The paper investigates GON and proposes a more generalizable approach. However, the experimental setup does not seem sufficiently comprehensive. I believe the authors provided excellent examples of GON in the introduction, but it is unclear whether experiments on related datasets are feasible. Of course, obtaining relevant data may be challenging.", "other_comments_or_suggestions": "The full name of GON is mentioned twice in the paper.", "questions_for_authors": "None", "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": "MIV3u7dOaI", "reviewer": "Reviewer_99JV", "summary": "The paper N2GON presents a new approach to graph learning, with a focus on the Graph-of-Net structure and a position-aware neural network model. The comprehensive experimental evaluation and detailed methodology are significant strengths. However, the paper could be further improved by including runtime comparisons, expanding the related work section, and correcting some grammatical errors.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1. It is recommended to add runtime comparisons.**\n\n>R1: Thank you for your comment. We conducted the runtime experiments and the results (the average elapsed time per epoch), summarized in the table below, indicate that on benchmark graphs, our runtime is comparable with that of SOTA baselines, while on biomedical datasets our method is significantly more efficient than traditional algorithms.\n\n*Table I. Running Time per epoch on Benchmark Graph Datasets (in seconds)*\n\n| Algorithms | Cora  | Ciesear | PubMed | Cornell | Texas | Actor | Squirrel | Chameleon |\n| :---: | :---: | :-----: | :----: | :-----: | :---: | :---: | :------: | :-------: |\n|    Ours    | 0.055 |  0.062  | 0.239  |  0.039  | 0.043 | 0.09  |  0.071   |   0.055   |\n|   H2GCN    | 0.005 |  0.005   | 0.016  |  0.005  | 0.004 | 0.011 |  0.088   |   0.043   |\n|   DAGNN    | 0.012 |  0.013  |  0.01  |  0.011  | 0.019 | 0.01  |  0.012   |   0.013   |\n|   HopGNN   | 0.01  |  0.01   | 0.024  |  0.012  | 0.009 | 0.013 |  0.013   |   0.012   |\n\n*Table II. Running Time per epoch on Biological Datasets (in seconds)*\n\n|   Algorithms   | PEP-MHC |  PPI   |  TCR  |   MTI   |\n| :------------: | :-----: | :----: | :---: | :-----: |\n|      Ours      |  2.00   | 0.502  | 1.17  |  1.15   |\n| ConjointTraid |  82.08  | 31.96  | 6.01  | 175.96  |\n|   Quasi-Seq    | 117.15  | 30.91  | 5.97  | 174.99  |\n|      ESPF      | 131.97  | 25.99  | 7.02  | 177.98  |\n|      CNN       | 1023.49 | 310.96 | 76.96 | 1103.97 |\n|  Transformer   | 351.97  | 88.49  | 25.29 | 1040.97 |\n\n\n**Q2. More discussion on position-awareness in graph learning should be included.**\n\n>R2: Thank you for the feedback. In our work, *position-awareness* is achieved by leveraging the Personalized PageRank algorithm, which computes the topological influence of node graphs on each other, implicitly revealing their *relative positions* in the network. In essence, once all pairwise similarities are determined, each node inherently carries information about its relative position within the similarity network structure.\n\n**Q3. Providing dataset details (e.g., node/edge counts) for clarity.**\n\n>R3: Thank you for your valuable suggestion. We have now added detailed descriptions of benchmark dataset  in the table below. \n\n|   Datasets   | **Texas** | **Wisconsin** | **Actor** | **Squirrel** | **Chameleon** | **Cornell** | **Citeseer** | **Pubmed** | **Cora** |\n| :----------: | :-------: | :-----------: | :-------: | :----------: | :-----------: | :---------: | :----------: | :--------: | :------: |\n|  **#Nodes**  |    183    |      251      |   7,600   |    5,201     |     2,277     |     183     |    3,327     |   19,717   |  2,708   |\n|  **#Edges**  |    295    |      466      |  26,752   |   198,493    |    31,421     |     280     |    4,676     |   44,327   |  5,278   |\n| **#Classes** |     5     |       5       |     5     |      5       |       5       |      5      |      7       |     3      |    6     |\n\n\n**Q4. **Discuss more real-world applications (e.g., drug discovery) and potential limitations to better illustrate GON's impact.**\n\n>R4: Thank you for your valuable suggestions. We will further expand the discussion of GON’s practical applications in the manuscript. For example, in drug discovery, GON can model drug molecules as graphs of atoms while representing proteins as residue graphs, capturing the binding patterns between them through a global network. In bioinformatics, GON enables multi-scale analysis of protein interaction networks, such as identifying functional modules at the residue level. For social networks, GON can model user communities (e.g., user-centric social subgraphs) and the relationships across communities, though real-world deployment must address privacy concerns and data sparsity issues. Common challenges include the cost of data construction (e.g., the need for expert annotations for molecular graphs) and computational overhead.\n   \n**Q5. There are some grammatical errors and typos.**\n\n>R5: Thank you for your detailed feedback. We will review the manuscript and corrected all the mentioned grammatical errors and typos.\n\n**Q6. Could you clarify if Algorithm 1 uses full-batch training or mini-batch sampling?**\n\n>R6: Thank you for your valuable feedback. For the GON datasets, we use full-batch training—processing the entire data at once—since it is feasible to handle these datasets in one go. In the revised manuscript, we will clarify this point.\n\n**Q7. How is this coefficient set for the loss in the algorithm?**\n\n>R7: Thank you for raising this important point. In our work, we deliberately did not introduce a weighting coefficient between the constraint loss ($L_{\\text{con}}$​) and the NLL loss. We found that both loss components naturally operate on comparable scales, allowing us to simply sum them without additional tuning. This design choice not only simplifies the loss function but also reduces the number of hyperparameters, streamlining the training process.", "claims_and_evidence": "Yes, the claims made in the paper are well-supported by the evidence.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria align with the problem's core challenges.", "theoretical_claims": "N/A.", "experimental_designs_or_analyses": "The experimental designs and analyses in the paper are sound.", "relation_to_broader_scientific_literature": "The key contribution of this paper extends traditional graph models by representing each node as a graph itself, enabling a more sophisticated multi-level representation GON. It builds on prior work in graph, which have been used to model complex relationships in social and biological networks. However, existing graph models do not fully capture the hierarchical dependencies present in real-world systems. GON provides a more flexible and general framework, making it a valuable work in graph representation learning.", "essential_references_not_discussed": "N/A.", "other_strengths_and_weaknesses": "Pros\n1. I appreciate the motivation behind this paper. The concept of representing each node as a graph within a larger network is innovative and extends traditional graph structures.\n2. The proposed algorithm integrates both intra-graph and inter-graph connections, and incorporating PPR to capture the relative position of node graphs is interesting.\n3. The experimental results also seem to validate the effectiveness of the proposed model.\n\nCons\n1. Although the paper provides a complexity analysis in the appendix, it lacks experimental comparisons in terms of runtime performance. Including an intuitive comparison of computational efficiency could enhance the paper.\n2. The related work section is not comprehensive enough. More discussion on position-awareness in graph learning should be included.\n3. In the experimental section, although multiple datasets are mentioned, the paper does not provide sufficient details on their specific characteristics (e.g., the number of nodes, edges, and class distribution).", "other_comments_or_suggestions": "1. It would be helpful to expand the discussion on real-world applications, particularly in complex domains like drug discovery, bioinformatics, or social networks. While the paper covers some biomedical datasets, offering a broader perspective on potential use cases (including possible limitations in these domains) would give readers a more complete understanding of the impact of GON.\n\n2. There are some grammatical errors and typos in the paper:\n\nLine 23: \"an\" should be changed to \"a\".\nLine 60 (left column): \"applicable\" should be removed.\nLine 123 (right column): \"are\" should be \"is\".\nLine 155 (right column): \"possess\" should be in plural form.\nLine 267 (left column): \"usally\" should be corrected to \"usually\".\nLine 261 (right column): \"is\" should be \"are\".", "questions_for_authors": "In addition to the weaknesses mentioned earlier, I have the following specific questions:  \n\n> 1. In Algorithm 1, the phrase \"Sample all node graphs\" seems contradictory. \"Sample all\" suggests sampling the entire set, whereas \"Sample mini-batch\" would indicate a subset. Could you clarify whether full-batch training is used or update the terminology accordingly?  \n> 2. The paper does not specify the weighting coefficient between the constraint loss \\( L_{\\text{con}} \\) and the NLL loss. How is this coefficient set and adjusted in the algorithm?", "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": "4owqL1r3Jp", "reviewer": "Reviewer_bLhx", "summary": "The paper introduces a novel framework called Graph-of-Net (GON), which extends traditional graph structures by modeling each node as a graph itself, creating a multi-level perspective on relationships between objects. This approach enables the capture of both the internal structure of individual nodes and the broader network of dependencies. To learn node representations within GON, the paper proposes a position-aware neural network model, which processes both intra-graph and inter-graph connections. The model incorporates dual encoders and graph constructors to build and refine a constraint network, where nodes are adaptively arranged based on their positions, as determined by the network’s constraint system.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1. The term \"Position Awareness\" is not strictly defined in the paper. A formal definition could be added for clarity.**\n\n>R1:  Thank you for highlighting the need for a more formal definition of the term \"Position Awareness.\" In our work, we use \"Position Awareness\" to refer to the model’s ability to capture and integrate the relative placement and connectivity of nodes within the overall network structure. Position Awareness is the property of a network model whereby each node's representation explicitly encodes its structural context within the graph. This encoding captures not only the node’s intrinsic features (e.g., labels or attributes) but also its topological characteristics—such as how it is connected to other nodes, and its overall influence in the network. In our framework, this is operationalized by leveraging a scoring mechanism (i.e., via the Personalized PageRank algorithm) that quantifies the relative importance or influence of nodes. \n>\n>By assigning a score that measures the likelihood of reaching one node from another through random walks with restarts, our approach formalizes a node’s “position” within the network. This score effectively differentiates nodes that might share similar labels but occupy distinct roles depending on their connectivity pattern and centrality in the graph. These scores are then integrated into the node representations, ensuring that position-dependent information contributes to the final embedding. As a result, the model is better equipped to capture nuanced relationships that go beyond simple label similarity.\n>\n>We will include this formal definition in the revised manuscript to enhance the clarity and rigor of our contribution.\n\n**Q2. The distinction between GON and existing hierarchical graph structures (e.g., Hierarchical Graph Networks, Hypergraphs) is not clearly quantified.**\n\n>R2: Thank you for your comment. We distinguish our Graph-of-Nets (GON) framework from other hierarchical graph structures as follows: 1) **Explicit Two-Level Representation:** In GON, each node is a complete graph that retains its full internal structure. This is different from many hierarchical graph networks, which typically process standard graphs where the nodes are represented as vectors rather than graphs, potentially losing fine-grained structural details. 2) **Preservation of Intra-Node Complexity:**  Instead of merging all details into a single representation, GON processes each graph independently (using specialized graph neural networks) and then integrates these detailed embeddings into the higher-level network. This two-stage approach effectively captures both local (intra-node) and global (inter-node) relationships.\n\nWe will add these distinctions in the revised manuscript.\n   \n**Q3. In the description below Equation (1) in the algorithm section, \"funcitons\" should be corrected to \"functions.\"**\n\n>R3: Thank you for catching the typo. We appreciate your attention to detail, and we will correct in the revised manuscript.\n\n**Q4. The similarity function ( \\text{sim}(\\cdot, \\cdot) ) in Equation (6) lacks a clear explanation of its basis (e.g., if using cosine similarity, the vector normalization step should be specified).**\n\n>R4: Thank you for your comment. We clarify that in Equation (6), we directly compute the cosine similarity on the final representations without any additional normalization. We will update the manuscript to clearly state this.\n\n**Q5. \"node graph\" → It is recommended to standardize this term as \"node-graph\" (with a hyphen).**\n\n>R5: Thank you for your comment. We appreciate your suggestion to standardize the term. We will update the manuscript to use \"node-graph\" consistently.\n    \n**Q6. The paper would benefit from additional discussion on the future research directions. For example, investigating how GON could be integrated with other deep learning techniques (e.g., reinforcement learning, meta-learning) to adapt to new or evolving graphs could open up interesting lines of inquiry.**\n\n>R6: Thank you for the insightful suggestion. We plan to explore several integration approaches with GON in our future work. For example, integrating reinforcement learning could allow for dynamic graph adaptation, where RL agents iteratively rewire inter-graph connectivity in applications like drug discovery and evolving recommender systems. Additionally, we are considering meta-learning approaches to pre-train GON encoders on diverse tasks—enabling rapid adaptation to new scenarios, especially when encountering limited labeled data. Moreover, we can extend GONs to handle temporal dynamics by incorporating architectures for gradually updating both intra- and inter-graph connections in evolving systems. We believe these strategies will further enhance GON’s flexibility and robustness across various challenging, real-world applications.", "claims_and_evidence": "The claims made in the paper are generally well-supported by clear and convincing evidence. The novelty of the Graph-of-Net (GON) structure is justified through the discussion and examples from general networks and biological systems. The position-aware neural network model is supported by the detailed methodology, including the dual encoders and graph constructors, which capture intra- and inter-graph relationships. The effectiveness of GON is empirically validated through extensive experiments on 16 network datasets, outperforming state-of-the-art methods.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria are well-suited for the problem.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "Yes, I examined the soundness and validity of the experimental design and analyses, and overall, they are largely systematic.", "relation_to_broader_scientific_literature": "The paper successfully bridges gaps in hierarchical graph representation learning. Its innovations—particularly the GON structure and PPR constraints—address challenges in modeling multi-scale systems, positioning it as a meaningful contribution to the broader graph learning literature.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strengths\n\n1. The concept of treating each node as a graph is intereting. The GON structure extends traditional graph nodes into subgraphs (Graph-as-Node), enabling a shift from single-level (node-edge) modeling to multi-level modeling (subgraph internal structure + global network topology). This design is important in biological networks (such as protein-protein interactions).\n\n2. The paper is well-written and easy to follow.\n\n3. The experimental results are promising. The datasets are comprehensive and well-executed, covering a variety of domains including social networks, citation networks, and biomedical networks.\n\nWeaknesses\n\n1. The term \"Position Awareness\" is not strictly defined in the paper. A formal definition could be added for clarity.\n\n2. The distinction between GON and existing hierarchical graph structures (e.g., Hierarchical Graph Networks, Hypergraphs) is not clearly quantified.", "other_comments_or_suggestions": "1. In the description below Equation (1) in the algorithm section, \"funcitons\" should be corrected to \"functions.\"\n\n2. The similarity function ( \\text{sim}(\\cdot, \\cdot) ) in Equation (6) lacks a clear explanation of its basis (e.g., if using cosine similarity, the vector normalization step should be specified).\n\n3. \"node graph\" → It is recommended to standardize this term as \"node-graph\" (with a hyphen).\n\n4. The paper would benefit from additional discussion on the future research directions. For example, investigating how GON could be integrated with other deep learning techniques (e.g., reinforcement learning, meta-learning) to adapt to new or evolving graphs could open up interesting lines of inquiry.", "questions_for_authors": "See Weaknesses.", "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": "HVIotb1giE", "reviewer": "Reviewer_eUvt", "summary": "This paper introduces Graph-of-Net (GON), a novel graph structure where each node is itself a graph, enabling multi-level modeling of complex systems that involve hierarchical relationships. Examples include biological networks (e.g., protein-protein interactions, where individual proteins are represented as graphs within a larger network) and citation networks (where papers, modeled as text graphs, are interconnected). The authors propose N2GON, a position-aware neural network designed to learn node representations in GONs by jointly modeling intra-graph (within-node) and inter-graph (between-node) connections.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1. The Introduction's definition of Graph-of-Net (GON) is vague and needs a precise mathematical formulation of each node-as-graph (size, structure, features) and clarification of how it differs from traditional graphs in modeling hierarchical, multi-dimensional relationships.**\n\n>R1: Thank you for your insightful comments. In our framework, GON is designed as a hierarchical graph where each node is not a single data point but an entire graph. Formally, let the top-level graph (network) be defined as $\\mathcal{G}^N = ( \\mathcal{V}_G, \\mathcal{E}_G )$, where $\\mathcal{V}_G$ is the set of graph nodes and $\\mathcal{E}_G$ denotes the edge set. For each node $v_G \\in \\mathcal{V}_G$, we associate a graph: $\\mathsf{G}_v = (V, E, X)$. Here, $V$ represents the nodes within the graph, $E$ is the set of edges among these nodes, and $X$ is the feature matrix corresponding to the graph nodes. The size of each graph is determined by $|V|$, which depends on the inherent structure or the domain-specific construction of the graph. The graph structure, represented by $E$, captures the internal relationships among the nodes in the graph. This structure may vary depending on the level of detail or domain-specific insights desired. Each graph is equipped with a feature representation $X$ that encodes the characteristics of the nodes in $V$. The formation of these features can be based on raw data attributes or results from a prior processing step.\n>\n>Traditional graph models represent data as a general graph where each node corresponds to an atomic data point. In contrast, GON captures multi-dimensional relationships by explicitly modeling two levels of interaction. This dual-level representation allows GON to be particularly effective for complex hierarchical data, as it provides the capacity to model nested relationships and capture both local and global patterns within the data.\n\n**Q2. In some formulas (e.g., the PPR algorithm), the derivation process could benefit from more detailed explanations. Providing a more comprehensive derivation would aid in understanding.**\n\n>R2: The Personalized PageRank algorithm provides a way to measure how “close” or “important” one node is relative to another within a network. Imagine a random walker who starts at a specific node in the network. Instead of wandering the network entirely at random, the walker follows a rule: at each step, they decide either to move to one of the neighboring nodes or to jump back to the starting node. This jump-back mechanism ensures that the influence of the starting node remains strong throughout the walk.\n>\n>What makes PPR particularly useful is that it considers not only the direct connections between nodes but also the broader network structure. In simple terms, it captures the idea that even if two nodes share the same label or initial property, they can have varying levels of relatedness depending on how they are connected within the network.\n>\n>By adopting this method, our approach goes beyond simply saying, \"nodes with the same label are similar.\" Instead, we are able to gauge the subtle nuances in how closely nodes are related based on both their labels and their positions within the graph structure. This allows our model to better capture complex relationships and provides a more refined way of measuring similarity among node graphs.\n\n**Q3. It would be helpful to clarify how GON manages the hierarchical structures within the graphs of nodes.**\n\n>R3: Thank you for the valuable feedback. In a Graph-of-Net (GON), each high-level node represents an entire graph that can have its own structure and detailed relationships. Instead of treating every graph as a monolithic object, we decompose the problem into two levels: 1) At the lower-level, we focus on extracting meaningful representations from the individual graphs. We utilize graph neural networks to process each graph, effectively summarizing its properties into a fixed-size embedding. 2) Once each graph has been transformed into a representation, these embeddings serve as the nodes for the higher-level graph. The interrelations among these nodes are then modeled, combining both the abstract representation of the graphs and the positional information within the larger network. The two-stage process helps us manage the inherent complexity of graphs that reside within nodes. We will add above description in subsequent revisions of the paper.", "claims_and_evidence": "The claims made in the paper are generally supported by clear evidence.", "methods_and_evaluation_criteria": "The methods and evaluation criteria are largely appropriate for the problem of multi-level graph learning, with strengths in architectural design and empirical breadth.", "theoretical_claims": "n/a", "experimental_designs_or_analyses": "The experimental designs and analyses presented in the paper generally appear to be sound and robust, supporting the claims made about the Graph-of-Net (GoN) model.", "relation_to_broader_scientific_literature": "1. The key contributions of the paper, namely the introduction of the Graph-of-Net (GoN) structure and the position-aware neural network model, build upon several foundational ideas and methods from the broader graph learning and neural network literature. The paper situates itself within a well-established body of work, while introducing novel elements that extend existing techniques.\n2. The idea of nodes being graphs themselves shares conceptual similarities with hypergraphs and heterogeneous graphs. In these graph structures, edges (or hyperedges) can connect more than two nodes, and the nodes can represent different types of entities or relationships. GoN extends this idea by introducing a more general notion where nodes are not just connected through edges but are, in fact, entire graphs with their own structure.", "essential_references_not_discussed": "n/a", "other_strengths_and_weaknesses": "**Strengths:**\n1. The paper is well-written and easy to understand.\n2. The proposed Graph-of-Net (GoN) structure is an innovative contribution. By representing each node as a graph, GoN enables multi-level representations of individual nodes.\n3. The \"position-aware neural network\" mechanism introduced in the paper is a meaningful enhancement to model capabilities. This mechanism allows the model to not only focus on node features but also process interactions and dependencies between nodes.\n4. The paper provides thorough experimental validation on multiple datasets, covering a range of domains such as social networks, citation networks, and biomedical networks. The experimental results are impressive.\n\n**Weaknesses:**\n1. In the introduction, the definition of Graph-of-Net (GoN) is somewhat vague. Although a general conceptual understanding is provided, a more precise mathematical definition may be needed. For instance, the definition mentions that \"each node is itself a graph,\" but the theoretical aspects of how the \"graph\" size, structure, and features are defined might not be sufficiently clear. Further clarification of how GoN differs from traditional graphs, especially in terms of modeling multi-dimensional relationships in hierarchical node structures, would be helpful.\n2. In some formulas (e.g., the PPR algorithm), the derivation process could benefit from more detailed explanations. For new readers, the PPR algorithm may not be immediately familiar, and providing a more comprehensive derivation would aid in understanding.", "other_comments_or_suggestions": "1. The explanation of the GoN concept could be further strengthened, especially with regard to how the complexity of \"graphs within nodes\" is handled during the modeling process. For example, it would be helpful to clarify how GoN manages the hierarchical structures within the graphs of nodes.", "questions_for_authors": "Refer to the \"Other Strengths and Weaknesses\" part.", "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:

Meta-review: This paper introduces ​​N2GON​​, a novel neural network framework designed for ​​Graph-of-Net (GON)​​, a hierarchical graph structure where each node is itself a graph. The proposed model leverages ​​position-aware learning​​ to capture both intra-graph (within-node) and inter-graph (between-node) relationships, integrating node labels and topological constraints via dual encoders and a refined constraint network. The authors demonstrate the effectiveness of N2GON through extensive experiments on ​​16 datasets​​ spanning social networks, citation networks, and biomedical applications, showing significant improvements over state-of-the-art baselines. 

The paper received generally positive reviews, with reviewers acknowledging its ​​novelty​​, ​​theoretical grounding​​, and ​​empirical effectiveness​​. While initial concerns were raised, all four reviewers expressd that the responses have addressed their concerns, and consider this paper worthy of acceptance.
Decision: Accept (poster)