Graph Isomorphism Network (GIN) Overview
Many high-performance graph neural networks rely on empirical intuition, heuristics, and experimental trial-and-error in designing their structures. Theoretical understanding of the properties and limitations of graph neural networks is lacking, and “How Powerful are Graph Neural Networks?” by Xu et al. provides a framework for such discussions.
The GIN proposed in this paper represents feature vectors of neighboring nodes as multisets (allowing overlapping elements) and considers neighborhood aggregation in a graph neural network as an aggregation function of multisets. In this model, graph neural networks need to represent the aggregation results of different multisets differently in order to have a highly expressive graph neural network. Furthermore, this paper theoretically characterizes the discriminative power (i.e., the extent to which an aggregate function can distinguish between different multisets) by examining the functions of different multisets. It is shown that the higher the discriminative power of a multiset function, the higher the expressive power of its graph neural network.
In “How Powerful are Graph Neural Networks?“, the following theoretical approaches are listed in relation to the WL homomorphism test described in “Overview of PATCHY-SAN and Examples of Algorithms and Implementations“.
- In identifying graph structures, graph neural networks are at best as good as WL isomorphic tests.
- Clarify the conditions for neighborhood aggregation and graph readout functions for graph neural networks to be as good as the WL isomorphic test.
- Show graph structures that are not identifiable by common graph neural networks such as GCNs and GraphSAGE, and characterize the graph structures that these graph neural networks can identify.
- Propose a simple structure, Graph Isomorphism Network (GIN), and show that it has the same level of discriminative power as the WL isomorphism test.
The Weisfeiler-Lehman graph isomorphism test (WL isomorphism test) is an algorithm often used to determine whether a given graph is isomorphic. A multiset is an extension of a set, which (unlike a set) allows elements of the same value to occur more than once.
The learning of node representations in graph neural networks is done by aggregating the representations of neighboring nodes in the graph, and the discriminative power of graph neural networks depends on the nature of the aggregation. For a graph neural network to have high representational capability, the aggregation should be such that the multiset is maintained, i.e., the aggregation should be a monojective multiset function. However, projecting different graphs onto different embedding spaces means solving the (difficult) graph isomorphism problem. Here, it is desirable that isomorphic graphs project to the same representation and nonisomorphic graphs project to different representations, but GIN characterizes the representational power of graph neural networks by a slightly weaker criterion. It is a heuristic WL isomorphism test, and with a few exceptions, such as regular graphs, it is generally known to correctly identify graphs.
The graph identification capability of any aggregate-based graph neural network is at best limited to the identification capability of the WL homomorphism test. The conditions for a graph neural network to be as good as the WL homomorphism test are that the aggregation of neighboring nodes and the representation generation (readout) of the graph from the node representation must be monojective.
A graph neural network that satisfies these conditions generalizes the WL homomorphism test by learning to embed subtrees in a low-dimensional space, and is able to identify not only different graph structures, but also similarities between them. Graph neural networks that satisfy these conditions generalize the WL homomorphism test by learning to embed subtrees into a lower-dimensional space, and can not only identify different graph structures, but also project similar graphs into similar embeddings (embeddings) to understand the dependencies of the graph structures.
Graph Isomorphism Network (GIN) is a generalized model of the WL homomorphism test and has the greatest discriminative power among graph neural networks. Graph neural networks that do not satisfy this condition are the GCN described in “Overview of Graph Convolutional Neural Networks (GCN), Algorithms, and Examples of Implementations” and the GraphSAGE described in “Overview of GraphSAGE, Algorithms, and Examples of Implementations. Algorithms and Examples of Implementations“. These tests have lower discriminative power than the WL homomorphic test and may not be able to distinguish even simple graphs. Nevertheless, aggregation by average, such as GCN, works well in many node classification tasks.
Pooling by mean and max in multiset is not unidirectional, so the order of expressive power is sum, mean, and max, in that order. This is because the sum of a multiset can distinguish multisets, while the mean can only distinguish their proportions and distributions, and the maximum considers a multiset to be just an aggregate. Aggregation by average works well for tasks where information about the distribution of the graph is more important than its exact structure, or where the characteristics of nodes are diverse and rarely appear more than once.
In summary, the Graph Isomorphism Network (GIN) is a neural network model for learning isomorphism of graph structures, designed to encode features of graph structures and use that encoding to determine isomorphism, which basically means that the graph The model ignores isomorphism by labeling and ordering, and focuses on the graph structure itself. Such graph isomorphism is an important approach in many fields, such as chemical structures, social networks, protein-protein interaction networks, and the problem of determining whether two graphs have the same structure.
For GIN tools and libraries, they are available from the author’s git page.
Algorithms related to Graph Isomorphism Network (GIN)
The Graph Isomorphism Network (GIN) is a neural network model for learning graph isomorphism, but GIN itself is not an algorithm for solving graph isomorphism problems. Instead, GIN provides a framework or architecture for learning graph isomorphism problems.
In general, there are several approaches to algorithms for solving graph isomorphism problems. Typical ones include the following.
1. the Eichnholz algorithm (Aho-Hopcroft-Ullman Algorithm): This uses recursive depth-first search (DFS) to verify graph isomorphism. The algorithm compares two graphs to see if they have the same number of vertices and the same number of edges, and then uses a DFS search to find correspondences between the vertices. For more details, see “Overview of the Aho-Hopcroft-Ullman Algorithm and Examples of Algorithms and Implementations“.
2. Weisfeiler-Lehman Algorithm: This algorithm labels a graph based on the surrounding structure of nodes and uses the information in the labels to verify isomorphism. The algorithm is generally fast and has been applied to large data sets of graphs. See also “Overview of the Weisfeiler-Lehman Algorithm and Related Algorithms and Examples of Implementations” for more details.
3. Monte Carlo Method: This approach estimates the likelihood of isomorphism by comparing pairs of randomly generated vertices. This method is efficient for large graphs, but its stochastic nature requires caution in accuracy. See also “Overview and Implementation of Markov Chain Monte Carlo Methods” for more information.
These algorithms are the classical methods for dealing with graph isomorphism problems, while neural network-based approaches such as GIN generally perform better than these classical methods and, by using trained models, can be applied to a wider range of graph isomorphism problems. It is believed that.
Application of Graph Isomorphism Network (GIN)
The Graph Isomorphism Network (GIN) has been applied to a variety of graph-related problems. The following are examples of these applications.
1. Chemical Structure Analysis: GIN is used for graphs that represent the chemical structure of molecules. Molecular structures are represented as graphs, and GIN is used to predict molecular isomorphisms and chemical properties, and to evaluate the similarity of compounds. This will be useful in areas such as drug discovery and materials science.
2. Social Network Analysis: GIN is being applied to social media and Internet network structures such as social networks and web graphs. 3. biological network analysis
3. biological network analysis: GIN is applied to graphs representing biological systems, such as protein-protein interaction networks, gene expression networks, and metabolic pathway networks. This enables functional analysis of biological networks, prediction of protein-protein interactions, and elucidation of gene function.
4. graph classification: GIN is used to take the structure of a graph as input and predict whether the graph belongs to a particular class. This is applied to tasks similar to document categorization and image identification, but specific to graph structure.
These applications demonstrate that GIN is widely used in learning and analyzing graph structures, and its flexibility and versatility make it a powerful tool for addressing problems in a variety of domains.
Graph Isomorphism Network (GIN) Implementation Example
Examples of Graph Isomorphism Network (GIN) implementations are commonly done using Python and the machine learning library PyTorch. The following is a basic example of implementing a GIN using PyTorch.
import torch
import torch.nn as nn
import torch.nn.functional as F
from torch_geometric.nn import MessagePassing
from torch_geometric.utils import add_self_loops, degree
class GINConv(MessagePassing):
def __init__(self, in_channels, out_channels):
super(GINConv, self).__init__(aggr="add")
self.mlp = nn.Sequential(
nn.Linear(in_channels, out_channels),
nn.ReLU(),
nn.Linear(out_channels, out_channels)
)
self.eps = nn.Parameter(torch.Tensor([0]))
def forward(self, x, edge_index):
edge_index, _ = add_self_loops(edge_index, num_nodes=x.size(0))
out = self.mlp((1 + self.eps) * x + self.propagate(edge_index, x=x))
return out
def message(self, x_j):
return x_j
class GIN(nn.Module):
def __init__(self, num_features, hidden_channels, num_classes):
super(GIN, self).__init__()
self.conv1 = GINConv(num_features, hidden_channels)
self.conv2 = GINConv(hidden_channels, hidden_channels)
self.fc = nn.Linear(hidden_channels, num_classes)
def forward(self, x, edge_index):
x = F.relu(self.conv1(x, edge_index))
x = F.relu(self.conv2(x, edge_index))
x = F.dropout(x, p=0.5, training=self.training)
x = global_mean_pool(x, batch) # Aggregate features of the entire graph
x = self.fc(x)
return F.log_softmax(x, dim=-1)In this example, a custom message passing layer called GINConv is defined and used to build the GIN model. It also uses the PyTorch Geometric library to effectively process graph data.
Graph Isomorphism Network (GIN) Challenges and Measures to Address Them
The Graph Isomorphism Network (GIN) has several challenges, and methods to address them are also being considered. They are described below.
1. Lack of invariance to label substitutions:
Challenge: GIN is not invariant to label substitutions in graphs. That is, if the same graph is labeled in different ways, different feature representations may be generated.
Solution: To ensure label invariance, GIN has various methods. For example, sorting graph labels in a stable order, or combining methods such as the Weisfeiler-Lehman Algorithm (Weisfeiler-Miller-Terman Algorithm) can be effective.
2. lack of robustness to graph size:
Challenge: GIN is not robust to large graphs. In particular, memory usage and computational complexity may increase, and training and inference efficiency may decrease.
Solution: Methods such as mini-batch processing, sampling techniques, or graph dimensionality reduction can be used to improve the performance of GIN for large graphs, and methods such as approximation algorithms and parallel processing can also be applied.
3. lack of domain adaptability:
Challenge: GIN is effective for certain domain-specific tasks, but its performance may degrade when applied to other domains.
Solution: Methods such as data augmentation for domain adaptation, transition learning, and domain-specific feature engineering can be used to improve the performance of GIN, and fine tuning of pre-trained models can also be an effective method.
Reference Information and Reference Books
For more information on graph data, see “Graph Data Processing Algorithms and Applications to Machine Learning/Artificial Intelligence Tasks. Also see “Knowledge Information Processing Techniques” for details specific to knowledge graphs. For more information on deep learning in general, see “About Deep Learning.
Reference book is
“Graph Neural Networks: Foundations, Frontiers, and Applications“等がある。
“Introduction to Graph Neural Networks“
“Graph Neural Networks in Action“
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