IsoRank（Isomorphism Ranking）
IsoRank (Isomorphism Ranking) is one of the algorithms used to align different networks. It uses network isomorphism (graph isomorphism) to calculate the similarity between two different networks and estimate the correspondence of nodes based on it. IsoRank is used in areas such as data integration between different networks, network comparison, bioinformatics, and social network analysis. The main features and key points of IsoRank are described below.
1. use of network isomorphisms:
IsoRank performs mapping based on network isomorphism. In other words, if two networks are isomorphic, it is assumed that a node correspondence between them is possible.
2. computation of similarity scores:
IsoRank calculates similarity scores between nodes. These scores evaluate the similarity between nodes within the same network and between different networks.
3. solving optimization problems:
The correspondence problem is modeled as an optimization problem, which aims to find correspondences that maximize similarity scores. Typically, this problem is NP-hard and an approximation algorithm is used rather than an exact solution.
4. ranking of scores:
IsoRank ranks node correspondences and ranks the most similar correspondences higher. The ranking allows the reliability and quality of the correspondence to be evaluated.
5. Comparison of different networks:
IsoRank can be used to compare different networks and help identify common structures. They are suitable, for example, for comparing protein interaction networks and social networks.
Implementations of IsoRank are available through research papers and open source projects and are provided in various programming languages. In actual applications, it is necessary to adjust the parameters of IsoRank and select evaluation criteria.
Specific procedures for IsoRank
The following is a general procedure for IsoRank. The procedure may vary depending on the specific application and variations, but the basic idea is the same.
1. input network definition:
Define two different networks (usually graphs) to be mapped. These networks consist of nodes and edges. For example, when comparing two bioinformatics networks, each network may be a protein interaction network, a metabolic pathway network, etc.
2. node similarity score calculation:
For each pair of nodes in the two networks, a similarity score is calculated. The similarity score is calculated based on the structural similarity between nodes, usually based on the number of common neighbors or other graph features, which evaluates the similarity of each node to its equivalent node in the other network.
3. ranking similarity scores:
The computed similarity scores are ranked, placing node pairs with high similarity at the top. This identifies the most promising candidate responses.
4. optimization problem formulation:
The node correspondence problem is formulated as an optimization problem. Typically, this problem has a maximization objective function, where the objective function aims to find the correspondence that maximizes the similarity score of the corresponding nodes.
5. application of the optimization method:
Optimization methods are applied to solve the formulated optimization problem. Typically, integer linear programming (ILP), Greedy methods, iterative optimization algorithms, etc. are used to find the optimal correspondence in this step.
6. establishing the correspondence:
Once the optimal correspondence is found, the correspondence of the nodes is established based on it. Once the correspondence is established, the correspondence between nodes in different networks is known.
7. evaluation of the correspondence:
An evaluation metric is used to assess the quality of the correspondence. This allows the adequacy of the correspondence to be evaluated and adjustments to the correspondence to be made if necessary.
Examples of IsoRank implementations
It is difficult to provide specific examples of IsoRank implementations because they depend on programming languages and libraries. However, it will be possible to demonstrate a general procedure for implementing IsoRank. The following is an example implementation that demonstrates the basic idea of IsoRank using Python. Note that this implementation is a simple one, and actual applications will require various optimizations and extensions.
import numpy as np
# Define adjacency matrices for two networks
# These matrices represent connection information between nodes in different networks
network1 = np.array([[0, 1, 0], [1, 0, 1], [0, 1, 0]])
network2 = np.array([[0, 1, 1], [1, 0, 0], [1, 0, 0]])
# Initialize node similarity matrix
# This matrix holds similarity scores, initially setting all values to zero
similarity_matrix = np.zeros((len(network1), len(network2)))
# Similarity Score Calculation
for i in range(len(network1)):
for j in range(len(network2)):
# Calculate the similarity score between nodes here and set to similarity_matrix
similarity_score = some_similarity_function(network1[i], network2[j])
similarity_matrix[i, j] = similarity_score
# Seek the response that maximizes the similarity score (optimization)
from scipy.optimize import linear_sum_assignment
row_ind, col_ind = linear_sum_assignment(-similarity_matrix)
# Show correspondence
for i, j in zip(row_ind, col_ind):
print(f'Node {i} in network1 is aligned with Node {j} in network2')In this example implementation, the adjacency matrix of the two networks is defined and similarity scores between nodes are calculated. For optimization, a Hungarian algorithm is used to find the best correspondence.
Challenge for IsoRank
IsoRank is a powerful network mapping algorithm, but there are several challenges and limitations. The main challenges of IsoRank are described below.
1. increased computational cost:
IsoRank requires computation on every pair of nodes in two networks to compute similarity scores between nodes. Therefore, the computational cost is high for large networks, creating scalability issues.
2. handling noise and mismatch:
Real-world networks can contain noise and inconsistencies, and IsoRank does not provide a method to deal with these. Therefore, the presence of noise and inconsistencies may affect the quality of the correspondence.
3. accuracy dependence:
The performance of IsoRank is highly dependent on accurately calculating similarity scores between nodes. Therefore, the quality of the correspondence may vary depending on the metrics and algorithms used to calculate similarity scores.
4. application to very large networks:
Especially when applying IsoRank to very large networks, memory and computational resource constraints become an issue, requiring the application of efficient algorithms and parallel processing.
5. response evaluation:
IsoRank does not provide built-in evaluation metrics to assess the quality of the response. Response evaluation is application-dependent and requires the selection of appropriate metrics.
6. application to dynamic networks:
IsoRank focuses on correspondence for static networks, and its application to dynamic networks (i.e., networks that change over time) presents challenges.
IsoRank’s Strategy for Addressing Challenges
The following measures can be considered to address the IsoRank issue
1. reduction of computational cost:
- Graph sampling: To reduce computational cost in large networks, graph sampling techniques should be introduced. Specifically, random sampling and sampling of important nodes can be considered.
2. handling of noise and discrepancies:
- Robust similarity measures: Use robust similarity measures to deal with noise and disagreement. For example, a similarity measure that is not sensitive to outliers could be selected.
- Outlier detection: Detect nodes and discrepancies that contain noise and apply methods to exclude them from the mapping process.
3. accuracy improvement:
- Advanced similarity measures: Improve the quality of the correspondence by using more accurate similarity measures and features. Possible extensions include taking into account graph isomorphisms and edge attributes.
4. handling of very large networks:
- Distributed processing: Leverage distributed processing frameworks to deal with large networks. Specifically, Spark and Hadoop could be used.
- Approximation Algorithms: Instead of seeking advanced optimal solutions, use approximation algorithms to reduce computational cost.
5. evaluation of response:
- Selection of evaluation criteria: Select evaluation criteria appropriate for the application and evaluate the quality of the response. Evaluation criteria may include general quality metrics or metrics specific to a particular application.
6. application to dynamic networks:
- Time-varying model: Develop a method to model time variation for dynamic networks and update the response at each time step.
7. integration of user interactions:
- User feedback: Implement mechanisms to collect feedback from users and adjust the results of responses. They may allow users to provide the correct response.
Reference Information and Reference Books
Detailed information on relational data learning is provided in “Relational Data Learning“, “Time Series Data Analysis, “Graph data processing algorithms and their application to Machine Learning and Artificial Intelligence tasks“, Please refer to that as well.
Reference books include “Relational Data Mining”
“Inference and Learning Systems for Uncertain Relational Data“
“Graph Neural Networks: Foundations, Frontiers, and Applications“
“Non-negative Matrix Factorization Techniques: Advances in Theory and Applications“
“An Improved Approach On Distortion Decomposition Of Magnetotelluric Impedance Tensor“
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