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The Secret of Network Similarity Revealed! What is the Magical Difference Between Structural, Arithmetic, and Conventional Equivalence?
In today's data-driven era, network analysis is increasingly used. Whether in social media, business networks, or ecosystems, understanding the similarities between different nodes is important to drive decision making and management. Similarity in networks usually occurs when two nodes (or other more complex structures) belong to the same equivalence class. There are three basic approaches when building network similarity metrics: structural equivalence, automorphism equivalence, and conventional equivalence.
There is a hierarchical relationship between these three kinds of equivalence: any set of structural equivalences is also both an automorphism and a conventional equivalence, and any set of automorphism equivalences is also a conventional equivalence.
Structural equivalence means that two nodes have the same neighbors and the connection patterns between them are exactly the same. Automorphism equivalence holds that if by relabeling the nodes, a graph can be generated that is indistinguishable from the original in terms of distance, then the two nodes are automorphically equivalent. Finally, general equivalence is when two nodes are considered equivalent if they have a relationship with other similar nodes, although not necessarily a direct relationship.
Structural Equivalence
In a network, two vertices are structurally equivalent if they share many of the same neighbors. For example, if node A has a certain set of connections to a particular set of nodes, then other nodes similar to A should also have the same connection pattern. Here are a few key points about structural equivalence:
For example, two banks may be geographically close but have very different linkage patterns and therefore cannot be considered structurally equivalent, yet they have some structural similarity because they both operate in the same financial sector. degree of institutional equivalence.
Structural equivalence measures
For the measurement of structural equivalence, we can use several indicators:
Cosine Similarity: Calculates the number of common neighbors of two nodes and compares it to the degree of the nodes.Pearson correlation coefficient: Measures the similarity of two nodes by comparing them to the number of common neighbors that would occur in a random network.Euclidean distance: Although this is a measure of dissimilarity, it provides an intuitive understanding of the differences between nodes.
Automorphism Equivalence
The formal definition of automorphism equivalence is: if all nodes can be relabeled so that swapping u and v does not affect the distances of all other nodes in the graph, then two nodes are automorphism equivalent. In an organizational chart, this is important for employees who have similar tasks but do not share a direct relationship.
In a corporate organization, roles in the central office are automorphically equivalent if they can be interchanged without changing the overall operating model.
General Equivalence
General equivalence is defined as follows: two nodes are considered general equivalent if they have similar relationships with other similar nodes. This is a looser judgment of similarity, such as the relationship between mothers in SES (family structure), even if their partners and children are different, they have similar interaction patterns with the community or other family members they participate in together.
General equivalence emphasizes the relationship with other equivalent nodes rather than the direct adjacency relationship.
Conclusion
By understanding structural, arithmetic, and conventional equivalence, we can gain a more complete understanding of the underlying patterns and similarities in network data. This not only helps academic research, but also provides guidance for practical applications. Can you imagine how future network analysis will further reveal the hidden meanings in data?
