Logical Patterns in Knowledge Graphs
Understanding logical patterns in knowledge graphs is crucial for capturing the structure and meaning of real-world relationships. Three important logical patterns in relations are symmetry, inversion, and composition. Each pattern describes a different way that entities can be connected through relations.
Symmetry means that if a relation holds from entity A to entity B, it also holds from B to A. For example, the relation is sibling of is symmetric: if Alice is a sibling of Bob, then Bob is a sibling of Alice. In a knowledge graph, this is represented by both (Alice, is_sibling_of, Bob) and (Bob, is_sibling_of, Alice) being true.
Inversion refers to pairs of relations that are logical opposites. If a relation holds from entity A to B, its inverse holds from B to A. For instance, parent of and child of are inverse relations: if Alice is a parent of Bob, then Bob is a child of Alice.
Composition involves chaining relations. If one relation connects A to B and another connects B to C, then their composition connects A to C. For example, if is parent of connects Alice to Bob and is parent of connects Bob to Carol, then the composition is grandparent of connects Alice to Carol.
Embedding models attempt to capture these logical patterns in vector space. The ability of a model to represent symmetry, inversion, or composition depends on its scoring function and how it encodes entities and relations.
- DistMult can represent symmetric relations because its scoring function multiplies the same components for head and tail entities, resulting in the same score when they are swapped;
- ComplEx can also model symmetric relations, but it is not limited to only symmetric ones due to its use of complex-valued embeddings;
- TransE generally cannot represent symmetry, as its scoring function is based on vector translation, which is directional.
- ComplEx can model inversion by using the conjugate of the relation embedding;
- RotatE can represent inverse relations by rotating in opposite directions in the complex plane;
- TransE can model inversion because the negative of a relation vector represents the inverse.
- TransE can represent compositionality, as the sum of relation vectors can approximate the composition of relations;
- RotatE can model composition, since the product of two rotations equals the composition of their corresponding relations;
- DistMult and ComplEx have limited ability to model composition because their scoring functions do not naturally handle chaining of relations.
1234567891011121314151617181920import numpy as np # Suppose 'is_sibling_of' is a symmetric relation # Entity embeddings alice = np.array([1.0, 2.0, 3.0]) bob = np.array([4.0, 5.0, 6.0]) # Relation embedding (for DistMult, often just a vector) is_sibling_of = np.array([0.5, 1.0, 0.5]) # DistMult scoring function: score = sum(head * relation * tail) def distmult_score(head, relation, tail): return np.sum(head * relation * tail) score_alice_bob = distmult_score(alice, is_sibling_of, bob) score_bob_alice = distmult_score(bob, is_sibling_of, alice) print("Score (Alice, is_sibling_of, Bob):", score_alice_bob) print("Score (Bob, is_sibling_of, Alice):", score_bob_alice) # The scores should be equal if the relation is symmetric
1. Which model can represent symmetric relations?
2. What is an example of a compositional relation?
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Understanding logical patterns in knowledge graphs is crucial for capturing the structure and meaning of real-world relationships. Three important logical patterns in relations are symmetry, inversion, and composition. Each pattern describes a different way that entities can be connected through relations.
Symmetry means that if a relation holds from entity A to entity B, it also holds from B to A. For example, the relation is sibling of is symmetric: if Alice is a sibling of Bob, then Bob is a sibling of Alice. In a knowledge graph, this is represented by both (Alice, is_sibling_of, Bob) and (Bob, is_sibling_of, Alice) being true.
Inversion refers to pairs of relations that are logical opposites. If a relation holds from entity A to B, its inverse holds from B to A. For instance, parent of and child of are inverse relations: if Alice is a parent of Bob, then Bob is a child of Alice.
Composition involves chaining relations. If one relation connects A to B and another connects B to C, then their composition connects A to C. For example, if is parent of connects Alice to Bob and is parent of connects Bob to Carol, then the composition is grandparent of connects Alice to Carol.
Embedding models attempt to capture these logical patterns in vector space. The ability of a model to represent symmetry, inversion, or composition depends on its scoring function and how it encodes entities and relations.
- DistMult can represent symmetric relations because its scoring function multiplies the same components for head and tail entities, resulting in the same score when they are swapped;
- ComplEx can also model symmetric relations, but it is not limited to only symmetric ones due to its use of complex-valued embeddings;
- TransE generally cannot represent symmetry, as its scoring function is based on vector translation, which is directional.
- ComplEx can model inversion by using the conjugate of the relation embedding;
- RotatE can represent inverse relations by rotating in opposite directions in the complex plane;
- TransE can model inversion because the negative of a relation vector represents the inverse.
- TransE can represent compositionality, as the sum of relation vectors can approximate the composition of relations;
- RotatE can model composition, since the product of two rotations equals the composition of their corresponding relations;
- DistMult and ComplEx have limited ability to model composition because their scoring functions do not naturally handle chaining of relations.
1234567891011121314151617181920import numpy as np # Suppose 'is_sibling_of' is a symmetric relation # Entity embeddings alice = np.array([1.0, 2.0, 3.0]) bob = np.array([4.0, 5.0, 6.0]) # Relation embedding (for DistMult, often just a vector) is_sibling_of = np.array([0.5, 1.0, 0.5]) # DistMult scoring function: score = sum(head * relation * tail) def distmult_score(head, relation, tail): return np.sum(head * relation * tail) score_alice_bob = distmult_score(alice, is_sibling_of, bob) score_bob_alice = distmult_score(bob, is_sibling_of, alice) print("Score (Alice, is_sibling_of, Bob):", score_alice_bob) print("Score (Bob, is_sibling_of, Alice):", score_bob_alice) # The scores should be equal if the relation is symmetric
1. Which model can represent symmetric relations?
2. What is an example of a compositional relation?
Thanks for your feedback!
