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发布于:2019-5-15 20:15:35  访问:52 次 回复:0 篇
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(b, c) d(b, a) + d(a, c) d(a, c
Inequality 00+0 11+1 01+1 10+We have shown that a binary indicator SHP2 IN-1COA function for an equivalence relation satisfies properties P1, P2, P4, and P5. Graph isomorphism (an equivalence relation amongst two graphs) is a unique case of graph automorphism (an equivalence relation within a single graph) in which we declare that the two graphs are certainly a single graph that‘s not connected. This distance independence is absent from prioximity-based node similarity measures.ACM Trans Knowl Discov Information.(b, c) d(b, a) + d(a, c) d(a, c). The only technique to reconcile d(a, c) d(b, c) and d(b, c) d(a, c) is if d(a, c) = d(b, c). The following corollary emphasizes a basic difference in between function similarity measures journal.pone.0159456 and proximity-based similarity measures: function similarity does not reduce merely mainly because distance increases. Corollary 2. (Distance Independence): For each finite k, there exists a graph and a pair of nodes a and b such that the distance involving a and b is a minimum of k, and sim(a, b) = 1 Proof: Let G(V, E) be a linear path graph with n edges and n + 1 vertices, exactly where n k. We label the vertices so that V = 0, 1, ..., n, plus the edge set E = (0, 1), (1, 2), ..., (n-1, n). It is clear that nodes that are precisely the same distance from the endpoints are automorphically equivalent and therefore role-equivalent. That‘s, j n-j, for all 0 j n/2. We make a final observation. Our axiomatic function similarity model can just as simply obtain similarities in between two or extra graphs, for the reason that automorphism is usually extended beyond single graphs. Graph isomorphism (an equivalence relation between two graphs) is a unique case of graph automorphism (an equivalence relation inside a single graph) in which we declare that the two graphs are certainly a single graph that is not connected. This distance independence is absent from prioximity-based node similarity measures.ACM Trans Knowl Discov Data. Author manuscript; available in PMC 2014 November 06.Jin et al.Page3.1. Binary-Valued and Real-Valued Part Similarity Measures Every equivalence relation features a corresponding binary-valued indicator function: I(u, ) = 1 iff u . Otherwise, I(u, ) = 0. This indicator function is often a admissible axiomatic role similarity metric. Theorem 1. (Binary Admissibility) Given any equivalence relation that also satisfies automorphism confirmation (P3), its binary indicator function is definitely an admissible similarity metric. Proof: Binary values satisfy the Range requirement (P1). Any equivalence relation satisfies symmetry (P2) and transitivity (P4), by definition. We prove that this indicator function satisfies the triangle inequality(P5), namely, d(a, c) d(a, b) + d(b, c), exactly where d(a, b) = 1 - sim(a, b), by taking into consideration all attainable class assignments for any, b, and c:NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author ManuscriptCase 1 2 3Description All in the identical class All in distinctive classes a and c in the exact same class b and a single other within the very same classDistances d(a, c) = d(a, b) = d(b, c) = 0 d(a, c) = d(a, b) = d(b, c) = 1 d(a, c) = 0 d(a, b) = 0 or d(b, c) =Tri.
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