给定下图，节点 v1 和 v2 之间的杰卡德相似度是多少？

Given the following graph, what is the Jaccard Similarity between nodes v1 and v2?

Question

给定下图，节点 v1 和 v2 之间的杰卡德相似度是多少？

Given the following graph, what is the Jaccard Similarity between nodes v1 and v2?

BlackTom AI · Accepted Answer

Jaccard similarity = |N(v1) ∩ N(v2)| / |N(v1) ∪ N(v2)|. For node-pair similarity in a graph we typically compare their neighbor sets; when the two nodes are directly connected it is common to exclude the two nodes themselves from each other’s neighbor sets so the score measures shared neighbors rather than the existence of the direct edge.

1) Relevant concepts
- Neighbor set N(x) = set of nodes adjacent to x (not including x).
- Jaccard = size of intersection of neighbor sets divided by size of their union.
- For adjacent nodes v1 and v2 we remove v2 from N(v1) and v1 from N(v2) before computing, to focus on common neighbors.

2) Identify the neighbor sets (from the drawing)
- N(v1) = {v2, v3}
- N(v2) = {v1, v3, v4}

3) Remove the other node from each neighbor set (since v1 and v2 are adjacent)
- N'(v1) = N(v1) \ {v2} = {v3}
- N'(v2) = N(v2) \ {v1} = {v3, v4}

4) Compute intersection and union (showing intermediate steps)
- Intersection: N'(v1) ∩ N'(v2) = {v3} ∩ {v3, v4} = {v3} → |intersection| = 1
- Union: N'(v1) ∪ N'(v2) = {v3} ∪ {v3, v4} = {v3, v4} → |union| = 2

5) Jaccard calculation
- J = |intersection| / |union| = 1 / 2 = 1/2

6) Why the correct answer is correct and why other options are wrong
- 1/2 is correct because there is exactly one shared neighbor (v3) and, after excluding the two nodes themselves, there are two distinct neighbors in total (v3 and v4). Options like 1/3 or 2/5 would require different counts of intersection or union (e.g., an intersection of 1 with union 3 gives 1/3), and 2/3 or 3/4 would require more common neighbors than the graph shows. None of those match the true counts of 1 and 2 used above.

Conclusion: The Jaccard similarity between v1 and v2 (using neighbor sets excluding each other) is 1/2, which matches the provided answer.

给定下图，节点 v1 和 v2 之间的杰卡德相似度是多少？ Given the following graph, what is the Jaccard Similarity between nodes v1 and v2?单项选择题

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