• Entities have transitivity: if \( a \) and \( c \) are the same things, \( a \) and \( d \) are not the same things, then \( a \) and \( c \) can not be the same things.
• We already have probabilities \( p \) for each pairs telling they are matching or not, e.g. \( p(r_a, r_b) = 0.46 \)

Only six pairs is needed for querying to Oracle:

\( (r_a, r_b) \), \( (r_a, r_c)\), \((r_d, r_e)\), \((r_a, r_d)\), \((r_a, r_f )\), \((r_d , r_f )\)

Previous:

\( (r_a,r_b) \), \( (r_a,r_c) \), \( (r_d,r_e)\), \((r_a,r_d)\), \((r_a,r_f)\), \((r_d,r_f)\).

Another ordering:

\( (r_a, r_d)\), \((r_a, r_f )\), \((r_d, r_f )\), \((r_d, r_e)\), \((r_a,r_b)\), \((r_a,r_c)\).

Considering pairs with probabilities:

Matching:

\( p(r_d, r_e) = 0.80\), \(p(r_b, r_c) = 0.60\), \(p(r_a, r_c) = 0.54\), \( p(r_a, r_b) = 0.46\).

Non-matching: \(p(r_a, r_d) = 0.84\), \(p(r_d, r_f) = 0.81\), \(p(r_c, r_e) = 0.72\), \(p(r_a, r_e)=0.65\), \(p(r_c, r_d)=0.59\), \(p(r_e, r_f) = 0.59\), \(p(r_a, r_f)=0.55\), \(p(r_b, r_d) = 0.51\), \(p(r_b, r_e) = 0.46\), \(p(r_c, r_f) = 0.45\), \(p(r_b, r_f) = 0.29\)

A total of 7 pairs will be queried \((r_a,r_d)\), \((r_d,r_f)\), \((r_d,r_e)\), \((r_c,r_e)\), \((r_b,r_c)\), \((r_a,r_f)\), \((r_a,r_c)\), if using minimizing # queries strategy

In an entity graph,

with \( E^+ \) as the ground truth of positive edges,

and \(E_T^+\) as the positive edges we can get with \( t = |T| \) queries and \(T\) is the set of the query.

Then the progressive recall will be defined as

\( precall(t) =\sum_{t'=1}^t recall(t') \), and \(precall^+(t)=\sum_{t'=1}^t recall^+(t')\).

A \(recall\) of \(T\) can be defined as \( recall(t) = \frac{|E^+_T|}{|E^+|} \), and \( recall^+(t) = \frac{|E^+_{T^+}|}{|E^+|} \)

For matching edges, with probability \((1- \frac{\alpha}{n} )\), \( \alpha \gt 0 \) the score is above 0.7, and the remaining probability mass is distributed uniformly in \([0, 0.7)\).

For non-matching edges, with probability \( (1-\frac{\beta}{n}) \) , \( \beta \gt 0 \), the score is below 0.1 and the remaining n probability mass is distributed uniformly in \( (0.1,1] \).

\( O(log^2n)-approximation\) under edge noise model

\( \Omega(n)-approximation\) under progressive recall

A total of 7 pairs will be queried \((r_a,r_d)\), \((r_d,r_f)\), \((r_d,r_e)\), \((r_c,r_e)\), \((r_b,r_c)\), \((r_a,r_f)\), \((r_a,r_c)\).

\( O(log^2n)-approximation\) under edge noise model

\( \Sigma(\sqrt n) \) under progressive recall

A total of 7 pairs will be queried \((r_e,r_d)\), \((r_a,r_d)\), \((r_c, r_e)\), \((r_c, r_a)\), \((r_f, r_d)\). \((r_f, r_a)\), \((r_b, r_c)\).

A maximized PRecall strategy \(S^*\) is:

1. Presume we already know the ground truth
2. For each cluster \(C_k\) in the graph (which is a same real-world entity), we ask \(|C_k|\) questions to the Oracle.
3. Questions are asked in the sequence of \( C_1, C_2, ..., C_k \), where \( |C_k| \) has a decreasing order.
4. Finally asking \( \binom k2 \) non-matching questions.
5. Ideally, \(s^*\) will ask \(n - k + \binom k2 \) questions

\( b_e \) is the benefit of a edge between \(u\) and \(v\), calculated as \( |c_T(u) | * | c_T(v) |* p(u,v) \)

\(b_v\) is the benefit of a node when adding this node to the \(P\) set.

\(b_{vc}(v,c) = p_v(v,c) * |c|\)

\(b_v(v,P) = \max b_{vc}(v,c)\)

A total of 6 pairs will be queried \((r_a,r_d)\), \((r_d,r_f)\), \((r_d,r_e)\), \((r_c,r_e)\), \((r_b,r_c)\), \((r_a,r_c)\).

A total of 6 pairs will be queried \((r_a,r_d)\), \((r_d,r_f)\), \((r_d,r_e)\), \((r_c,r_e)\), \((r_b,r_c)\), \((r_a,r_c)\).

\(b_e(r_a,r_c)\) = 2 * 1 * 0.54 = 1.08

\(b_e(r_a, r_c)\) = 1 * 1 * 0.55 = 0.55

if \(w = 1\):

if \( \tau = n\) and \(\theta = 0\) then \(s_{hybrid} = s_{vesd} \)

if \(\tau = 0\) or \(\theta = n\), then \(s_{hybrid} = s_{wang} \)

n: num of records

k: num of entities (clusters)

k': num of non-singleton entities (clusters)

\(|c_1|\): largest clusters

\(t^*_r\): size of the spanning forest

ER: dirty or clean-clean

origin: real or synthetic of data

R: # record pairs

\( n' \): \( \binom n2 = \lfloor R \rfloor \)

\(s_{edge}\)

\(s_{hybrid}\)

\(s_{hybrid} \) is better than \(s_{edge}\) with same parameter, but...