There are two very important types of relationships between products: substitute and complementary. Substitute products are those that are interchangeable while complementary products are those that might be purchased in addition. For example, when a user is looking at a T-shirt, substitute products are other different types of T-shirts and complementary products can be shorts, hoodies or jackets. Swing is designed for substitute relationships and Surprise is designed for complementary relationships.
Swing
The basic idea of Swing is that if user $A$ and user $B$ who both clicked item $i$ also clicked item $j$, then item $i$ and item $j$ are considered similar. The less similar $A$ and $B$ are, i.e. the smaller the number of items bought both by $A$ and $B$ is, the more similar $i$ and $j$ are.
$$ s(i,j) = \sum_{u\in U_i \cap U_j}\sum_{v\in U_i \cap U_j}\frac{1}{\sqrt{|I_u|}} * \frac{1}{\sqrt{|I_v|}} * \frac{1}{\alpha + |I_u \cap I_v|} $$
$\frac{1}{\sqrt{|I_u|}}$ and $\frac{1}{\sqrt{|I_v|}}$ are weights to reduce the impact of active users and $\alpha$ is the smoothing factor.
Surprise
Category-level
The probability of $c_j$ is a related category for $c_i$ is defined as:
$$ \theta_{i,j} = p(c_{i,j}|c_j) = \frac{N(c_{i,j})}{N(c_j)} $$
where $N_{c_j}$ is the total purchases in category $c_j$, and $N_{c_{i,j}}$ is the number of times $c_j$ is purchased after $c_i$.
Product-level
For each top related category, we can find top related products by:
$$ s_1(i,j) = \frac{\sum_{u\in U_i \cap U_j} 1/(1+t_{uj} - t_{ui})}{|U_i|\times |U_j|} $$
where $t_{ui}$ and $t_{uj}$ denote the time that user $u$ bought item $i$ and item $j$. As the order in the product pairs is important, e.g. we can recommend phone cases to a user who just bought a new phone but we shouldn’t recommend phones to the user who just bought a new phone case, thus $t_{uj} \ge t_{ui}$.
Cluster-level
Since the item-level user co-purchasing data is extremely sparse, we can calculate the relevance score for items at the cluster level to alleviate this problem.
Let $L(i)$ denote the cluster that item $i$ belongs to, which is calculated by the relevance score of items from Swing. Then the cluster-level relevance score can be calculated by:
$$ s_2(i,j) = s_1(L(i), L(j)) $$
Compute Surprise score
The final Suprise score of a product pair is:
$$ s_{i,j} = \omega * s_1(i,j) + (1 - \omega) * s_2(L(i),L(j)) $$
where $\omega$ is the mannually set combination weight.
References
[1] Xiaoyong Yang et al. “Large Scale Product Graph Construction for Recommendation in E-commerce.” ArXiv, abs/2010.05525.
[2] Ruyi Luo et al. FunRec - Swing