# Ranking Function

A ranking function is an function whose function range is an ordinal scale (can be ordered).

**AKA:**Ordinal-Output Rank Function.**Context:****range:**an Ordinal Ranking.- It can range from being an Integer-Valued Ranking Function to being a Real-Valued Ranking Function.
- It can range from being an Abstract Ranking Function to being a Ranking Function Structure.
- It can be (often) used to place a Partial Order on a set of Objects.
- It can be an input to a Ranking Task (solved by a ranking system).
- It can range from being a Heuristic Ranking FUnction to being a Trained Ranking Function.

**Example(s):**- [math]f_A[/math](3.2) ⇒ Large, where Function Range([math]f_A[/math]) = (Small < Medium < Large).
- [math]f_B[/math](9.2,Red,True,0,Null) ⇒ Small, where Function Range([math]f_B[/math]) = (Small < Medium < Large).
- an Information Retrieval Ranking Function.
- a Constant Ranking Function.
- a Term Frequency Ranking Function.
- a TF-IDF Ranking Function.
- a Graph Node Ranking Function (e.g. based on node centrality, node prestige, node diversity)
- a Graph Edge Ranking Function.
- a Okapi BM25 Ranking Function.
- a Predictive Ranker, such as a probabilistic ranking function.
- an Ordinal Random Variable.

**Counter-Example(s):**- Class-Output Function/Classification Function, such as [math]f[/math](9.2,Red,True,0,Null) ⇒ E.coli.
- a Number-Output Function
- an Ordinal-Input Function.

**See:**Ranking Algorithm, Partial Order, Ordinal Function, Ranking Model Learning Algorithm.

## References

### 2009

- (Liu, 2009) ⇒ Tie-Yan Liu. (2009). “Learning to Rank for Information Retrieval.” In: Foundations and Trends in Information Retrieval Journal, 3(3). doi:10.1561/1500000016
- QUOTE: Learning to rank for Information Retrieval (IR) is a task to automatically construct a ranking model using training data, such that the model can sort new objects according to their degrees of relevance, preference, or importance.

### 2006

- (Agarwal, 2006) ⇒ Shivani Agarwal. (2006). “Ranking on Graph Data.” In: Proceedings of the 23rd International Conference on Machine Learning (ICML 2006) doi:10.1145/1143844.1143848

### 2004

- (Podelski & Rybalchenko, 2004) ⇒ Andreas Podelski, and Andrey Rybalchenko. (2004). “A complete method for the synthesis of linear ranking functions. In: Verification, Model Checking, and Abstract Interpretation.
- QUOTE: … A heuristic-based approach for discovery of ranking functions is described in [DGG00]. It inspects the program source code for ranking function candidates. This method restricted to programs where the ranking function is exhibited already in the source code. ...

### 2003

- (Fagin et al., 2003b) ⇒ Ronald Fagin, Ravi Kumar, D. Sivakumar. (2003). “Comparing top k lists.” In: Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms.