Modeling Set Relationships

This section will talk about how to model relative cardinality between different types of entities. Before reading this, you should already be familiar with data mapping and cardinality functions.

Suppose you have two different sets of things in your virtual dataset:

#zoom:0.75
#direction:right
#.data: fill=#FFFFFF visual=frame
[entity type A|
[<data> "foo"]
[<data> "bar"]
[<data> "baz"]
]

#zoom:0.75
#direction:right
#.data: fill=#FFFFFF visual=frame
[entity type B|
[<data> "X"]
[<data> "Y"]
[<data> "Z"]
]

It may be the case that there is no association between elements of the first set to the second set, as illustrated above.

You could have a many-to-many association from set A to set B, like this:

#zoom:1.0
#direction:down
#.data: fill=#FFFFFF visual=frame
[<data> foo]
[<data> bar]
[<data> baz]
[<data> X]
[<data> Y]
[<data> Z]
[foo] -> [X]
[foo] -> [Y]
[foo] -> [Z]
[bar] -> [X]
[bar] -> [Y]
[bar] -> [Z]
[baz] -> [X]
[baz] -> [Y]
[baz] -> [Z]

In any case, the associations are themselves part of the data set. They must be modeled as such. The way that we do this in virtual dataset is to group our functions and thus their value relationships in the same way that data should be grouped.

Isolating the values that we use for for uniqueness is a key step. Assume for now, that you are using a simple long value to identify each unique entity or instance of your virtual dataset. For example, foo is entity 0, bar is entity 1, and baz is entity 2. The same applies for X, Y and Z: X is 0, Y is 1, and Z is 2.

Let’s make this clear in terms of the value groupings that we would need to simulate for a full association between all type-A entities and all type-B entities. Here are the tuples: (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2).

Another way to say this is that “All members of set A are associated with All members of Set B”.

A simple modulo function that does only range adjustment for the sizes of these two populations is simply “mod 3”. You could draw many many samples via a mapping function, but as long as the mapping functions are the same, you would get the same identifier value for each entity in the tuple. You could only ever get (0,0), (1,1), and (2,2). This does not suffice for our nine different associations above.

Let us look at the associations directly, and work backwards to find the appropriate functions that will maintain the correspondence.

Here are the associations, enumerated by sample number (coordinate value). The discrete correspondence is captured for A and B in a more visually obvious way in the columns to the right.

sample=#  association  A map   B map

sample=0  (0,0)        0->0    0->0
sample=1  (0,1)        1->0    1->1
sample=2  (0,2)        2->0    2->2
sample=3  (1,0)        3->1    3->0
sample=4  (1,1)        4->1    4->1
sample=5  (1,2)        5->1    5->2
sample=6  (2,0)        6->2    6->0
sample=7  (2,1)        7->2    7->1
sample=8  (2,2)        8->2    8->2

sample=9  (0,0)        9->0    9->0
sample=10 (0,1)       10->0   10->1
sample=11 (0,2)       11->0   11->2
...       ...         ...     ...

In this case, we can see that the functions must do more than simply correct for the possible number of unique identifiers in each population. They must also have some type of relationship that adjust for the relative rates of progression through the possible values. For every A value, we need to see one each of B. In other words, we need to see a new A value about ¹⁄₃ as frequently as we see a new B value.

For A, we need a function like “divide by 3, then modulo by 3”. For B, simply “modulo by 3”. By dividing the sampling input by a number, we reduce its rate of progression.

Although we are looking at correspondence between discrete values (the counting numbers), it is easy to demonstrate on a coordinate plane as well using standard plotting tools.

First, by using a floor function to visually illustrate the discrete mapping:

And second, the continuous analog, to show the relative slope and repeating patterns at every 9th interval:

So What?

From all this we can see one of the most basic strategies for establishing a one-to-many relationship. The basic rule is that the many side of the relationship must have exactly that many more values per coordinate range than the one side. It is usually just a matter of adjusting the relative rates of progression along the coordinate line, as illustrated above.

Also, the example above is very basic. It shows a regular repeating pattern with a full one-to-many association. How do we extrapolate this to other more advanced examples? This is a more advanced topic that will be explored in the future. For now, refer to the recipes section for some ideas.