Assuming acid strength relates directly to number of \(\mathrm{O}\) atoms bonded to the central atom, rank \(\mathrm{H}_{2} \mathrm{~N}_{2} \mathrm{O}_{2}\left[\mathrm{or}(\mathrm{HON})_{2}\right], \mathrm{HNO}_{3}\) (or HONO \(_{2}\) ), and \(\mathrm{HNO}_{2}\) (or HONO) in order of decreasing acid strength.

In any given compound, the central atom is the one to which all other atoms are bonded. Identifying the central atom is the first and crucial step in understanding the structure and behavior of a molecule.
In the exercise provided, we have three different acids: \(\text{H}_{2}\text{N}_{2}\text{O}_{2} \( \text{or}(\text{HON})_{2}\)\), \(\text{HNO}_{3} \( \text{or}\text{HONO}_{2}\)\), and \(\text{HNO}_{2} \( \text{or}\text{HONO}\)\). For each of these acids, the central atom is the one that forms bonds with oxygen atoms.
For example: \ - In \(\text{H}_{2}\text{N}_{2}\text{O}_{2}\), the central atom is nitrogen (N).
\ - In \(\text{HNO}_{3}\), the central atom is also nitrogen (N).
\ - In \(\text{HNO}_{2}\), the central atom remains nitrogen (N).
This identification is necessary to further analyze how the molecule interacts and to count the number of bonds with other atoms.

The strength of an acid can often be determined by the number of oxygen atoms bonded to the central atom. This is because oxygen atoms are highly electronegative and can stabilize the negative charge that arises when the acid loses a proton \(\text{H}^{+}\).
By counting the oxygen atoms, we can get a good indication of the acid's strength. In our given examples:
\- \(\text{H}_{2}\text{N}_{2}\text{O}_{2}\) has 2 oxygen atoms.
\- \(\text{HNO}_{3}\) has 3 oxygen atoms.
\- \(\text{HNO}_{2}\) has 2 oxygen atoms.
From this analysis, \(\text{HNO}_{3}\) has the highest number of oxygen atoms bonded to its central atom, indicating it potentially has the highest acid strength.

Once we've identified the central atom and counted the number of oxygen atoms, we can rank the acids by their strength. The general rule is: the more oxygen atoms bonded to the central atom, the stronger the acid.
In our examples, we see:
\- \(\text{HNO}_{3}\) with 3 oxygens is the strongest.
\- \(\text{HNO}_{2}\) with 2 oxygens is next.
\- \(\text{H}_{2}\text{N}_{2}\text{O}_{2}\) with 2 oxygens is the weakest among the given acids.
Even though \(\text{HNO}_{2}\) and \(\text{H}_{2}\text{N}_{2}\text{O}_{2}\) both have 2 oxygen atoms, \(\text{HNO}_{2}\) is typically considered stronger than \(\text{H}_{2}\text{N}_{2}\text{O}_{2}\) due to the specific arrangement and bonding of atoms which influences the molecule's overall stability when it loses a proton.