Let Alice and Bob compute boolean function $f(x_1,\dots,x_{2n})$.
Select a random subset $\mathcal I\subseteq\{1,\dots,2n\}$ of cardinality $n$ and let $\mathcal J=\{1,\dots,2n\}\backslash\mathcal I$.
Let Alice get variables $x_i$ where $i\in\mathcal I$ and Bob get $x_j$ where $j\in\mathcal J$.
Let communication complexity of this function under this partition be $CC_{\mathcal I,\mathcal J}(f)$
Define $$cc_{min}(f)=\min_{\substack{\mathcal I\subseteq\{1,\dots,2n\}\\\mathcal J=\{1,\dots,2n\}\backslash\mathcal I\\|\mathcal I|=|\mathcal J|=n}}CC_{\mathcal I,\mathcal J}(f)$$ $$cc_{max}(f)=\max_{\substack{\mathcal I\subseteq\{1,\dots,2n\}\\\mathcal J=\{1,\dots,2n\}\backslash\mathcal I\\|\mathcal I|=|\mathcal J|=n}}CC_{\mathcal I,\mathcal J}(f)$$
Is there a term for $cc_{max}(f)-cc_{min}(f)$ and $\frac{cc_{max}(f)}{cc_{min}(f)}$?
Is the related concepts introduced and studied anywhere?
I am also interested in scenarios where $|\mathcal I|\neq\mathcal J|$ under condition $||\mathcal I|-\mathcal J||\leq \log n$.