Let $G=(V,E)$ be an $n$-vertex simple undirected graph. A *signing* of the graph is a function $s:E \to \{1,-1\}$, and $(G,s)$ is a *signed graph*. That is, we label each edge of the graph with $1$ or $-1$. The sign function, defined on edges, can be naturally extended to paths; the sign of a path is the product of the signs of the edges on the path. A path is said to be *balanced* if its sign is $1$. In other words, the path has an even number of $-1$-labelled edges.

The signed graph $(G,s)$ is said to be *balanced* if every closed path is balanced, and is called imbalanced otherwise. For instance, the trivial signing where we label each edge by $1$ is clearly a balanced signed graph. However, there are non-trivial signings that can be balanced too; for example, a bipartite graph where all edges are labeled $-1$.

There is much literature on balanced signed graphs and ways to balance them. However, I am curious about the polar opposite: "horribly imbalanced graphs", say (though this notion too is a different kind of "balance"). Let us call a graph "horribly imbalanced" if half the closed paths are balanced. More precisely, define:

A signed graph $(G,s)$ is said to be "horribly imbalanced" if for every $k$, exactly half the number of closed paths of length $k$ are labeled $1$.

Now it is possible that the total number of closed paths of length $k$ is odd, so maybe exactly "half the number" is hoping for too much. This looks related to discrepancy theory, so lets say the avg label of closed walks of length $k$ is $0$, with some error. My questions:

Is such a notion meaningful? Has it been studied anywhere already?

Lets assume, for simplicity, that the graph is regular. Then is there some bound on the error margin in terms of $n$, $k$ and the degree?

Given a regular graph, can we appropriately sign the edges so that the resulting signed graph becomes horribly imbalanced this way?

Thanks.