I'm not a mathematician or a math student, this is simply a problem I stumbled upon while working on a personal project, so excuse me if I seem ignorant
First, I have the equation (1) $$p_{ij} = \frac{1}{2}\left( \tanh{(-\frac{\theta_i + \theta_j}{2} + 2Kp_{ij} )} + 1 \right)$$ where $1 \leq i,j \leq N$. $N$ and $K$ are known quantities and $\theta_i, \theta_j$ are unknown.
The $N$ equations are $$C_i = \sum_{j\neq i}p_{ij}^* = \frac{N-1}{2} + \frac{1}{2}\sum_{j\neq i} \tanh{\left( - \frac{\theta_i + \theta_j}{2} + 2Kp_{ij}^* \right)}$$ where $C_i$ is a known quantity and $p_{ij}^*$ is the solution to the equation (1) given above. I want to solve for $\theta_i, \theta_j$.
I don't necessarily need to solve this exactly, it can be done numerically. I first tried to solve this problem by expanding the tanh to first order, which simply gives me a system of $N$ linear equations, but this wasn't very useful.
I feel as if this system of equations is only solvable if instead of the $2Kp_{ij}^*$ in the argument of the tanh, I had $2K\sum_{j\neq i}p_{ij}$. This way, I would have the value of $\sum_{j\neq i}p_{ij}^* = C_i (\textrm{given})$ for every $i$, I could plug this value into the tanh, and then solve for $\theta_i$.
Is this problem solvable? Or do I not have enough information. To me it seems that I have $N$ unknown quantities ($\theta_i$) and $N$ equations, which should be doable. or am I missing something?