**Questions 2 \& 3**
This paper has two graphs (i.e., the benefit graph $\mathcal{G_b}$, and the competing graph $\mathcal{G_c}$) that are assumed to be known to a FL manager; $\mathcal{G_b}$ is obtained by collaborative model training, while the information on $\mathcal{G_c}$ is reported by clients to the FL manager (or the central server). Below, we detail the ways of obtaining them and how the local model information is leveraged in the graph part.
Firstly, data heterogeneity/complementarity between clients is characterized by a benefit graph $\mathcal{G_b}$. Like [5,30], this paper uses the hypernetwork technique in [23] to evaluate the data complementarity, thus obtaining the benefit graph. In the graph part, the local model information is mainly leveraged in the calculation of $\mathcal{G_b}$. We also refer readers to Section 4.2 of Ref. [5] for the technical details on the way of obtaining $\mathcal{G_b}$.
Specifically, there are $n$ clients. Each client $v_{i}$ has a risk/loss function $\ell_{i}$: $\mathbb{R^n}\rightarrow \mathbb{R_+}$. Given a learned hypothesis $h\in$ $\mathcal{H}$, let the loss vector $\mathbf{\ell}(h)=[\ell_{1}, \dots, \ell_{n}]$ represent the utility loss of the $n$ clients under the hypothesis $h$. The hypothesis $h$ is considered a Pareto solution if there is no other hypothesis $h^{\prime}$ that dominates $h$, i.e., $\nexists h^{\prime}\in \mathcal{H}, s.\,t.\, \forall i: \ell_{i}(h^{\prime})\leqslant \ell_{i}(h) \text{ and } \exists j: \ell_{j}(h^{\prime}) < \ell_{j}(h). $
Let $r=(r_{1}, \dots,r_{n})$ $\in \mathbb{R^n}$ denote a preference vector which denotes the weight of the objective local model loss that is normalized with $\sum_{k=1}^{n}{r_{k}}=1$ and $r_{k}\geq 0, \forall k\in \{1, \dots, n\}$. The hypernetwork $HN$ takes $r$ as input and outputs a Pareto solution $h$, i.e., $h\gets HN(\phi, r)$, where $\phi$ denotes the parameters of the hypernetwork [23]. For each client $v_{i}$, linear scalarization can be used. Like [5,30], an optimal preference vector $r_{i}^{\ast}=\left(r_{i,1}^{\ast}, r_{i,2}^{\ast}, \dots, r_{i,n}^{\ast}\right)$ is determined to generate the hypothesis $h_{i}^{\ast}$ that minimizes the loss with the data $\hat{\mathcal{D_i}}$. This is expressed as
$h_{i}^{\ast}=HN(\phi, r_{i}^{\ast}), \text{ where } r_i^* = argmin_r \hat{\mathcal{L_i}}(HN(\phi, r)).$
For each client $v_{i}$, the value of $r_{i,j}^{\ast}$ is used as an estimate to the weight of $v_{j}$ to $v_{i}$ [5,30]. $r_{1}^{\ast}, r_{2}^{\ast}, \dots, r_{n}^{\ast}$ define a directed weighted graph, i.e., the benefit graph $\mathcal{G_b}$.
Secondly, following the standard technical practices in literature, the central server is assumed to be trustable, as explained above. In real worlds, the central server may represent an impartial and authoritative third-party (e.g., the industry association) [5]. Then, FL-PTs can report their competitive relationships to the third-party in person and confidentiality agreements can be signed between the third-party and FL-PTs.
The above information will be clarified in the final version of our manuscript.