the *interest set* of the agent. In what follows, we point out that the $n$-th component of the field has a one-to-one correspondence with the $n$-th agent: this one-to-one correspondence is best illustrated by visualizing the agents to be (geographically) distributed in a field with $\theta^{\ast}_{n}$ representing the state of the field at the location of the $n$-th agent. Formally, the interest set of an agent is represented as $\mathcal{I}_{n}$. The interest set could vary from one agent to another. The interest sets can be arbitrary but need to satisfy the following assumption:

[A3]{} \[as:3\] *The set of agents physically coupled with agent $n$ is a subset of the interest set of agent $n$, i.e., $\widetilde{\mathcal{I}}_{n}\subset\mathcal{I}_{n}$*.

We assume without loss of generality that $\widetilde{\mathcal{I}}_{n}$ and hence $\mathcal{I}_{n}$ is non-empty for all $n$. (For illustration, see below the example after Assumption \[as:6\]). We number the nodes (equivalently, components of $\btheta$) in the interest sets of agents in increasing order. Thus, the interest set $\mathcal{I}_{n}$ at an agent $n$ can be considered to be a vector with dimension $\left|\mathcal{I}_{n}\right|$. For example, $\mathcal{I}_{n}(r)=p$ indicates that agent $p$ is the $r$-th agent in increasing order in the interest set $\mathcal{I}_{n}$. We also have that 