Consider a network modeled by a connected, undirected graph $G = (V, E)$, where each vertex $v_i\in \mathrm{V\; represents\; a\; node,\; and\; each\; edge }$$(v_i, v_j) \in E$ represents a connection between nodes.

 Let $x_i(\mathrm{t)\; be\; the\; state\; of\; vertex }$$v_i$at time t, which evolves according to the following ordinary differential equation (ODE): 

This equation models the change in state of node $v_i$as a result of interactions with its neighbors.

Which of the following statements is true regarding the behavior of this system over time?

(a) The system represents a diffusion process, and all nodes will eventually reach the same state.

(b) The system will exhibit chaotic behavior for sufficiently large graphs.

(c) The solution to the system depends on the initial states of only the boundary nodes.

(d) The state of each node will remain constant over time if it starts in the same state as its neighbors.

(e) The system represents a decay process where all node states approach zero over time.

Original idea by Thalia Anastácia da Silva Araujo.