Complete-network maximizer conjecture for homogeneous finite networks

Fix parameters $p,q$ and a positive integer $M$, and let

$${\cal G}(p,q,M):=\{{\cal N}\mid {\cal N}\text{ has }M\text{ nodes and satisfies the homogeneity condition in }\{p_j\}\text{ and }\{q_j\}\}.$$

For ${\cal N}\in{\cal G}(p,q,M)$, write $f(t;{\cal N})$ for its expected adoption level, and let $f_{\rm complete}(t;p,q,M)$ be the adoption level on the homogeneous complete network. Complete-network maximizer conjecture.

$$\sup_{{\cal N}\in{\cal G}(p,q,M)} f(t;{\cal N})=f_{\rm complete}(t;p,q,M).$$

The conjecture asks whether the fastest diffusion among homogeneous networks with a fixed number of nodes is attained by the homogeneous complete network. The supplied text presents it as a new question in the open-problems section, so its resolution is not given here.