Finite Counterexamples to the Monotonicity Conjecture of Amir, Baldasso and Beilin for Median and Majority Dynamics

Amir, Baldasso and Beilin conjectured (Oberwolfach Report 4/2021, Conjecture 1; Stoch. Proc. Appl. 155 (2023), Conjecture 3.1) that in median dynamics on any locally finite graph (i.i.d. uniform opinions in \([0,1]\), independent rate-one Poisson clocks, a ringing vertex takes the median of its neighbours' opinions, its own opinion being added when its degree is even) the probability that the opinion at a fixed vertex lies in \([0,\alpha]\) is non-decreasing in time for every \(\alpha>\frac12\). Equivalently, in majority dynamics started from i.i.d. Bernoulli\((p)\) opinions, \(f_p(t)\)\({}=\mathbb{P}_p[\xi_t(x)\)\({}=1]\) is non-decreasing for every \(p\ge\frac12\). They proved this for complete graphs and for \(\mathbb{Z}\). We show that the conjecture is false as stated. On a connected graph with six vertices and eight edges, all of odd degree, we determine \(f_p\) at one vertex for every \(p\): with \(r=p(1-p)\), \[ f_p(t)-f_p(\infty)=r(2p-1)e^{-t}\Bigl[\tfrac9{16}rt-\tfrac{4+r}4 -\tfrac7{24}re^{-2t}+r\bigl(\tfrac{71}{200}+\tfrac{7t}{80}\bigr)e^{-4t}\Bigr]. \] Hence, for every \(p\in(\frac12,1)\), \(f_p(t)\)\({}>f_p(\infty)\) for all \(t\)\({}>\frac{16}{9r}\)\({}+\frac{26}{27}\), and \(f_p\) is eventually strictly decreasing; the conjecture fails at this vertex for every \(\alpha\in(\frac12,1)\). The formula is obtained by a finite computation in exact rational arithmetic over the \(64\) configurations of the graph, and one instance is certified separately by rational bounds which do not use it. On a connected \(3\)-regular graph on ten vertices, an exact closed form shows that \(f_p(t)\)\({}>f_p(\infty)\) for all large \(t\), for every \(p\in(\frac12,1)\), with a larger overshoot (\(f(3)\)\({}-f(\infty)\)\({}>3.47\)\({}\cdot10^{-3}\) at \(p=\frac7{10}\)). All degrees in both examples are odd, so neither depends on the rule for ties at even degree. For infinite graphs we obtain only disconnected counterexamples. The conjecture for vertex-transitive graphs, in particular for \(\mathbb{Z}^d\) \((d\ge2)\) and for regular trees, and for connected infinite graphs is not settled, and we have no certified counterexample among regular graphs of degree at least four. This is an unrefereed note.