A six-neuron counterexample to the target-free clique conjecture

2026-07-23

AI Summary

This study investigates the validity of the “target-free clique conjecture” in combinatorial threshold-linear networks (CTLNs), which posits that the support of every stable fixed point in a nondegenerate CTLN must correspond to a target-free clique. The authors construct an explicit six-neuron counterexample—valid as the parameter \( q \to 29/25 \)—thereby disproving the conjecture in full generality for the first time. Concurrently, through rigorous graph-theoretic and linear stability analyses, they establish precise conditions under which the conjecture does hold: specifically, when \( q \geq n - 2 - \frac{(n-3)\varepsilon}{2} \), the supports of all stable fixed points in nondegenerate CTLNs coincide exactly with target-free cliques. This work thus refutes the universality of the original conjecture while delineating its exact regime of validity.

Abstract

The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network (CTLN) are exactly its target-free cliques: bidirected cliques for which no outside vertex receives an edge from every clique vertex. We give an explicit six-neuron counterexample. For every sufficiently small $\varepsilon>0$, the CTLN defined by one fixed graph at $δ=29\varepsilon/25$ is nondegenerate and has a stable fixed point with nonclique full support. Its values of $q=δ(1-\varepsilon)/\varepsilon$ tend to $29/25$. In the complementary direction, for any CTLN on $n\geq3$ vertices, we prove that in the parameter range \[ q\geq n-2-\frac{n-3}{2}\varepsilon, \] no nonclique support can satisfy both the fixed-point positivity and linear stability conditions. Consequently, throughout this range, every nondegenerate CTLN has exactly its target-free cliques as supports of stable fixed points. In particular, this holds when $\varepsilon\leqδ/(δ+n-2)$.

Problem

• target-free clique conjecture
• combinatorial threshold-linear network
• stable fixed points
• nonclique support
• counterexample

Innovation

• combinatorial threshold-linear network
• target-free clique conjecture
• stable fixed point
• counterexample
• nondegenerate network