Breaking the Illusion of Artificial Consensus: Clone-Robust Weighting for Arbitrary Metric Spaces

TL;DR

This work provides a general construction of clone-robust weighting functions that applies to arbitrary metric spaces, is entirely independent of the underlying topology, and admits efficient computation.

Abstract

Independent media are central to democratic decision-making, yet recent technological developments, such as social media, pseudonymous identities, and generative AI, have made them more vulnerable to coordinated influence campaigns--usually referred to as Coordinated Inauthentic Behavior. By automatically generating large numbers of similar messages and news reports, such campaigns create an illusion of widespread support, and exploit the tendency of human observers and aggregation mechanisms alike to treat frequency as evidence of credibility or consensus. Clone-robust weighting functions offer a solution to this problem by assigning influence in a way that is insensitive to arbitrary duplication or near-duplication, as measured by a metric. This axiomatic framework rests on three principles: symmetry (equivalent elements are treated equally), continuity (weights vary smoothly under perturbations), and clone-robustness (adding duplicates or near-duplicates does not distort the overall distribution). We provide a general construction of clone-robust weighting functions that applies to arbitrary metric spaces, is entirely independent of the underlying topology, and admits efficient computation. Our approach identifies radius graphs as a natural invariant under cloning, and builds on graph weighting functions that satisfy a basic locality condition. We explore the resulting design space, starting with a simple family that satisfies the core axioms, and then identify explainability as a guiding criterion for navigating this design space. To this end, we introduce sharing coefficients that enable meaningful comparison and interpretation of different constructions, but require additional axiomatic principles. We then consider alternative constructions based on clique-covers, and unveil approaches using clique-partitions that are grounded in information-theoretic principles.

Figures (4)

• Figure 1: Impact of adding a vertex $y$ that is an approximate clone of $x$. By the triangle inequality, each distance $d(y,z)$ with $z\in S$ lies within an interval of length $2d(x,y)$ centered at $d(x,z)$. Thus, for all radii $r \ge 0$ outside the blue forbidden intervals, the vertices $x$ and $y$ are equivalent in $G_r(S \cup \{y\})$.
• Figure 2: Illustration of the class-uniform weighting $w^{\mathrm{CU}}$ on a small graph. Nodes are grouped into $\vert V/{\equiv_G}\vert =5$ equivalence classes: $\{0,1,2\}$ in blue, $\{3\}$ in orange, $\{4,5\}$ in green, $\{6\}$ in pink, and $\{7\}$ in purple. Each class receives the same total weight, distributed uniformly among its vertices.
• Figure 3: Consider the set $S =\{x,y_1,y_2,z\}.$ Since $y_1$ is closer to $x$ than $z$, it shares more potential voters with $x$ than $z$ does, as noted by the volume inequality $\operatorname{Vol( B_r(x)\cap B_r(y_1) )} \geq \operatorname{Vol ( B_r(x)\cap B_r(z) )}.$ However, the presence of a clone $y_2$ dilutes $y_1$'s sharing coefficient and $\chi_{g_r,S}(x,y_1) \leq \chi_{g_r,S}(x,z).$
• Figure 4: Simple instance of graph $G$ showing that $w^{\mathrm{CU}}$ does not satisfy Axioms \ref{['axi:non_neg_chi_vertex']} and \ref{['axi:vertex_shar_sym']}.

Theorems & Definitions (17)

• Definition 1: Clone-robust Weighting Function
• Definition 2: Graph weighting Functions
• Example : Uniform distribution over equivalence classes
• Theorem 1
• Lemma 1
• Lemma 2
• Definition 3: Sharing coefficient of $g_r$
• Lemma 3: Sharing Domination
• Definition 4: Sharing coefficient of $f_\nu$
• Definition 5: Sharing Coefficient for Graph Weighting Function