Modeling generative models as nodes in a directed weighted graph shows cross-model synthetic-data flow jointly sets the convergence threshold and diversity contraction of self-consuming training systems
Synopsis
This work models multiple generative models that consume each other's synthetic data as nodes in a directed weighted graph, derives sufficient conditions for local asymptotic stability of the retraining dynamics via the Jacobian and a comparison matrix, and gives an upper bound on system diversity at the fixed point, showing that stability is jointly governed by local amplification factors and graph propagation while synthetic-data consumption induces a topology-dependent smoothing effect that contracts inter-model heterogeneity; experiments on an 8-mode Gaussian dataset and CIFAR-10 across several generative model classes and interaction topologies align with the theoretical predictions.
Figure 1 : From networked self-consuming models to graph abstraction. (1) Left: A generative ecosystem in which each model is trained on its own real-data distribution and a weighted mixture of synthetic data produced by other models. (2) Right: Abstraction of the system as a directed, weighted interaction graph that captures effective synthetic-data propagation among models. (3) Top right: An example local retraining rule, where Model 3 3 is trained at iteration t t on p 3 data + λ 3 ∑ j = 1 K w 3 j p 𝜽 j t − 1 p^{\mathrm{data}}_{3}+\lambda_{3}\sum_{j=1}^{K}w_{3j}p_{\boldsymbol{\theta}_{j}^{t-1}} .
arXivInterpretation
The paper introduces a unified framework representing networked self-consuming generative models as a directed weighted graph: nodes are models, edge weights are the probability that a synthetic sample generated by one model is consumed by another, the adjacency matrix is row-stochastic and self-loops are allowed, and each model updates its parameters each round on a mixture of real data and synthetic data from itself and its parents. Prior analyses of self-consuming training were largely limited to a single isolated model or to highly simplified interactions between two models; this framework accommodates arbitrary directed weighted graphs with heterogeneous models and heterogeneous real-data distributions, and subsumes the single-model setting as a special case. The framework is given as formal definitions in the problem formulation, and the paper notes that with one model consuming only its own data it recovers the single-model self-consuming dynamics; a two-model heterogeneous example shows the coupled dynamics can admit a fixed point distinct from the uncoupled solution.
The paper provides a local stability criterion for networked retraining dynamics: analyzing the Jacobian around a fixed point, it shows that if the spectral radius of a comparison matrix is below one, the fixed point is locally asymptotically stable and iterates converge at a linear rate; the criterion separates local amplification factors, set by each model's synthetic-data mixing weight and local curvature, from graph propagation effects. This condition strictly generalizes the stability threshold of prior single-model analyses and accommodates arbitrary directed weighted graphs; a norm-based sufficient condition is also given whose threshold depends jointly on cross-model sensitivity and the spectral norm of the interaction graph. The results are stated as a theorem and corollary with proofs; the paper notes the norm-based condition relies on a global worst-case characterization of the graph and can therefore be conservative, and that different graphs may share the same norm value yet behave qualitatively differently.
The paper characterizes topology effects through cycle gains of directed cycles: the spectral radius is lower-bounded by the cycle gain of any simple directed cycle, so the network is primarily governed by the dominant cycle with the largest geometric-mean gain, with cycle length entering only through that geometric mean; if the system is stable, every simple directed cycle must have cycle gain below one, and so must every self-loop. This moves stability from a global norm scalar to locatable structural features, showing that a single dominant cycle can bottleneck the whole system regardless of the total number of cycles or their lengths. The results are stated as a proposition and corollary with proofs; in a controlled closed-form system the paper reports four systems sharing the same interaction norm but with exact convergence factors ranging from 0.582 to 0.900, and empirical convergence rates matching the spectral radius to three decimal places.
The paper gives an upper bound on system output diversity at the fixed point: fixed-point representations result from applying a linear smoothing operator to real-data centroids, mapping initial data diversity to final output diversity through a collapse coefficient; with no cross-model interaction diversity is preserved, stronger synthetic-data mixing strengthens averaging, graph topology determines which disagreements are reduced, and a representation-error term contributes topology-independent diversity. Prior work focused mostly on collapse in single or two-model settings; this result explicitly links diversity contraction to interaction graph structure, and in the symmetric-interaction special case the diversity bound is characterized by algebraic connectivity, with higher connectivity producing stronger collapse. The results are stated as a theorem and corollary with proofs; on CIFAR-10 with models trained on disjoint or partially overlapping data subsets, higher synthetic mixing and denser interaction graphs both reduce diversity, and systems with stronger interaction weights show faster diversity decline.
Perspective
The framework targets retraining systems in which multiple generative models are treated as graph nodes and synthetic-data flows are described by a row-stochastic matrix; it applies to local stability near a fixed point and to diversity at equilibrium rather than providing global existence and uniqueness guarantees. It can be used directly to compare long-term behavior under different interaction topologies and synthetic mixing strengths, for example to identify dominant feedback cycles, assess how connectivity drives homogenization, and support mitigation ideas such as controlling high-gain cycles or allocating real data to high-amplification nodes. The intended audience is researchers and platform operators who want to understand long-term risks in multi-model synthetic-data ecosystems, under the premise that the system does evolve near some fixed point.
The analysis rests on standard local regularity assumptions around a fixed point, whereas practical deep generative models involve finite samples, stochastic optimization and nonconvex objectives, so a gap remains between the theory and large-scale training. The diversity result is an upper bound; sharper characterizations, and how to separate meaningful specialization from instability-induced diversity, remain open questions, and the paper notes that when the fixed point is generally unknown, increased diversity may stem from instability and is hard to quantify. In addition, the experiments validate qualitative trends, and how specific choices of interaction graph and mixing ratio affect the conclusions still needs examination across more topologies and larger-scale settings.