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Stability and Diversity of Networked Self-Consuming Generative Ecosystems

Xiukun Wei, Yang Zhang, Xueru Zhang

arXiv:2610.09409v1cs.LGcs.CY

TL;DR

Prior work on self-consuming models has focused largely on isolated models or simplified two-model interactions, despite synthetic data entering future training pipelines. This paper models multiple interacting models with a directed, weighted graph and analyzes their retraining dynamics. It derives stability conditions and finds topology-dependent amplification and diversity contraction.

  • Problem

    Prior self-consuming-model analyses have largely focused on isolated models or simplified interactions between two models, while synthetic data is increasingly incorporated into future training pipelines.

    Source: §1 · p. 1

  • Method

    The paper models models as nodes in a directed, weighted graph whose edges represent cross-model synthetic-data flows, then analyzes retraining dynamics and fixed points.

    Source: §1 · p. 2, §2 · p. 3

  • Results

    The analysis derives fixed-point stability conditions and finds that cross-model feedback can induce topology-dependent amplification and diversity contraction.

    Source: §1 · p. 2, §6 · p. 9

  • Takeaways & Limitations

    The graph-based dynamical framework provides a first step toward understanding how synthetic-data mixing and interaction topology jointly govern multi-model ecosystem stability and diversity.

    Source: §6 · p. 9

    The analysis studies local dynamics around an arbitrary fixed point when one exists, rather than establishing fixed-point existence globally.

    Source: §3.1 · p. 4

Abstract

from arXiv · show

The widespread deployment of generative AI has made it increasingly difficult to distinguish synthetic content from real data. Consequently, synthetic data is inevitably incorporated into the training pipelines of future model generations, forming a self-consuming training loop. Prior work has studied the effects of such recursive self-consuming training, but analyses have largely been limited to isolated models, where a model consumes only its own synthetic data, or to simplified interactions between two models. This paper takes a first step toward understanding networked self-consuming generative models, in which multiple models consume synthetic data generated by one another through complex interaction pathways. We introduce a theoretical framework representing models as nodes in a directed, weighted graph, with edge weights governing the flow of synthetic data among models. Using this framework, we analyze the long-term behavior of networked models under retraining dynamics, establishing conditions for convergence and characterizing the resulting fixed points. We further investigate how the system's long-term stability and diversity are shaped by each model's access to real data, cross-model data consumption, and the structure of the interaction graph.

1 Introduction

Prior analyses largely cover isolated or simplified two-model systems, leaving the stability and diversity of complex model networks unexplored. This paper models synthetic-data interactions as a directed, weighted graph and analyzes stability, diversity, and empirical behavior across network settings.

1 Introduction

  • Self-consuming training is motivated by synthetic data becoming difficult to distinguish from real data and appearing in future training pipelines, while prior work reports drift, collapse, tail-coverage loss, and possible bias amplification.Examples of cross-model synthetic-data use include LLM training and text-to-image systems.Source: §1 · p. 1
  • Existing analyses focus mainly on isolated models or simplified two-model systems, leaving arbitrary network interactions and their effects on stability and diversity uncharacterized.The paper identifies the need for a framework that accommodates arbitrary interaction graphs and examines how topology and synthetic-data mixing affect system behavior.Source: §1 · p. 2
  • The framework represents models as nodes in a directed interaction graph, with weighted edges capturing cross-model synthetic-data flows.It is used to study retraining dynamics, long-term stability, and the effects of real-data access, cross-model consumption, and graph topology.Source: §1 · p. 2
  • Under standard regularity conditions and assuming a fixed point exists, stability conditions combine local amplification factors with global propagation through arbitrary directed, weighted graphs.The resulting condition generalizes the stability threshold of prior single-model analyses [4] [13].Source: §1 · p. 2
  • At the fixed point, cross-model synthetic-data consumption induces topology-dependent smoothing that contracts inter-model heterogeneity.The analysis considers models with varying real-data distributions.Source: §1 · p. 2
  • Experiments on a synthetic 8-Gaussian dataset and CIFAR-10 align with the theoretical predictions across multiple model classes and interaction topologies.The paper presents these experiments as empirical validation of its analysis.Source: §1 · p. 3

2 Problem Formulation

The formulation allows heterogeneous models with distinct real-data distributions to exchange synthetic data through a directed, weighted interaction graph. It then describes retraining as an iterative process that mixes real data with synthetic samples from connected models.

2 Problem Formulation

  • Models may have heterogeneous architectures and distinct real-data distributions while sharing a data space.Each model is parameterized separately and induces its own generated-data distribution.Source: §2 · p. 3
  • A directed edge from model j to model i indicates that i may consume synthetic data generated by j, and its weight represents the probability of that source being used.Self-loops are allowed, so a model may reuse its own generated data.Source: §2 · p. 3
  • At initialization, each model starts from a pretrained state on its own real data, and subsequent rounds update model parameters through iterative training.The update for each model is represented by an update operator; the synthetic-data mixing weight controls its relative contribution.Source: §2 · p. 3
  • The formulation targets system-level long-term behavior because synthetic data can propagate through complex, heterogeneous pathways and influence multiple models.The paper frames convergence, fixed points, and the effects of graph structure and synthetic-data strength on stability and diversity as its motivating questions.Source: §2 · pp. 3–4

3 Convergence & Stability Analysis

The section analyzes local retraining dynamics around fixed points and gives conditions for stability that combine model-level amplification with network structure. Cycle structure further refines which interaction patterns constrain stability.

3.1 Fixed Point and Technical Assumptions

  • The uncoupled solution is a fixed point when models share homogeneous real-data distributions and their per-model maximum-likelihood estimates induce a common generative distribution.In general multi-model settings, that uncoupled solution need not be a fixed point.Source: §3.1 · p. 4
  • The analysis studies local dynamics around an arbitrary fixed point when one exists, rather than establishing global fixed-point existence for general networks.The coupled retraining map and flexible interaction graph make global existence and characterization more challenging.Source: §3.1 · p. 4
  • The local amplification factor βi quantifies how strongly synthetic-data perturbations are amplified by model i’s retraining dynamics.Its value depends on local curvature and cross-model sensitivity around the fixed point.Source: §3.1 · p. 5

3.2 Convergence of networked self-consuming models.

  • If αi > Liεi for every model and ρ(D) < 1, the fixed point is locally asymptotically stable and nearby iterates converge to it.The comparison matrix D combines local amplification factors with the interaction graph.Source: §3.2 · p. 5, §3.1 · p. 5
  • The sufficient norm condition βmax∥W∥2 < 1 guarantees stability, but can be conservative because it compresses graph structure into a worst-case scalar.The threshold also reflects cross-model sensitivity Mi and graph amplification ∥W∥2; increases in either lower the maximum allowable synthetic-data weight λi.Source: §3.2 · p. 5
  • Every simple directed cycle must have cycle gain below one for ρ(D) < 1; in particular, each self-loop must satisfy βiwii < 1.A dominant cycle can bottleneck the system, and cycle gain reflects average amplification per retraining step around that cycle.Source: §3.2 · p. 6

4 Diversity Analysis

The section measures diversity through model outputs in a shared feature space and characterizes their fixed-point representations. Synthetic-data mixing smooths inter-model differences in a topology-dependent way, while representation error also contributes to output diversity.

4 Diversity Analysis

  • The analysis represents each model by its expected feature-map output and measures system diversity as variance across models’ output representations.This shared representation space enables comparison even when model architectures or parameter dimensions differ.Source: §4 · p. 6
  • At a fixed point, output representations result from applying a linear smoothing operator to real-data centroids, and a bound quantifies attenuation of diversity along the non-consensus subspace.The collapse coefficient maps real-data diversity to final output diversity.Source: §4 · p. 7
  • With no synthetic-data mixing or cross-model edges, the operator does not average across models; if outputs match their training mixtures, output diversity equals real-data diversity.Larger synthetic-data weights increase averaging but can also amplify the contribution from imperfect synthetic training data.Source: §4 · p. 7
  • The smoothing operator reduces disagreements that vary rapidly across strongly connected regions more than differences across weakly connected regions.For undirected interactions with homogeneous synthetic weights and zero feature discrepancy, higher algebraic connectivity gives a smaller diversity upper bound.Source: §4 · p. 7
  • Feature discrepancy contributes a topology-independent term to output diversity, reflecting model error rather than meaningful inter-model disagreement.The discrepancy measures the gap between a model’s fixed-point output representation and its training-mixture representation.Source: §4 · pp. 6–7

5 Experiments

Experiments test whether the theory’s qualitative predictions persist under realistic training conditions. On CIFAR-10, stability and diversity vary with synthetic-data mixing and interaction-graph structure.

Experiments

  • Each retraining round uses generated and mixed training datasets of 10,000 samples; FID measures distributional quality and feature-based D measures cross-model diversity.The study uses DDPM and CFM models on real-data experiments and reports CIFAR-10 results here.Source: §5 · pp. 8–9

Stability experiments

  • Across all systems, increasing synthetic-data mixing raises FID, while neither interaction norm nor graph structure alone determines stability.The spectral-radius criterion combines both ingredients.Source: §5 · p. 9
  • With real data present, shorter interaction cycles can mitigate FID degradation, unlike synthetic settings where short cycles may accelerate instability.The paper attributes instability to which models participate in dominant feedback paths rather than cycle length alone.Source: §5 · p. 9

Diversity experiments

  • Increasing α and using denser interaction graphs generally reduce diversity, while isolated retraining is an exception because instability can drive its diversity.For System 1, diversity does not decrease with α.Source: §5 · p. 9
  • Systems 2 and 3 share topology but differ in edge weights; System 3 shows a faster, more pronounced decline in diversity, indicating interaction strength matters.Systems 2 and 3 also show faster diversity decay than System 4 across most α > 0.Source: §5 · p. 9

Experiments

  • Stability experiments use full CIFAR-10 for every model, whereas diversity experiments use disjoint or partially overlapping subsets to represent differing data access.The four designed graphs support controlled comparisons of structure, edge weights, and cycle configuration.Source: §5 · pp. 8–9, Table 1 · p. 8

6 Conclusion

The paper’s graph-based dynamical framework examines how synthetic-data mixing and interaction topology jointly shape networked self-consuming models. It identifies topology-dependent amplification of instability and contraction of diversity.

6 Conclusion

  • A graph-based dynamical framework models networked self-consuming generative systems and their long-term behavior.Source: §6 · p. 9
  • Synthetic-data mixing and interaction topology jointly govern system stability and diversity.Source: §6 · p. 9
  • Cross-model feedback can produce topology-dependent amplification and diversity contraction, clarifying long-term behavior in multi-model ecosystems.Source: §6 · p. 9

A Limitations and Future Directions

The paper identifies limits in prior self-consuming-model analyses and outlines the scope and implications of its networked framework. Its results are local and conditional, while several broader guarantees and sharper diversity analyses remain future directions.

A Limitations and Future Directions

  • Stability results are local around an existing fixed point; global existence and uniqueness for heterogeneous, nonconvex retraining systems remain future directions.The analysis also relies on local regularity assumptions, while practical deep generative models involve finite-sample data, stochastic optimization, and nonconvex objectives.Source: A Limitations and Future Directions · p. 14

B Broader Impact

  • The framework could inform diagnosis and mitigation of long-term risks, but its analysis could also be misused to steer ecosystems toward narrower or biased outputs.The paper calls for transparent data tracing, flow audits, and evaluation of downstream effects across user groups and content domains.Source: B Broader Impact · p. 14

C.1 Single-model Self-Consuming Loop

  • Single-model studies report recursive distribution drift and collapse, while this framework recovers that setting when each model consumes only its own outputs.The cited studies describe accumulated approximation errors and loss of distribution tails, and divergence without sufficient fresh real data.Source: §C.1 · p. 15

C.2 Multi-Model Interactive Ecosystems

  • Prior multi-model analyses mostly examine small model pairs or simplified interactions, leaving larger, heterogeneous network behavior underexplored.This work models synthetic-data flows across models and analyzes factors including synthetic-data ratio, topology density, and model heterogeneity.Source: §C.2 · pp. 15–16

C.3 Mitigation Strategies for Model Collapse

  • The study finds that real data stabilizes retraining against distributional drift, and says existing mitigation strategies could be incorporated into its framework.The cited approaches include retaining real data, curation or verification, and methods that filter or correct generated samples.Source: §C.3 · p. 16

D.1 Proofs of Characterization of Fixed Points

  • Under α_i > L_iε_i, the appendix establishes local uniqueness of model updates and characterizes fixed points through each model’s stationarity condition.The fixed-point characterization follows from the local objective having a unique local maximizer under the stated condition.Source: §D.1 · p. 17

D.2 Proof of the Uncoupled Solution as a Fixed Point

  • With homogeneous real-data distributions and a shared target distribution, the common target parameters form a fixed point for any nonnegative synthetic weights and row-stochastic interaction matrix.This is the appendix’s uncoupled-solution fixed-point result.Source: §D.2 · p. 17

D.3 Example: Heterogeneous Fixed Point Different from the Uncoupled Solution

  • In a two-model example with heterogeneous data, the coupled dynamics have a locally stable fixed point that differs from the uncoupled solution.For the stated example, the Jacobian spectral radius is less than one.Source: §D.3 · p. 18

D.4 Proofs of Theorem 3.7

The appendix bounds the local Jacobian by a graph-weighted matrix and uses its spectral radius to establish local stability and convergence. It also gives a sufficient norm-based criterion and a necessary constraint on directed-cycle gains.

D.4 Proofs of Theorem 3.7

  • If ρ(D) < 1 under α_i > L_iε_i, a fixed point is locally asymptotically stable and nearby retraining iterates converge to it at an asymptotic linear rate.The proof bounds Jacobian blocks by β_iw_ij and shows ρ(J) ≤ ρ(D), with D = diag(β_1, …, β_K)W.Source: §D.4 · pp. 19–20

D.5 Proofs of Corollary 3.8

  • β_max∥W∥_2 < 1 is a sufficient condition for local asymptotic stability; the appendix also recovers a single-model stability threshold when K = 1.The sufficient condition implies ρ(D) < 1, while the single-model case reduces to λ_1 < α_1/(2L_1ε_1).Source: §D.5 · p. 21

D.6 Proofs of Proposition 3.10 and Corollary 3.11

  • If ρ(D) < 1, every simple directed cycle must have gain below one, including each self-loop.The self-loop condition is β_iw_ii < 1.Source: §D.6 · p. 23

D.7 Proofs of Proposition 4.4

The appendix represents fixed-point model outputs through real-data representations and a residual term, then bounds system diversity under additional assumptions. For symmetric interactions, homogeneous synthetic weight, and zero feature discrepancy, the bound depends on graph connectivity.

D.7 Proofs of Proposition 4.4

  • At a fixed point, each model’s output representation decomposes into a real-data contribution, a weighted average of other models’ representations, and a residual.The matrix form uses M = I + Λ(I − W), which is invertible under the stated row-stochastic interaction structure.Source: §D.7 · pp. 23, 25

D.8 Proofs of Theorem 4.5 and Corollary 4.6

  • For an undirected graph, homogeneous synthetic weight, and zero feature discrepancy, output diversity is bounded by data diversity scaled by 1/(1 + λμ_2)^2.Here μ_2 is the algebraic connectivity, the second-smallest eigenvalue of L.Source: §D.8 · pp. 26–27

E Closed-Form Spectral Validation of the Networked Dynamics

An exactly analyzable four-node system validates the Jacobian spectral criterion and shows that topology-aware quantities distinguish dynamics that interaction norms alone cannot. Complementary synthetic-Gaussian experiments provide a low-dimensional illustration alongside the large-scale image experiments.

E Closed-Form Spectral Validation of the Networked Dynamics

  • The controlled affine system has an exact fixed point and constant Jacobian, making its asymptotic convergence factor directly measurable as ρ(J).It uses four nodes with heterogeneous local sensitivities and isolated, ring, and fully connected topologies.Source: E Closed-Form Spectral Validation of… · pp. 27–28
  • The numerical validation initializes near the exact fixed point, runs 250 iterations, and estimates the asymptotic contraction rate from the latter half of each trajectory.The controlled system complements neural-model experiments under finite data, imperfect optimization, and model misspecification.Source: E Closed-Form Spectral Validation of… · p. 28
  • The empirical convergence rate matches ρ(J) to three decimal places in every system, and ρ(D) upper-bounds ρ(J).This directly verifies the spectral criterion in the controlled system.Source: E Closed-Form Spectral Validation of… · p. 28, Table 2 · p. 28
  • With ∥W∥2 = 1 in all four systems, exact convergence factors range from 0.582 to 0.900, so interaction norm alone cannot distinguish their dynamics.Topology-aware spectral quantities capture how heterogeneous local sensitivities propagate through the network.Source: E Closed-Form Spectral Validation of… · p. 28
  • In dense System 4, the sufficient threshold is 0.556 versus the exact Jacobian threshold 0.833, while the bound is tight in isolated System 1.The comparison-matrix certificate is topology-aware but can be conservative.Source: E Closed-Form Spectral Validation of… · p. 28

F Experiments with Synthetic Gaussian Data

  • The synthetic-Gaussian experiments offer a low-dimensional empirical check using two-dimensional mixtures of eight isotropic Gaussian modes.Homogeneous real-data assignments test stability, while heterogeneous mode subsets test diversity.Source: F Experiments with Synthetic Gaussian… · p. 29
  • The Gaussian study uses DDPM, CFM, and GAN models, warm-starts each on its real-data distribution, then retrains on fixed-size real/synthetic mixtures.Each round uses 10,000 training samples; results average three random seeds, and diversity is measured by system-level D.Source: F Experiments with Synthetic Gaussian… · p. 29
  • Figures 4 and 5 report Wasserstein distance across interaction graphs and synthetic mixing strengths for the Gaussian stability experiments.Figure 4 maps each graph's data flows alongside model-level distances to real samples.Source: Figure 4 · p. 30, Figure 5 · p. 31

F.1 Stability Experimental Results

The homogeneous-data experiments show that stability depends jointly on synthetic mixing, interaction structure, and model-specific stability. Graph topology affects how local instability is amplified or contained, beyond what interaction norms alone capture.

F.1 Stability Experimental Results

  • Model C has weaker baseline stability, and its direct participation in high-gain cycles in System 5 accompanies higher WD and variance across models than in System 6.The effect is attenuated in System 6, where Model C is less involved in the interaction cycles.Source: §F.1 · p. 31
  • In isolated System 1, increasing α consistently raises WD for every model, most strongly for Model C.This matches the predicted amplification of local instability under stronger synthetic mixing without cross-model interactions.Source: §F.1 · p. 31
  • System 5 has the highest WD mean and variance despite not having the longest cycle, consistent with stability depending on dominant cycle gain rather than cycle length alone.The comparison is among Systems 1, 3, and 5 at the same synthetic mixing strength.Source: §F.1 · p. 31
  • System 3 shows faster WD growth and greater variance than System 2 at the same α, although only System 2 satisfies the norm-based stability condition.Differences among Systems 1, 2, and 4 also show that norm-based conditions are conservative.Source: §F.1 · p. 31
  • Stability depends jointly on synthetic mixing strength, interaction structure, and unstable models; topology shapes amplification beyond interaction norms alone.The experiments use eight interaction graphs across four models trained on the same real-data target.Source: §F.1 · p. 31

F.2 Diversity Experimental Results

With heterogeneous real-data access, stronger synthetic mixing generally reduces cross-model diversity and encourages homogenization. However, elevated diversity can also reflect unstable collapse rather than meaningful differentiation, and topology and unstable-model participation shape the outcome.

F.2 Diversity Experimental Results

  • System-level D tracks diversity across retraining rounds, while generated samples provide qualitative views of stability and diversity across heterogeneous-data graphs.Figure 8 varies synthetic mixing strength; Figures 9 and 10 show samples at α values of 1.0 and 0.2.Source: §F.2 · p. 31, Figure 8 · p. 34, Figure 9 · p. 35
  • Across interaction graphs, increasing α generally reduces diversity as models align despite heterogeneous real-data access.When synthetic data dominates training, this alignment results in homogenization.Source: §F.2 · p. 34
  • In nearly isolated System 1, large α can raise diversity through independent model collapse rather than meaningful differentiation.Large run-to-run variance supports this interpretation; the theory characterizes stability, not semantic diversity quality.Source: §F.2 · p. 34
  • Diversity collapses rapidly in densely connected System 4 even at moderate α, while Systems 2 and 4 behave differently despite similar interaction norms.The contrast indicates structural effects beyond sufficient but conservative norm-based conditions.Source: §F.2 · p. 34
  • When weakly stable Model C participates in System 5 cycles, elevated diversity accompanies large variance and system-level instability; the effects are attenuated when it is weakly coupled or excluded from dominant cycles.The observations are consistent with a single unstable node dominating global behavior when embedded in dominant interaction cycles.Source: §F.2 · p. 34

G Additional Results on CIFAR-10 Datasets

The CIFAR-10 appendix adds precision and recall to characterize image-generation quality and describes model variants with distinct training objectives. It also presents qualitative samples at specified mixing strengths for heterogeneous-data settings.

G Additional Results on CIFAR-10 Datasets

  • Precision and recall complement FID by distinguishing reduced sample fidelity from loss of distributional coverage.A precision drop suggests less realistic or less data-aligned images, while a recall drop indicates fewer covered modes or classes.Source: G Additional Results on CIFAR-10… · p. 37
  • The CIFAR-10 models include VLB and Hybrid diffusion variants with shared UNet backbones but different objectives, plus OT-CFM and iCFM variants with different flow-matching couplings.The diffusion objectives differ in likelihood and regression components; the flow variants differ in how intermediate targets and probability mass are coupled.Source: G Additional Results on CIFAR-10… · p. 37, Table 5 · p. 37
  • The image experiments use publicly released pretrained checkpoints and retain each model's original training configuration while changing training steps per round.Dataset specifications and step counts are provided in Table 5.Source: G Additional Results on CIFAR-10… · p. 37, Table 5 · p. 37
  • Additional qualitative samples are shown at r = 10 for α = 1.0 and α = 0.2 to illustrate sample quality under different synthetic mixing levels.The stability figures also show graph flows and model-level precision and recall across mixing strengths.Source: G Additional Results on CIFAR-10… · p. 37, Figure 11 · p. 38, Figure 12 · p. 38

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