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arxiv: 2604.12685 · v1 · submitted 2026-04-14 · ⚛️ physics.soc-ph · cs.SY· eess.SY· math.OC

Signed DeGroot-Friedkin Dynamics with Interdependent Topics

Yangyang Luan , Muhammad Ahsan Razaq , Xiaoqun Wu , Claudio Altafini This is my paper

Pith reviewed 2026-05-10 14:25 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cs.SYeess.SYmath.OC
keywords DeGroot-Friedkin dynamicssigned influence networkssocial powerinterdependent topicslogic matricesdominant left eigenvectormulti-topic dynamicsglobal convergence
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The pith

When agents' logic matrices share a common dominant left eigenvector, signed multi-topic DeGroot-Friedkin dynamics reduce exactly to a scalar map whose limits fall into pluralistic, mixed, or vertex-dominant social power types.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how signed interpersonal repulsions and cross-topic self-appraisals together shape the evolution of agent social power in networks with interdependent topics. It establishes that a single structural property—all agents' logic matrices sharing the same dominant left eigenvector—makes the full multi-dimensional dynamics collapse into an ordinary one-dimensional DeGroot-Friedkin iteration. Under this condition the system always converges globally to one of three explicitly characterized limit configurations, and the ordering of powers by interaction centrality is preserved in two of the three cases. The reduction also remains locally stable under small heterogeneous changes to the logic matrices, while the paper separately describes how the picture changes when the shared-eigenvector property is absent.

Core claim

The central claim is that the signed multi-topic dynamics admit an exact reduction to an explicit scalar DeGroot-Friedkin map precisely when every agent's logic matrix possesses the same dominant left eigenvector. This reduction supplies a complete classification of the limiting social-power vectors into pluralistic, mixed, and vertex-dominant families. Global convergence holds for all three families, and the interaction-centrality ordering is preserved in the pluralistic and mixed families. The paper further establishes local robustness under small perturbations of the logic matrices and clarifies the loss of the reduction and classification when the common-eigenvector condition fails.

What carries the argument

The common dominant left eigenvector shared by all agents' logic matrices; it supplies the exact coordinate change that collapses the vector-valued signed dynamics onto a scalar DeGroot-Friedkin map.

If this is right

  • Social-power limits are always one of the three classified types: pluralistic, mixed, or vertex-dominant.
  • Global convergence to the limit occurs from any initial condition in all three cases.
  • In pluralistic and mixed limits the relative ordering of agents' social powers matches their ordering by interaction centrality.
  • Small heterogeneous perturbations of the logic matrices leave the local behavior unchanged.
  • When the common-eigenvector condition is lost the exact scalar reduction and the three-type classification cease to hold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • In applied settings, verifying the shared-eigenvector property from topic-interdependence surveys could serve as a practical test for whether power formation will simplify to the scalar case.
  • The three-type classification offers a concrete prediction that could be checked against longitudinal data on multi-issue opinion leadership.
  • The reduction technique may generalize to other signed multi-dimensional network models once an analogous common structural vector is identified.
  • Numerical exploration of the non-shared-eigenvector regime could reveal whether approximate reductions or different convergence behaviors appear.

Load-bearing premise

All agents' logic matrices for topic interdependence must share exactly the same dominant left eigenvector.

What would settle it

A numerical simulation or real-world measurement of the full multi-topic signed dynamics on a small network whose logic matrices lack a shared dominant left eigenvector, checking whether the social-power trajectories still follow the predicted scalar map and converge to one of the three classified limits.

Figures

Figures reproduced from arXiv: 2604.12685 by Claudio Altafini, Muhammad Ahsan Razaq, Xiaoqun Wu, Yangyang Luan.

Figure 1
Figure 1. Figure 1: Illustration of equilibrium branches for ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical illustration of the baseline dynamics and perturbation response. Panels (a)–(c): Evolution of self-weights [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
read the original abstract

This paper investigates DeGroot-Friedkin (DF) dynamics over signed influence networks with interdependent topics. We propose a multi-topic signed framework that combines repelling interpersonal interactions with cross-issue self-appraisal, examining how antagonism and topic interdependence shape the evolution of agent-level social power. When the logic matrices (for topic interdependence) of all agents share a common dominant left eigenvector, we identify structural conditions under which the original dynamics admit an exact reduction to an explicit scalar DF map. This yields a complete classification of limiting social power configurations into pluralistic, mixed, and vertex-dominant types. In all three cases, the dynamics are globally convergent, and in the first two the ordering induced by the interaction centrality is preserved. We further show local robustness under small heterogeneous perturbations of the logic matrices. We also clarify what changes when this common-eigenvector structure is lost. These results extend signed social power dynamics beyond the standard nonnegative scalar setting and shed light on the robustness and scope of centrality-based social power formation in multi-topic signed influence systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. This paper investigates DeGroot-Friedkin dynamics over signed influence networks with interdependent topics. It proposes a multi-topic signed framework combining repelling interpersonal interactions with cross-issue self-appraisal. When all agents' logic matrices share a common dominant left eigenvector, the dynamics admit an exact reduction to an explicit scalar DF map. This enables a complete classification of limiting social power configurations into pluralistic, mixed, and vertex-dominant types. The paper proves global convergence in all cases, with ordering preservation (induced by interaction centrality) in the first two types. It also establishes local robustness under small heterogeneous perturbations of the logic matrices and clarifies the changes when the common-eigenvector condition fails.

Significance. If the reduction and classification hold, this work significantly extends signed social power dynamics beyond the standard nonnegative scalar setting to multi-topic interdependent systems. The exact reduction to a scalar DF map, the exhaustive classification into three limiting types, the global convergence proofs, and the ordering-preservation results provide a clean theoretical framework. The local robustness analysis and discussion of the condition's failure further clarify the scope and applicability of centrality-based social power formation in signed networks with topic interdependence. These are substantive contributions to opinion dynamics on signed graphs.

minor comments (1)
  1. The abstract states the common-eigenvector condition clearly but does not indicate the dimension or structure of the logic matrices; a brief parenthetical note on their size (e.g., number of topics) would improve immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. We are pleased that the referee finds the reduction to scalar DF maps, the classification of limiting configurations, the global convergence results, and the robustness analysis to be substantive contributions.

Circularity Check

0 steps flagged

No significant circularity; minor self-citation not load-bearing

full rationale

The derivation proceeds from the structural hypothesis that all agents' logic matrices share a common dominant left eigenvector, under which an exact reduction to a scalar DF map is shown via matrix algebra and eigenvector properties. This yields the classification into pluralistic/mixed/vertex-dominant types and global convergence proofs, all of which are self-contained mathematical consequences rather than redefinitions or fitted quantities renamed as predictions. No load-bearing step reduces by construction to prior fitted inputs or to a self-citation chain that itself lacks independent verification; the common-eigenvector condition is explicitly stated as an assumption whose failure is also analyzed. Normal citations to prior DF literature (including possible self-citations by Altafini) provide context but do not carry the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard signed-graph and matrix assumptions plus one key domain assumption that enables the reduction; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Logic matrices of all agents share a common dominant left eigenvector
    This condition is explicitly required for the exact reduction to scalar DF dynamics and the classification of limiting configurations.

pith-pipeline@v0.9.0 · 5498 in / 1299 out tokens · 45757 ms · 2026-05-10T14:25:07.384746+00:00 · methodology

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