2406.09907
Balance with Memory in Signed Networks via Mittag-Leffler Matrix Functions
Yu Tian, Ernesto Estrada
correcthigh confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves 0 ≤ Kα(G) ≤ 1 and Kα(G)=1 iff G is structurally balanced by decomposing Tr(Eα(A)) into positive/negative closed-walk contributions and invoking complete monotonicity of Eα(−x) for 0<α≤1, see Eq. (4.10)–(4.12) and Theorem 4.7 . The candidate solution reaches the same result via a termwise series inequality Tr(|A|^k)−Tr(A^k)≥0 and, for the equality case, uses a diagonal signature S with A=S|A|S to show trace equality when balanced. Both are correct; the approaches are related but not the same.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper correctly extends walk-based balance indices to the Mittag–Leffler setting and grounds the construction in a fractional nonconservative diffusion model. The main bound and equality condition are sound. A few minor clarifications (explicit positivity of the denominator; optional inclusion of a simple conjugation argument for the equality case) would enhance rigor and readability.