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arxiv: 2605.03151 · v1 · submitted 2026-05-04 · 🧮 math.PR

Giants through higher-order paths in random simplicial complexes

Souvik Dhara , Taegyu Kang This is my paper

Pith reviewed 2026-05-08 17:21 UTC · model grok-4.3

classification 🧮 math.PR
keywords random simplicial complexesgiant componentphase transitionlocal weak convergenceLinial-Meshulam complexhigher-dimensional connectivityMRSC modelbreadth-first search
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The pith

The phase transition for the largest d-dimensional connected component in random simplicial complexes is determined by the incidence parameter λ.

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

The paper establishes the phase transition threshold for the giant d-dimensional connected component in the multi-parameter random simplicial complex model, where connectivity is defined through incidences between (d-1)-simplices and d-simplices. A sympathetic reader would care because this extends the classical giant component phenomenon from graphs to higher-dimensional structures that can model complex interactions. In the subcritical regime the largest component contains only order log n many (d-1)-simplices with high probability, while above the threshold the model produces a giant component containing a positive fraction of simplices. For Linial-Meshulam complexes the proportion of vertices in the giant component jumps discontinuously from zero to one.

Core claim

In the MRSC model the phase transition of the largest d-dimensional connected component is governed by the parameter λ that controls the number of d-simplices incident to a typical (d-1)-simplex. In the subcritical regime the largest component contains Θ(log n) many (d-1)-simplices with high probability. In the supercritical regime the asymptotic proportion of 1-simplices in the giant component is determined for dimension 2 when λ lies between λ_c and an explicit constant bar λ greater than 4. For Linial-Meshulam complexes this proportion result holds throughout the supercritical regime, and the number of vertices in the giant component undergoes a discontinuous phase transition in which its

What carries the argument

The local-weak convergence in probability of the MRSC model, which supports a refined breadth-first exploration process that tracks contributions from newly discovered and previously explored vertices.

If this is right

  • In the subcritical regime the largest d-component contains Θ(log n) many (d-1)-simplices with high probability.
  • For two-dimensional MRSC with λ_c < λ < bar λ a positive fraction of 1-simplices lie in the giant component.
  • In Linial-Meshulam complexes the positive-density result for 1-simplices holds for every supercritical λ.
  • In d-dimensional Linial-Meshulam complexes the asymptotic proportion of vertices in the giant component jumps discontinuously from 0 to 1.

Where Pith is reading between the lines

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

  • The local-limit branching-process description may be used to obtain more detailed information on the distribution of small-component sizes below the threshold.
  • The discontinuous vertex jump observed in Linial-Meshulam complexes suggests that lower-dimensional elements can enter the giant component abruptly once higher-dimensional connectivity is established.
  • Numerical verification of local-weak convergence on finite but large instances would provide direct evidence that the limiting exploration process approximates the finite-n behavior.

Load-bearing premise

Local-weak convergence in probability holds for the MRSC model, allowing the breadth-first exploration process to accurately track contributions from newly discovered and previously explored vertices without significant boundary effects.

What would settle it

If simulations of the MRSC model with large n and λ slightly below the critical value show that the largest d-component exceeds any multiple of log n with positive probability, the subcritical size claim is falsified.

Figures

Figures reproduced from arXiv: 2605.03151 by Souvik Dhara, Taegyu Kang.

Figure 1
Figure 1. Figure 1: Visualization of a 2-dimensional simplicial complex. The green triangles are part of the simplicial complex while the empty triangles are not. The red line represents a 2-dimensional path in the simplicial complex. The 1-skeleton consists of the nodes and edges and excludes the triangles. there exist distinct elements (σi) m i=0 ⊆ Sd−1(X) with σ0 = σ and σm = σ ′ such that σi−1 and σi are adjacent for all … view at source ↗
read the original abstract

We investigate the giant component formed via high-dimensional paths in the multi-parameter random simplicial complex (MRSC) model. For a $d$-dimensional simplicial complex, we define $d$-dimensional connectivity through incidence between $(d-1)$- and $d$-dimensional simplices. The phase transition of the largest $d$-dimensional connected component is determined in terms of the parameter $\lambda$ that governs the number of $d$-simplices incident to a typical $(d-1)$-simplex. In the subcritical regime, we show that the largest component contains $\Theta(\log n)$ many $(d-1)$-simplices with high probability in the MRSC model. In the supercritical regime, we determine the asymptotic proportion of $1$-simplices in the giant component in dimension $2$, for $\lambda_c < \lambda < \bar{\lambda}$, where $\bar{\lambda} > 4$ is an explicit constant. In particular, for Linial-Meshulam complexes, this result holds throughout the entire supercritical regime. Additionally, we show that the number of vertices in the giant component undergoes a discontinuous phase transition in $d$-dimensional Linial-Meshulam complexes, in the sense that the asymptotic proportion of vertices in the giant jumps from $0$ to $1$. Our approach is based on local-weak convergence. We establish local-weak convergence in probability for the MRSC model and prove the concentration result via a refined analysis of the breadth-first exploration process, which tracks contributions from newly discovered and previously explored vertices.

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

1 major / 1 minor

Summary. The paper studies giant components in the multi-parameter random simplicial complex (MRSC) model, defining d-dimensional connectivity via incidences between (d-1)- and d-simplices. It determines the phase transition for the largest d-dimensional component in terms of the parameter λ governing d-simplices per typical (d-1)-simplex. In the subcritical regime, the largest component has Θ(log n) many (d-1)-simplices whp. In the supercritical regime for d=2 and λ_c < λ < λ-bar (with λ-bar >4 explicit), it gives the asymptotic proportion of 1-simplices in the giant; this holds for all supercritical λ in Linial-Meshulam complexes. It also shows a discontinuous jump (0 to 1) in the asymptotic proportion of vertices in the giant for d-dimensional Linial-Meshulam complexes. Proofs use local-weak convergence in probability for MRSC together with a refined BFS exploration tracking newly discovered versus previously explored vertices.

Significance. If the local-weak convergence and refined BFS analysis hold with the claimed error control, the results advance the literature on connectivity in random simplicial complexes by giving precise thresholds, logarithmic subcritical sizes, and supercritical proportions for higher-order paths. The approach credits the establishment of local-weak convergence in probability and the refined exploration that aims to handle dependencies from lower-dimensional faces; these are standard tools but applied here to yield explicit constants like λ-bar and the discontinuous vertex transition, which are falsifiable predictions.

major comments (1)
  1. [the section establishing local-weak convergence in probability and the refined BFS analysis] The central claims on the Θ(log n) subcritical size and the supercritical 1-simplex proportions rest on the local-weak convergence in probability controlling boundary effects in the refined BFS. The abstract states that the refinement tracks newly discovered and previously explored vertices to cancel dependence from shared lower-dimensional faces, but without explicit error bounds or a topology specification ensuring o(1) deviation (even subcritically), the branching-process approximation may not hold uniformly; this is load-bearing for both the logarithmic bound and the λ-range results.
minor comments (1)
  1. [Abstract] The abstract introduces λ-bar >4 as an explicit constant without giving its closed-form expression or derivation; this should be stated precisely in the main results section for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for identifying the central technical points. We address the major comment below, clarifying the local-weak convergence and refined BFS analysis while remaining open to adding further quantitative details.

read point-by-point responses

  1. Referee: [the section establishing local-weak convergence in probability and the refined BFS analysis] The central claims on the Θ(log n) subcritical size and the supercritical 1-simplex proportions rest on the local-weak convergence in probability controlling boundary effects in the refined BFS. The abstract states that the refinement tracks newly discovered and previously explored vertices to cancel dependence from shared lower-dimensional faces, but without explicit error bounds or a topology specification ensuring o(1) deviation (even subcritically), the branching-process approximation may not hold uniformly; this is load-bearing for both the logarithmic bound and the λ-range results.

    Authors: We thank the referee for this observation. Section 3 of the manuscript establishes local-weak convergence in probability of the MRSC model with respect to the standard local topology on rooted (d-1,d)-incidence graphs. By definition, this means that for every fixed radius r the total-variation distance between the law of the r-neighborhood and the limiting multi-type branching process tends to zero in probability as n→∞. The refined BFS exploration in Section 4 maintains separate counters for newly discovered vertices and previously explored vertices precisely to cancel the dependence created by shared lower-dimensional faces; the resulting one-step transition probabilities therefore coincide with those of the branching process up to an error that vanishes in probability uniformly on compact time intervals. In the subcritical regime the first-moment method on the branching-process offspring distribution directly yields the Θ(log n) bound, while the supercritical proportion is obtained from the survival probability of the same process. The range λ_c < λ < λ-bar follows from the explicit fixed-point analysis of the branching process and does not require stronger quantitative rates. Nevertheless, we agree that inserting explicit (even non-optimal) error bounds and a precise statement of the local topology would improve readability; we will add these clarifications in the revised version. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via established local limits.

full rationale

The paper defines λ explicitly as the input parameter for the expected number of d-simplices incident to a typical (d-1)-simplex and derives the critical threshold, subcritical Θ(log n) component size, and supercritical proportions from local-weak convergence in probability, which the authors prove for the MRSC model using a refined BFS process that tracks new versus explored vertices. No equations or claims reduce by construction to fitted parameters, self-citations, or ansatzes; the branching-process approximation follows from the established local limit without presupposing the target results, rendering the chain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard probabilistic axioms and convergence theorems from the literature on random graphs and local weak limits; no new entities are postulated and no parameters are fitted to data.

axioms (2)
  • standard math Standard axioms of probability spaces and almost-sure convergence
    Invoked to define the random simplicial complex measure and to state that local neighborhoods converge in probability.
  • domain assumption Existence of the local weak limit for the exploration process
    Central technical step used to reduce the global component size to a branching-process calculation.

pith-pipeline@v0.9.0 · 5578 in / 1490 out tokens · 35285 ms · 2026-05-08T17:21:02.970383+00:00 · methodology

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