{"id":"fc0c6e8b-b156-4ce3-8aff-ffeb3a30d133","arxiv_id":"2607.23091","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On infinite weighted graphs, porous-medium solutions extinguish in finite time in the fast-diffusion range m < 2/ν, smooth into every ℓ^q for m > 2/ν, and satisfy an exact mass balance under stochastic completeness at infinity.","lead":"This paper proves that nonlinear diffusion on infinite network graphs is well-posed and that solutions either extinguish in finite time or smooth out, depending on a critical exponent; it also proves an exact mass-balance law when absorption (killing) is present. It matters because it extends the qualitative theory of porous-medium and fast-diffusion equations from Euclidean space and manifolds to graphs, including non-locally finite networks, recovering the same critical exp","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Extinction and mass-balance theorems are conditional on the ℓ1-mild solution imported from companion [10]; if its m-accretive realization or finite-resolvent convergence fails on non-locally finite graphs with unbounded killing, Theorems 4.6, 4.9, and 5.5 lose their well-defined solution object.","rationale":"The reader's weakest assumption identifies exactly the load-bearing point: the ℓ1-mild solution, its uniqueness, and the finite-resolvent approximation are imported from the companion preprint [10]. I traced the main theorems to this import: Theorem 4.1, Lemma 4.4(b), and Remark 5.4 all explicitly depend on [10]; Lemma 4.5 and Theorems 4.6, 4.9, and 5.5 then operate on this imported object. I found no internal algebraic or analytical error in the in-paper arguments: the exhaustion construction of Section 3 is self-contained and plausible, the energy inequality algebra in Lemma 4.5 checks out, the scalar comparison lemma A.2 is sound, and the mass-balance proof in Theorem 5.5 is coherent given Lemma 5.3 and the resolvent properties imported from [10]. The concern is therefore not that a displayed estimate is miscomputed, but that the entire framework of Sections 4 and 5 is conditional on results that are not verified in the manuscript. This is precisely the kind of structural dependence that justifies a conditional verdict rather than acceptance: the claims may be correct, but the reader cannot certify them without independent verification of [10]. The proposed concrete test targets the most failure-prone part of that dependency: finite-resolvent convergence and exhaustion-independence in a non-locally finite, unbounded-killing setting, which is exactly the advertised generality of the paper.","tokens_in":49262,"tokens_out":19529,"duration_ms":194586,"concrete_test":"Independently verify the key imported claims for a non-locally finite graph with unbounded killing, e.g., X=N∪{o}, w(o,n)=n^{-2}, κ(n)=n, µ≡1, with the porous nonlinearity φ(s)=s|s|^{m-1}. Fix an exhaustion by finite connected sets and prove that the zero-Dirichlet resolvent solutions (id+λ∆_{D,n}Φ)u_n = π_n g converge in ℓ1 to a limit Jλg that is independent of the exhaustion, order-preserving, and solves the pointwise resolvent equation for the maximal operator L. If this convergence or independence fails, Theorem 4.1 has no foundation and Theorems 4.6, 4.9, and 5.5 are unsubstantiated. If it holds for this example and for a second example with locally finite but unbounded degree, the main concern is materially weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims are all proved for the 'ℓ1-mild solution', but that object is not constructed in this paper. Theorem 4.1 (existence, uniqueness, positivity, stability) is imported from [10, Theorems 3.11 and 4.6]; Lemma 4.4(b) (finite zero-Dirichlet resolvents converge in ℓ1 to Jλ) is imported from [10, Lemma 3.8]; and Remark 5.4 again routes the choice of the operator A to [10, Theorem 3.11]. Since [10] is an unpublished companion preprint by overlapping authors, Sections 4 and 5 rest on non-independently-verified foundations. The load-bearing point is structural, not a local derivation error: if the m-accretive restriction A is not unique, or if the finite-resolvent convergence fails for non-locally finite graphs or with unbounded killing, then Theorems 4.6, 4.9, and 5.5 are statements about an object that may depend on arbitrary choices or may not exist. The in-paper proofs of Lemma 4.5, Theorem 4.6, Theorem 4.9, and Theorem 5.5 appear internally consistent conditional on these imported inputs, but they cannot rescue the solution concept itself. The manuscript explicitly flags this reliance (Remarks 4.2 and 5.4), yet the conditionality is not resolved within the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the generalized porous medium equation (GPME) on infinite weighted graphs, with emphasis on the case where the nonlinearity is the signed power φ(s)=s|s|^{m-1}. In Section 3, for arbitrary bounded initial data and continuous ℓ∞-valued forcing, the authors use finite-subgraph comparison principles and a monotone exhaustion argument to construct lower and upper extremal global pointwise solutions; under bounded degree and local Lipschitz nonlinearity they obtain uniqueness in ℓ∞. In Section 4, assuming a ν-Sobolev inequality, they derive energy estimates for ℓ1-mild solutions and obtain quantitative finite-time extinction for 0<m<2/ν and ℓ1-ℓq smoothing for m>2/ν, recovering the Euclidean critical exponent on Cayley graphs of polynomial growth. In Section 5, under stochastic completeness at infinity, they prove an exact generalized mass balance with arbitrary killing term, reducing to conservation of mass when κ=0, for mild solutions and for suitable classical and bounded pointwise solutions. The main results are Theorems 4.6, 4.9, and 5.5.","tokens_in":49561,"tokens_out":20079,"duration_ms":196230,"significance":"If the underlying mild-solution theory is sound, the paper delivers genuinely quantitative extinction and smoothing estimates with explicit constants, and an exact mass-balance law allowing arbitrary killing and non-locally finite graphs. The energy estimate in Lemma 4.5 is carefully derived, the exponent algebra is consistent, and the limit passages in Theorem 5.5 are handled with appropriate care. The recovery of the Euclidean critical exponent m_c=(N-2)/N on Cayley graphs of polynomial growth is a strong and appealing feature. No parameters are fitted: the critical exponent emerges from the Sobolev exponent and the estimates are explicit. However, the existence, uniqueness, and approximation of the ℓ1-mild solution — the very object on which Theorems 4.6, 4.9, and 5.5 act — are imported entirely from the unpublished companion preprint [10]. The in-paper derivations are internally consistent conditional on that import, but the contribution is not self-contained and the central theorems inherit any potential defect in [10].","major_comments":[{"comment":"The ℓ1-mild solution used throughout Sections 4 and 5 is not constructed in this paper. Theorem 4.1 is imported from [10, Theorems 3.11 and 4.6], Lemma 4.4(b) from [10, Lemma 3.8], and Remark 5.4 again selects the operator A via [10, Theorem 3.11]. Consequently Theorems 4.6, 4.9, 5.5 and Corollary 5.7 are statements about an object whose existence and approximation properties are assumed from an unpublished companion preprint by overlapping authors. This is load-bearing. Please either include the necessary statements and proofs in an appendix, update to a published reference, or explicitly reformulate the theorems as conditional on the construction in [10].","section":"§4.1, Theorem 4.1 and Lemma 4.4; Remarks 4.2 and 5.4"},{"comment":"The uniqueness statement in Theorem 4.1 is only uniqueness for the selected m-accretive restriction A, not uniqueness for the formal Cauchy problem as defined in Definition 2.9. Since A⊆L, an ε-approximate solution for A is also an ε-approximate solution for L, so the theorem does not rule out different limits arising from different choices of A. Remark 5.4 acknowledges that A need not coincide with L and that the resolvent limit is independent of the exhaustion only for nonnegative data. Theorems 4.6 and 4.9 allow signed initial data, so 'the mild solution' is not shown to be independent of the choice of A. Please clarify how A is selected and state/prove the needed independence, or restrict the theorems accordingly.","section":"Definition 2.9; Theorem 4.1; Remark 5.4"}],"minor_comments":[{"comment":"There is a notation clash: u0 is used both for the initial datum and for the minimal positive global pointwise solution. Please use different symbols, e.g. u̲ for the solution.","section":"Corollary 4.8"},{"comment":"The parenthetical proof of (SC∞) from (BD) only establishes h=0 for 0<λ<(2D)^{-1}. The citation [43, Corollary 27] covers all λ, but the inline argument should say so, or add the standard reduction from small λ to all λ.","section":"Theorem 5.10(ii)"},{"comment":"The convexity inequality |b|^q-|a|^q ≥ q a^{q-1}(b-a) uses the signed-power convention; this is correct but might be unfamiliar. A one-line reference to the convention in Section 4.2 would help.","section":"Lemma 4.5, Step 1"},{"comment":"The notation δ_q, γ_q is introduced in (4.5), but the reader would benefit from a brief statement that δ_q>0 for all q≥α and δ_q<1 iff m<2/ν, which is already in the text. No substantive issue.","section":"Throughout Section 4"},{"comment":"In the stochastically incomplete example, the identity u^A(t)=P_t1-1 is clear, but it rests on the minimal heat semigroup monotone convergence; please state the cited [64] and [44, Chapter 7] near the display for completeness.","section":"Remark 3.6(2)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central defect is structural: the object of study in Sections 4 and 5 is an ℓ1-mild solution whose construction is entirely delegated to the companion preprint [10]. This is not a local derivation error, but it means the main theorems cannot be independently verified from the submitted manuscript alone. The in-paper arguments are otherwise careful and the results are substantial. I recommend major revision rather than rejection because the dependency is fixable — by making the relevant portions of [10] self-contained in an appendix or by updating to a published version. If the companion cannot be included, the theorems should be reframed as conditional on the m-accretive realization of [10]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper delivers a credible quantitative extinction/smoothing dichotomy and a mass-balance law for the GPME on weighted graphs, and it is honest about what it does not prove. The (Sν)-based energy framework yields explicit extinction times for m<2/ν and ℓ1-ℓq smoothing for m>2/ν, and the recovery of the Euclidean critical exponent on Cayley graphs is a nice touch. Theorem 3.4 on extremal pointwise solutions is self-contained and the proof is careful, with the Dini/dominated convergence steps checked. Lemma 4.5's exponent algebra holds up, and the signed-power inequality in Lemma A.1 is proved in both sign cases. The paper also flags the borderline case m=2/ν and shows the mass balance does not characterize SC∞ (Remark 5.6).\n\nThe soft spot is structural, and the stress-test note is right: the ℓ1-mild solution is not constructed here. Existence, uniqueness, and the finite-resolvent approximation are imported from the companion preprint [10] (Theorems 4.1, Lemma 4.4(b), Remarks 4.2 and 5.4). If the m-accretive realization or the resolvent convergence fails on non-locally finite graphs with unbounded killing, then Theorems 4.6, 4.9, and 5.5 lose their well-defined solution object. The authors acknowledge this reliance explicitly, but the conditionality is not resolved within the paper. Lemma 5.3 also leans on external results from [32]/[23], though that seems standard. I found no internal contradiction or derivation error; the in-paper proofs are consistent conditional on those imports.\n\nThis paper is for researchers in nonlinear diffusion on graphs and discrete analysis. It deserves a serious referee, not a desk reject, but the referee should have access to [10] and verify that the imported results hold in the claimed generality. If the companion preprint checks out, this is a strong contribution; if not, the central theorems are statements about a possibly arbitrary or nonexistent object. I would engage with it, but I would condition my acceptance on the companion preprint being made available and verified.","headline":"Solid, honest extension of nonlinear diffusion theory to graphs, but the main theorems lean on an unpublished companion preprint for the definition of the solution object.","tokens_in":50159,"tokens_out":1955,"would_cite":true,"duration_ms":20532,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K55","35R02","47H06","05C63"],"pacs":[],"model":"deepseek-v4-flash","headline":"Fast diffusion on weighted graphs extinguishes in finite time when m < 2/ν and smooths when m > 2/ν; under stochastic completeness at infinity, mass loss is charged exactly to the killing term.","keywords":["generalized porous medium equation","filtration equation","graph Laplacians","finite-time extinction","fast diffusion","Sobolev inequality on graphs","stochastic completeness at infinity","mass conservation"],"falsifier":"Run the implicit Euler scheme (2.4) on a large finite box in ℤN (N ≥ 3) with φ(s)=s|s|^{m−1}, m just below 2/ν, and check whether the measured ℓq norm ever stays strictly above the truncated power [∥u0∥q^{q(1−δq)} − K(1−δq)M^{−γq}t]_{+}^{1/(1−δq)} on the right of (4.15); a violation would disprove the energy inequality (4.8) or the constant K_{m,q,ν}. A second, qualitative check: on the stochastically incomplete birth–death graph of Remark 5.6 with φ(s)=s|s|, the paper predicts exact mass conservation for every nonnegative ℓ1 mild solution — observing any ℓ1 deficit there would delimit the cla","tokens_in":49031,"feed_emoji":"⏳","tokens_out":29512,"duration_ms":226810,"temperature":0.7,"pith_summary":"This paper studies the generalized porous medium equation — ∂t u + Δ(φ(u)) = f — on infinite weighted graphs, where the Laplacian may carry arbitrary edge weights, a killing term, and vertices with infinitely many neighbors. Its central aim is to show that two qualitative features of nonlinear diffusion on Euclidean space survive in this discrete setting with the same critical exponents: finite-time extinction for fast diffusion, and exact accounting of mass. The authors prove, under a ν-Sobolev inequality (a bound on the ℓν norm of compactly supported functions by their Dirichlet energy), that ℓ1 solutions with porous-medium nonlinearity φ(s) = s|s|^{m−1} vanish in finite time when m < 2/ν, with an explicit extinction time, and become instantly smooth for m > 2/ν — recovering the Euclidean threshold mc = (N−2)/N on Cayley graphs of polynomial growth N. They also prove an exact generalized mass balance under stochastic completeness at infinity (the only bounded λ-harmonic functions are trivial, so no heat can leak to infinity): the only mass removed is the mass absorbed by the killing term, and when there is no killing, mass is conserved. Along the way, every graph admits minimal and maximal pointwise solutions for arbitrary bounded initial data, with no assumptions beyond the standing ones. A sympathetic reader would care because these are quantitative, checkable statements — explicit extinction times and an exact conservation law — for a class of graphs that earlier discrete theory did not reach.","feed_headline":"Fast diffusion on graphs dies out in finite time","feed_subtitle":"One Sobolev exponent splits extinction from smoothing, and the critical value matches Euclidean fast diffusion.","key_machinery":"The load-bearing mechanism is a single differential energy inequality (4.8), proved at the implicit-Euler level: ∥u(t)∥q_q + K_{m,q,ν} M^{−γq} ∫ₛᵗ ∥u(τ)∥q^{qδq} dτ ≤ ∥u(s)∥q_q. Its exponents obey δq < 1 exactly when m < 2/ν and δq > 1 exactly when m > 2/ν, so feeding (4.8) into a scalar Gronwall-type comparison (Lemma A.2) yields extinction in the first regime and smoothing in the second — one identity, two theorems. The proof substitutes for the missing chain rule the signed-power inequality (A.1), (b^σ−a^σ)(b^τ−a^τ) ≥ c_{σ,τ} |b^{(σ+τ)/2} − a^{(σ+τ)/2}|², and interpolates between ℓ1 and the Sobolev exponent. The second mechanism is the no-flux identity (Lemma 5.3): under (SC∞), v, Δv ∈ ℓ1∩","core_discovery":"Central to the paper is a dichotomy governed by one ratio. For ∂t u + Δ(u|u|^{m−1}) = 0 with ℓ1 data on a graph satisfying a ν-Sobolev inequality (ν>2): if m < 2/ν, every mild solution extinguishes in finite time, via bound (4.15); if m > 2/ν, every ℓ1 solution instantly enters every ℓq, with ∥u(t)∥q ≤ C t^{−θ} M^σ. On Cayley graphs of polynomial growth of order N ≥ 3 the threshold is mc = (N−2)/N, the Euclidean exponent. The second pillar is exact mass accounting: on graphs stochastically complete at infinity, every nonnegative ℓ1-mild solution obeys ∥u(t)∥1 + ∫₀ᵗ Σx κ(x)φ(u(s,x))ds = ∥u0∥1, so every lost unit is charged to the killing term; the same law holds for classical and bounded poin","pith_inferences":["The open borderline m = 2/ν should interpolate between the two regimes — likely power-law decay with logarithmic corrections, as on the Euclidean critical line — and pinning that rate on ℤN would complete the dichotomy; the paper stops at m ≠ 2/ν.","Because the proofs never use local finiteness, the same Sobolev-energy machinery should transfer to the nonlocal operators the paper cites as motivation (fractional Laplacians, Dirichlet-to-Neumann maps); the authors flag but do not carry out that transfer.","The birth–death example in Remark 5.6 shows the mass balance can hold without stochastic completeness, so (SC∞) is sufficient but not necessary; characterizing, for a fixed φ, exactly which graphs satisfy the balance would sharpen the scope of Theorem 5.5.","For signed classical solutions the accounting law requires the killing contribution to be absolutely integrable in time; finding natural hypotheses that force that integrability, rather than assuming it, would extend the balance to genuinely sign-changing evolutions."],"forward_implications":["On every graph with a ν-Sobolev inequality (ν>2), the fast-diffusion range 0 < m < 2/ν gives finite-time extinction of ℓ1 mild solutions with an explicit extinction-time bound; at the critical choice q = α the bound is ∥u(t)∥α_α ≤ [∥u0∥α^{1−m} − K(1 − 2/ν)t]_{+}^{ν/(ν−2)}.","In the complementary range m > 2/ν, every ℓ1 initial datum instantly produces an ℓq solution for every q > 1, with the explicit smoothing bound ∥u(t)∥q ≤ C t^{−θ} M^σ.","On Cayley graphs of polynomial volume growth of order N ≥ 3, the threshold is exactly mc = (N−2)/N — the Euclidean critical exponent (1/2 on the Heisenberg-type group, N = 4) — and the dichotomy is stable under adding any killing term.","Under stochastic completeness at infinity, every nonnegative ℓ1 mild solution satisfies the exact balance ∥u(t)∥1 + ∫₀ᵗ Σx κ(x)φ(u(s,x))ds = ∥u0∥1 with arbitrary killing; when κ = 0 this is conservation of mass, and on polynomial-growth Cayley graphs it holds as pure mass conservation throughout mc < m < 1.","For arbitrary bounded initial data, minimal and maximal global pointwise solutions exist on every graph (no local finiteness or bounded degree required); under bounded degree and locally Lipschitz nonlinearity the ℓ∞ Cauchy problem is globally well-posed, and the minimal nonnegative pointwise solution also extinguishes for m < 2/ν, even for data not in ℓ1."],"fun_headline_variants":["Graph diffusion: exponent m=2/ν decides extinction vs smoothing","Euclidean critical exponent found on graphs for fast diffusion","Mass balance on graphs: every lost unit is killing term","Graph fast diffusion: extinction iff m below 2/ν","One exponent on graphs: below 2/ν means finite-time death"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the ℓ1 solution theory used by the main theorems is already in place — existence, uniqueness, order-preserving resolvents, and convergence of finite-graph resolvents are all imported from the companion preprint [10] (Theorem 4.1, Lemma 4.4(b), Remarks 4.2 and 5.4), not proved here — so if any of those unpublished results fails on non-locally finite graphs or with unbounded killing, the extinction, smoothing, and mass-balance theorems lose thei","fun_headline_variants_meta":{"raw":{"variants":["Graph diffusion: exponent m=2/ν decides extinction vs smoothing","Euclidean critical exponent found on graphs for fast diffusion","Mass balance on graphs: every lost unit is killing term","Graph fast diffusion: extinction iff m below 2/ν","One exponent on graphs: below 2/ν means finite-time death"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000633,"raw_usage":{"total_tokens":2794,"prompt_tokens":817,"completion_tokens":1977,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":1890}},"tokens_in":561,"tokens_out":1977,"duration_ms":13539,"temperature":1.0,"reasoning_tokens":1890,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T03:37:54.836658+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the implicit Euler scheme (2.4) on a large finite box in ℤN (N ≥ 3) with φ(s)=s|s|^{m−1}, m just below 2/ν, and check whether the measured ℓq norm ever stays strictly above the truncated power [∥u0∥q^{q(1−δq)} − K(1−δq)M^{−γq}t]_{+}^{1/(1−δq)} on the right of (4.15); a violation would disprove the energy inequality (4.8) or the constant K_{m,q,ν}. A second, qualitative check: on the stochastically incomplete birth–death graph of Remark 5.6 with φ(s)=s|s|, the paper predicts exact mass conservation for every nonnegative ℓ1 mild solution — observing any ℓ1 deficit there would delimit the cla","supporting_citations":[],"review_version":1}