{"id":"8bcb17e1-116b-4e1c-9fa7-bd50cb9f4b8b","arxiv_id":"2508.18544","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A three-scale urn model of origin-destination trips converges to a universal mixed-Poisson law in the large sparse limit, yielding analytic formulas for occupancy, vacancy, coverage, and overflow.","lead":"What this paper did: it treats trips in an origin-destination network as balls thrown into bins and shows that three mathematical descriptions, a counting model, a probabilistic model, and a continuous graphon model, agree in the large sparse limit. Why read it: it produces closed-form formulas for unvisited locations, coverage time, and capacity overflow that could help planners generate synthetic mobility networks and assess congestion.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Mass-mismatch in the continuum link of the central equivalence: Eq. (6) gives the empirical graphon expected L1-mass ||κ||_1 independent of ℓ, while Theorem 2 fixes it at ℓ||κ||_1 = n/N; the fixed-mass graphon equivalence and Theorem 5 are not established as written.","rationale":"The reader's CONDITIONAL verdict is reasonable, but the sharpest place to apply pressure is not primarily the self-averaging assumption in Remark 2. Self-averaging of degree distributions is a real condition, but it is a standard concentration result for this class of models and is downstream of the ensemble equivalence. The more immediate obstacle is the dimensional inconsistency in Theorem 2: the object whose LDP is stated has the wrong mass relative to the constraint imposed on it. This invalidates the specific proof of the central equivalence, and because Theorem 5 explicitly invokes Theorem 2, the paper's derivation of the universal mixed-Poisson law is incomplete. I do not claim the final formula is false: the mixed-Poisson occupancy law is consistent with known inhomogeneous random graph theory, and the Monte Carlo agreement in Section 5 is evidence for the formula under the paper's own normalization. However, those simulations use the same ν0 normalization as the continuum calculation, so they cannot detect the missing factor of ℓ in the graphon bridge; they test internal consistency, not the contested equivalence. The concrete mass check above is cheap and decisive. With the scaling repaired, the rest of the framework (Theorem 1, Proposition 4, and the occupancy, vacancy, coverage, and overflow asymptotics) could likely be made rigorous; hence the paper should remain conditional rather than being rejected outright, but the central proof obligation is specifically to fix the normalization in Eq. (6) and re-derive Theorem 2.","tokens_in":32813,"tokens_out":13665,"duration_ms":144057,"concrete_test":"Set κ ≡ 1 and choose n = N/2, so ℓ = n/N = 1/2. Under Eq. (6), compute E||\\tilde A_N||_1 exactly for finite N or asymptotically: it equals ||κ||_1 = 1, while Theorem 2's constraint requires ℓ||κ||_1 = 1/2. This one-line check falsifies the mass normalization. Then re-derive the fixed-mass graphon equivalence with the corrected per-pair law a_ij ~ Binomial(ℓ, κ/N), so that E||\\tilde A_N||_1 = ℓ||κ||_1, and verify whether a speed-N LDP with rate D(·||ℓκ) follows from Sanov/Gärtner-Ellis; if the speed is N^2 or the rate picks up an additional factor, Theorem 2 needs a substantial rewrite rather than a minor erratum.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The bridge that makes the three granularities equivalent is Theorem 2, and Theorem 5 explicitly invokes it to derive the universal mixed-Poisson occupancy law. The bridge has a concrete scaling inconsistency. In Eq. (6), the empirical graphon is defined with a_ij^(ℓ) ~ Binomial(ℓ, κ/(ℓN)), so E[a_ij^(ℓ)|κ] = κ/N. Therefore E||\\tilde A_N||_1 = (1/N) Σ_{i,j} κ(x_i,y_j)/N → ||κ||_1, with no factor of ℓ. But Theorem 2 imposes the fixed-mass constraint ||\\tilde A_N||_1 = ℓ||κ||_1 = n/N. These two masses agree only when ℓ = 1 (i.e. n = N); in the sparse regime with n = O(N) and ℓ = n/N, they differ by a factor ℓ. The proof of Proposition 3 also sends ℓ → ∞ to pass from binomials to Poissons, although the fixed-mass constraint makes ℓ = n/N = O(1) in the same regime. Since the rate functional I(\\tilde A) = D(\\tilde A || ℓκ) in Theorem 2 is the relative entropy against a reference field of mass ℓ||κ||_1, while the empirical graphon has mass ||κ||_1, the variational identity P_C(A|n) ≍ exp[-N D(\\tilde A||ℓκ)] cannot hold as stated. This is a problem in the argument for the central claim, not an edge case: Theorem 5's normalization ν_x = (1/(N||κ||_1)) ∫ κ(x,y) φ(y) dy is inherited from this defective conditioning step. The final mixed-Poisson result is plausibly recoverable from standard inhomogeneous-random-graph theory, but the paper's proof of the 'renormalization theorem' does not currently go through.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a unified three-scale framework for directed origin-destination mobility networks, modeled as balls into bins. The three scales are a fine-grained multivariate hypergeometric (enumerative) ensemble, a coarse-grained product-of-binomials (canonical) ensemble, and a continuum graphon ensemble with latent variables. The central claim is that, in a large sparse regime, the three ensembles are asymptotically equivalent, and that the occupancy/degree distribution converges to a universal mixed Poisson law. The paper also derives formulas for vacancy, coverage, and overflow, and reports Monte Carlo simulations with gravity-like kernels, using parameters imported from the first author's earlier empirical study [12].","tokens_in":33256,"tokens_out":12851,"duration_ms":140857,"significance":"If the main equivalence theorem were established, the framework would provide a useful bridge between exact combinatorial allocation models and continuum graphon analysis for synthetic OD-network generation and congestion diagnostics. The paper has several strengths: the model formulation is original and clearly connected to inhomogeneous random graph theory; the simulations in Section 5 cover both coarse- and fine-grained generative algorithms; and the quantitative predictions are tested against the analytical formulas rather than merely illustrated. Importantly, the parameters alpha, beta, eta, and lambda are not fitted to the asymptotic formulas in this paper but are taken from [12], so the simulation agreement is not circular. The main obstacle is that the proof of the central renormalization theorem contains a normalization inconsistency in the graphon mass, and the statement of Theorem 1 contains contradictory sparsity conditions. The final mixed-Poisson law may be recoverable from standard inhomogeneous-random-graph results, but the paper's proof as written does not establish it.","major_comments":[{"comment":"The mass of the empirical graphon is inconsistent with the fixed-mass constraint in Theorem 2. Equation (6) defines a_ij^(ell) ~ Binomial(ell, kappa(x_i,y_j)/(ell N)), so E[a_ij] = kappa(x_i,y_j)/N and therefore E||A_tilde_N||_1 = (1/N) sum_{i,j} E[a_ij] -> ||kappa||_1, with no factor of ell. Theorem 2, however, imposes ||A_tilde_N||_1 = ell||kappa||_1 = n/N. These two masses agree only in the special case ell = 1, which is incompatible with the sparse regime n = O(N) and ell = n/N used elsewhere. The proof of Theorem 2 compounds the problem by setting ell := n/N while omitting the factor 1/||kappa||_1 when equating D(A/n || pi) and D(A/N || ell kappa). Since Theorem 5 explicitly derives the normalization nu_x = (1/(N||kappa||_1)) integral kappa(x,y) phi(y) dy from Theorem 2, the proof of the central mixed-Poisson statement, Eq. (12), is not valid as written. The authors need to choose one consistent graphon normalization, either rescaling A_tilde_N or changing the definition of the binomial multiplicity, and then re-derive the rate functional and the constraint.","section":"Section 2.3, Eq. (6), Theorem 2, and Theorem 5"},{"comment":"The stated assumptions of Theorem 1 are internally inconsistent. The coarse-grained regime requires ell >> 1, p_ij << 1, and ell sum_{i,j} p_ij << 1, but Corollary 1.1 gives ell = E[n] / sum p_ij, so ell sum_{i,j} p_ij = E[n], which is of order n and hence >> 1 under the fine-grained condition n >> 1. The proof in Appendix A.1 actually uses the condition ell sum p_ij^2 << 1, which is plausibly what was intended. The fine-grained condition 'n >> 1, N -> infinity, and n << n' is also not meaningful; it should presumably be n << N^2 or n << NM. As printed, the central equivalence theorem has no consistent set of hypotheses, and the statement must be corrected before the theorem can be cited.","section":"Theorem 1 statement and Corollary 1.1"},{"comment":"The overflow formula in Proposition 6 is off by one. For a nonnegative integer-valued load Y, E[(Y-C)_+] = sum_{k=C+1}^infty P(Y >= k) = sum_{k=C+1}^infty sum_{j=k}^infty P(Y = j). The displayed formula uses an inner sum starting at j = k+1, which computes sum_{k=C+1}^infty P(Y >= k+1) = E[(Y-C-1)_+]. The outer summation upper limit N is also unjustified, since the Poisson variable Y is not bounded by the number of bins N; the sum should run to infinity. Because the overflow formulas in Section 5 are compared quantitatively with simulations, this indexing error affects the claimed numerical agreement and must be corrected.","section":"Proposition 6 and Eq. (19)"},{"comment":"The mixed-Poisson degree distribution is presented as describing the typical outcome of a single synthetic network, but the justification is delegated to an unproved self-averaging assumption. Remark 2 states that the empirical degree distribution converges in probability to the ensemble average and then says the proof is omitted because it 'closely parallels the standard case.' This is load-bearing: the continuum predictions in Eq. (12) and the simulations in Section 5 concern individual realizations, not ensemble averages. The paper should either prove concentration of the empirical occupancy distribution under the stated kernel conditions or explicitly invoke and verify a known theorem for inhomogeneous random graphs, such as the Bollobas-Janson-Riordan convergence theorem, with the required integrability conditions checked for the Pareto/exponential kernels used in the paper.","section":"Remark 2 and Section 3.2 (self-averaging)"}],"minor_comments":[{"comment":"The sentence 'in the particular case of simulations with parameters eta = 2 and beta = 5/6' should refer to alpha = 5/6, since the power-law exponent mu = 1 + (eta - 1)/alpha is derived from alpha; as printed, the beta notation is inconsistent with the preceding definition beta = 3/4.","section":"Section 5, occupancy paragraph"},{"comment":"The summation index in Eq. (8) is written as m, although the sum runs over destination bins; this should be N (or the bin index set), and the 'n << n' condition in the same remark repeats the typo from Theorem 1.","section":"Remark 1, Eq. (8)"},{"comment":"The definition of proportionate stratified sampling appears inconsistent: after defining weights w_h = M_h/M, the text states sum_h w_h = n, but the weights should sum to 1; the sample-size allocation should be n_h = w_h n with sum_h n_h = n.","section":"Footnote 1"},{"comment":"The remark says the degree distribution is a 'compound Poisson law,' but the displayed formula Eq. (10) is a mixed Poisson law (Poisson mean randomized by latent type). These are different notions, and the terminology should be corrected.","section":"Section 3.2, Remark 2"},{"comment":"The step replacing E[(Z(x)-C(x))_+] by max(lambda(x)-C(x),0) for a Poisson variable is not generally valid when lambda(x) is close to C(x); it requires a stated asymptotic regime (lambda large and boundary-layer contribution negligible). The proof should quantify the error, since the formula is then used to produce Eq. (28).","section":"Section 4.2, Proposition 7"},{"comment":"The paper refers to 'purpose-built code' and reports Monte Carlo results over S = 50 realizations, but no code or data repository is provided, and the figures do not show error bars or confidence bands, which would help verify the claimed agreement with the analytical predictions.","section":"Reproducibility"}],"recommendation":"major_revision","confidential_remarks":"The central obstacle is the graphon normalization in Theorem 2; if the authors repair the scaling and re-state Theorem 1 consistently, the paper is likely publishable after revision. The self-averaging concern in Remark 2 should be addressed by a precise invocation of standard inhomogeneous-random-graph convergence results rather than an informal mean-field statement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before you spend time on this. It is a synthesis of urn/occupancy methods and inhomogeneous random graphs applied to origin-destination mobility, and it contains a few genuinely new closed-form expressions (Lambert-W coverage time, overflow with logarithmic correction) that I did not find verbatim in the cited literature. But the proof of the central \"renormalization theorem\" has a scaling inconsistency that the authors need to fix; as written, it does not go through.\n\nWhat is good: the three-scale framing is pedagogically useful. The paper correctly recognizes that the fine-grained multivariate hypergeometric ensemble and the conditioned product-of-binomials ensemble are asymptotically multinomial, and that the continuum limit gives a mixed Poisson degree law. The occupancy, vacancy, coverage, and overflow formulas are clearly derived from the latent-variable kernel, and the power-law exponent mu = 1 + (eta-1)/alpha is a nice check. The simulations match the analytic curves at the level of visual agreement, which is fine for a methods paper, though error bars and code would help.\n\nWhere it is soft: Theorem 1 states \"ell sum p_ij << 1\" which is impossible alongside n = ell sum p_ij >> 1; presumably a typo for ell sum p_ij^2 << 1. Proposition 6's overflow sum computes E[(Y-C-1)_+] rather than E[(Y-C)_+], an off-by-one that is easily fixed. The coverage-time result (Prop. 5) relies on a \"balanced regime\" assumption that is not stated formally in the proposition. More serious: the stress-test note is right. Eq. (6) defines a_ij^(ell) ~ Binomial(ell, kappa/(ell N)) with mean kappa/N, so the empirical graphon mass converges to ||kappa||_1, not ell||kappa||_1. Theorem 2 then imposes the fixed-mass constraint ||tilde A||_1 = ell||kappa||_1 = n/N, and the proof of Proposition 3 sends ell to infinity as if ell were a growing Poissonization parameter. These two uses of ell are incompatible. The standard route would be to Poissonize with intensity ell kappa/N where ell = n/(N||kappa||_1), and then condition on total mass n/N. As written, the large-deviation bridge between the discrete and continuum ensembles is not established. The final mixed-Poisson law is a classical result in inhomogeneous random graph theory, so it is likely recoverable with a corrected proof, but the paper's derivation does not yet supply it.\n\nWho is it for: applied network scientists and urban mobility modellers who want compact formulas for synthetic network generation and congestion screening. It deserves a serious referee: the errors are fixable, the new closed forms are useful, and the synthesis is competent. I would send it to referees, with instructions that the proof of Theorem 2/Proposition 3 and the overflow indexing need careful checking.","headline":"A useful synthesis of urn and random-graph methods for OD mobility, with a few new closed-form formulas and a real scaling error in the central equivalence proof that should be fixed before publication.","tokens_in":33850,"tokens_out":7440,"would_cite":false,"duration_ms":72624,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","60C05","90B20","91D10"],"pacs":["89.75.-k","05.40.-a","89.65.Lm"],"model":"deepseek-v4-flash","headline":"This paper claims that, in the large sparse limit, three ways of generating origin-destination mobility networks—exact enumeration, independent edge probabilities, and continuum graphons—are asymptotically equivalent and share one…","keywords":["origin-destination networks","balls-into-bins models","inhomogeneous random graphs with latent variables","occupancy and load problems","human mobility modeling","mixed Poisson distribution","graphon large deviations","overflow and congestion"],"falsifier":"For a sequence with $N\\to\\infty$, $n/N$ held fixed and $n/N^2\\to 0$, generate many single realizations of the coarse-grained ensemble with the calibrated kernel $\\kappa(x,y)=x^{5/6}y^{3/4}$, $\\rho(x)\\propto x^{-2}$, and $\\phi(y)=0.2e^{-0.2y}$, and measure the variance across realizations of the empirical occupancy fraction $N_k/N$ for each $k$. If that variance does not tend to zero as $N$ grows, or if the ensemble average drifts from $P_n(k)=\\int (n\\nu_x)^k e^{-n\\nu_x}\\rho(x)\\,dx/k!$, the self-averaging step fails and the law is not typical of single networks.","tokens_in":32540,"feed_emoji":"🚇","tokens_out":10774,"duration_ms":104450,"temperature":0.7,"pith_summary":"The paper tries to establish that human mobility flows can be described as a balls-into-bins allocation—trips are balls, destination locations are bins—and that the resulting origin-destination network has the same large-scale statistics at three levels of description: an exact enumerative ensemble, a probabilistic independent-edge ensemble, and a continuum graphon ensemble. Under the large-sparse regime these three representations converge to a single asymptotic law: the number of visits to a destination with attractiveness $x$ is Poisson with mean $n\\nu_x$, so the degree (visit) distribution is the mixture $P_n(k)=\\int (n\\nu_x)^k e^{-n\\nu_x}\\rho(x)\\,dx/k!$. If correct, the equivalence lets planners generate synthetic OD networks at the cheapest scale and still reproduce exact constrained statistics, and it turns vacancy, coverage, and overflow into closed-form formulas. The paper verifies the asymptotics with Monte Carlo simulations of fine- and coarse-grained ensembles using a gravity-like kernel calibrated on empirical OD data.","feed_headline":"One law governs trip counts in mobility networks across scales","feed_subtitle":"Balls-into-bins, random-graph and graphon OD models agree in the sparse limit, giving closed-form congestion metrics.","key_machinery":"The load-bearing object is the urn allocation of distinguishable trip-balls into destination-bins, expressed at three resolutions. The fine-grained ensemble is the multivariate hypergeometric allocation with capacities $m_{ij}$; the coarse-grained ensemble is the product of binomials conditioned on the total number of links; the continuum ensemble is the empirical graphon with kernel $\\kappa(x,y)$ and fixed mass $n/N$. The identity that carries the argument is the asymptotic equality of their configuration probabilities, $P_C(A\\mid n)\\sim n!\\prod_{i,j}\\pi_{ij}^{a_{ij}}/a_{ij}!\\sim P_m(A)$, together with the large-deviation equivalence $P_C(A\\mid n)\\leftrightarrow P^{\\aleph}(\\tilde A\\mid\\|\\tilde A\\|_1=n/N)\\asymp e^{-N D(\\tilde A\\|\\ell\\kappa)}$, which lets integrals replace combinatorics in the occupancy and load formulas.","core_discovery":"The central discovery is an equivalence result: in the sparse limit with many nodes, the conditional coarse-grained ensemble of independent edge probabilities, the fine-grained multivariate hypergeometric ensemble, and a continuum graphon with fixed total mass describe the same random graph. The coarse-grained probability of a configuration with $n$ links is $n!\\prod_{i,j} \\pi_{ij}^{a_{ij}}/a_{ij}!$ up to vanishing corrections, and the continuum limit replaces sums by integrals with a large-deviation rate functional given by the relative entropy $D(\\tilde A\\|\\ell\\kappa)$ between the empirical graphon and the kernel. Consequently the occupancy problem—how many destinations receive exactly $k$ visits—is asymptotically a mixed Poisson law (Theorem 5), with the per-destination rate $n\\nu_x$ set by the destination's attractiveness and the normalized kernel. The same equivalence yields expressions for the fraction of empty destinations, the stopping time until a target number of destinations remain unvisited, and expected overflow beyond capacity; in the calibrated gravity case the degree tail is a power law with exponent $\\mu=1+(\\eta-1)/\\alpha\\approx 2.2$.","pith_inferences":["I infer that the mixed-Poisson occupancy law is not special to mobility: any sparse inhomogeneous random graph with an integrable kernel and self-averaging latent variables should show the same degree law, so the toolkit transfers to trade flows, migration, or communication networks.","I infer a model-selection recipe the paper does not develop: given an observed OD matrix $A$, minimize the relative entropy $D(A/N\\|\\ell\\kappa)$ over kernel parameters, since that rate functional ranks how atypical a flow matrix is under the model and could replace or complement gravity-model calibration.","I infer from the overflow threshold $x^\\star(n)=(n\\nu_0/C_0)^{1/(1-\\alpha)}$ that when capacity scales linearly with attractiveness but arrival rate scales sublinearly, congestion appears first at low-attractiveness destinations rather than hubs—a policy-relevant consequence the paper leaves implicit."],"forward_implications":["In the sparse regime the fine-grained capacity-constrained ensemble can be replaced by the coarse-grained conditional multinomial ensemble without changing the large-scale statistics, so synthetic OD networks can be generated by lightweight categorical sampling instead of exact hypergeometric enumeration.","The visit (in-degree) distribution of gravity-like mobility networks has a fat tail with exponent $\\mu=1+(\\eta-1)/\\alpha$; with the calibrated exponents $\\eta=2$ and $\\alpha=5/6$ this predicts $\\mu\\approx 2.2$, in agreement with the simulations.","The expected fraction of never-visited destinations after $n$ trips decays as $\\rho_0 a^{1-\\eta}/(\\alpha \\nu_0 a^\\alpha)\\cdot e^{-\\nu_0 a^\\alpha n}/n$, so accessibility gaps close exponentially fast once the least attractive destination is regularly reached.","The coverage time until exactly $b$ destinations remain empty is $E[T_b]=(\\nu_0 a^\\alpha)^{-1}W(N\\rho_0 a^{-(\\eta-1)}/(\\alpha b))+O(N)$, reducing the coupon-collector problem to a Lambert-$W$ evaluation.","Expected overflow beyond capacity grows linearly in total trips with a negative logarithmic correction; for the calibrated case $E[R]\\sim n(1+O(n/\\log n))$ for large $n$, giving a closed-form congestion forecast."],"supporting_citations":[{"why":"Supplies the balls-into-bins/urn formulation of configuration models on which the fine-grained ensemble is built.","marker":"[9]"},{"why":"Provides the empirical latent-variable densities, gravity-like kernel, and parameter values used in the validation simulations.","marker":"[12]"},{"why":"Underpins the inhomogeneous random graph and graphon theory used for the continuum occupancy limit.","marker":"[17]"},{"why":"Supplies the canonical-ensemble and generating-function machinery behind the coarse-grained degree derivation.","marker":"[27]"},{"why":"Gives the Poisson limit law for occupancy and vacancy that the asymptotic formulas use.","marker":"[19]"},{"why":"Source of the generating-function treatment of occupancy distributions in urn models.","marker":"[38]"},{"why":"Standard random-graph reference from which the mixed-Poisson degree distribution is adapted.","marker":"[33]"},{"why":"Provides the second-order binomial-to-Poisson approximation used to bound the error in occupancy formulas.","marker":"[41]"}],"fun_headline_variants":["Trip counts obey one law across all model scales","Sparse OD networks: bins, graphs, graphons agree","Universal Poisson law governs mobility trip counts","Three-scale equivalence: one mixed-Poisson law","From balls to graphons: one law for OD networks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central claim rests on the self-averaging assumption that a single large network behaves statistically like the average over many networks; if fluctuations around the ensemble average stay sizable in the sparse heterogeneous regime, the mixed-Poisson formulas describe only averages over realizations, not typical synthetic networks.","fun_headline_variants_meta":{"raw":{"variants":["Trip counts obey one law across all model scales","Sparse OD networks: bins, graphs, graphons agree","Universal Poisson law governs mobility trip counts","Three-scale equivalence: one mixed-Poisson law","From balls to graphons: one law for OD networks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000293,"raw_usage":{"total_tokens":1704,"prompt_tokens":937,"completion_tokens":767,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":692}},"tokens_in":553,"tokens_out":767,"duration_ms":7492,"temperature":1.0,"reasoning_tokens":692,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:58:27.085598+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a sequence with $N\\to\\infty$, $n/N$ held fixed and $n/N^2\\to 0$, generate many single realizations of the coarse-grained ensemble with the calibrated kernel $\\kappa(x,y)=x^{5/6}y^{3/4}$, $\\rho(x)\\propto x^{-2}$, and $\\phi(y)=0.2e^{-0.2y}$, and measure the variance across realizations of the empirical occupancy fraction $N_k/N$ for each $k$. If that variance does not tend to zero as $N$ grows, or if the ensemble average drifts from $P_n(k)=\\int (n\\nu_x)^k e^{-n\\nu_x}\\rho(x)\\,dx/k!$, the self-averaging step fails and the law is not typical of single networks.","supporting_citations":[{"cited_title":"Casiraghi, V","cited_arxiv_id":null,"evidence_quote":"Supplies the balls-into-bins/urn formulation of configuration models on which the fine-grained ensemble is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the empirical latent-variable densities, gravity-like kernel, and parameter values used in the validation simulations."},{"cited_title":"Bollob´ as, S","cited_arxiv_id":null,"evidence_quote":"Underpins the inhomogeneous random graph and graphon theory used for the continuum occupancy limit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the canonical-ensemble and generating-function machinery behind the coarse-grained degree derivation."},{"cited_title":"Sevast’Yanov, Poisson limit law for a scheme of sums of dependent random variables, Theory of Probability & Its Applications 17 (4) (1973) 695–699","cited_arxiv_id":null,"evidence_quote":"Gives the Poisson limit law for occupancy and vacancy that the asymptotic formulas use."},{"cited_title":"Johnson, S","cited_arxiv_id":null,"evidence_quote":"Source of the generating-function treatment of occupancy distributions in urn models."},{"cited_title":"Bollob´ as, Random graphs, in: Modern graph theory, Springer, 1998, pp","cited_arxiv_id":null,"evidence_quote":"Standard random-graph reference from which the mixed-Poisson degree distribution is adapted."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the second-order binomial-to-Poisson approximation used to bound the error in occupancy formulas."}],"review_version":2}