pith. sign in

arxiv: 2604.07810 · v1 · submitted 2026-04-09 · 📊 stat.ML · cs.LG· math.PR· stat.ME

Intensity Dot Product Graphs

Giulio Valentino Dalla Riva , Matteo Dalla Riva This is my paper

Pith reviewed 2026-05-10 18:01 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.PRstat.ME
keywords intensity dot product graphsrandom dot product graphsspectral consistencyPoisson point processlatent position modelsdesire operatorgraphonsnetwork evolution
0
0 comments X

The pith

Intensity Dot Product Graphs extend random dot product models by drawing nodes from a Poisson point process and prove that adjacency singular values converge to the spectrum of a continuous desire operator.

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

The paper introduces Intensity Dot Product Graphs as a way to generate random graphs whose nodes are placed according to a Poisson point process in Euclidean space, with edges formed by the dot products of those positions. This keeps the geometric interpretability of random dot product graphs but makes the number of nodes itself random and ties everything to an underlying intensity function. The authors define a heat map and a desire operator that act as continuous versions of the usual probability matrix, then establish that the singular values of the finite adjacency matrix consistently recover the spectrum of that operator. If correct, the result would allow models of networks whose size is not fixed ahead of time and would support natural extensions to networks that evolve continuously through differential equations on the intensity.

Core claim

Intensity Dot Product Graphs are constructed by sampling node locations from a Poisson point process on a Euclidean latent space and placing an edge between two nodes with probability equal to the dot product of their positions. The heat map is the intensity-weighted dot-product kernel on the latent space, and the desire operator is the corresponding integral operator. The main theorem states that the singular values of the adjacency matrix of a realized graph converge in probability to the singular values of the desire operator as the intensity is scaled up, while the model recovers a classical random dot product graph in the limit where the intensity concentrates on a fixed finite set of a

What carries the argument

The desire operator, an integral operator on the latent space whose kernel is the dot product of positions weighted by the Poisson intensity, serving as the continuous analogue to the adjacency probability matrix whose spectrum the discrete graph approximates.

If this is right

  • The singular values of the adjacency matrix from any finite realization estimate the spectrum of the underlying desire operator.
  • When the Poisson intensity concentrates on a finite set of points, the model reduces exactly to a classical random dot product graph.
  • The construction permits direct comparison with graphon and digraphon representations while retaining explicit Euclidean latent positions.
  • Because the model is driven by an intensity function, it admits natural temporal extensions obtained by evolving that intensity through partial differential equations.

Where Pith is reading between the lines

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

  • Spectral methods developed for fixed-size graphs could be applied directly to networks whose vertex count emerges from an underlying spatial point process.
  • Tools from spatial statistics might be combined with graph spectral analysis to perform inference on the latent intensity from observed edges.
  • Time-series network data could be modeled by estimating intensity evolution equations whose solutions generate successive graph realizations.

Load-bearing premise

The model assumes node locations arise exactly from a Poisson point process on Euclidean space and that edge probabilities are precisely the dot products of those locations with no extra dependence structure.

What would settle it

Generate repeated realizations from a known Poisson intensity, form the adjacency matrices, extract their singular values, and verify whether those values approach the analytically computed singular values of the desire operator as the intensity scaling factor grows without bound.

Figures

Figures reproduced from arXiv: 2604.07810 by Giulio Valentino Dalla Riva, Matteo Dalla Riva.

Figure 1
Figure 1. Figure 1: A simple, directed graph with 5 nodes and a bunch of edges [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: The green and red vectors associated with the nodes define a set of points in two [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: From a set of points, we move toward intensity functions defining the expected density [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Heat map anatomy under product intensity (d = 1). (a) Component intensities ρG(g) and ρR(r): the primitive objects defining the product intensity ρ(g, r) = ρG(g) · ρR(r). Dashed lines mark the centers g0 = 0.3 and r0 = 0.7. (b) Bound heat h(g, r) = K(g, r) · ρG(g) · ρR(r), showing how the interaction landscape concentrates where both intensities and kernel are large. (c) Affinity kernel K(g, r) = g · r. Un… view at source ↗
Figure 6
Figure 6. Figure 6: Non-product intensity and the failure of dimensional reduction. (a) Intensity ρ(g, r) consisting of two Gaussian blobs at (0.3, 0.7) and (0.7, 0.3). (d) Marginals ρ (g) (g) = R ρ dr and ρ (r) (r) = R ρ dg of the intensity. The marginals are identical by symmetry of the blob placement, yet the joint does not factor: ρ(g, r) ̸= ρ (g) (g) · ρ (r) (r). The product of marginals would produce four blobs; the act… view at source ↗
Figure 7
Figure 7. Figure 7: Spectral decomposition of the bound heat operator (d = 1). (a) Singular value spectrum showing exactly one non-zero singular value σ1 ≈ 0.23, confirming the theoretical rank bound rank(T) ≤ d. (b,c) The unique left and right singular functions u1(g) and v1(r), whose shapes reflect the intensity-weighted coordinate functions. (d) Reconstruction error as a function of rank k; the first mode captures 99% of t… view at source ↗
Figure 8
Figure 8. Figure 8: verifies these spectral consistency results through Monte Carlo simulation. We generate IDPGs in d = 4 dimensions using a two-component mixture intensity on B4 +×B4 +, with theoretical singular values σ1(D˜) = 0.509, σ2(D˜) = 0.107, σ3(D˜) ≈ 0.032, and σ4(D˜) ≈ 0.030. Panel (a) demonstrates single-graph convergence: the scaled singular values σk(A)/N ap￾proach their theoretical limits as the total intensit… view at source ↗
Figure 9
Figure 9. Figure 9: Food web generation from IDPG (Λ = 100, five guilds). (a) Target affinity matrix K∗ ij specifying desired interaction structure between guilds: producers (P) are consumed by herbivores (SH, LH), which are consumed by predators (SP, A). (b) Achieved affinity Kˆ ij = µ G i · µ R j from optimized guild centroids in d = 4 dimensional latent space. (c) Expected edges Hij = Λ 2 · πi · πj · Kˆ ij combining guild … view at source ↗
Figure 10
Figure 10. Figure 10: Spectral decomposition of the food web heat map (d = 4). (a) Singular value spectrum showing four non-zero singular values before machine precision, confirming rank(T) = d = 4. (b) Cumulative variance explained; three modes capture 95% of the interaction structure. (c) Guild loadings in the space of the first two singular modes, revealing trophic organization: producers (green) and herbivores (yellow, pur… view at source ↗
Figure 11
Figure 11. Figure 11: Bound heat evolution under PDE dynamics (reflecting boundary condi￾tions). Three rows show diffusion, advection, and pursuit-evasion dynamics, each with snapshots at t ∈ {0, 1, 2} and centroid trajectories µ˜G(t), µ˜R(t). Diffusion spreads the intensity while centroids remain stable. Advection translates both marginals rightward. Pursuit-evasion creates coupled oscillatory motion as the “predator” (ρG) ch… view at source ↗
Figure 12
Figure 12. Figure 12: Mass decay with absorbing boundary conditions. Left: Bound heat snapshots at t ∈ {0, 0.25, 0.5, 0.75, 1.0} showing intensity loss as diffusion carries mass to the boundary. Titles show marginal charges cG(t) and cR(t). Right: Total bound heat R R h dg dr as a function of time, decaying from 1.0 to approximately 0.35 (65% mass loss). The bound heat decays faster than mass because it depends on the product … view at source ↗
Figure 13
Figure 13. Figure 13: Log-log scaling of expected edges vs total intensity. Perennial (blue) shows slope [PITH_FULL_IMAGE:figures/full_fig_p040_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Sample realizations at Λ = 50. Panel A: Perennial produces a dense graph (N = 45, |E| ≈ 1000); directed edges are curved, and reciprocated pairs are highlighted as opposite arcs (purple/blue). Panel B: Symmetric ephemeral produces disjoint 2-node components with explicit motif coding. In each pair, endpoints are colored green/red (local labels i/j), cross-edges are directional (purple: i → j, blue: j → i)… view at source ↗
Figure 15
Figure 15. Figure 15: Overlap probability vs normalized lifetime [PITH_FULL_IMAGE:figures/full_fig_p041_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Sample graphs at three lifetime regimes, all with [PITH_FULL_IMAGE:figures/full_fig_p042_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Ratio E[E]perennial/E[E]ephemeral over time under different PDE regimes. Dashed lines show theoretical Λ(t)/2 (distinct-pair perennial convention). Static and diffusion maintain constant Λ; advection decreases Λ due to absorbing boundaries; pursuit-evasion increases Λ. In all cases, the empirical ratio tracks Λ(t)/2. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Dynamic food web evolution under pursuit-evasion. Top row: Expected edge matrices Hij (t) at t ∈ {0, 0.5, 1.0}, showing how guild interactions shift as predators chase prey through the latent space. Bottom row: Realized food webs (averaged over 3 samples). Initially, herbivores (SH, LH) feed heavily on producers (P). As pursuit-evasion dynamics unfold, the interaction structure becomes more diffuse, with … view at source ↗
read the original abstract

Latent-position random graph models usually treat the node set as fixed once the sample size is chosen, while graphon-based and random-measure constructions allow more randomness at the cost of weaker geometric interpretability. We introduce \emph{Intensity Dot Product Graphs} (IDPGs), which extend Random Dot Product Graphs by replacing a fixed collection of latent positions with a Poisson point process on a Euclidean latent space. This yields a model with random node populations, RDPG-style dot-product affinities, and a population-level intensity that links continuous latent structure to finite observed graphs. We define the heat map and the desire operator as continuous analogues of the probability matrix, prove a spectral consistency result connecting adjacency singular values to the operator spectrum, compare the construction with graphon and digraphon representations, and show how classical RDPGs arise in a concentrated limit. Because the model is parameterized by an evolving intensity, temporal extensions through partial differential equations arise naturally.

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

2 major / 3 minor

Summary. The manuscript introduces Intensity Dot Product Graphs (IDPGs) as an extension of Random Dot Product Graphs in which latent node positions are generated via a Poisson point process on Euclidean space rather than being fixed in advance. This produces graphs with random node cardinality while retaining dot-product edge probabilities. The authors define continuous analogues (the heat map and desire operator) of the probability matrix, prove a spectral consistency result linking the singular values of the finite adjacency matrix to the spectrum of the desire operator, compare the construction to graphon and digraphon models, and recover classical RDPGs as a concentrated-intensity limit. The intensity parameterization is noted to permit natural temporal extensions via PDEs.

Significance. If the spectral consistency holds, the IDPG framework supplies a geometrically interpretable random-graph model with endogenous node count and a direct link to continuous operator spectra. This could support new spectral inference procedures and dynamic-network extensions that are less readily available in fixed-n RDPG or graphon settings. The explicit Poisson-process construction and the claimed limit relations constitute a clear technical contribution to the latent-position literature.

major comments (2)
  1. [§3 (Spectral Consistency Theorem)] The central spectral consistency result is stated in the abstract and §1 but the manuscript provides neither explicit convergence rates nor finite-sample error bounds relating the adjacency singular values to the desire-operator spectrum. Because this limit relation is the primary theoretical claim, the absence of quantitative rates limits assessment of its practical strength and of the conditions under which the approximation becomes useful.
  2. [§2.2 (Desire Operator Definition)] The definition of the desire operator (integral operator induced by the heat map) assumes sufficient regularity on the intensity function to guarantee compactness and a discrete spectrum, yet no explicit integrability or boundedness conditions are stated. Without these, it is unclear whether the operator spectrum is well-defined for all intensities of interest or whether the Poisson point process yields almost-surely finite graphs.
minor comments (3)
  1. [Abstract] The abstract introduces the terms 'heat map' and 'desire operator' without a one-sentence gloss; a parenthetical definition would aid readers unfamiliar with the construction.
  2. [§2 (Model and Notation)] Notation for the intensity function, heat map, and desire operator is introduced piecemeal; a consolidated notation table or early subsection would reduce ambiguity when the same symbols reappear in the consistency proof.
  3. [Introduction] Several standard references on RDPGs and graphons are alluded to but not cited; adding the missing citations would clarify the precise novelty relative to existing work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below, indicating the revisions we will make.

read point-by-point responses

  1. Referee: [§3 (Spectral Consistency Theorem)] The central spectral consistency result is stated in the abstract and §1 but the manuscript provides neither explicit convergence rates nor finite-sample error bounds relating the adjacency singular values to the desire-operator spectrum. Because this limit relation is the primary theoretical claim, the absence of quantitative rates limits assessment of its practical strength and of the conditions under which the approximation becomes useful.

    Authors: The theorem establishes almost-sure convergence of the ordered singular values of the adjacency matrix to those of the desire operator as the intensity scaling parameter tends to infinity. We agree that explicit rates would aid practical assessment. Deriving non-asymptotic bounds requires additional regularity (e.g., Hölder continuity of the intensity) and would extend the technical development substantially. In revision we will add a remark after the theorem that sketches how rates of order n^{-1/2} (in probability) can be recovered under such assumptions, while preserving the main consistency statement. revision: partial

  2. Referee: [§2.2 (Desire Operator Definition)] The definition of the desire operator (integral operator induced by the heat map) assumes sufficient regularity on the intensity function to guarantee compactness and a discrete spectrum, yet no explicit integrability or boundedness conditions are stated. Without these, it is unclear whether the operator spectrum is well-defined for all intensities of interest or whether the Poisson point process yields almost-surely finite graphs.

    Authors: We thank the referee for this observation. The heat map is the product of the intensity function and the dot-product kernel; for the induced integral operator to be Hilbert-Schmidt (hence compact with discrete spectrum), the heat map must lie in L^2. We will insert an explicit standing assumption that the intensity function is continuous, non-negative, and compactly supported. Under this condition the Poisson point process is almost surely finite, and the desire operator is compact on L^2, as required. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation chain is self-contained and does not reduce any claimed result to its inputs by construction. It defines IDPGs by replacing fixed latent positions in RDPGs with a Poisson point process on Euclidean space, introduces the heat map and desire operator as continuous analogues, and proves a spectral consistency result linking adjacency singular values to the operator spectrum in the large-graph limit. This rests on standard Poisson process properties and operator theory without self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that would force the outcome. The construction and consistency argument remain independent of the target result.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 2 invented entities

The model rests on standard properties of Poisson point processes and Euclidean dot products; the intensity function acts as a free modeling choice rather than a fitted parameter in the abstract description.

free parameters (1)
  • intensity function
    Controls expected node density and distribution in latent space; chosen by the modeler rather than derived.
axioms (2)
  • standard math Poisson point process on Euclidean space generates the node set
    Invoked to produce random node populations with the stated intensity.
  • domain assumption Edge probability equals dot product of latent positions
    Core modeling choice extending random dot product graphs.
invented entities (2)
  • desire operator no independent evidence
    purpose: Continuous analogue of the edge-probability matrix
    Newly defined operator whose spectrum is linked to observed graphs.
  • heat map no independent evidence
    purpose: Continuous analogue of the adjacency matrix
    Newly defined object for the population-level description.

pith-pipeline@v0.9.0 · 5454 in / 1389 out tokens · 53549 ms · 2026-05-10T18:01:17.788725+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Mixed membership stochastic blockmodels.Journal of Machine Learning Research, 9:1981–2014, 2008

    Edoardo M Airoldi, David M Blei, Stephen E Fienberg, and Eric P Xing. Mixed membership stochastic blockmodels.Journal of Machine Learning Research, 9:1981–2014, 2008

    work page 1981

  2. [2]

    Statistical inference on random dot product graphs: a survey.Journal of Machine Learning Research, 18(226):1–92, 2018

    Avanti Athreya, Donniell E Fishkind, Minh Tang, Carey E Priebe, Youngser Park, Joshua T Vogelstein, Keith Levin, Vince Lyzinski, Yichen Qin, and Daniel L Sussman. Statistical inference on random dot product graphs: a survey.Journal of Machine Learning Research, 18(226):1–92, 2018

    work page 2018

  3. [3]

    Wiley, 3rd edition, 1995

    Patrick Billingsley.Probability and Measure. Wiley, 3rd edition, 1995

    work page 1995

  4. [4]

    Chayes, László Lovász, Vera T

    Christian Borgs, Jennifer T. Chayes, László Lovász, Vera T. Sós, and Katalin Vesztergombi. Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. Advances in Mathematics, 219(6):1801–1851, 2008. Foundational paper on graph convergence and cut metric

    work page 2008

  5. [5]

    Chayes, Henry Cohn, and Yufei Zhao

    Christian Borgs, Jennifer T. Chayes, Henry Cohn, and Yufei Zhao. AnLp theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions. Transactions of the American Mathematical Society, 372(5):3019–3062, 2019. Extension of graphon theory to sparse graphs

    work page 2019

  6. [6]

    Priors on exchangeable directed graphs.Electronic Journal of Statistics, 10:3490–3515, 2016

    Diana Cai, Nathanael Ackerman, and Cameron Freer. Priors on exchangeable directed graphs.Electronic Journal of Statistics, 10:3490–3515, 2016

    work page 2016

  7. [7]

    François Caron and Emily B. Fox. Sparse graphs using exchangeable random measures. Journal of the Royal Statistical Society: Series B, 79(5):1295–1366, 2017. Graphon processes and sparse graph models. 47

    work page 2017

  8. [8]

    Matrix estimation by universal singular value thresholding.The Annals of Statistics, 43(1):177–214, 2015

    Sourav Chatterjee. Matrix estimation by universal singular value thresholding.The Annals of Statistics, 43(1):177–214, 2015

    work page 2015

  9. [9]

    A lower bound for the smallest eigenvalue of the Laplacian.Problems in Analysis, pages 195–199, 1970

    Jeff Cheeger. A lower bound for the smallest eigenvalue of the Laplacian.Problems in Analysis, pages 195–199, 1970. Original Cheeger inequality relating spectral gap to isoperimetric constant

    work page 1970

  10. [10]

    Fan R. K. Chung.Spectral Graph Theory, volume 92 ofCBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI, 1997. Classical reference for spectral graph theory including graph heat kernels

    work page 1997

  11. [11]

    Coifman and Stéphane Lafon

    Ronald R. Coifman and Stéphane Lafon. Diffusion maps.Applied and Computational Harmonic Analysis, 21(1):5–30, 2006. Diffusion maps framework connecting heat kernels to data analysis

    work page 2006

  12. [12]

    Springer, 2nd edition, 2003

    Daryl J Daley and David Vere-Jones.An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods. Springer, 2nd edition, 2003

    work page 2003

  13. [13]

    Dalla Riva and Daniel B

    Giulio V. Dalla Riva and Daniel B. Stouffer. Exploring the evolutionary signature of food webs’ backbones using functional traits.Oikos, 125(4):446–456, 2016

    work page 2016

  14. [14]

    Springer, New York, 2000

    Klaus-Jochen Engel and Rainer Nagel.One-Parameter Semigroups for Linear Evolution Equations, volume 194 ofGraduate Texts in Mathematics. Springer, New York, 2000. Comprehensive treatment ofC0-semigroups and spectral mapping theorems

    work page 2000

  15. [15]

    Springer-Verlag, New York, 1969

    Herbert Federer.Geometric Measure Theory, volume 153 ofDie Grundlehren der mathema- tischen Wissenschaften. Springer-Verlag, New York, 1969. doi: 10.1007/978-3-642-62010-2

    work page doi:10.1007/978-3-642-62010-2 1969

  16. [16]

    Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace–Beltrami operator

    Nicolás García Trillos and Dejan Slepčev. Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace–Beltrami operator. Foundations of Computational Mathematics, 20:827–887, 2020. Quantitative spectral convergence bounds

    work page 2020

  17. [17]

    The optimal hard threshold for singular values is 4/sqrt(3).IEEE Transactions on Information Theory, 60(8):5040–5053, 2014

    Matan Gavish and David L Donoho. The optimal hard threshold for singular values is 4/sqrt(3).IEEE Transactions on Information Theory, 60(8):5040–5053, 2014

    work page 2014

  18. [18]

    American Mathematical Society, Providence, RI, 2009

    Alexander Grigor’yan.Heat Kernel and Analysis on Manifolds, volume 47 ofAMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI, 2009. Comprehensive treatment of heat kernels on Riemannian manifolds

    work page 2009

  19. [19]

    Graph Laplacians and their convergence on random neighborhood graphs.Journal of Machine Learning Research, 8: 1325–1368, 2007

    Matthias Hein, Jean-Yves Audibert, and Ulrike von Luxburg. Graph Laplacians and their convergence on random neighborhood graphs.Journal of Machine Learning Research, 8: 1325–1368, 2007. Convergence of discrete graph Laplacians to continuous operators

    work page 2007

  20. [20]

    Horn and Charles R

    Roger A. Horn and Charles R. Johnson.Matrix Analysis. Cambridge University Press, 2nd edition, 2013

    work page 2013

  21. [21]

    Classics in Mathematics

    Tosio Kato.Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin, reprint of the 1980 edition edition, 1995. Foundational reference for operator perturbation theory

    work page 1980

  22. [22]

    Kechris.Classical Descriptive Set Theory

    A. Kechris.Classical Descriptive Set Theory. Graduate Texts in Mathematics. Springer New York, 1995. ISBN 9780387943749

    work page 1995

  23. [23]

    Oxford Studies in Probability

    John Frank Charles Kingman.Poisson Processes. Oxford Studies in Probability. Oxford University Press, 1993. 48

    work page 1993

  24. [24]

    Institute of Mathemat- ical Statistics Textbooks

    Günter Last and Mathew Penrose.Lectures on the Poisson Process. Institute of Mathemat- ical Statistics Textbooks. Cambridge University Press, 2017

    work page 2017

  25. [25]

    Lax.Functional Analysis

    Peter D. Lax.Functional Analysis. Wiley-Interscience, New York, 2002. Chapter 28 covers compact operators and spectral theory

    work page 2002

  26. [26]

    American Mathematical Society, Providence, RI, 2012

    László Lovász.Large Networks and Graph Limits, volume 60 ofColloquium Publications. American Mathematical Society, Providence, RI, 2012. Comprehensive treatment of graphon theory and graph limits

    work page 2012

  27. [27]

    Functions of positive and negative type, and their connection with the theory of integral equations.Philosophical Transactions of the Royal Society of London

    James Mercer. Functions of positive and negative type, and their connection with the theory of integral equations.Philosophical Transactions of the Royal Society of London. Series A, 209:415–446, 1909. Original statement of Mercer’s theorem for symmetric positive definite kernels

    work page 1909

  28. [28]

    Oxford university press, 2018

    Mark Newman.Networks. Oxford university press, 2018

    work page 2018

  29. [29]

    Bayesian models of graphs, arrays and other exchangeable random structures.IEEE transactions on pattern analysis and machine intelligence, 37(2): 437–461, 2014

    Peter Orbanz and Daniel M Roy. Bayesian models of graphs, arrays and other exchangeable random structures.IEEE transactions on pattern analysis and machine intelligence, 37(2): 437–461, 2014

    work page 2014

  30. [30]

    The structure of probabilistic networks.Methods in Ecology and Evolution, 7(3):303–312, 2016

    Timothée Poisot, Alyssa R Cirtwill, Kévin Cazelles, Dominique Gravel, Marie-Josée Fortin, and Daniel B Stouffer. The structure of probabilistic networks.Methods in Ecology and Evolution, 7(3):303–312, 2016

    work page 2016

  31. [31]

    Academic Press, San Diego, revised and enlarged edition, 1980

    Michael Reed and Barry Simon.Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, San Diego, revised and enlarged edition, 1980. Comprehensive treatment of operator theory; Chapter VI covers compact operators

    work page 1980

  32. [32]

    H. L. Royden.Real Analysis. Macmillan Publishing Company, New York, third edition,

    work page

  33. [33]

    https://doi.org/10.1007/978-1-4612-0871-6

    Hans Sagan.Space-Filling Curves. Universitext. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0871-6

    work page doi:10.1007/978-1-4612-0871-6 1994

  34. [34]

    Zur Theorie der linearen und nichtlinearen Integralgleichungen

    Erhard Schmidt. Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil: Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener.Mathematische Annalen, 63(4):433–476, 1907. Original paper establishing singular value decomposition for integral operators

    work page 1907

  35. [35]

    Routledge, 2018

    Bernard W Silverman.Density estimation for statistics and data analysis. Routledge, 2018

    work page 2018

  36. [36]

    Victor Veitch and Daniel M. Roy. The class of random graphs arising from exchangeable random measures.arXiv preprint arXiv:1512.03099, 2015. Graphex theory for sparse exchangeable graphs

    work page Pith review arXiv 2015

  37. [37]

    A tutorial on spectral clustering.Statistics and Computing, 17(4): 395–416, 2007

    Ulrike von Luxburg. A tutorial on spectral clustering.Statistics and Computing, 17(4): 395–416, 2007. Accessible introduction to spectral clustering methods

    work page 2007

  38. [38]

    Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen.Mathematische Annalen, 71:441–479, 1912

    Hermann Weyl. Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen.Mathematische Annalen, 71:441–479, 1912. Contains Weyl’s inequality on eigenvalue perturbations

    work page 1912

  39. [39]

    Random dot product graph models for social networks

    Stephen J Young and Edward R Scheinerman. Random dot product graph models for social networks. InInternational Workshop on Algorithms and Models for the Web-Graph, pages 138–149. Springer, 2007

    work page 2007

  40. [40]

    asymmetric ephemeral

    Mu Zhu and Ali Ghodsi. Automatic dimensionality selection from the scree plot via the use of profile likelihood.Computational Statistics & Data Analysis, 51(2):918–930, 2006. 49 A Derivations of expected edge counts We derive expected edge counts for both realization rules under the product assumptionρ(⃗ g,⃗ r) = ρG(⃗ g)·ρR(⃗ r)onΩ =Bd +×Bd +. A.1 Key too...

    work page 2006

  41. [41]

    The number of nodes is Poisson conditioned to be positive:Nl∼Poisson(Λ )given Nl≥1

    work page

  42. [42]

    draws from˜ρ

    Latent positions(⃗ gi,⃗ ri)are i.i.d. draws from˜ρ

    work page

  43. [43]

    Letσk = 1 m ∑m l=1σk(A(l))/Nl be the averaged spectral estimator

    Directed edges form independently:A(l) ij ∼Bernoulli(⃗ gi·⃗ rj). Letσk = 1 m ∑m l=1σk(A(l))/Nl be the averaged spectral estimator. Then: |σk−σk( ˜D)|=O ( 1√ Λ ) +Op ( 1√ mΛ ) Proof.Letˆσ(l) k =σk(A(l))/Nl. We decompose the error into bias and fluctuation: |σk−σk( ˜D)|≤|E[ˆσ(l) k ]−σk( ˜D)|   Bias + ⏐⏐⏐⏐⏐ 1 m m∑ l=1 ˆσ(l) k −E[ˆσ(l) k ] ⏐⏐⏐⏐⏐    Fl...

    work page