Intensity Dot Product Graphs

Giulio Valentino Dalla Riva; Matteo Dalla Riva

stat.MLApr 9, 2026

Intensity Dot Product Graphs

Authors: Giulio Valentino Dalla Riva, Matteo Dalla Riva

Organizations: Baffelan OU, Noumea, New Caledonia · Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università di Padova, Vicenza, Italy

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 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.

Figures & tables

Figure 3: The green and red vectors associated with the nodes define a set of points in two metric spaces. We define two matrices: one green, whose rows will be the coordinates of the green points, one red, whose columns will be the coordinates of the red points. Their matrix multiplication gives all the connection probabilities.

Figure 4: From a set of points, we move toward intensity functions defining the expected density of individuals in the latent space. Regions of higher intensity will contain more individuals on average. The green coordinate describes propensity to propose connections; the red coordinate describes propensity to accept them.

Figure 5: Heat map anatomy under product intensity ( $d=1$ ). (a) Component intensities ${\color[rgb]{0,0.6,0}\bm{\rho_{G}}}(g)$ and ${\color[rgb]{0.7,0,0}\bm{\rho_{R}}}(r)$ : the primitive objects defining the product intensity ${\color[rgb]{0,0,0.7}\bm{\rho}}(g,r)={\color[rgb]{0,0.6,0}\bm{\rho_{G}}}(g)\cdot{\color[rgb]{0.7,0,0}\bm{\rho_{R}}}(r)$ . Dashed lines mark the centers $g_{0}=0.3$ and $r_{0}=0.7$ . (b) Bound heat $\overline{h}(g,r)=K(g,r)\cdot{\color[rgb]{0,0.6,0}\bm{\rho_{G}}}(g)\cdot{\color[rgb]{0.7,0,0}\bm{\rho_{R}}}(r)$ , showing how the interaction landscape concentrates where both intensities and kernel are large. (c) Affinity kernel $K(g,r)=g\cdot r$ . Under product intensity, raw heat $\mathcal{H}$ and bound heat $\overline{\mathcal{H}}$ have the same shape, differing only by a scalar factor: $\mathcal{H}={\color[rgb]{0,0,0.7}\Lambda}\cdot\overline{\mathcal{H}}$ .

Figure 6: Non-product intensity and the failure of dimensional reduction. (a) Intensity ${\color[rgb]{0,0,0.7}\bm{\rho}}(g,r)$ consisting of two Gaussian blobs at $(0.3,0.7)$ and $(0.7,0.3)$ . (d) Marginals $\rho^{(g)}(g)=\int{\color[rgb]{0,0,0.7}\bm{\rho}}\,dr$ and $\rho^{(r)}(r)=\int{\color[rgb]{0,0,0.7}\bm{\rho}}\,dg$ of the intensity. The marginals are identical by symmetry of the blob placement, yet the joint does not factor: ${\color[rgb]{0,0,0.7}\bm{\rho}}(g,r)\neq\rho^{(g)}(g)\cdot\rho^{(r)}(r)$ . The product of marginals would produce four blobs; the actual joint has only two. (b,c,e,f) Slices of the 4D raw heat $h(g_{s},r_{t}\mid r_{s},g_{t})$ at four combinations of inactive coordinates. Different slices have different shapes because the inactive coordinates do not factor out. Under product intensity (Figure 5 ), all slices would be proportional, enabling the 2D bound heat summary.

Figure 7: Spectral decomposition of the bound heat operator ( $d=1$ ). (a) Singular value spectrum showing exactly one non-zero singular value $\sigma_{1}\approx 0.23$ , confirming the theoretical rank bound $\operatorname{rank}(\overline{T})\leq d$ . (b,c) The unique left and right singular functions $u_{1}(g)$ and $v_{1}(r)$ , whose shapes reflect the intensity-weighted coordinate functions. (d) Reconstruction error as a function of rank $k$ ; the first mode captures 99% of the Hilbert–Schmidt norm.

Figure 8: Numerical verification of spectral consistency. A two-component mixture intensity in $d=4$ dimensions with theoretical singular values $\sigma_{k}(\tilde{D})=(0.509,0.107,0.032,0.030)$ . (a) Single-graph convergence: scaled singular values $\sigma_{k}(A)/N$ approach theoretical limits as $\Lambda\to\infty$ . The noise floor $1/\sqrt{\Lambda}$ (dotted) bounds the estimation precision for small singular values. (b) Finite-intensity bias scales as $\mathcal{O}(1/\sqrt{\Lambda})$ , with $\sigma_{1}$ showing negligible bias. (c) Multi-graph averaging at $\Lambda=300$ : standard deviation decreases as $1/\sqrt{m}$ while bias remains fixed.

Product of Mixtures	Mixture of Products
$[\sum_{m}\rho_{G,m}]\times[\sum_{m}\rho_{R,m}]$	$\sum_{m}[\rho_{G,m}\times\rho_{R,m}]$
${\color[rgb]{0,0.6,0}\vec{g}}$ and ${\color[rgb]{0.7,0,0}\vec{r}}$ independent	${\color[rgb]{0,0.6,0}\vec{g}}$ and ${\color[rgb]{0.7,0,0}\vec{r}}$ coupled by species
Cross-species mixing in $({\color[rgb]{0,0.6,0}\vec{g}},{\color[rgb]{0.7,0,0}\vec{r}})$	No cross-species mixing

Table 8

Figure 9: Food web generation from IDPG ( ${\color[rgb]{0,0,0.7}\Lambda}=100$ , 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 $\hat{K}_{ij}={\color[rgb]{0,0.6,0}\bm{\mu^{G}_{i}}}\cdot{\color[rgb]{0.7,0,0}\bm{\mu^{R}_{j}}}$ from optimized guild centroids in $d=4$ dimensional latent space. (c) Expected edges $\mathcal{H}_{ij}={\color[rgb]{0,0,0.7}\Lambda}^{2}\cdot\pi_{i}\cdot\pi_{j}\cdot\hat{K}_{ij}$ combining guild abundances with interaction propensities. (d) Realized food web (averaged over 5 samples) showing trophic structure: arrow width proportional to edge count, vertical position indicates trophic level.

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 $\mathrm{rank}(\overline{T})=d=4$ . (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, purple) separate from predators (red, black). (d) Comparison of singular value spectra between the realized heat map and the target affinity matrix.

Figure 11: Bound heat evolution under PDE dynamics (reflecting boundary conditions). Three rows show diffusion, advection, and pursuit-evasion dynamics, each with snapshots at $t\in\{0,1,2\}$ and centroid trajectories $\tilde{\mu}_{G}(t)$ , $\tilde{\mu}_{R}(t)$ . Diffusion spreads the intensity while centroids remain stable. Advection translates both marginals rightward. Pursuit-evasion creates coupled oscillatory motion as the “predator” ( ${\color[rgb]{0,0.6,0}\bm{\rho_{G}}}$ ) chases the “prey” ( ${\color[rgb]{0.7,0,0}\bm{\rho_{R}}}$ ). Reflecting boundaries preserve total mass throughout.

Figure 12: Mass decay with absorbing boundary conditions. Left: Bound heat snapshots at $t\in\{0,0.25,0.5,0.75,1.0\}$ showing intensity loss as diffusion carries mass to the boundary. Titles show marginal charges ${\color[rgb]{0,0.6,0}c_{G}}(t)$ and ${\color[rgb]{0.7,0,0}c_{R}}(t)$ . Right: Total bound heat $\int\int\overline{h}\,dg\,dr$ 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 ${\color[rgb]{0,0.6,0}c_{G}}\cdot{\color[rgb]{0.7,0,0}c_{R}}$ .

Figure 13: Log-log scaling of expected edges vs total intensity. Perennial (blue) shows slope $\approx 1.97$ (theory: 2), ephemeral (red) shows slope $\approx 1.01$ (theory: 1). Dashed lines show theoretical predictions.

Figure 14: Sample realizations at ${\color[rgb]{0,0,0.7}\Lambda}=50$ . Panel A: Perennial produces a dense graph ( $N=45$ , $|E|\approx 1000$ ); 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: $i\to j$ , blue: $j\to i$ ), and self-loops are shown separately (dark green: $i\to i$ , dark red: $j\to j$ ). Both panels use identical intensity parameters; the difference arises purely from the realization rule.

Figure 15: Overlap probability vs normalized lifetime $\eta/W$ . Empirical estimates (points) match the theoretical curve $p_{\text{overlap}}=(2/u^{2})(u-1+e^{-u})$ where $u=W/\eta$ . The transition from ephemeral ( $p_{\text{overlap}}\approx 0$ ) to perennial ( $p_{\text{overlap}}\approx 1$ ) spans roughly two orders of magnitude in $\eta/W$ .

Figure 16: Sample graphs at three lifetime regimes, all with ${\color[rgb]{0,0,0.7}\Lambda}=50$ . Left: $\eta/W=0.1$ (near ephemeral) shows sparse edges. Center: $\eta/W=1$ (intermediate) shows moderate connectivity. Right: $\eta/W=10$ (near perennial) shows dense connectivity approaching the $\mathbf{R}_{\infty}$ limit. Directed edges are plotted as curved arrows, so reciprocal interactions are visible as paired arcs. For visualization clarity, we display a fixed fraction of realized edges and color each by sampled establishment time within its overlap interval (shared colorbar).

Figure 17: Ratio $\mathbb{E}[E]_{\text{perennial}}/\mathbb{E}[E]_{\text{ephemeral}}$ over time under different PDE regimes. Dashed lines show theoretical ${\color[rgb]{0,0,0.7}\Lambda}(t)/2$ (distinct-pair perennial convention). Static and diffusion maintain constant ${\color[rgb]{0,0,0.7}\Lambda}$ ; advection decreases ${\color[rgb]{0,0,0.7}\Lambda}$ due to absorbing boundaries; pursuit-evasion increases ${\color[rgb]{0,0,0.7}\Lambda}$ . In all cases, the empirical ratio tracks ${\color[rgb]{0,0,0.7}\Lambda}(t)/2$ .

Figure 18: Dynamic food web evolution under pursuit-evasion. Top row: Expected edge matrices $\mathcal{H}_{ij}(t)$ at $t\in\{0,0.5,1.0\}$ , 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 increased cross-guild connections and modified trophic flow. Edge weights reflect changing expected edge counts.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

$\kappa_{i}$	Interpretation	Std. dev. $\sigma_{i}$
30	Normal variation	$\approx 0.18$
100	Tight	$\approx 0.10$
500	Very tight	$\approx 0.045$
1000	Nearly fixed	$\approx 0.032$

Appendix

Table 19

Notation	Meaning
$G$	A graph
$V$	The set of nodes of a graph
$E$	The set of edges of a graph: (ordered) pairs of nodes
$i$ , $j$	Two nodes in a graph, or individuals
$s$ , $t$	Generally, the source and target of an interaction, connection, edge, …
$i\to j$	The edge given by the ordered pair $(i,j)$

Appendix

Table 20

Graphon Theory

See all 9 papers

Mar 5, 2026

Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

Can we learn the differential equations governing the evolution of a temporal network? We investigate this within Random Dot Product Graphs (RDPGs), where each network snapshot is generated from latent positions evolving under unknown dynamics. We identify three fundamental obstructions: gauge freedom from rotational ambiguity in latent positions, realizability constraints from the manifold structure of the probability matrix, and trajectory recovery artifacts from spectral embedding. We develop a geometric framework based on principal fiber bundles that formalizes these obstructions. We characterize invisible dynamics as exactly the skew-symmetric generators, and show the realizable tangent space has dimension

nd - d(d-1)/2

. An holonomy dichotomy emerges: polynomial dynamics have commuting generators, stationary eigenvectors, and trivial holonomy, making gauge alignment purely statistical; Laplacian dynamics satisfy a non-commutativity criterion producing nontrivial holonomy, with curvature weighted by

1/(λ_ι+ λ_γ)

linking gauge sensitivity to the spectral gap. In

d=2

this yields full restricted holonomy

\mathrm{SO}(2)

; for

d \ge 3

generic full

\mathrm{SO}(d)

remains conjectural. Cram'er--Rao lower bounds reveal that the same spectral gap controlling curvature and injectivity simultaneously controls Fisher information, so geometric and statistical difficulty are inextricable. We prove an identifiability principle: symmetric dynamics cannot absorb skew-symmetric gauge contamination, so dynamics structure can resolve gauge ambiguity. We demonstrate this constructively with anchor-based alignment and a UDE pipeline recovering vector fields from noisy graph sequences. Yet finite-sample interactions between noise, gauge, and dynamics expressiveness remain beyond the asymptotic theory. We frame this gap as an open challenge.

Giulio Valentino Dalla Riva

Latent Dynamics Modeling Dynamic Graph Learning System Identification Latent Variable Models

Ask PDF

Jul 15, 2026

Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs

Recovering a latent potential from observed flow on a directed graph (a discrete Poisson problem with Dirichlet boundaries) is ill-posed, and the standard fix backfires: ridge regularization shrinks toward a gauge-meaningless origin, collapsing and reversing the recovered ordering (

+0.81\to-0.42

rank correlation against a planted ground truth). The gauge-invariant graph Dirichlet energy removes the hazard and delivers parameter-insensitivity: the estimate is stable across four orders of magnitude in

λ

, whereas ridge inverts the ordering for every

λ>0

. We prove the reduced solve is SPD and preserves dynamic range exactly where ridge collapses it, and localize absorbing boundaries from flow alone via a Poisson residual. The

H^1

seminorm is classical; what is new is the gauge diagnosis, the parameter-insensitivity it buys, and an ablation showing the result is robust to the extraction method. On three public clickstream corpora the gauge-invariant estimate retains

28

41\%

of the interior dynamic range while ridge collapses to as little as

0.2\%

. The same gauge invariance carries into graph neural networks -- neutralizing the constant mode per layer prevents the oversmoothing that collapses a deep directed GCN -- linking this classical inverse problem to a central question in graph learning.

Mohammad Forouhesh

Amirkabir University of Technology, Iran

Oversmoothing Inverse Problems

Ask PDF

Oct 3, 2026

Path Laplacian Encodings for Directed Graphs

Directed graphs naturally model many real-world systems in which interactions are asymmetric, such as citation networks, web graphs, and information-flow networks. However, graph learning methods commonly rely on message passing with symmetrized graph representations or positional encodings that only partially exploit edge directionality. We introduce PathLapPE, a novel spectral positional encoding (PE) derived from the path Laplacian on directed graphs. PathLapPE provides node- and edge-level features that encode directional higher-order structure and can be incorporated into standard graph learning architectures. Empirical results on node- and graph-level benchmark tasks show that PathLapPE yields consistent improvements across several architectures, especially when combined with direction-aware message passing. Compared with magnetic Laplacian positional encodings, a widely studied spectral positional encoding for directed graphs, PathLapPE does not require additional fine-tuning of directionality hyperparameters while offering competitive runtime and performance.

Lydia Mezrag, Semih Cantürk, Michael Perlmutter +2

Université de Montréal · Mila – Quebec AI Institute · Boise State University +1

Graph Positional Encoding Positional Encoding Graph Neural Networks

Ask PDF

Abstract

Graphon Theory

Explore similar work

Random Dot Product Graphs as Dynamical Systems: Limitations and Opportunities

Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs

Path Laplacian Encodings for Directed Graphs