# Intensity Profile Projection: A Framework for Continuous-Time Representation Learning for Dynamic Networks

arXiv: https://arxiv.org/abs/2306.06155 · HTML: https://mlsanity.com/paper/2306.06155/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 74 references that are in the map.
- First posted on arXiv 9 Jun 2023.

## Builds on (most cited first, 30 of 31)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding](https://mlsanity.com/paper/1712.09005/) (2017)
- [Statistical inference on random dot product graphs: a survey](https://mlsanity.com/paper/1709.05454/) (2017)
- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel](https://mlsanity.com/paper/1309.4111/) (2013)
- [Dynamic Network Embedding Survey](https://mlsanity.com/paper/2103.15447/) (2021)
- [Optimality of Spectral Clustering in the Gaussian Mixture Model](https://mlsanity.com/paper/1911.00538/) (2019)
- [dynnode2vec: Scalable Dynamic Network Embedding](https://mlsanity.com/paper/1812.02356/) (2018)
- [Statistical clustering of temporal networks through a dynamic stochastic block model](https://mlsanity.com/paper/1506.07464/) (2015)
- [Latent Space Models for Dynamic Networks](https://mlsanity.com/paper/2005.08808/) (2020)
- [Dynamic stochastic blockmodels for time-evolving social networks](https://mlsanity.com/paper/1403.0921/) (2014)
- [The multilayer random dot product graph](https://mlsanity.com/paper/2007.10455/) (2020)
- [Limit theorems for eigenvectors of the normalized Laplacian for random graphs](https://mlsanity.com/paper/1607.08601/) (2016)
- [Manifold structure in graph embeddings](https://mlsanity.com/paper/2006.05168/) (2020)
- [Understanding Regularized Spectral Clustering via Graph Conductance](https://mlsanity.com/paper/1806.01468/) (2018)
- [Latent space models for multiplex networks with shared structure](https://mlsanity.com/paper/2012.14409/) (2020)
- [Spectral embedding for dynamic networks with stability guarantees](https://mlsanity.com/paper/2106.01282/) (2021)
- [A semiparametric extension of the stochastic block model for longitudinal networks](https://mlsanity.com/paper/1512.07075/) (2015)
- [A Mutually Exciting Latent Space Hawkes Process Model for Continuous-time Networks](https://mlsanity.com/paper/2205.09263/) (2022)
- [Piecewise-Velocity Model for Learning Continuous-time Dynamic Node Representations](https://mlsanity.com/paper/2212.12345/) (2022)
- [On estimation and inference in latent structure random graphs](https://mlsanity.com/paper/1806.01401/) (2018)
- [Spectral embedding of weighted graphs](https://mlsanity.com/paper/1910.05534/) (2019)
- [Spectral clustering on spherical coordinates under the degree-corrected stochastic blockmodel](https://mlsanity.com/paper/2011.04558/) (2020)
- [The Block Point Process Model for Continuous-Time Event-Based Dynamic Networks](https://mlsanity.com/paper/1711.10967/) (2017)
- [Out-of-sample extension of graph adjacency spectral embedding](https://mlsanity.com/paper/1802.06307/) (2018)
- [Spectral Clustering for Multiple Sparse Networks: I](https://mlsanity.com/paper/1805.10594/) (2018)
- [Mutually exciting point process graphs for modelling dynamic networks](https://mlsanity.com/paper/2102.06527/) (2021)
- [Continuous Latent Position Models for Instantaneous Interactions](https://mlsanity.com/paper/2103.17146/) (2021)
- [Block modelling in dynamic networks with non-homogeneous Poisson processes and exact ICL](https://mlsanity.com/paper/1707.02780/) (2017)

## Built on by (most cited first, 5 of 5)

- [A Primer on Temporal Graph Learning](https://mlsanity.com/paper/2401.03988/) (2024)
- [Multiresolution Analysis and Statistical Thresholding on Dynamic Networks](https://mlsanity.com/paper/2506.01208/) (2025)
- [Generalized Grade-of-Membership Estimation for High-dimensional Locally Dependent Data](https://mlsanity.com/paper/2412.19796/) (2024)
- [Unsupervised Attributed Dynamic Network Embedding with Stability Guarantees](https://mlsanity.com/paper/2503.02859/) (2025)
- [Non-Negative Matrix Factorization for Event Data](https://mlsanity.com/paper/2606.06205/) (2026)

## Abstract (arXiv)

We present a new representation learning framework, Intensity Profile Projection, for continuous-time dynamic network data. Given triples (i,j,t), each representing a time-stamped (t) interaction between two entities (i,j), our procedure returns a continuous-time trajectory for each node, representing its behaviour over time. The framework consists of three stages: estimating pairwise intensity functions, e.g. via kernel smoothing; learning a projection which minimises a notion of intensity reconstruction error; and constructing evolving node representations via the learned projection. The trajectories satisfy two properties, known as structural and temporal coherence, which we see as fundamental for reliable inference. Moreoever, we develop estimation theory providing tight control on the error of any estimated trajectory, indicating that the representations could even be used in quite noise-sensitive follow-on analyses. The theory also elucidates the role of smoothing as a bias-variance trade-off, and shows how we can reduce the level of smoothing as the signal-to-noise ratio increases on account of the algorithm `borrowing strength' across the network.