Euclidean mirrors and first-order changepoints in network time series

TL;DR

It is proved that a spectral estimate of the associated Euclidean mirror localizes these changepoints, even when the graph distribution evolves continuously, but at a rate that changes.

Abstract

We describe a model for a network time series whose evolution is governed by an underlying stochastic process, known as the latent position process, in which network evolution can be represented in Euclidean space by a curve, called the Euclidean mirror. We define the notion of a first-order changepoint for a time series of networks, and construct a family of latent position process networks with underlying first-order changepoints. We prove that a spectral estimate of the associated Euclidean mirror localizes these changepoints, even when the graph distribution evolves continuously, but at a rate that changes. Simulated and real data examples on organoid networks show that this localization captures empirically significant shifts in network evolution.

Figures (10)

• Figure 1: Piecewise linearity in a real data example, reproduced from athreya2025euclidean. Left panel shows the iso-mirror estimation for a time series of organizational communication networks generated each month from January 2019 to January 2021. A linear trend observed prior to April 2020, after which COVID-19 work-from-home protocols were introduced. Right panel shows iso-mirror estimation on time series of brain organoid connectivity networks sampled roughly once per week. Estimated iso-mirror exhibits piecewise linearity with slope change at day 188.
• Figure 2: A schematic view of a latent position process time series of graphs. Trajectories follow a latent position process, so marginal distributions of $X_t$ are dependent between times. Each IID sample from the LPP provides a row to each matrix $\mathbf{X}_t$ across time. Conditioned on the latent position matrices, the adjacency matrices are observed conditionally independently for each time. Solid lines denote dependence, while dotted lines represent (conditional) independence.
• Figure 3: The first three dimensions of CMDS for the true distance matrix $\mathcal{D}_{\varphi_m}$ for Model \ref{['no-changepoint']}, where $c=0.1$, $\delta=\frac{0.9}{m}$, and $m=40$, $p=0.4$. The left panel, the first dimension of CMDS, shows an approximately linear function of time increments $m$. The second and third dimensions, by contrast, are sinusoidal, but range over an order-of-magnitude smaller scaling. We provide analytic formulas for these in Appendix \ref{['Sec:beyond-1st-dim']}.
• Figure 4: The first three dimensions of CMDS for the true distance matrix $\mathcal{D}_{\varphi_m}$ for Model \ref{['changepointmodel']}, where $c=0.1$, $\delta=\frac{0.9}{m}$, and $m=40$, $p=0.4$, $q=0.2$, and $t^{*}_m=20$. The first dimension, given in the leftmost panel, shows approximate piecewise linearity with a slope change at $t^{*}_m$. The second and third dimensions also exhibit a change in behavior at $t_m^*$, where the frequency of the cosine and sine curves changes abruptly, but the range in both sinusoidal terms is an order of magnitude smaller; analytical descriptions for these curves are provided in Appendix \ref{['Sec:beyond-1st-dim']}.
• Figure 5: CMDS results on first 3 dimensions for Model \ref{['changepointmodel']} with same setting as in Figure \ref{['fig:cmds_changepoint']}. The black dots are the numerical CMDS result on $\mathcal{D}^{(2)}_{\varphi_m}$. The blue dots are estimation from one realization of time series of graphs with $n=1500$. Blue dots precisely aligns with the black dots in the first dimension. In the second dimension, the alignment of the estimated mirrors is less accurate, and the discrepancy increases further in the third dimension. For more details, see the final section of the Appendix.

Theorems & Definitions (61)

• Definition 2.1: Order notation
• Definition 2.2: Convergence with high probability
• Definition 2.3: Random dot product graph
• Remark : Orthogonal nonidentifiability in RDPGs
• Definition 2.4: Inner product distribution
• Definition 2.5: Latent position process
• Definition 2.6: Latent position process network time series
• Definition 2.7
• Definition 2.8
• Definition 2.9: Approximate Euclidean realizability with mirror $\psi$