Multiresolution Analysis and Statistical Thresholding on Dynamic Networks
Authors: Raphael Romero
Tijl De Bie
Nick Heard
Alexander Modell
Venue: NeurIPS 2025 poster ()
Keywords: Dynamic Networks
Multiresolution Analysis
Changepoint Detection
Primary area: general_machine_learning
TL;DR: This paper introduces a framework for multiresolution change detection in dynamic networks using wavelet decomposition and low-rank approximation.

Abstract: Detecting structural change in dynamic network data has wide-ranging applications. Existing approaches typically divide the data into time bins, extract network features within each bin, and then compare these features over time. This introduces an inherent tradeoff between temporal resolution and the statistical stability of the extracted features. 
   Despite this tradeoff, reminiscent of time–frequency tradeoffs in signal processing, most methods rely on a \emph{fixed temporal resolution}.  Choosing an appropriate resolution parameter is typically difficult, and can be especially problematic in domains like cybersecurity, where anomalous behavior may emerge at multiple time scales.
   We address this challenge by proposing ANIE ($\textbf{A}$daptive $\textbf{N}$etwork $\textbf{I}$ntensity $\textbf{E}$stimation), a multi-resolution framework designed to automatically identify the time scales at which network structure evolves, enabling the joint detection of both rapid and gradual changes.
   Modeling interactions as Poisson processes, our method proceeds in two steps: (1) estimating a low-dimensional subspace of node behavior, and (2) deriving a set of novel *empirical affinity coefficients* that measure change in interaction intensity between latent factors and support statistical testing for structural change across time scales.
   We provide theoretical guarantees for subspace estimation and the asymptotic behavior of the affinity coefficients, enabling model-based change detection. Experiments on synthetic networks show that ANIE adapts to the appropriate time resolution, and is able to capture sharp structural changes while remaining robust to noise. Furthermore, applications to real-world data showcase the practical benefits of ANIE’s multiresolution approach to detecting structural change over fixed resolution methods.
    An open-source implementation of the method is available at [https://github.com/aida-ugent/anie].

Multiresolution Analysis and Statistical Thresholding on Dynamic Networks
Raphaël Romero
Tijl De Bie Ghent University www.::eeiieellieeeeeee tijl.ddebieugent..cm be
Nick Heard Imperial College London
Ghent University
raphael. raphaeel.rmmroougentt.ooo
wweeeeeimneeriaaa ac
Alexander Modell Imperial College London a. prodell@imperial. ac uk
Abstract
Detecting structural change in dynamic network data has wide-ranging applica- tions. Existing approaches typically divide the data into time bins, extract net- work features within each bin, and then compare these features over time. This introduces an inherent tradeoff between temporal resolution and statistical stabil- ity of the extracted features. Despite this tradeoff, reminiscent of time frequenyy tradeoffs in signal processing, most methods rely a fixed temporal resolution. Choosing an appropriate resolution parameter is typically difficult, and can be especially problematic in domains like cybersecurity, where anomalous behav- ior may emerge at multiple time scales. We address this challenge by propos- ing ANIE (Adaptive Network Intensity Estimation), work designed to automatically identify the time scales at which network structure evolves, enabling the joint detection of both rapid and gradual changes. Modeling interactions as Poisson processes, our method proceeds mating -iw-diiensional subspace of node behavior, and (2) deriving set of novel empirical affinity coefficients that quantify change in interaction intensity between latent factors and support statistical testing for structural change across time scales. We provide theoretical guarantees for subspace estimation and the asymptotic behavior of the affinity coefficients, enabling model-based change de- tection. Experiments on synthetic networks show that ANIE adapts to the ap- propriate time resolution, and able capture sharp structural changes while remaining robust to noise. Furthermore, applications to real-world data showcase the practical benefits of ANIE 's multiresolution approach detecting structural change over fixed resolution methods. An open-source implementation of the method available at https: //gtttut.tttt/// com/ aida- aedlaaauent/aee annie.
multi-resolution frame-
two steps: (1) esti-
Introduction
Understanding dynamic networks, namely datasets taking the form of sequences of interaction events (u,v, between nodes and v at timestamp has wide-ranging applications in domains such as contact tracing[ [6]] cybersecurity[34] and urban mobility studies [3, 21]. Despite this domain di- versity. temporal networks commonly exhibit two fundamental types of structure: cross-sectional structure, where the network seen a graph evolving over time, and longitudinal structure, where the data at its finest resolution is best modeled as collection of point processes [33, 30, 29].
At its core, change detection in such networks involves understanding how these two types of struc- ture interact. However, doing so involves an inherent tradeoff. On one hand, identifying cross-
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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sectional structure-such as communities-requires aggregating events over a sufficiently wide time window to achieve statistical stability. On the other hand, imposing certain resolution of anal- ysis may obscure transient events which occur at higher temporal resolutions. This mirrors the time-frequency tradeoff in signal processing, where narrow time windows reveal high- seggfrreueeccy details but miss low-frequency trends, and wide windows improve frequency resolution at the cost of temporal localization.
In practice, the choice of an appropriate time resolution is a challenge which manifests in a variety of ways, such as selecting the number of timesteps at which to evaluate dynamic node embeddings [41, 42], or selecting bandwidth in order to derive smooth temporal signals from the dynamic network [29]. Often this challenge is resolved by selecting a resolution which seems to correspond to some characteristic period or frequency derived a priori from domain knowledge [17, 20, 29]. However in applications such as cybersecurity [33, 35, 13), where time-localization of anomalous event is essential, or more broadly community detection [43], where node behaviors may align at different resolution levels, such an arbitrary choice of resolution is not satisfactory.
To resolve this paradox, we highlight that cross-sectional structure in real-world networks typically manifests at several resolution levels simultaneously. For instance in social networks, community events of few hours coexist with gradually evolving friendship structures (weeks to months). Sim- ilarly, in cybersecurity, malicious activity might include both rapid bursts of suspicious connections and slowly evolving patterns designed to evade detection [35]. On the other hand, in bike-sharing networks [11], interaction patterns exhibit daily rhythms (commuting). weekly cycles (workday vs. weekend usage), and seasonal trends (weather effects).
In this paper, we introduce ANIE (Adaptive Network Intensity Estimation), a novel approach for detecting changes in dynamic networks across multiple temporal resolutions. Our approach takes inspiration in recent work in multiresolution analysis of point process [7, 45, 14, 12], and more generally wavelet analysis [28], and adapts them to the network domain.
statistical signal processing prob- lem, where the goal is to recover edge-level temporal signals from noisy dynamic network obser- vations. In Section 4, present new statistical method for multi-resolution change detection, supported by theoretical guarantees. In Section 5, we evaluate our method on both synthetic and real-world datasets, demonstrating that ANIE outperforms fixed-resolution approaches by effec- tively capturing changes at multiple time scales in dynamic networks.
Contributions In Section 3, we formulate change detection
2 Related Work
The proposed work lies at the intersection of several fields, which we briefly overview below.
Change Detection in Dynamic Networks The task of understanding the temporal evolution of dynamic network structure has been approached from various angles. One common approach is to view it as a dimensionality reduction task where the goal is to construct time-varying statistical summaries using for instance node embeddings [41, 29]. dynamic extensions of spectral clustering [48, 31, 44, 29], latent space models [41, 42], and tensor factorization methods [25, 39, 49, 2, 17, 46), which represent temporal structure through time-evolving latent factors. A related line of work focuses specifically on detecting change points, often in an online setting, by comparing network summaries across time windows [20, 13]. These methods typically rely on fixed time intervals, which assumes short-term stationarity. We note that [50] operates in an online setting using an adapted CUSUM statistic, making direct comparison with our proposed offline method difficult.
Wavelet analysis has been proposed as a prin- cipled approach to addressing the frequecyy tradcoff in signal processing [28, 47], and has proven effective in estimating the intensity of single point processes [9, 15, 14, 12, 45, 51, 23], where key features of the intensity function often appear at multiple resolution levels. To our knowl- edge, our work is the first to integrate these wavelet- based point process analysis with a low-rank decomposition of cross-sectional network structure.
Wavelets and Point Process Intensity Estimation
Functional Data Analysis Functional Data Analysis (FDA) has been widely used to analyze data with a continuous time dimension [40], and has been extended to multivariate settings [18]. Recent work has also adapted FDA to point process observations [37]. This work is the first to explicitly apply similar techniques to analyzing the temporal structure of dynamic networks.

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3
Multiresolution Change Detection in Dynamic Networks
This section gives some context our proposed method, by casting the problem of detecting signif- icant change in dynamic networks as a network intensity estimation problem.
3.1 A Low-Rank Poisson Process Model
The work considers dynamic network data, represented by an ordered sequence of interaction events 3 {(um, Um, )m) where the m-th event represents an interaction between nodes
Um U' belonging to set of nodes U - {l....; .N at time f. and the timestamps are provided in increasing order < 4 < <... < LM < T. We represent this data more concisely using a matrix of counting measures Y (Yan)a.vEter named the adjacency measure, and defined on the Borel o-algebra B() of the time interval T = [o, T). For any Borel set I C T,, the element (Z) Jz d'uuv (t) Eier.. (t(t) E of the matrix (I) counts the number of interactions between nodes u and e that occurred within time interval I. We model the edge-level interactions as arising from independent Inhomogeneous Poisson Processes. Mathematically, this means that there exists matrix of absolutely continuous intensity measures A (I + Auc (I)) u.v such that for any Borel set I T. the count of interactions between any node pair u on I is distributed as (uo(I) ~ Poisson (Au. (I)). We denote this using the shorthand notation ~ PoissonProcesss (A). This work relies eritically nodes may be explained by means of a measure of affinity between unobserved latent factors over time. We formalize this intuition in the following definition.
low-rank assumption, where we assume that the interactions between
Definition 3.1 (Common Subspace Independent Processes (COSIP)). A dynamic network Y is said to follow the COSIP model, i.e. Y ~ COSIP(U, S) Y ~ PoissonPrrcess(()) and for all bore- lian I C T, A(T) = US(I)UT where U E R is a subspace matrix whose D columns are orthonormal, S(I) E RDx is low-dimensional matrix measure, named the affinity measure, and D is latent dimension, or rank of the model.
This model extends the COSIE model from [4] to the continuous time setting. A special case of COSIP is the Dynamic Stochastic Block Model (DSBM), where U E {0. 10.1 assignment matrix, and S(I) specifies interaction rates between blocks. The COSIP model doesn't restrict the subspace matrix U to be binary. but assumes that the dynamic network distribution globally has low-rank. While the model is defined in terms of the measures A and S, both of them are assumed to admit respective densities (() and S(() with respect to Lebesgue measure on T,
NxD a community
such that for any Borel set C T, A(I) f (t) and S(I) = fz Stt)dt. We refer to them as the intensity function and the affinity function respectively.
3.2 Change Detection as an Intensity Estimation Problem
A naive approach to estimating the intensity function (t) is to use a histogram-based estima- tor such as (tu B B 1z, (t) (Ts)) where {Is 1 1 is a partition of the time interval [0, 27. However, such a naive edge-level estimator will tend to reflect not only meaningful structural changes, but also spurious fluctuations due to sparsity and edge-level randomness. In contrast, under the COSIP model, the observed data Y is viewed as a noisy observation of a latent intensity mea- sure A, whose density Att) decomposed into a node-level subspace matrix U and time-varying affinity function (t)), and the intensity function is expressed as sum over pairs of latent factors, thus borrowing strength across all node pairs:
Ago (1)
Σ D.VE[D]=
U Ung Spg(1). (t).
Crucially, this formulation unifies the two central goals of our work. First, estimating U reveals the network's ' cross-sectional structure- set of latent factors that capture how nodes align in their behavior over time. Second, estimating the time-varying affinity S(t) entails the identification of structural change points corresponding to features- for instance abrupt shifts or singularities- in the temporal signal S(L). this way, detecting changes in network structure naturally framed as the problem of identifying meaningful temporal features of the affinity function. As discussed in the introduction, such features often manifest at multiple resolution levels, motivating the use of wavelets for their detection.
3

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2
Signal
Nodes
Time
(b) Difference between inter and intra-
(a) Longitudinal plot of the interactions.
community rate over time.
-7.5 -5.
(c) Step 1: Hierarchy of net- (d) Step Low-rank approx- (c) Step 3: Coefficients after work Haar wavelet coefficients.
imation of the wavelet coeffi-
statistical thresholding.
cients.
Visualization of the ANIE approach with the Haar wavelet dynamic stochastic block model: (a) shows the raw interaction data over time; (b) illustrates the intensity gap between intra- community and inter-community node-pairs; (c) shows the wavelet decomposition of the dynamic network, with each row representing a time scale and each leaf corresponding to a specific time location; (d) shows the low-rank approximation of the wavelet coefficients; finally (e) illustrates the denoising step where statistical thresholding is applied to the coefficients, separating the noise (bottom coefficients in (d)) from the signal (the top right coefficients (d)).
Figure 1:
4 ANIE Adaptive Network Intensity Estimation
We now introduce ANIE (Adaptive Network Intensity Estimation), a novel method estimating the intensity measure of dynamic networks under the COSIP model by detecting significant changes in the affinity function. The method takes as input a dynamic network represented by its corresponding adjacency measure Y, and outputs a subspace matrix U and an adaptive intensity estimate (t). A full algorithmic description of the procedure provided in the appendix.
4.1 Function Spaces and Basis Decomposition
ANIE leverages an orthonormal functional basis { of the set of square-integrable functions LT(T). For any measure i on T and function f. we denote by p(f) = ff f(t)du(t) the projection of µa onto When f admits density x(t) that can be decomposed as x(t) = - 3 (t) in this be obtained using projection 3h u(σ²). In
basis, orthonormality implies that the coefficients particular for Poisson Process Y with intensity measure the coefficients Y(0²) provide unbiased estimates of 3h, specifically [[ considered in this paper. As such, we denote A(0*) for the coefficients of the intensity on the basis and ((²)) for their empirical estimates.
3b.. This is also valid for the matrix Poisson Process Y
Choice of Basis While any basis of (T) can be used, we illustrate our method using wavelet basis functions, which are known for their effectiveness in adaptive denoising [45, 28, 15). For non-orthonormal basis { ] spanning 1.2 (T), we can orthonormalize it using the Gram matrix
G = (Jr (t) R.1=1" Indeed, defining & (t) A
(t)) the rows of the vector

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{(t) - (1)/24(t) form an orthonormal basis that can be used directly in our framework. As a result our proposed method is highly flexible and variants of it can be derived using any functional bases in (T) used functional data analysis [40]. For example, orthonormal bases include the Fourier basis, wavelet bases (Haar, Daubechies) and Legendre polynomials. On the other hand, non-orthonormal bases include B-splines, natural and cubic splines, classical polynomial bases (which can be orthonormalized as needed using the previous remark).
Haar Wavelet Basis We use the Haar wavelet basis to illustrate the multi-resolltion capabilities
of ANIE This basis consists of the family {ff U f | j 2 0, k = O,.... ........'... 2 - -1}. where the scaling function f(t) 1 (o.1((t) and each 43,& scaled and translated version of the mother wavelet (t) = 1 | /2) (t) [1//2.1] (t), given by (y,x(t) = 22'2t - k). Here, j controls the scale (resolution) and k E (0...... .00.....2. 2 - - the location for a given scale j. As shown in [45], for a dyadic interval Σ kx, 2 [k + 1)] of width 2 the coefficients (1 measure the scaled difference in the number of events between its left and right halves:
Y(13,,) = 2- 1/2 (25)+122) ((])+128+1)]
These coefficients capture changes in the empirical event intensity across scales and locations, re- sulting in a hierarchy of coefficient matrices as shown on Figure 1. Positive values in these matrices indicate fewer events in the right subinterval, while negative values indicate more. As commonly done in wavelet analysis, we use a finite subset of this basis up to a maximum scale J.
Throughout the paper, we use (66) (indexed by b) to denote generic basis function. For the Haar basis, this set includes the scaling function f and all wavelet functions dyj, with j 2 0 and k E (O...., 2 - 13, with b serving as single unified index over them.
Note:
4.2 First stage: Low-Rank Decomposition
Basis Decomposition The first step of ANIE decomposes the adjacency measure the basis {o')
Tb=1 resulting in empirical coefficients: ((0h) e J (t)d(()) = (EreE...'01 (0())) a.v
RNXN where Eu,y is the set of interaction times between nodes u and v. This computation is effi- cient: for each node pair, we evaluate the basis function at each interaction time and sum the results. Notably, the resulting coefficient matrices inherit the sparsity of the adjacency measure. Moreover, by Campbell's theorem [5]. these coefficients unbiased estimates of the coefficients of the true intensity: E[Y = A(6").
Global Subspace Estimation The estimation of the subspace U follows closely the UASE [21] strategy. The empirical coefficients are first arranged into a unfolded matrix
X (V (M ()T] N
whose rows represent each node's relational behaviors over time. Despite their high dimensionality, these behaviors typically exhibit low low-dimensional structure due to two alignment factors: cross- sectional alignment (often reflecting community structure) and longitudinal alignment (reflecting structure in nodes' activity patterns). For example, in social network, nodes may interact with the same communities but different times, placing them in related but distinct behavioral subspaces. To capture these dominant modes of variation, we apply Truncated Singular Value Decomposition (TSVD) to matrix X, extracting the D singular vectors corresponding to the largest singular values into a matrix Ü E RNDD, As a note, this step may be viewed as partial Tucker decomposition of the 3-mode tensor {Y... (V")( }. where the first two modes are the node indices and the third mode is the basis index, namely a SVD of the mode-2 unfolding of the tensor [39]. Under suitable assump- tions, this subspace estimation procedure is consistent, as formalized in the following theorem.
Theorem 4.1 (Subspace Estimation Consistency ). Suppose that Y ~ COSIP(U,S) and that there exists fixed matrix-function R(t) = EB_ Ch Ct E RDD,, and a sparsity factor pN S 1 satisfying N Npn= w(log" (N)), such that S(t) == RN)( In addition, suppose that
1
1. The matrix A (CO) co has full rank.
2. The subspace matrix U satisfies the incoherence condition
||1||,,00 = oVV
log(N) Npv
5

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Then, there exists an orthogonal matrix Q such that
|UQ - Ul|2 - OF
(I)
VN
A proof of Theorem 4.1 is provided in the appendix.
4.3 Second stage: Denoising through statistical thresholding
The first stage of ANIE outputs estimated subspace matrix Ü E RNXD, which encodes the cross- sectional structure by representing each node in terms of its projection onto the dominant latent factors of node behavior. Based on this, it is natural to consider that each pair p. q of these latent factors will be subject to change over time. This change can be quantified directly by combining the coefficients of all the node pairs and weighing them by their respective nodes' affinity with the latent factors, which we do here:
Definition 4.1 (Empirical affinity coefficients). The empirical affinity coefficients are defined as the collection of x D matrices
$(0h)= = -" E RDDD, E [B]
(2)
The empirical affinity coefficients play central role in approach and offer several advan- tages. First, they have well-defined statistical properties: their expectation is [[S))(dd))
Up E., Uap U (rgAav(o')) and their variance is Var[Sp,(d'']] = En, Ú U U. results follow directly from the distributional properties of point process projections [23]. More- over, when using Haar wavelet functions Ø² = L2z.. they have a straightforward interpretation: each (1)a)) captures changes in interaction intensity between latent factors p and q at specific scales and locations. Large values indicate potential structural changes over the support of di For in- stance, with the Haar wavelet, positive (respectively negative) coefficients correspond (respectively increasing) interaction affinity between factors and over the interval Zj Extreme values reflect strong structural changes between latent factors. Finally, due the fact that they bor- row strength across node pairs, the next theorem shows that they are asymptotically normal under suitable conditions, enabling statistical testing.
These
decreasing
Theorem 4.2 (Asymptotic normality of the empirical affinity coefficients). Suppose that there exists sequences aN,, BN, PN such that for all u, [NV, p E [D] and
<xx aN Aur S AN
and U
NN N
up
which satisfy
PN N
3N
3/2
+0
as - 0c.
ON
Then, the standardized version of the empirical affinity coefficients Sp defined in 4.1 converge to standard normal distribution N- + x.. More specificully:
Spy(0") ElS pe (d')] 4, N),, Var[S
as +C.
(3)
The proof of Theorem uses the Lyapunov Central Limit Theorem applied the family of indepen- dent variables (Ü.,p Y (ád)) for given b, and is included in the appendix.
Multiple statistical testing for change in the network structure
As result of Theorem 4.3, the task of identifying changes in the network structure can be formulated as a multiple statistical testing problem, where the null hypotheses are that the latent factors p and q are not significantly associated with the wavelet functions σ²: H = [[S) = O. To carry out these tests, we may define the following Z-scores, by replacing the expectation by O, and the variance by its empirical estimate:
(pg(
Zpg(e) (0")
where Varlšpp
02 Ú2 8a0 Ba ((ob)2). (4)
up U208
V VarlS Var(Spp

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Under the null hypothesis that [d") = 0 (indicating no correlation with o"), these Z-scores follow approximately a standard normal distribution (O 1) for large N. This is particularly rele- vant when using wavelets 6 = gggk= as the null hypothesis corresponds a locally constant intensity between a pair of factors, and 9. on the support of the wavelet. To determine which coefficients are statistically significant, we compare [ with a threshold. Since we conduct B x D D simultaneous tests (one for each coefficient), we must account for multiple comparisons. We control procedure [8] at a significance level
the False Discovery Rate (FDR) using the menjjamiii-Hohhiee (typically 0.05), resulting in a binary significance mask M E {0, 1}. The final denoised inten- sity estimate is then constructed using only the coefficients determined to be statistically significant. We note that there exist more approaches for thresholding wavelet coefficients [24, 15, 45] which we didn't 't explore in this work but could improve the accuracy of the thresholding stage.
4.4 Parameter Selection and Computational Efficiency
The rank can be determined by examining the scree plot of singular values of the matrix X. In turn, the choice of significance level reflects how concervativee/areesive the thresholding should be. A threshold a = 0.0 will lead to a constant signal, since all the detail coefficients will be classified as noise. Conversely, threshold
1.0 will classify all the coefficients signal, leading to a noisy estimate. Typically, a is set to 0.05, as is common in multiple testing scenarios. However, the choice of can be adjusted based the specific application and desired level of significance. Our method can be made time and memory efficient by leveraging sparsity in three key ways: (1) coefficients ((0") naturally inherit the sparsity of the original adjacency measure Y, (2) ecis''s sparse SVD implementation can be employed compute dominant singular vectors without constructing dense matrices, and (3) thresholding operates only on the compact D D affinity matrices S(o") rather than the full N N matrices Y(o²). A timed experiment reporting computation times for varying numbers of nodes is provided in the appendix. All our experiments were run on MacBook Air with a MI chip and 8GB of RAM.
5 Experiments
The proposed ANIE method is evaluated on two tasks. First, we generate synthetic Erdös-Renyi (ER) and Stochastic Block Model (SBM) datasets, and measure the performance of our method in estimating a known network intensity from an observed dynamic network. Second, in order to demonstrate the practical utility of our method, we apply it to the task of detecting change points in a real-world dataset of message interactions, and compare our method with two existing methods: Laplacian Anomaly Detection (LAD) [20] and Tensorsplat [26].
5.1 Intensity Estimation on Synthetic Datasets
Dataset We test our wavelet-based approach on synthetic datasets designed specifically to test tem- poral adaptivity. We simulate networks with both Erdös-Rényí (ER) and Stochastic Block Model (SBM) structures, where intensity functions show complex temporal patterns. For ER-blocks, every node pair shares the same intensity. ( = Aboocka (t), based on the "blocks" test function from [15], which features blocks of varying widths. For SBM, we generate two-community network with piecewise constant intensities: intra- cora--communty intensities are significantly higher than inter- community ones, except over an interval where they are equal. This setting tests our method's ability to detect sharp intensity changes. For the Erdös- Renyi model, we generate various networks with a number of nodes ranging from 50 to 1000. In contrast for the SBM model, we generate networks with 50 to 2500 nodes.
Experimental Setting We compare our ANIE-Haar approach against non-adaptive IPP estimators from [29]. IPP first constructs naive intensity estimate A, then applies low-rank denoising (t) O Ü()ŪTT We consider two variants: IPP-KDE, which constructs Á using kernel density estimation, and IPP-Hist, which uses histogram- estimation. Note that IPP-Hist is equivalent to ANIE with the Haar wavelet without thresholding. We use the Mean Integrated Squared Error
N² E(u,))ENN Au.(t) - (t) dt. This metric averages the intensity estimation error over node pairs. In our setting we average the error over patches of N- 100 nodes (i.e. 1002 node pairs). We report mean and standard error over 10 runs.
(MISE) as our primary metric: MISE = 1
7

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(g) ER-blocks dataset
(h) SBM dataset
Figure 2: Comparison of intensity estimation methods ER-blocks and SBM datasets. The first. two rows show the estimated intensity functions for different methods, while the last row shows the MISE vs. number of nodes for both datasets.
Results The experimental results, summarized Figure 2 demonstrate two essential advantages of ANIE. A first notable advantage is multi-scale abilities. As can be seen in Figure 2c that the pro- posed approach allows capturing perturbations of the underlying intensity which occur at different temporal resolutions, while staying robust noise. In contrast, in order accurately capture the same perturbations, non-adaptive methods such as IPP-Hist and IPP-RBF pay the price of overfitting noise, leading spurious oscillations in the flat regions of the intensity function, 2b, 2a. A second, related key advantage is sharp change localization: the ANIE approach precisely identifies structural change points in the empirical affinity coefficients (Figure 2f), while staying robust to noise. In this setting, the intensity function between node pairs is piecewise constant, with abrupt changes in the intensity function. Our method effectively captures these changes, while other methods like IPP-Hist and IPP-RBF require a small bandwidth, and thus overfit to noise, in order to capture the same changes.
be seen in
5.2 Case Study: Multi-scale Anomaly Detection the UCI Messages Dataset
Dataset This experiment evaluates the practical utility of our ANIE method for detecting change- points real-world interaction datasets. To this end, we apply ANIE to the UCI Messages dataset, as done in previous work [20]. This dataset contains 59,835 messages sent among 1,899 users over a period of 196 days between April and October 2004. Fach message interaction is represented as directed edge with weight corresponding to the number of characters in the message.

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Frequency
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(c) Tensorsplat
(d) ANIE multiscale anomaly score (Ours)
Figure 3: Comparison of anomaly detection methods on the UCI dataset. Left column show respec-
tively (a) message count over time, (b) LAD anomaly score [20], and (c) Tensorsplat [26]. Right
column (d) shows the multi-scale anomaly scores from our ANIE method. The two main events identified in [32] are highlighted with vertical dashed lines.
Experimental Setting We compare ANIE method against two existing approaches for anomaly detection in temporal networks. Laplacian Anomaly Detection (LAD) [20] bins the data into tem- poral snapshots, embeds each graph using its Laplacian cigenvalues, and detects anomalies by mon- itoring changes in these successive spectral representations. The Tensorsplat [26] anomaly score is calculated as Tensorsplat(t) (t) T(t 1)ll where T(t) is the the row of the third matrix in the PARAFAC decomposition of the seed-resoluttin adjacency tensor. In contrast, for given scale the ANIE multi-multiscale anomaly score scale j is defined the Frobenius norm of the 2 empirical affinity for all k O...., 2 1, (k) - [(00),)) Using piecewise constant interpolation, we then convert this discrete time series into a continuous function (t) defined on the interval [0, 1] which we plot on Figure 3d. We use the Haar Wavelet basis in this example.
Results Figure 3 shows anomaly detection results on the UCI Messages dataset. As can be seen, ANIE successfully identifies major structural changes in the network, notably at the end of the spring term (day 60) and the start of the fall term (day 150). We first validate that this second event is not detectable by simply counting message volume over time. Tensorsplat seems follow to great extent the message count over time. This is likely due to the fact that in the PARAFAC decompo- sition, the third factor indirectly encodes the interaction rate into the latent activity vectors. Unlike LAD, which produces single aggregated anomaly score, ANIE provides multi-resolution scores that distinguish arge-scale structural changes from finer scale temporal oscillations. As shown in Figure 3d, the two main events described in [32] lead to change observed over different time scales. possible explanation for the discrepancy between resolutions that coarse-scale changes (green and blue curves) capture the formation and dissolution of social groups at the end and beginning of the academic year, whereas finer-scale changes (yellow) reflect short-term fluctuations driven by the academic calendar, such as classes, group projects, or other time-limited activities.

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Discussion and Conclusion
Due to the continuous nature of dynamic networks, tools from functional data analysis such as basis expansions combined with low-rank approximations appear to be a natural fit for analyzing dynamic networks. In this work, have presented ANIE, method that uses low-rank approximation to estimate the global structure of the data. It then employs a multi-resolution, wavelet-based approach test for significant changes in network structure at different resolution levels. In doing so, addresses the time-freyuency trade-offs inherent to fixed-bandwidth and discretized approaches.
The proposed methodology comes with several limitations. First, the testing strategy relies an estimate of the variance of the empirical affinity coefficients for computing the Z-scores (Equa- tion 4). The robustness of the thresholding procedure is therefore highly sensitive to this variance estimation error, particularly at high resolutions. Exploring alternative thresholding strategies thus appears to be a promising direction for future work. Moreover, in this study, have focused on the Haar wavelet basis due to its convenient interpretation as an adaptive histogram. However, in appli- cations requiring smoother intensity estimates, alternative bases such Daubechies wavelets [28] or B- B-spline-based approaches like Splinets [38, 27] could be employed. Additionally, our method could be extended by incorporating tensor factorization techniques such as PARAFAC [17], applied to the sparse tensor of empirical coefficients. Lastly, an interesting avenue would be to explore a hy- decomposition, building on recent work in PCA for point processes [37], potentially leading to a deeper theoretical understanding of the opportunities and limitations of functional analysis on dynamic networks.
brid
Acknowledgments and Disclosure of Funding
The research leading to these results has received funding from the Special Research Fund (BOF) of Ghent University (BOF20/BB//11)), from the Flemish Government under the "Onderzoek- sprogramma Artificiéle Intelligentie (AI) Vlaanderen" programme. from the FWO (project no. 60,9816N, 3G042220, G073924N). Funded by the European Union (ERC, VIGILIA, 101142229). Views and opinions expressed are however those of the author(s) only and do not necessarily re- flect those of the European Union the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. Modell and Heard acknowledge support from EPSRC NeST Programme grant EP/X002195/I.
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20

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Appendix to the paper "Multiresolution Analysis and Statistical Thresholding on Dynamic Networks"
A Full algorithm
Algorithm I describes the full pipeline of our proposed Adaptive Network Intensity Estimation (ANIE) method.
Algorithm Adaptive Network Intensity Estimation
Input: Dynamic network Y, orthonormal basis {d)) Jbmi of (T), rank D, significance level
I: Basis decomposition: Decompose the dynamic network the functional basis
v(o") / o" (f)d E NNK,, b= B
2: Low rank estimation: Form the concatenated matrix and compute truncated SVD
= [M (Y(")"" NNXN
X ÚEVT (keeping D largest singular values)
Calculate the empirical affinity coefficients
S(0") (" e RDDD
And their associated sample variance estimates
Var[Spa(0111 Σ U2 Yo You ((d")3) e R
3: Statistical thresholding: For each p, E [D] and b E [B]. compute
(p9)
Z-score Z
and associated p-value p/r, 2(1 ((1Zpgg)) ()
VarlSpp
where i is the standard normal cumulative density function. Then apply multiple-testing cor- rection to obtain the corrected p-values (lp..- and finally the thresholded coefficients using
Spp (d'), if the coefficient is significant, i.c. pr a, 10,
T (Spg(00))
otherwise.
4: Reconstruction: Compute thresholded intensity estimate
Σ Th Em(sss)oo or
A(I) Ú
Output: Low-rank subspace U, significance mask pq (ib <n). intensity estimate A(t).
B
Proof of theorem 4.1
Recall that
X = [( |T( RNNN
and then by the properties of Poisson processes, we have that
EX [06')) [
R NNNN
21

---

Observe that
A(00) - J, ((t) Npw / UR(t) 8² -NNN ((,R1)) UT.
addition, since the basis functions of
bb are orthonormal, we have that
1 d(t) 1
dt Σ C" (f (tod)()))))) C.
Σ cd 0y
1=
It follows that A(o") UCNUCCU and
NPx
C -=
Therefore, there exists an orthogonal matrix 0, E RDDDD such that the left (orthonormal) singular values corresponding to the non-zero singular values of EX, which we denote σi 2 ... an, are given by the columns of UO
Let Ù (ū....., D be the matrix whose columns contains the left (orthonormal) singular vectors of X corresponding to the D largest eigenvalues of X, which we denote o1 2 ÔD.
Then, by Wedin's sin theorem [10, Theorem 2.9] we have, providing - | S (1
1/v2)(aD DD41) that there exists an orthogonal matrix O₂ E RDxD such that
ú U0,0%
X EX
(5)
OD - DD+1
By assumption, the matrix A
Σ (Ch) (M)TCO Ch has full rank, and therefore 01...., .D ((Np)).
b=1
The matrix EX has rank 1 and so 0D+1 = O. Therefore ap 0D+1 PNN
To complete the proof, will suffice to show that | EX| Op Op(vNNN), after which we can subsequently hightmultiply equation by Q (2, T conclude the proof.
To do so, we will prove the following concentration inequality, which we prove Section B.1. Lemma 1. Let Y denote the counting measure of an inhomogeneous Poisson process with finite intensity measure A on [0, 1). Let o E ((0, and let L be a value such that ((t) S L. Then
82
P( - -06)) > 1) < 2 exp
2(A(Ø) + +1133)
In particular, for 2 A(o).
3t
P (( - A(6)) > t) exp
8L
In particular, since ||U||2,00 O(Vlog(N)/NpN) by assumption, we have that ( (log(NV) and therefore by Lemma we have that (o") - (o') (log( By union bound, this holds simultaneously for all E (1....N),. {1, {1...., B).
To obtain a bound on X - |X|12, we observe that this equal to |El|a where the symmetric dilation of X - EX [19, Theorem 7.3.3]. I.e.
E (xx_
X-EX
(x EX)
0
We then apply the following concentration inequality for random symmetric matrices to which Corollary 3.12 of Bandeira and Van Handel [6].
Lemma (Corollary 3.12 of Bandeira and Van Handel [6]). Let M an N x N symmetric matrix
whose entries maj are independent random variables which obey
E(miy) 0,
|mal < L,
Σ E(mi)) Sv
22

---

for all j. There exists a universal constant C > 0 such that for any 2 0,
P 2 2 2 + & N exp (-1
t²
(-
We then apply Lemma to E, conditional Eue (log(N) (which holds with overwhelming probability due to the above derivation) with = (llog(N)) and v = =((NNN)) O(N (since B is assumed be fixed). We obtain
X
EX||2
|B|2 Op(vNPN + log (n)) Op(vNNx)
where the final inequality follows from the assumption that Npx (n)log"n)). This completes the proof.
B.1 Proof of Lemma
For given N E IN, let x(N) ..... (N)
denote N independent Poisson random variables with rates
A = A
( n (
n NN)
n
I, N.
Then, by the definition of inhomogeneous Poisson process we have that
Y(o) Jim
X (
In addition, by a property Poisson random variables, we have that
M
( lim
(M.N)
M -0000
where r ((M,N) ..... M with success probabilities (N) /M. Therefore
(M,N) are independent and identically Bernoulli random variables
M
N
Y(δ)=
lim
lim
Σ
(M.N) m.n
Z (M.N) (M,.) n m.
M-+00 -
m
m=
Observe that
X (N)
EZ(,NN)
M
N
and define Em,
(M.N)
mm
(M.N)
which independent zero- mean random variables. < L and for sufficiently large M
E
m.m
(M.N)
Then, we have that
(M.N)
2 (
o (M.N)
Σ
Σ
E
E
(EM.M)
(M.N)
mm
Note that taking limits on both sides we have that limas lim (M A(0). Now, by Bernstein's 's inequality, we have that
Σ
M
Σ
N
E(M,N) m.n
2) s 2eep
{-
t2
((.(M,N) + Lobmax /3)
m=1 m=1
Taking M + 00 and N on both sides, obtain the desired bound.
23

---

C Proof of Theorem 4.2
Lemma 3 (Lyapunov's Central Limit Theorem (CLT) ) Let ;,, be independent random variables with finite mean and variance. for some å > 0:
[|x, - F[X,]|
0
where s = Σ [ar is the cumulated variance, then the sum Sn = Σ X; converges in distri-
bution normal distribution.
Sn - E(Sm) 4, NO)
8
Proof. The proof of Theorem 4.2 applies Lemma the family of centered random variables
Uu. up
Úe.q (o)) A (d))),
Puv
which are the 22 terms in the sum forming the numerator of Equation Section 4.3:
Sp - E[Sp = Σ Puv.
The variables Pao satisfy E[puv] = O, and their variance is given by
Ú2 Varlpur] Ú v.p n.a VarlY
U U2 up Mu (d)
This last equality follows from the fact that Poisson process projection Sn dt(d) (t) can be viewed weighted sum of independent infinitesimal Poisson increments d (t) Poisson (Nw (t) each having Poisson variance Auv (t)dt and weighted by d²()). These weights get squared in the variance, leading to:
Var ( ((or(t)Manot)) (t))AAnno = Auv (3)
We will verify the Lyapunov condition with d 1. Namely, goal is to show that
1
Σ
Ellp 3 0 N oo,
where the cumulated variance in the denominator is defined
Σ
0 02 U2.q. ( upp
NN
To do so, we upper-bound the third absolute moments and lower-bound the cumulated variance.
Lower bounding the cumulated variance.
We have that
Aue ((05))) = Φ (1)
2 aN f (t))) dt
aN 110'11
ON-
where the inequality follows by assumption.
24

---

Substituting this inequality into the previous equation yields
NN
2 ON
Σ
U u.P
U
o.p
Σ
U
Σ
U 0gg
ON
aap
KEIN]
PE(N]
= || hp
ON
ON,
where the final equality holds since the columns of U have unit norm. This shows that the cumulative variance is lower bounded by the factor aN. namely the lower bound on the intensity function.
Upper bounding the third moment of Puv. Now that we have lower bounded the cumulated variance, need upper bound the third moment of Pue- We have
E [IPwel "] = = 1
E [ Aue (6+)
By assumption, |Üapp [ rgl (UN/ (n)), and by the Cauchy-Schwee Schwwrtz incquality, we have
E [1 (0")) Anc (0 = E [oo - Ano (o"')")) (Trl""") - Awo (0"))
I
E (Ø") - Ano (db)) (Nuo( Aun (6b)))))
172
mi
The two factors the right-hand side are the second and fourth central moments of Y (Ø)), which denote as m2 and m4 respectively. These moments relate to the so-called cumulants K2 and K4 of V Buo (o"), as we have:
m2 R2
m4 K4 3x2
where K2 is the second cumulant and Ka is the fourth cumulant of the random variable (dt)) We will use Campbell's theorem from [22] to express R2 and R4. For the Poisson Processes Yau with intensity Age and any measurable function o, the cumulant generating function is given by Campbell's theorem (Equation 3.6 from [22]) by:
/ (cho00)) -1)
(A) log(E [exp aN (+))) 101))
Aun
By expanding (1) - that (d*)*Au are all compactly supported and continuous (hence integrable), we get:
Σ
(t) and applying the lincarity of the integral and the fact MT
k=1
x S d'()** (tt))Annot))d (t)
k())=
K(A) - Σ
By evaluating the second and fourth derivatives of K(A) at A O. we obtain the second and fourth cumulants
/ (t)2 (t) dt, K4 = Auo (co = / ((()) (t) dt.
Ki = Anu ((0b)))
By assumption, (t) s SN. and since &² are fixed, we have
122 / 2 (1)2 (t) s BN (1)1) BN 110b113 BNN,
/ (t)) (t) s 8N / dt e NM = 0(3x). (0(()*
25

---

Therefore
[I")v(d") - Ano (d m2 my V (1e + +333) & VBN (IN =
As result, we have
P [P =0 =0
JuN N
83/2
PN
Summing over the 22 terms, we obtain
03/2
Σ (N
Ellpur = O
N
Dividing this expression by sy gives
Σ
Ellpwo | 0
MN
3
NN
3/2
N +
N
ON
[N
which vanishes by assumption.
This shows that the Lyapunov condition is satisfied, and we can apply the Lyapunov CLT (Lemma 3) to conclude that N -> 00,
Σ
Pun
Σ
Úu. Ün. (d') Auo(("))
N
M, (N
4 10.
SN
Σ
U2 U2 a.p Auo ((bb)2)
N.
D Experimental Setup
D.1 Synthetic Data Generation
We generate two types of synthetic networks to evaluate our methods: the Erdös-Rényi (ER) blocks model and Dynamic Stochastic Block Model (DSBM). Their time-varying intensity functions are shown in Figure 4.
0.05
Erdos-Renyl
0.30
0.04
0.25
0.03
Intra rate
Inter rate
0.02
0.20
0.01
0.15
0.00
0.2
0.6
0.8
1.0
0.0
0.4
0.6
0.8
1.0
ER-Blocks model
Dynamic SBM model
Figure 4: Intensity functions for the synthetic network models. (a) ER-blocks uses a piecewise-
constant intensity with abrupt jumps. (b) DSBM distinguishes intra-community (blue) and inter- community (orange) intensities, with mid-experiment perturbation.
26

---

D.1.1
Erdös-Rényi (ER) Blocks Model
In this model, the intensity between every node pair is the same, and defined as the following piecewise-constant function:
K
K(x) sign(x)
Auo (1) = Σ
hx K(t 1n),
1m
with
(4 = {0.10, 0.13, 0.15, 0.23, 0.25, 0.40, 0.44, 0.65, 0.76, 0.78, 0.81},
(hg (4, 5. 3, 4, 5, -.2, 2.1, 4.3, -3.1, 5.1, -422}.
This model, adapted from the synthetic example from [15], simulates network with single com- munity, where the interaction intensity only depends on time, and not on other latent factors such as community assignments.
D.1.2 Dynamic Stochastic Block Model (DSBM)
In this model, the nodes are partitioned into two communities C₁ and Cz. This time the intensity function varies depending on whether the node pair belongs to the same community (genering intra- community interactions) or different communities iiner-commmunty interactions):
Aww {f (t). u, E C1 or u, C2, (iinte(()) otherwise.
As shown in Figure 4b, both the intra and inter community intensities piecewise-constant func- tions. The intra-community intensity is set much higher than the inter-community intensity except on interval [0.5, 0.75) where both intensities equal. This model simulates the temporary fusion of two communities into single one.
D.2 Hyperparameter selection
We give more details the methods used in the intensity estimation experiment and the sociated hyperparameters. We experimented with various parameters for the different method, and selected the ones which yielded the lowest MISE.
Table 1: Hyperparameter selection for different methods
Method
Parameter
ER-blocks dataset
SBM dataset
IPP-KDE
Bandwidth
0.005
0.05
IPP-Hist
Number of bins (M)
128
64
ANIE (ours)
Resolution level (J)
8
6
Significance level (a:)
0.05
0.05
D.3 Resources
Hardware used for the experiments All the experiments on a MacBook Air with an Apple MI chip with CPU cores and 8GB of RAM.
Fitting time of ANIE We report the fitting time of ANIE vs the number of nodes in Figure for different levels for the maximum resolution of the Haar basis (which modulates the size of the orthonormal basis). We observe that the fitting time of ANIE is quadratic in the number of nodes, and scales exponentially with the number levels. This underlines a limitation: grained change, the number of levels / must be large, which leads to an exponentially large number of coefficients to process. However, some optimizations could be made such as parallelizing the computation of the coefficients, using more efficient algorithm to compute the truncated SVD.
order to capture fine
27

---

10
- ==
J=
Fit time (s)
10a
10-2
200
400
1000
1200
1400
1600
1800
Number of nodes
Figure 5: Fitting time of ANIE number of nodes for different values of J.
E Effect of the Hyperparameters of ANIE
In order to better illustrate the effect the resolution on the estimation error, we ran the ANIE method simplified SBM dataset with different values. In this simplified setting, parameter- ized the model such that a resolution the ANIE method with different values of J and compared the estimation error of the linear and thresholded estimators.
sufficient to capture the intensity function. We then
Effeet of the number of levels J Figure shows the effect of the number of levels J on the estimation error. We observe that the linear estimator performs well for small values of J, but its performance degrades as J increases. In contrast, the thresholded estimator maintains a low estimation error across all values of demonstrating its robustness overfitting.
Low Rank+Thresh
40
Low Rank
30
Error
20
10
2
8
J
Figure 6: Estimation error vs number of levels for the linear and the thresholded estimator
F LAD Implementation
We use the publicly available implementation https://github. cetalll.co//hheennniie com/shenyaanHhuangLAAD to compare method with LAD. We use the default parameters of the implementation.
28

---

G Case Studies: London Bike Dataset and Enron email network
Both dataset described below may be viewed temporal networks, where continuous interactions represent trips between bike stations in London, or email exchanges between Enron employees. For both the datasets, we plot two discrete wavelet scaleograms which we here refer to as the naive and reconstructed scaleograms, by computing for each level j, the Frobenius norm of either the naive co- efficient estimates (+(g)))) ]I p or the empirical affinity coefficients ||((t)a)) [ p. It can be shown that the latter corresponds to the Frobenius norm of low-rank reconstruction of the naive coefficients. Specifically, using the orthonormality of U, we have (((()))))) || = Ü ((")))) [[.. both cases, typically observe that the wavelet power of the reconstructed affinities between latent fac- tors is more concentrated in low- www-frquenccy bands, while the naive per-edge estimates (1x(()) exhibit more energy in the high-frequency bands.
G.1 London Bike Dataset
The London Bike dataset, published by Transport for London [1], has an inherent dynamic network structure which has been previously studied for instance in [36, 41]. We consider week of data, from 1st to 8th of May 2017, and mark each starting trip from docking station u to docking station time t instantaneous event. This results in 219515 interactions between 780 nodes (bike stations). As shown on Figure 7. plotting the naive and reconstructed wavelet scalcograms for this dataset allows to identify periods and time scales during which significant structural changes occur, and to contrast these with changes that are merely due to fluctuations in exchange intensity between individual pairs of bike stations. Moreover, the estimate obtained using ANIE enables us to visualize the bike stations in a low-dimensional space using a t-SNE plot, where stations that connect similar neighborhoods at similar times appear close together. Coloring the bike stations by London borough reveals that stations from some boroughs, such as Tower Hamlets or Newham, tend to cluster tightly. In contrast, stations from boroughs like Westminster, Kensington and Chelsea, or Hammersmith and Fulham are more dispersed, indicating greater diversity in their patterns of use.
G.2 Enron Email Dataset
For the Enron dataset, shown Figure 8, find that both the naive and reconstructed scale- ograms capture changes in 2001, which marked the buildup to the company 's bankruptcy.
These changes are concentrated in specific frequency band, illustrating the ability of our method to identify the time scale at which structural shifts occur. Additionally, we observe that, in the raw scaleogram (Figure 8b), measuring change purely at the edge-level, events following 2001 are associated with changes across a wider range of frequency bands. In contrast, the reconstructed, denoised scaleogram (Figure 8c) shows less variation during this later period, suggesting that many of the post-2001 edge-level fluctuations do not correspond to substantial structural changes in the network, or at least not to the same extent during the major 2001 events.
29

---

7
800
Scale (1)
5
600
ccoeff
400
3
200
0.0
0.2
0.4
0.6
0.8
1.0
Normalized time
(a) Raw wavelet scaleogram [+b
7
250
200
Scale (1)
5
coefff
150
100
3
50
0.0
0.2
0.4
0.6
0.8
1.0
Normalized time
(b) Denoised wavelet scaleogram [$(W5N)
Boroughs
Camder
Creydon
Hackney
Hommersmith and Fulham
Islington
Kensington and Chelee
Lambeth
Southwark
Tower Hamlets
Wandsworth
Westminster
-10
-20
-10
(c) 1-SNE embedding of rows estimated subspace matrix Ü colored by the boroughs of London
Figure 7: Visualization of the ANIE results the the London Bike dataset: and denoised wavelet scaleograms, and t-SNE embedding of the estimated latent factors.
30

---

Event
Date
Description
Nov 1999
Enron launched
Feb 2001
Jeffrey Skilling takes over as CEO
3
14 Aug 2001
Kenneth Lay takes over as CEO after Skilling resigns
Nov 2001
Enron restates 3rd quarter earnings revealing massive losses
5
29 Nov 2001
Dynegy deal collapses, ending Enron's 's last hope for rescue
6
10 Jan 2002
Department of Justice confirms criminal investigation begun
7
23 Jan 2002
Kenneth Lay resigns as CEO amid investigations
Feb 2002
Lay implicated in plot to inflate profits and hide losses
9
24 Apr 2002
U.S. House passes accounting reform package in response Enron scandal
(a) Timeline of key events in the Enron scandal.
2
68
1
3
5
7
6000
5000
Scale ()
5
4000
Σ coeff
3000
2000
3
1000
2
1998-11 1999-04 1999-09 2000-01 2000-06 2000-11 2001-04 2001-09 2002-01
2002-06
Normalized time
(b) Raw wavelet scaleogram [*))))))
2
68
1
3
15
7
8000
6
5
6000
$a i
coefff)
4000
2
2000
1
0
1998-11 1999-04 1999-09 2000-01 2000-06 2000-11 2001-04 2001-09 2002-01 2002-06
Normalized time
(c) Denoised wavelet scaleogram
Figure 8: Discrete wavelet scalcogram analysis of the Enron email dataset. The top table lists the key events annotated in the scalograms below.
31


Reviews:
[{"review_id": "RjqrlVNqWp", "reviewer": "Reviewer_n6aj", "summary": "For the problem of detecting structural changes in dynamic network data, the authors propose a multi-resolution framework called ANIE (Adaptive Network Intensity Estimation). Instead of binning over fixed time intervals and comparing changes in graph structure, ANIE models interactions as Poisson processes by estimating a low-dimension subspace for node behavior, and then quantifying change in node behavior using empirical affinity coefficients. This allows for detection of both rapid and gradual changes, without static time binning.\n\nThe first stage of ANIE uses truncated SVD over empirical coefficients to estimate a global subspace matrix Subsequently, these are denoised through statistical thresholding. The empirical affinity coefficient are computed by combining node pair coefficients weighted by latent factor affinity. \n\nThe authors conduct empirical analysis using synthetically constructed networks, following Erdos-Renyi approach and stochastic block models and on real world UCI messages data. The proposed approach effectively captures perturbations of underlying intensity across various temporal resolutions, demonstrating robustness to noise.", "questions": "1. Elaborate in more detail on statistical thresholding for denosing.\n\n2. More empirical analysis on an expanded set of graph network datasets.\n\n3. The section on computational efficiency is relevant but short and vague. Please refine \"For a number of nodes below a few thousands, our method runs in a few seconds on a standard laptop.\" to provide more specific details of configurations and results.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The work is well written, the methodology is sound and the problem is well motivated. The experiments are conducted on a mix of synthetic and real world datasets and support the claims. The experiments are somewhat limited and could be made stronger with more network datasets.", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer n6aj for their time and thoughtful evaluation of our work. We are pleased that the reviewer found the manuscript well written, the motivation clear, and the problem well posed. Below, we provide a point-by-point response to the reviewer’s questions.\n\n> Elaborate in more detail on statistical thresholding for denoising.\n\nIn the context of wavelet analysis, statistical thresholding is a technique used to estimate a clean signal $\\\\hat{x}(t)$ from noisy observations $x(t)$. It generally involves three steps:\n\n1. *Analysis*: An isometric transformation  $ T $ is applied to the signal to obtain a new representation  $ w $ (e.g., via wavelet decomposition,  $ x \\\\mapsto w$), such that the underlying signal in  $ x $ is supported by only a few components of  $ w$. Examples of such transformations include the Fourier transform, wavelet transform, splinets, etc. A good transformation is one in which the signal present in  $ x $ is captured by only a small number of non-zero components in  $ w$. Book [1] discusses these sparse representations (in Chapter 1) and their applications to denoising (in Chapter 11).  \n\n2. *Statistical Thresholding*: Non-significant entries of $w$ are determined using statistical testing (hence the term statistical thresholding) where the null hypothesis is that they are 0. This results in a thresholded set of coefficients $\\\\hat{w}$ which only retains the signal components. A common approach is to assume that, under the null hypothesis $w^b = 0$, the coefficients $w^b$ follow a zero-centered Gaussian distribution with variance estimated from the data. This yields a set of $p$-values, which are compared to a confidence threshold. Since each component in $w$ corresponds to a hypothesis, multiple testing correction is typically applied to control the Family-Wise Error Rate (FWER) or the False Discovery Rate (FDR). We refer the reviewer to the seminal [2] for an example of statistical thresholding in this context.\n     \n3. *Synthesis*: Finally, the inverse transformation $T^{-1}$ is applied to obtain the denoised signal $\\\\hat{x}(t)$ from the denoised coefficients $\\\\hat{w}$.\n\nOur framework follows this three step approach, where\n1. The analysis is done by calculating the coefficients $\\\\mathbb{Y}(\\\\phi_{jk})$, estimating $\\\\hat{U}$ and then doing projection to get the empirical affinity matrix $\\\\hat{S}$.\n2. Thresholding is done globally: our procedure decides which of the $D\\\\times D\\\\times 2^J$ coefficients of $\\\\hat{S}$ are significant (i.e. rejecting the hypothesis that this coefficient is zero), and which are not.\n3. The synthesis, or reconstruction, step converts the thresholded wavelet coefficients back into signals. A low-rank approximation is then performed by multiplying the result on the left and right by $\\\\hat{U}$ and $\\\\hat{U}^T$, respectively, which serves as a form of denoising, leading eventually to a reconstructed intensity matrix.\n\n\n> More empirical analysis on an expanded set of graph network datasets.\n\nTo demonstrate the applicability of our method to a broader range of real-world datasets, we will include in the appendix of the revised version additional experiments on both the Enron email dataset and the London Santander Bike dataset (a dynamic graph dataset previously studied for instance in [3]), using data from May 1 to May 8, 2017.\n\n*Details of the additional experiments*\n\nFor both datasets, we plot two discrete wavelet scaleograms which we here refer to as the naive and reconstructed scaleograms, by computing for each level $j$, the Frobenius norm of either the naive coefficient estimates $\\lVert\\mathbb{Y}(\\phi_{jk})\\rVert_F$ or the empirical affinity coefficients $\\lVert\\hat{S}(\\phi_{jk})\\rVert_F$. It can be shown that the latter corresponds to the Frobenius norm of a low-rank reconstruction of the naive coefficients. Specifically, using the orthonormality of $\\hat{U}$, we have\n$\n\\lVert\\hat{S}(\\phi_{jk})\\rVert_F = \\lVert\\hat{U} \\hat{U}^T \\mathbb{Y}(\\phi_{jk}) \\hat{U} \\hat{U}^T\\rVert_F.\n$\nIn both cases, we typically observe that the wavelet power of the reconstructed affinities between latent factors is more concentrated in low-frequency bands, while the naive per-edge estimates $\\lVert\\mathbb{Y}(\\phi_{jk})\\rVert_F$ exhibit more energy in the high-frequency bands.\n\n\n\nFor the Enron dataset, we find that both the naive and reconstructed scaleograms capture changes in 2001, which marked the buildup to the company’s bankruptcy. These changes are concentrated in a specific frequency band, illustrating the ability of our method to identify the time scale at which structural shifts occur. Additionally, we observe that, at the edge level, events following 2001 are associated with changes across a wider range of frequency bands. In contrast, the reconstructed scaleogram shows less variation during this later period, suggesting that many of the post-2001 edge-level fluctuations do not correspond to substantial structural changes in the network, or at least not to the same extent as during the major 2001 events.\n\nIn the London Bike dataset, plotting the naive and reconstructed wavelet scaleograms allows us to identify periods and time scales during which significant structural changes occur, and to contrast these with changes that are merely due to fluctuations in exchange intensity between individual pairs of bike stations. Moreover, the $\\\\hat{U}$ estimate obtained using ANIE enables us to visualize the bike stations in a low-dimensional space using a t-SNE plot, where stations that connect to similar neighborhoods at similar times appear close together. Coloring the bike stations by London borough reveals that stations from some boroughs, such as Tower Hamlets or Newham, tend to cluster tightly. In contrast, stations from boroughs like Westminster, Kensington and Chelsea, or Hammersmith and Fulham are more dispersed, indicating greater diversity in their patterns of use.\n\n> The section on computational efficiency is relevant but short and vague. Please refine \"For a number of nodes below a few thousands, our method runs in a few seconds on a standard laptop.\" to provide more specific details of configurations and results.\n\n\nWe would like to draw attention to the timed experiment included in the appendix, where we report both the specifications of the machine used to run the experiments, along with the computation times of our method for varying numbers of nodes and different values of the hyperparameter $J$. However, we acknowledge that the sentence referenced by the reviewer should be made more specific. In the main body of the revised version, we will clarify the configuration used, report specific timing results, and include an explicit reference to the relevant experiment in the appendix.\n\n\n***References***\n\n[1] Mallat, S. G. A Wavelet Tour of Signal Processing: The Sparse Way. 3rd ed. Elsevier/Academic Press, 2009.\n\n[2] Donoho, David L., Iain M. Johnstone, Gérard Kerkyacharian, and Dominique Picard. “Density Estimation by Wavelet Thresholding.” The Annals of Statistics 24, no. 2 (1996): 508–39.\n\n[3] Sanna Passino, F., Che, Y., & Cardoso Correia Perello, C. (2024). Graph-based mutually exciting point processes for modelling event times in docked bike-sharing systems. Stat, 13(1), e660."}, {"review_id": "Drx26UznBz", "reviewer": "Reviewer_AekB", "summary": "This paper proposes ANIE (Adaptive Network Intensity Estimation), a multi-resolution framework designed to detect changes in dynamic networks across multiple temporal resolutions, including both rapid and gradual changes. The method is tested on synthetic and real-world datasets and shows improved performance over selected baselines.", "questions": "Can the method be extended to settings where the cross-sectional structure $U$ also evolves over time, such as dynamic community structures?\n\nAre there any theoretical guarantees or approximation-based discussions for controlling false alarm or miss detection probabilities under the multiple testing framework?\n\nIn addition to the baseline LAD method used for comparison, there appear to be other change-detection approaches, such as the statistics used in Ref [43] and related works; comparison with these methods—both empirically and conceptually—would strengthen the paper from a change-detection perspective.\n\nLine 240 typo: should be \"conservative.\"", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength:\nANIE effectively captures network changes at multiple temporal scales and demonstrates good empirical performance across diverse datasets.\n\nWeakness:\nThe method assumes a fixed cross-sectional structure $U$, which may limit its applicability in real scenarios, and lacks discussion of false alarm or miss detection probabilities under multiple testing.", "was_revised": true, "final_justification": "I have increased my score accordingly, as the authors' response addressed my earlier concern about the limited representation of the fixed cross-sectional structure.", "rebuttal": "We thank reviewer AekB for taking the time to review our paper. We hereby provide a point-by-point response to the questions.\n\n> Can the method be extended to settings where the cross-sectional structure also evolves over time, such as dynamic community structures?  \n\n*Note: Joint Response with reviewer HVGn.*\n\nThe assumption of a global low-dimensional subspace that is constant across time is a common one in the dynamic network modelling literature (see e.g. [1]–[6]).\n\nWhile it may seem restrictive, one should note that the dimension of the subspace $U$ may be larger than the rank of the intensity matrix at any given time, allowing for dynamic community structures. There's a sense in which the rank $D$ represents the diversity of behaviours of nodes across time.\n\nConsider the example of a two-community dynamic stochastic block model on the time interval $[0, T)$ with constant intra-community intensity $a$ and inter-community intensity $b$. This example is similar to those given in [2], [3], and [4]. Suppose that at time $T/2$, half of community one moves to community two. This model can be represented as $\\\\mathbf{\\\\Lambda}(t) = \\\\mathbf{U}\\\\mathbf{S}(t)\\\\mathbf{U}^\\\\top$ where:\n\n1. The $i$th row of $\\\\mathbf{U}$ is a one-hot vector with a one in position 1 if node $i$ belongs to community one, a one in position 2 if node $i$ starts in community one and moves to community two, and a one in position 3 if node $i$ is in community two.  \n     \n2. The matrix $\\\\mathbf{S}(t)$ is given by:  \n\n   $$ \\\\mathbf{S}(t) = \\\\begin{pmatrix} a(t) & a(t) & b(t) \\\\\\\\ a(t) & a(t) & b(t) \\\\\\\\ b(t) & b(t) & a(t) \\\\end{pmatrix}, \\\\qquad t \\\\in [0, T/2) $$\n\n   $$ \\\\mathbf{S}(t) = \\\\begin{pmatrix} a(t) & b(t) & b(t) \\\\\\\\ b(t) & a(t) & a(t) \\\\\\\\ b(t) & a(t) & a(t) \\\\end{pmatrix}, \\\\qquad t \\\\in [T/2, T) $$\n\nAt any given point in time, the intensity matrix has rank 2, and could be represented in the model on (different) 2-dimensional subspaces, yet we can represent them using a *global* subspace of dimension 3.\n\n\n> Are there any theoretical guarantees or approximation‑based discussions for controlling false alarm or miss detection probabilities under the multiple testing framework?\n\nWe thank the reviewer for this important question. We hereby clarify the meaning of false alarms or miss detections in our setting. We remind that following the subspace projection step, we have for each pair of latent factors $p,q$, scale $j$ and location $k$, a hypothesis\n$\\mathcal{H}_{j,k}^{p,q} =\\text{``} \\mathbf{S}\\_{p,q}(\\phi\\_{jk}) = 0 \\text{''}$, which indicates that the observed affinity coefficient may be attributed to noise rather than structural change between latent factors $p$ and $q$.\n\nIn this context, a false alarm (Type I error) occurs when  $ \\\\mathcal{H}\\_{j,k}^{p,q} $ is true but wrongly rejected, i.e. a noise fluctuation is misinterpreted as a genuine change. This manifests as spurious artefacts in the reconstructed affinity signal. \n\nA miss detection (Type II error) occurs when  $ \\\\mathcal{H}\\_{j,k}^{p,q} $ is false but not rejected, i.e. a real change goes undetected. This leads to missing important features (change points) in the signal. We summarize the two types of error in the following table. \n\n|  $ \\\\mathcal{H}_{j,k}^{p,q} $ | False (no change) | True (change) |\n| :---- | :---- | :---- |\n| Reject | Correct | Type I error |\n| Accept | Type II error | Correct |\n\nThe false positive rate (false alarm rate) is the probability of a Type I error; the false negative rate (miss detection rate) is the probability of a Type II error. In classical single‑hypothesis testing, one sets a significance level  $ \\\\alpha $ and chooses a threshold  $ \\\\tau_\\\\alpha $ such that $ P\\\\bigl(T\\\\>\\\\tau_\\\\alpha \\\\mid \\\\mathcal{H}_0\\\\bigr) \\\\le \\\\alpha $.\n\nIn our problem there are  $ D\\\\times D\\\\times(2^J-1) $ such hypotheses. We aim to control the total number of Type I (false alarms) and Type II (misses) errors. Two standard quantities that we can control are\n\n* FWER (Family‑Wise Error Rate): the probability of any Type I error among all tests. By testing each hypothesis at level  $ \\\\alpha/M $ (with  $ M $ total tests) instead of $\\alpha$ directly, a union bound gives\n    \n   $$\\\\mathrm{FWER} \\\\le M\\\\cdot\\\\frac{\\\\alpha}{M} = \\\\alpha.$$  \n    \n  This method often referred to as the Bonferroni correction [7] is known to be conservative and reduce detection power. In our case this manifests through some structural changes being ignored in the detection.  This is why, as mentioned in the paper, we opted for the FDR method.\n    \n* FDR (False Discovery Rate): the expected proportion of false alarms among all rejections. Let  $ R $ be the number of rejected hypotheses and  $ V $ the number of false rejections. Then, a transformation of the $p$-values known as the Benjamini–Hochberg procedure [8] ensures that\n   $$\\\\mathrm{FDR} = \\\\mathbb{E}\\\\Bigl[\\\\frac{V}{R\\\\vee1}\\\\Bigr] \\\\le \\\\alpha$$\n  This approach allows more rejections than FWER control, typically leading to greater statistical power. Since it produced better empirical results in our setting, it is the method we adopted.\n \nWhile family-wise error rate (FWER) and false discovery rate (FDR) provide theoretical guarantees for controlling Type I errors, these procedures do not directly address Type II errors or test power (defined as 1 minus the probability of Type II error). In this work, we do not provide a detailed empirical or theoretical analysis of power. However, we acknowledge that studying power across different signal-to-noise and sparsity regimes (either empirically via simulations or through asymptotic approximations) would be a valuable direction for future research.\n\nTo conclude the discussion on miss detection, it can be noted that the Z-test used in our setting is a likelihood ratio test. By the Neyman–Pearson lemma, such tests are known to be uniformly most powerful for simple hypotheses (see for instance chapter 3 of [9]). While our context may go beyond the classical setting of the lemma, this connection supports the intuition that our procedure is likely to exhibit reasonable power under appropriate conditions.\n\n\n> In addition to the baseline LAD method used for comparison, there appear to be other change-detection approaches, such as the statistics used in Ref [43] and related works; comparison with these methods—both empirically and conceptually—would strengthen the paper from a change-detection perspective.\n\nWe thank the reviewer for raising this point. In our change detection experiment, we focused on comparing with the most relevant offline spectral method, namely the Laplacian Anomaly Detection (LAD) method, given its conceptual alignment with our approach. In contrast, Ref [43] proposes an online technique based on an extension of the CUSUM statistic, which operates under a different setting and objective. In the revised related work section, we will articulate more clearly the conceptual differences between our method and [43].\n\nHowever, in response to this question and reviewer HVGn’s suggestion to strengthen the change-point detection evaluation, we will add comparisons with existing methods, focusing on those with publicly available implementations to ensure fairness and reproducibility. Specifically, we will include a comparison with TensorSplat [10], a widely used approach based on PARAFAC decomposition of the temporal adjacency tensor. \n\n> Line 240 typo: should be \"conservative.\"\n\nWe thank the reviewer for pointing out this typo and corrected it in the revised version.\n\n***References***\n\n[1] Jones, A., & Rubin-Delanchy, P. (2020). The multilayer random dot product graph. *arXiv preprint arXiv:2007.10455*.\n\n[2] Zhang, X., Xue, S., & Zhu, J. (2020, November). A flexible latent space model for multilayer networks. In *International Conference on Machine Learning* (pp. 11288–11297). PMLR.\n\n[3] Arroyo, J., Athreya, A., Cape, J., Chen, G., Priebe, C. E., & Vogelstein, J. T. (2021). Inference for multiple heterogeneous networks with a common invariant subspace. *Journal of Machine Learning Research*, *22*(142), 1–49.\n\n[4] Gallagher, I., Jones, A., & Rubin-Delanchy, P. (2021). Spectral embedding for dynamic networks with stability guarantees. *Advances in Neural Information Processing Systems*, *34*, 10158–10170.\n\n[5] Jing, B. Y., Li, T., Lyu, Z., & Xia, D. (2021). Community detection on mixture multilayer networks via regularized tensor decomposition. *The Annals of Statistics*, *49*(6), 3181–3205.\n\n[6] Agterberg, J., Lubberts, Z., & Arroyo, J. (2025). Joint spectral clustering in multilayer degree-corrected stochastic blockmodels. *Journal of the American Statistical Association*, (just-accepted), 1–23.\n\n[7] Dunn, O. J. (1961). Multiple Comparisons Among Means. Journal of the American Statistical Association, 56(293), 52–64.\n\n[8] Benjamini, Yoav, and Yosef Hochberg. “Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing.” Journal of the Royal Statistical Society: Series B (Methodological) 57, no. 1 (1995): 289–300.\n\n[9] Lehmann, E. L., and Joseph P. Romano. Testing Statistical Hypotheses. 3rd ed. Springer Texts in Statistics. Springer, 2005.\n\n[10] Koutra, D., Papalexakis, E. E., & Faloutsos, C. (2012). TensorSplat: Spotting Latent Anomalies in Time. Proceedings of the 16th Panhellenic Conference on Informatics, 144–149."}, {"review_id": "zuH9YTT7Vm", "reviewer": "Reviewer_Kwi7", "summary": "This paper proposes a point-process based dynamic network modeling approach, which models the temporal network at the scale of nodes. Moreover, the wavelet signal decomposition is transferred to deal with the local changes in the intensity functions. The theoretical analysis is provided to demonstrate the consistentcy of subspace estimation and experiments as well as case studies are conducted to evluate the performance of the proposed method.", "questions": "1）Is  the proposed method capable of online change-point detection?\n2)  In implementing the comparison method LDA,  does you perform the optimal-parameter search?\n3)  It is hard to see the degree of agreement between the predicted change-point and the ground truth  in Fig.3(b).", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The proposed method shows the adaptability to the time resolution of dynamic network and offers an subspace decomposition approach for low-rank computing.\n2. Wavelet is introduced to network dynamic analysis.\n3. The experiments on synthetic data and real data demonstrate the effectiveness of the proposed method, especially on precisely change-dynamics fitting.\n\nWeaknesses:\n1. The evaluation is limited to single change-point synthetic networks and one real-world networks. More datasets e.g., Enron email networks and Reddit that  include multiple change-points will improve the comprehensive performance evaluation of the proposed method.\n2. Quantitative measuring the success of hitting the change points and how much the method accurately identify the change points for multiple change-point case are required, besides the visualization of the results, e.g., Adjusted F1.\n3. Incorporating additional baseline methods will strengthen the comparative analysis, such as the representation learning/network comparsion based methods and time window base methods (as mentioned in the paper).", "was_revised": true, "final_justification": "The paper is well-written and demonstrates technical novelty. All my concerns have been adequately addressed, and I recommend acceptance.", "rebuttal": "We thank reviewer Kwi7 for their comments and are pleased that they appreciated the novelty of applying low-rank methods and wavelet analysis to dynamic networks. Below is a point-by-point response to the reviewer's questions.\n\n> Quantitative measuring the success of hitting the change points and how much the method accurately identify the change points for multiple change-point case are required, besides the visualization of the results, e.g., Adjusted F1.\n\nWe thank the reviewer for raising this point. Given that the considere UCI dataset contains only a limited number of two ground truth change points, quantitative metrics such as Adjusted F1 may not provide substantially more insight beyond the visualizations already included.\n\n> Is the proposed method capable of online change-point detection?\n\nThis is a good question. The ANIE method proposed in this paper is inherently offline, as it relies on analyzing a complete history of interactions to identify latent change points in network structure. However, extending ANIE to support online change-point detection represents an interesting and valuable direction for future research. Such an extension could for instance incrementally update the subspace estimates as new data becomes available, and apply multiresolution analysis to incoming batches of newly observed interactions to achieve online change detection.\n\n> In implementing the comparison method LDA, does you perform the optimal-parameter search?\n\nWe assume the reviewer is referring to the Laplacian Anomaly Detection (LAD) method, rather than LDA. For our comparison with LAD, we used the authors’ publicly available implementation on GitHub. To ensure fairness and reproducibility, we followed their dataset preprocessing steps and used the hyperparameters provided by the authors, without performing additional tuning. We would also like to note that hyperparameter optimization in this setting is particularly challenging, as change-point detection performance is difficult to quantify. According to the metrics reported in the LAD papers, their method achieves a 100% hit rate on the dataset, with the metric taking only three possible values (0%, 50%, or 100%). This discreteness makes it unsuitable as an objective for hyperparameter selection. Nevertheless, we verified our experimental results by reproducing the anomaly score profile presented in their original paper, and we used this verified result as the baseline for comparison.\n\n> It is hard to see the degree of agreement between the predicted change-point and the ground truth in Fig.3(b).\n\nWe agree that overlaying the signals from each level makes it difficult to visualize the ground-truth change-points. First, we would like to clarify what is shown in Fig. 3(b). \n\n- For each resolution level $j$, we compute a discrete time series $E_j(k)$ by taking the sum of absolute values of the empirical affinity matrix $\\hat{S}(\\phi_{jk})$, which aggregates the total magnitude of change across all pairs of latent factors on the dyadic interval $I_{j,k} = [2^{-j}k, 2^{-j}(k+1)]$:\n\n  $$\n  E_j(k) = \\lVert \\hat{S}(\\phi_{jk}) \\rVert_{1}.\n  $$\n\n- We then construct a continuous, piecewise-constant function $E_j(t)$ by setting it equal to $E_j(k)$ over each interval $I_{j,k}$, so that\n\n  $$\n  E_j(t) = E_j(k) \\quad \\text{for } t \\in I_{j,k}.\n  $$\n\n  Each line in the plot in Fig.3(b) corresponds to one such function $E_j(t)$.\n\nIn the revised version of the paper, we will clarify the explanation of the plot as described above. Additionally, we will improve Fig. 3(b) by splitting it into eight separate subplots, each representing a different resolution level. This will make it easier to see that some levels are more affected by the ground-truth change points than others. Finally, we will add an aggregate plot showing the sum across all resolution levels, namely $E(t) = \\sum_j E_j(t)$, which provides a one-dimensional representation of the amount change over time."}, {"review_id": "LF59Tzsgoh", "reviewer": "Reviewer_HVGn", "summary": "This paper proposes a novel method for change detection in evolving graphs. The main goal of the approach is to provide the flexibility to be able to capture both highly localized changes, which might correspond to bursts in time, as well as broader changes. The authors introduce a time-continuous model that they call COSIP (Common Subspace Independent Processes), which is based on a shared subspace and a time-dependent affinity function, which captures the time-dependent changes. Given this theoretical model, detecting changes in the network structure can then be reduced to extracting the temporal features of the matrix-valued affinity function. The authors thus provide a practical algorithm for estimating the two components of their model from data, and provide theoretical consistency guarantees for their estimators. One nice feature of this model is that it is quite flexible and, for example, the basis can be chosen depending on the specific scenario. Finally, the authors also show experimental evidence that their model produces meaningful results in both synthetic and real-world scenarios, by comparing their resulting method against different baselines.", "questions": "1. As a non-expert in this domain, I was surprised to see that, as the authors mentioned, “Note that, IPP-Hist is equivalent to Anie with Haar wavelet without thresholding”. I feel that this should have been mentioned earlier on, when describing the theoretical model, which is presented without any prior background. The authors do mention that the model extends the COSIE model [3] but do not reference IPP-Hist\n2. Since the model provided by the authors is an extension of the COSIE model, I would expect a comparison against that baseline in the experimental section. Why did the authors choose not to include such a comparison?\n3. I’m also wondering to what extent the assumption that the basis is shared (time-independent) in the model is realistic? I would imagine that in some datasets, there can be slow time-dependent changes that alter the entire global structure of the graph and those cannot be meaningfully captured by a universal basis, unless the size of the basis is very large. Have the authors witnessed such a scenario?", "limitations": "Yes, the authors discuss the limitations of the proposed approach in Section 6 of the paper. They mention that one limitation is the method's reliance on variance estimation for Z-score computation, which may compromise thresholding robustness, especially at high resolutions. While this highlights one particular limitation, I am also wondering about the generality and applicability of the proposed theoretical model, which would be worth discussing as well.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "First, I should say that I am not an expert in this domain and therefore it is not easy for me to judge the novelty or scope of the contribution of this paper. From my perspective, the main strength of the method is the rigorous model behind the approach, which, as the authors highlight, combines robustness against high-frequency noise, together with the ability to meaningfully capture even rapid temporal changes. In addition, the flexibility of the model (with the ability to choose the basis) and the theoretical analysis, including consistency guarantees are all strengths of the proposed approach.\n\n1. The main weakness is probably the relatively weak experimental evaluation and inconsistent choice of baselines. This includes:\nThe key comparisons in Figure 2 seem to only be given in a qualitative manner. There is no quantitative evaluation against IPP-Hist and IPP-KDE which are the two closest baselines.\n2. The evaluation on real data (Section 5.2) is given against the Laplacian Anomaly Detection (LAD) baseline and not against IPP-Hist baselines. This is somewhat strange, since LAD is now over 15 years old, so comparing *only* against such a baseline is very limiting.\n\n\nI also have questions about the novelty and relation to prior work mentioned in the Questions section. Generally speaking, the lack of consistency of the baselines against which the method, and the exposition which (at least to a non-expert like myself) does not make it entirely clear which exact components are new and which ones are borrowed from existing works, be a limitation of the proposed approach.", "was_revised": true, "final_justification": "The authors have addressed the main issues raised in my review in their rebuttal, and I'm happy to recommend this paper for acceptance.", "rebuttal": "We thank reviewer HVGn for their time and detailed comments. We were glad to hear the reviewer appreciated the rigorous model, the flexibility of the approach, and the presence of theoretical guarantees. \n\n> The key comparisons in Figure 2 seem to only be given in a qualitative manner. There is no quantitative evaluation against IPP-Hist and IPP-KDE which are the two closest baselines.\n\n\nWe would like to point out that Figs. 2(e) and 2(h) provide a quantitative evaluation of intensity estimation. In these plots, our method is compared against the described baselines using the Mean Integrated Squared Error (MISE) metric, across varying numbers of nodes. These figures are intended to complement the visual insights from Figs. 2(a)–2(f) with a quantitative comparison.\n\n> The evaluation on real data (Section 5.2) is given against the Laplacian Anomaly Detection (LAD) baseline and not against IPP-Hist baselines.\n\nThe IPP-Hist and IPP-KDE baselines where not included in the change detection experiment as these methods were not designed as change detection method but as intensity estimation and dimensionality reduction method. As a result we decided to compare with the change point detection method closest to our work, namely LAD.\n\n> This is somewhat strange, since LAD is now over 15 years old, so comparing only against such a baseline is very limiting.\n\nWe would like to clarify that the Laplacian Anomaly Detection (LAD) method used in our work was first introduced at KDD 2020 [1], with an extended version published in ACM TKDD in 2023 [2]. It appears there may be some confusion with a different method that uses the same acronym. In the revised version of the paper, we have incorporated the reviewer's suggestion, which also addresses a related question from reviewer AekB, and strengthened our evaluation of change detection by including an additional comparison with the TensorSplat method [3].\n\n> As a non-expert in this domain, I was surprised to see that, as the authors mentioned, “Note that, IPP-Hist is equivalent to Anie with Haar wavelet without thresholding”. I feel that this should have been mentioned earlier on, when describing the theoretical model, which is presented without any prior background. The authors do mention that the model extends the COSIE model [3] but do not reference IPP-Hist.\n\nThank you for your observation. To clarify, the statement refers to an overlap between very special cases of the IPP and ANIE methods. More precisely, it is established in the literature that estimating the intensity of a point process using the Haar wavelet basis at resolution level $J$ is exactly equivalent to histogram estimation with $2^J$ equal-width bins (each of width $2^{-J}$). However, this equivalence is limited to this particular configuration and does not mean that ANIE is generally a special case of IPP-Hist.\n\nIn fact, there are fundamental differences between the two methods. Notably, ANIE is a non-linear approach because it incorporates adaptive thresholding on the wavelet coefficients, whereas IPP-Hist does not. We will make sure to clarify this relationship in the manuscript to avoid any potential confusion.\n\n> Since the model provided by the authors is an extension of the COSIE model, I would expect a comparison against that baseline in the experimental section. Why did the authors choose not to include such a comparison?\n\nThank you for your question. We would like to clarify that the COSIP model, like COSIE, is not itself an estimation method, but rather a statistical modelused to analyze our proposed ANIE method. Neither COSIP nor COSIE provides an algorithm for intensity estimation.\n\nThe main similarity between COSIE and COSIP is that both assume a single subspace structure across the entire observation interval. However, COSIE is specifically designed for collections of discrete-time graphs defined on the same set of nodes, while our work addresses the continuous-time setting, which COSIE does not cover.\n\nAdditionally, the MASE method introduced in the COSIE paper can be extended only to discrete-time graphs and does not directly extend to the continuous-time context that is the focus of our work. For these reasons, we did not include COSIE/MASE in our quantitative comparisons. We will clarify this distinction in the revised manuscript.\n\n> I’m also wondering to what extent the assumption that the basis is shared (time-independent) in the model is realistic? I would imagine that in some datasets, there can be slow time-dependent changes that alter the entire global structure of the graph and those cannot be meaningfully captured by a universal basis, unless the size of the basis is very large. Have the authors witnessed such a scenario?\n\n*Note: Joint Response with reviewer AekB*\n\nThe assumption of a global low-dimensional subspace that is constant across time is a common one in the dynamic network modelling literature (see e.g. [4]–[9]).\n\nWhile it may seem restrictive, one should note that the dimension of the subspace $U$ may be larger than the rank of the intensity matrix at any given time, allowing for dynamic community structures. There's a sense in which the rank $D$ represents the diversity of behaviours of nodes across time.\n\nConsider the example of a two-community dynamic stochastic block model on the time interval $[0,T)$ with constant intra-community intensity $a$ and inter-community intensity $b$. This example is similar to those given in [6], [7], and [8]. Suppose that at time $T/2$, half of community one moves to community two. This model can be represented as $\\\\mathbf{\\\\Lambda}(t) = \\\\mathbf{U}\\\\mathbf{S}(t)\\\\mathbf{U}^\\\\top$ where:\n\n3. The $i$th row of $\\\\mathbf{U}$ is a one-hot vector with a one in position 1 if node $i$ belongs to community one, a one in position 2 if node $i$ starts in community one and moves to community two, and a one in position 3 if node $i$ is in community two.  \n     \n4. The matrix $\\\\mathbf{S}(t)$ is given by:\n $$ \\\\mathbf{S}(t) = \\\\begin{pmatrix} a(t) & a(t) & b(t) \\\\\\\\ a(t) & a(t) & b(t) \\\\\\\\ b(t) & b(t) & a(t) \\\\end{pmatrix}, \\\\qquad t \\\\in [0,T/2) $$\n   $$ \\\\mathbf{S}(t) = \\\\begin{pmatrix} a(t) & b(t) & b(t) \\\\\\\\ b(t) & a(t) & a(t) \\\\\\\\ b(t) & a(t) & a(t) \\\\end{pmatrix}, \\\\qquad t \\\\in [T/2,T) $$\nAt any given point in time, the intensity matrix has rank 2, and could be represented in the model on (different) 2-dimensional subspaces, yet we can represent them using a *global* subspace of dimension 3.\n\n***References***\n\n\n[1] Huang, Shenyang, Yasmeen Hitti, Guillaume Rabusseau, and Reihaneh Rabbany. “Laplacian Change Point Detection for Dynamic Graphs.” Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining (New York, NY, USA), KDD ’20, Association for Computing Machinery, août 2020, 349–58. \n\n[2] Huang, Shenyang, Samy Coulombe, Yasmeen Hitti, Reihaneh Rabbany, and Guillaume Rabusseau. “Laplacian Change Point Detection for Single and Multi-View Dynamic Graphs.” ACM Transactions on Knowledge Discovery from Data 18, no. 3 (2024): 1–32. Yasmeen Hitti, Guillaume Rabusseau, and Reihaneh Rabbany.\n\n[3] Koutra, D., Papalexakis, E. E., & Faloutsos, C. (2012). TensorSplat: Spotting Latent Anomalies in Time. Proceedings of the 16th Panhellenic Conference on Informatics, 144–149.\n\n[4] Jones, A., & Rubin-Delanchy, P. (2020). The multilayer random dot product graph. *arXiv preprint arXiv:2007.10455*.\n\n[5] Zhang, X., Xue, S., & Zhu, J. (2020, November). A flexible latent space model for multilayer networks. In *International Conference on Machine Learning* (pp. 11288–11297). PMLR.\n\n[6] Arroyo, J., Athreya, A., Cape, J., Chen, G., Priebe, C. E., & Vogelstein, J. T. (2021). Inference for multiple heterogeneous networks with a common invariant subspace. *Journal of Machine Learning Research*, *22*(142), 1–49.\n\n[7] Gallagher, I., Jones, A., & Rubin-Delanchy, P. (2021). Spectral embedding for dynamic networks with stability guarantees. *Advances in Neural Information Processing Systems*, *34*, 10158–10170.\n\n[8] Jing, B. Y., Li, T., Lyu, Z., & Xia, D. (2021). Community detection on mixture multilayer networks via regularized tensor decomposition. *The Annals of Statistics*, *49*(6), 3181–3205.\n\n[9] Agterberg, J., Lubberts, Z., & Arroyo, J. (2025). Joint spectral clustering in multilayer degree-corrected stochastic blockmodels. *Journal of the American Statistical Association*, (just-accepted), 1–23."}]

Author rebuttal:

Meta-review: This paper proposes a multi-resolution approach for network structure change detection and estimation. Instead of using a fixed time resolution, the proposed method uses a Poisson process model for the network structure as a function of $t$, and is therefore able to adapt to changes in network structure without pre-defining a timescale. A partial theoretical analysis is provided for the subspace estimation performance and asymptotic behavior of the algorithm.

The initial reviews of the paper is quite positive, and the discussion has resulted in a number of useful suggestions for improving the paper. Overall, the paper is above the acceptance threshold if the authors can incorporate these comments into their camera ready version.
Decision: Accept (poster)