Change Point Localization and Inference in Dynamic Multilayer Networks
Authors: Fan Wang
Kyle Ritscher
Yik Lun Kei
Xin Ma
OSCAR HERNAN MADRID PADILLA
Venue: ICLR 2026 Poster (2026)
Keywords: Dynamic networks
Multilayer networks
Change point inference
Tensor-based methods
Primary area: learning on time series and dynamical systems

Abstract: We study offline change point localization and inference in dynamic multilayer random dot product graphs (D-MRDPGs), where at each time point, a multilayer network is observed with shared node latent positions and time-varying, layer-specific connectivity patterns. We propose a novel two-stage algorithm that combines seeded binary segmentation with low-rank tensor estimation, and establish its consistency in estimating both the number and locations of change points. Furthermore, we derive the limiting distributions of the refined estimators under both vanishing and non-vanishing jump regimes. To the best of our knowledge, this is the first result of its kind in the context of dynamic network data. We also develop a fully data-driven procedure for constructing confidence intervals. Extensive numerical experiments demonstrate the superior performance and practical utility of our methods compared to existing alternatives.

Published as a conference paper at ICLR 2026
CHANGE POINT LOCALIZATION AND INFERENCE IN DYNAMIC MULTILAYER NETWORKS
Fan Wang
Kyle Ritscher
School Mathematics and Statistics University of Melbourne
Department of Statistics
University of California, Los Angeles kritscherég, wcla.elleeeo.ooo
fan. wang. 2@un me lb..bbb... edu. au
Yik Lun Kei
Xin Ma
Department of Statistics University of California, Santa Cruz ykei @ucsc.
Department Biostatistics Columbia University
xm21 @cume. . co l umbia , edd
Osear H. Madrid Padilla
Department of Statisties
University of California, Los Angeles oscar. macar..adriieeee. mandiiddstatttt ucla a. edu
ABSTRACT
We study offline change point localization and inference in dynamic multilayer random dot product graphs (D-MRDPGss) where each time point, a multilayer network is observed with shared node latent positions and time-varying, layer- specific connectivity patterns. We propose novel two-stage algorithm that com- bines seeded binary segmentation with low-rank tensor estimation, and establish consistency in estimating both the number and locations of change points. Fur- thermore, we derive the limiting distributions of the refined estimators under both vanishing and non-vanishing jump regimes. To the best of our knowledge, this is the first result of its kind in the context of dynamic network data. We also develop a fully data-driven procedure for constructing confidence intervals. Extensive nu- merical experiments demonstrate the superior performance and practical utility of our methods compared to existing alternatives.
INTRODUCTION
Statistical network analysis models entities nodes and their interactions as edges. While single- layer networks capture pairwise interactions efficiently, many real-world systems involve multiple types of interaction among the same set of nodes. Multilayer networks address this complexity by organizing these varied interactions into distinct layers over a common node set, enabling both the capture of heterogeneity and the identification of shared latent structures. practice, network structures often evolve over time. For instance, transportation networks may exhibit gradual diurnal variations sudden structural changes due accidents road closures. Detecting such sudden shifts and providing adaptive strategies, such as dynamic traffic signal control or rerouting recom- mendations, is crucial for efficient transportation management. These abrupt structural shifts are referred to as change points. This naturally falls in the territory of change point analysis.
Change point analysis is well-established area statistics concerned with detecting abrupt struc- tural changes in ordered data. It can be broadly classified into online and offline settings, depend- ing on whether data are analyzed sequentially as they are collected or retrospectively after the full dataset has been observed. In the context of dynamic networks, online change point detection has been studied in models such as inhomogeneous Bernoulli networks (e.g., Yu et al. 2021) and ran- dom weighted edge networks (c.g.,Chenn network models, including inhomogeneous Bernouli networks (c.g., |Wann et al. (021). stochastic block models (e.g., Xu and Lee 2022 Bhattacharjee et al. 2020) and random dot product graphs
Chen et al. 2024). Offline detection has been explored in various

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Published as a conference paper at ICLR 2026
Pedillaadiila al. 2022). More recently, Wang al. (2025) investigated online change point detec- tion in dynamic multilayer random dot product graphs (D-MRDPGs).
In this paper, study offline change point localization and inference for D-MRDPGs. Specifically, at each time point, we observe realization of an L-layered multilayer network, where nodes are associated with fixed but latent positions, and layer-specific weight matrices capture heterogeneous interactions across layers. These weight matrices are allowed to vary over time. Our goal to develop efficient procedures for localizing and inferring change points under this dynamic multilayer structure in the offline setting.
1.1 LIST OF CONTRIBUTIONS
The main contributions of this paper are summarized as follows.
First, to the best of our knowledge, this is the first study on offline change point detection in dynamic multilayer networks. We propose a novel two-stage procedure: (t) seeded binary segmentation with refined CUSUM statistics to generate coarse set of candidates, and (it) refinement via low-rank tensor estimation. We establish consistency for both the estimated number of change points and their locations.
Second, we derive the limit distributions of the refined estimators, revealing two distinct regimes depending whether the jump size is fixed or vanishes as the time horizon grows. To the best of our knowledge, these are the first such results the network literature. We further provide a data-driven procedure for constructing confidence intervals.
Finally, extensive numerical experiments demonstrate that our methods substantially outperform existing state-of-the-art algorithms.
1.2 NOTATION AND ORGANIZATION
For e N+ let [p] [p....,p]. For sequences {a, . ibnhne C. RR, write a, = (b,,) a, < Cb, for some constant > and all sufficiently large n, and a, = O(r,) both a, Olb,) and b, =Oaa). ). For sequence of random variables {X, Inent, X, = (a,), if
1im lim sup, P((x,, 2 Ma,) = 0. For sets and C", define the one-sided Hausdorff distance C(C'(C)= max.ec minwer Ic' c). with d(C'lC) x if either set is empty.
For E RPIXP2, let A; and A denote its ith row and jth column, and on (A) P1App (A) 2 0 its singular values. For tensors M, Q E X define (M, Q) E Dj=1 Et=1 M.j. Qij, and ||M]] (M. The mode-s matricization of M is de- noted by Ms(M) with Mı MMMM,, 1)pa+tig = M, and M,(M) E Re Pt. Tucker ra) are given by r, = rank(MM (M.)) For U, E Raxx,, the marginal multiplication ) telps) E R9 XP2XP3 with
ranks (ri, F2, (21,72,
operator 1 is defined M x U1 X2 and X3 defined analogously.
(E
The paper is organized as follows. Section introduces the D-MRDPG model, the two-stage local- ization procedure and thcoretical guarantees. Section 3 derives limiting distributions of the refined estimators and proposes data-driven method for confidence intervals. fection|4presee numerical experiments and Section (ooriiunnecnncudess. Proofs and auxiliary results are in the Appendix.
2
CHANGE POINT LOCALIZATION
2.1 PROBLEM FORMULATION
We consider the multilayer random dot product graph (MRDPG) model (Jones and Subin-Delance 2020), an extension of the random dot product graph (Young and Scheinerman| (007) to multilayer networks. Each layer is characterized by distinct weight matrix, while all layers share common set of latent positions. We focus on undirected edges, noting that the directed case is analogous.
Definition (Multilayer random dot product graphs, MRDPGs), Given a sequence of deterministic matrices Wo)Le C Rdxd, let (X,) C Rdd be fixed vectors satisfying X, W(y E 00, 1) for
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all i,j. [m],1 E [L]. An adjacency tensor A {0, 1juxxxx follows an MRDPG
b II
P[A]
II
Pl (1-11111 -al-PAa Prasi)
I=1 E1115515
b II
II
(x]W Aidd (1- X, W()) 1- A-As5a
We write A ~ MRDPG((X.) (W (Wo))ew) and denote the probability tensor by P E RoxaxL
We now extend this static model to a dynamic setting and introduce a change point framework.
Definition 2 (Dynamic multilayer random dot product graphs, D-MRDPGs). Let {x,);= C Rd be latent positions and (()(())) of mutually independent adjacency tensors {A(t))et() follows the dynamic MRDPGs if (t)) MRDPG((X.) (Wa) for [T]. We write {A(1)] ~ D-MRDPGs((X/ ((X))
C Rdxd be a weight matrix sequence. A sequence
(t)) and write (P(t) as the corresponding sequence of probability tensors.
Model Let (A())) C {0, 1)0mxL, follow D-MRDPGs as in Definition (i) Assume that there exist change points 0 TJo mh
< 7KK T +| such that for t E [T 1], { (n)())>)))) * { (Way( + 1)}-= if and only if E { the minimal spacing between two consecutive change points and assume A = (T)). (id) For each
K 1- Let A min +1](11 - -T--1) be
l [K) define the k-th jump size and normalized jump tensor as K& [IP()++1) P(m)) || and
T k
Kk (P(n++)) -P(m))), and let min K2 denote the smallest jump magnitude.
Model 1 allows abrupt changes in layer connectivity (via weight matrices), while keeping latent positions unchanged time. This framework is motivated by applications such as air transporta- tion networks (Section[4. 4.2). where nodes represent airports with relatively stable intrinsic attributes (e.g. geographical location and logistical capacity). In contrast, airline routing preferences, encoded in the weight matrices, may shift due to route optimization strategies or policy interventions. The framework can be further extended allow latent positions change the change points; see Appendix for details.
In Model Tt). we assume that the minimal spacing A between successive change points scales with the time horizon T, essentially bounding the number of changes K. This assumption can be relaxed (see Section 5 and Appendix G.I. Model D (i), the change magnitude quantified via the Frobenius norm of the difference between expected adjacency tensors. This metric is sufficiently general to accommodate both dense changes small but widespread deviations across many layers- and sparse changes large deviations concentrated in few layers. Throughout, allow all model parameters, including the number of nodes n, number of layers L, latent dimension d, jump size k and minimal spacing A to diverge with T.
2.2 CHANGE POINT LOCALIZATION ALGORITHM
In this section, we introduce two-stage procedure for offline change point localization in dynamic multilayer networks, detailed in Algorithm Stage I generates a coarse set of change point candi- dates using seeded binary segmentation and CUSUM statisties. Stage II refines them via localized scan statistics constructed using a tensor- tensor-based low-rank estimation technique. This approach builds on Wang et al. (202T) for single-layer networks and extends it to the multilayer setting.
For Stage I, we begin by defining the seeded intervals (Kovács et al. 2023) and CUSUM statistics Page 1954) for dynamic multilayer networks in Definitions 3 and
Definition 3 (Seeded intervals). Let J [C', logg for some sufficiently large absolute constant
Cj > For each E [J], define the collection of intervals Jj as J, = [([(i - 1)T2 I], [(i 1)T2-3 -2- i E [2] 1]}. The full collection of seeded intervals is defined as T Uy Jj.
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Definition 4 (CUSUM statistics). Given tensor sequence (B(t))et(T) and any s <s<t<es T, define the CUSUM statistics as
e-t
for u E [d]]ss, for u E [e]]]t]
(1)(1)= = Σ
w (u) where ())(())
V (e-s)(t-s) (-
(1)
1=s+1
(e-s)le-t)
Stage I implements a modified version of seeded binary segmentation (SBS), a computationally efficient algorithm introduced by bováas al. (022).. SBS leverages seeded intervals construct multiscale collection of candidate regions for detecting multiple change points. Within each interval,
the algorithm computes CUSUM statisties and retains time points where the statistic is maximized and exceeds predefined threshold, as preliminary change point estimators
We next define the refined scan statisties used in Stage II, based on tensor heteroskedastic principal component analysis (TH-PCA), a low-rank tensor estimation method proposed by Han et al. (2022) and detailed in Algorithm 2 in Appendix [
Definition 5 (Refined scan statistics), Ler (A' (A((t))E((T) and (B' (t())e be independent sequences
generated according to Definition [ Given {(bk, 8k, Cx [k)]]= for any k E [K] and t E (sk, Ck), we define the refined scan statistic as Des. (t) (psn. (bk)/ Psa.es (bw)llF, A sE (t)) where Pen.e (ba) TH-PCA
(bb), (d, mbr (): Vee bbe sk)/ sk),
/ee bx) Se)/(CK - 8k)) with TH-PCA detailed in Algorithm nition and mbg defined in Assumption (t).
B'
(-) defined Defi-
Stage II refines each preliminary change point estimate from Stage by locating the time point that maximizes the refined scan statistics within a local window around the initial estimate. This step em- ploys the TH-PCA procedure with an additional truncation step (see Algorithm Q Appendix (D to more accurately estimate the local expected CUSUM adjacency tensors, yielding provably improved localization accuracy.
The assumption of mutual independence among the four sequences Algorithm T imposed for theoretical convenience. In practice (and in numerical experiments in Section 4). Stage I and Stage are implemented using the same two split tensor sequences via the odd-even splitting approach. The computational cost is O(T L log" (T)) for Stage and O Lr log(n)) for Stage II. where r is the maximum input rank in TH-PCA, giving an overall cost O Lr log (TV))).
2.3 THEORETICAL GUARANTEES
This section establishes the theoretical guarantees of the proposed two-stage change point local- ization procedure (Algorithm!)) []. We begin by justifying the use of low-rank tensor estimation via TH-PCA (Algorithm 2) in Stage II through an analysis of the expected CUSUM-transforme and average adjacency tensors. While the expected averaged adjacency tensors introduced below are not used in this section, they are essential for deriving the limiting distributions in Section
For any tensors
< <
t
e
< T, define the expected CUSUM-transformee and average adjacency
pa. (1) E (B*(())
and p=-== =B(ss.), where B = s)-1
Σ B(I) (2) 1=+11
and B (-) is defined in Definition 4
Both tensors admit Tucker representations of the form patee)) (1) = S x Xx2X =x))=(()),.)))....)))"))) and Ps.e = Sx, X X2 xaQee, where X= (X,,..., X,) E Raxd and S E ddddddd with Stj, 1(1 (i (i- 1)d+}.. The matrices (+)(()) and Q are given by
(+)(()) = wf (u)Q(u), - ()-s)-1 s)
Q(t).
(3)
==++1
(=+11
where (u) is define ① and Q(u) Rlx with rows (Q(w)),
lE [4]
(4)
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Algorithm 1 Two-stage change point localization for D-MRDPGs
INPUT: Mutually independent sequences (emt) (A'(t) (B(t)em) (B((t) c
(0, 1]nxxxx, threshold т E IR=,, collection of seeded intervals J Initialise: 8 s 0, +- T, F.C+v 0
Stage I: Seeded Binary Segmentation, (B)(
for (a',8" E do
if = 3' C (s, el then
(a. = (la' + 64 16'''6'' a'),, [3' 68' ad)1
if 3 - a 2 2 then
br <- arg max <<<8 (A). B (t)) + (A (br)(
else di + -1
end if
else az <
end if
end for
I' arg maxzez OI
ag. > T then
C+-Cu (bz-), SBS (s. T.). (),,
end if'
Stage II: Local Refinement, LR(C)
(be [be) + C with 0 = bo < by < ... < bg < 0R+1 T
for 1 K do
(sk, [x] < (1(bx-1 + /2] [ [(bg + bx+1)/21]
?e + arg max, stecs Ee. (t)
D See Definition
end for
OUTPUT: 1k
To establish the low-rank structure of (t) and p (in terms of Tucker ranks, see Section 1.2 and to state theoretical guarantees for Algorithm D
we state some necessary assumptions below.
Assumption 1. Consider (XM.DDGs((X.)E. ([Wo(())w from Definition
2
(1) Let X (XI.....x,) E Rnxd Assume that rank(X) = d. o1(X)/oa(X) s C. and (x 2 Cgapyn with absolute constants C.. Cgap
T, let (=*(1)) E LLxxd be defined in 3. Denote mi." rank(Qu (t). Assume that 01 (t))(t))(mm)) (Que(t)) C. and miq (Q)e(t)) 2 Ogap with absolute constants Cgaps C2 >0.
(ii) For any 0 S s <
(iit) For any 0 8 8<6 S T, let Qae E LB be defined in 3 Denote m. rank (Q*te.
Assume that 01 (o (Q++)= < C. and omnic (Qrie) 2 Cgap with absolute constants
Cgap: C2 >0. O.
Assumption a imposes full-rank condition the latent position matrix X, requiring its smallest singular value to least order /n,, with all singular values of the same order. Since X represents latent positions rather than observed data, the full-rankness of X be interpreted condition on the knowledge of the intrinsic dimension d, ensuring that the input dimension to TH- PCA is no smaller than the true latent dimension d. Further discussion rank selection, see Wang et al. (025) and Section 4.1
Assumptions I (i) and (Mii) with (1i1) for Section 3 impose low-rank conditions the CUSUM and averaged forms of (Q()) where each Qtt) comprises the weight matrices (w (t) In Appendix E we show that, with high probability, each working interval (Sk, ex] or (Sg, én] contains exactly one change point m,. implying max m, m 5x.6%) } rank(Q((x)) + rank( for t E (8k, ex). This implicitly constraints the ranks of [k While this low-rank structure
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may not directly or transparently reflect the explicit model structure, such ambiguity is common in tensor-based models (e.g. Jing et al. 2021).
The signal-to-noise ratio (SNR) is commonly used to characterize the inherent difficulty of change point detection. We now state the SNR condition required for our theoretical guarantces.
Assumption (Signal-to-noise ratio condition). Assume that there exists a large enough absolute constans CSNR > such that KVA 2 CSNR log(T) / /LL/2 + dm + nd + Limmaxs where mmmax max rank (Q())) with Q(T))) defined
We compare Assumption 2 to its counterpart in Wang et al. (202T). When the sparsity parameter
CsNR log (TT) V for some < > Our assumption is consistent with this and extends it to the multilayer setting by accounting
1, their SNR condition (Assumption 3) becomes Kv A
for the additional complexity from multilayers and the low-rank structure of layers' weight matrices.
1 Theorem 1. Let Fik T==1 be the output of Algorithm Suppose the mutually independent adja-
cency tensor sequences (A(t))e((, (A'(t) he (B'(t) (e)) C. (0,1)
100,1)axxs 1111xxxxx.
are generated according to Definition 2 and satisfy Model Assumptions 1 ). (ii) and 2 Assume the threshold T is chosen such that cr, log3/2(T) < t < C7,2x² A, where Cr, Cr are sufficiently large and small absolute constants, respectively. We have that
Pf K = K and li - -7ul S cb. Ak E [K1] 2 11-00 CTC where Co. = C. (T)
2
and C.2 C,C are absolute constants.
Theorem iimplies that, with probability tending to 1 as T + oo, the estimated number of change points satisfies K K and the relative localization error vanishes: maxke]K] A A- -el < C,A-'x (Agg(") + 0 by Assumption
2 This establishes the consistency of Algorithm in both
detecting and localizing all change points.
Remark 1. Compared to Wang et al. (2027), which established minimax-optimat localization rates for single-layer networks, our work extends these guarantees to more complex multilayer settings without sacrificing accuracy. In contrast. Waag et al.) (2025) focused on the online setting and obtained localization rate of order x (d= + nd. + Lm log( A/ar,, where controls the Type-I ervor rate. Our approach, by comparison, achieves a substantially sharper rate of order -2 lTg(T)).
3 LIMITING DISTRIBUTIONS
Inference on change points is generally more challenging than establishing high-probability bounds on localization errors. To address this, we introduce a final refinement step. inspired by approaches such as those in Madrid Padilla et al. (2023): ; Xue et al. (2024): Xu et al. (2024).
Let (e())))(T) and (e()))e((T) be independent samples as defined in Definition Let (7k T k= be the output of Algorithm with 0 TM M
R T. For each k [K].
define the final estimators as
7i = arg min (x) = arg min
Σ IIA(1) n 1.
|A( P. 11 (5)
Su SUSACEE
Bq <CKCE
w=
u=t+1
where (3k, ēx] ((ik- + 7/2 (fh Mh 1)2] and
pi TH-PCA(B m (d, d, m m1. ) 1,,,),
(6)
with TH-PCA detailed Algorithm 2 B defined tion
and migh 1,me defined in Assump-
(i)).
Theorem 2. Let (A(t))err, herr,) (B(t) (B((t)em C (0. mxxu be mutually
independent adjacency tensor sequences generated according to D satisfying Model
Assumptions
I and Let { be defined in (5) with (ie == obtained from Algorithm 1
using
threshold satisfying condition stated in Theorem

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For k [K], if Ri 0. oc, then when T - oo, we have 17p m7ll Op(Nx 2) and
r 20 D arg min (r()), where Pk(r) = 0.
r r< 0, r= 0, r R0 20
(T mTK)
FER
T> O,
for E Z. Here, (r) and (r) are independent standard Brownian motions, and for any k' {k, k + 11. o.. = Var ((Vx, Ew (1))), where V. is the normalized jump tensor (Model 1 and Eg. (1) P(n) with (1))E i.d. MRDPG((X.)).. (1 (n)))))))))
Theorem establishes the localization emror bounds and limiting distributions for the refined change point estimators in the vanishing jump regime (K& - 0). In particular, it shows the uniform tightness 7/ (p()), which improves upon Theorem 1 by logarithmic factor and guarantees the existence of limiting distributions. To the best of our knowledge, Theorem 2 is the first to derive limiting distributions for change point estimators in network data. These limiting distributions are associated with a two-sided Brownian motion. Results for the non-vanishing jump regime (Ki +
Ple > 0) are deferred to Appendix A
3.1 CONFIDENCE INTERVAL CONSTRUCTION
Using Theorem construct data-driven (1-or a)) confidence intervals for The A E [K], in the vanishing regime, for user-specified confidence level a E (0, 1) as follows.
Step 1: Estimate the jump size and normalized jump tensor. Compute the estimated jump size
Ru | 00+ 1 ||f and the estimated normalized jump tensor Lt Rig (Pi
(.)))..)))) where P.. is defined in 66
Step 2: Estimate the variances. For each k (k, k + 1}, compute
2
( A(1) - -Ph)-1
k,k'
Thk
Step 3: Simulate limiting distributions. Let B E N+ and M E IR.. For each E
[B]. let
2d
VT
Σi=
Tr)
r < 0, r 0,
(b) arg min Pk(r,, where pk(r)
M.M-
r VT
28t,
|rrl
r > 0, 24
(b)
with independent standard Gaussian random variables {2,
(b) TM
Step 4: Construct the confidence interval. Let aa/2, /1-0/2 be empirical quantiles of {4) (b)
j
The (1 - a)) confidence interval for is given by
-i-a/2 1fAk # 0}. n7
da/2 R²
lf + 0)
The empirical performance of this procedure is evaluated in Section 4.1
NUMERICAL EXPERIMENTS
4.1 SIMULATION STUDIES
To evaluate the performance of our method (Algorithm) [ for change point detection and localiza- tion, we compare it to gSeg (Chen and Zhang 2013) and kerSeg (Song and Chen 2024). For the competitors, consider two input types: networks (nets.) and their layer- layer-wise Frobenius norms (frob.). For gSeg, we construct the similarity graph using the minimum spanning tree and apply the original edge-count scan statistics. For kerSeg, use the kernel-based scan statisties fGKCP)

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For both methods, set the significance level a 0.05. Our proposed method is referred as CPDmrdpg. Following/Waad et al. (002)) we use relatively large Tucker ranks as imputs to TH-PCA (Algorithm 2 for robustness, setting "1 72 = 15 and r3 L to compute the refined scan statistics (Definition Based on Theorem I we set the threshold T cr, er.Inv Log ( with Cr 0.1. We also assess the confidence intervals constructed utilizing the procedure in Section B.1, a capa- bility not supported by the competitors. We set 500 and M T as suggested by Xu et al.
2024).
To assess sensitivity to tuning parameters, we vary the threshold constant Cr,1 E {0.05, 0.08, 0.10, 0.12, 0.15, 0.20, 0.25) and input ranks (10, 15, 20}. We further conduct additional simulations to evaluate the robustness of our method under temporal dependence, high-frequency change points and randomly located changes, In addition, we compare our approach with existing dynamic mul- tilayer network approaches (Wang et al. 22025), which are designed for online settings, as well as with deep-learning- baaeedd approaches (Li et al. 2024). All results are reported in Appendix [..]
Performance is quantified using the following metrics: (i) Absolute error: |R K| where R and K denote the numbers of estimated and true change points, respectively; (ti) One-sided Haus- dorff distances (see Section 1.2): d(c]c) and d(C(C) where C and C denote the sets of estimated and true change points, respectively; (iit) Time segment coverage: C(G, G') = T "' |A] max |A n A'I/IA A where 9 and g' denote the partitions of the time span [1, T into intervals between consecutive true and estimated change points, respectively.
Throughout, we set the time horizon to T = 200 and the number of layers to L = 4, and consider node sizes n E {50, 100). Each setting is evaluated over 100 Monte Carlo trials. We consider two network models: the Dirichlet distribution model (DDM) and the multilayer stochastic block model (MSBM), with structural changes specified in each scenario. In the DDM, we generate latent i..d. positions { (( ~ Dirichlet(1a) with d = 5 and 1a E F denoting the all-one vector. For each time t, we sample weight matrices (W() C Rddd with entries (W (Way(t))... ~ Uniform((p:)) + 1)/(4)) (p.L + + 1)/(4))). The edge probabilities are given by P (t.ja(t) = X,T W (t), and the adjacency entries are sampled as (t,,(t) ~ Bernoulli (t.). In the MSBM, the edge probability tensor Paj,((t) E [0, 1| is defined as (t),((t) - Pı, if nodes i, E B. for some c [Ce]. and p2,1 otherwise, where [BoJce(c.] partitions the nodes into C. communities. The connection probabilities are drawn from P1,1 ~ Unifom((3.-11))/(11 (3L+ ()T()) and p2,1 ~ Uniform((21. + (4T,),,, (2I. + ()T()) The adjacency tensor (() E (0. 1 is then sampled A (1) ind. Bernoulli (!).
Scenario 1. We consider the DDM with K 2 change points at (70, 140), yielding 3 time segments [A We set P: = 2 for & E A1 Us,,, and p: = 3 with reversed layer order for t E A2-
Scenario 2. We consider the MSBM with K = 5 change points at t {20, 80, 160, 180), be evenly-sized communities and specify the changes as follows: C, = 4 for 1 E Ai, C. 2 for 1 E A2, C. 4 for / E As, C. = 4 with reversed layer order for / E A4, C. = 3 for E As and C. = 4 for E As-
sulting in 6 time segments (A.))_1- We let (B,()))
Scenario 3. We consider the MSBM with K = 3 change points at {50, 100, 150), yielding 4 time segments {A The number of communities is fixed at C, = 3 but in the first layer, the the community sizes vary across segments (0.3n, 0.4n, 0.3n) in Ai U A4, (0.4n, 0.3n, 0.3n) in A and (0.5n, 0.3n, 0.2n) in Ag. The remaining layers retain equal-sized communities.
Scenario 4. We consider the MSBM with K- 5 change points resulting in 6 time segments {A The number of communities is fixed at Ct = with equal-sized partitions, while the connection probabilities vary across segments. Specifi- cally, for
E {20, 60, 80, 160, 180),
0.1, we let P1,1 ~ Uniform (0.5 (0.21 + de .c].0.5 [0.25 + òz c)) and P2, ~ Uniform (0.21 + 8, e,, 0.25 + d, e), where d, = 0 for t E Ai U. 8, = 1 for E A U.A As and 6, 2 for t E As.
The changes in Scenarios and follow Model [ while those in Scenarios 2 and 3 do not, allow- ing us to assess the robustness of our methods. Table Presents results all four scenarios. Across most scenarios, our method demonstrates the strongest overall performance, nearly accurately esti- mating both the number and locations of change points, and remaining robust even when Model I
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Table 1: Means of evaluation metrics for Scenarios 1-4.
30
100
Scenario
Method
Kl
d(Cic)
d(C())
C(G.G")
|R - K
d(OkC) 4
d(c(C)
C(G.6")
CPDmniip
0.01
0.00
0.42
99.86%
0.00
0.00
0.00
100%
gSeg (nets.)
1.09
Inf
Inf
52.82%
1.12
Inf
52.62%
1
kerSeg (nets.)
0.10
0.00
3.12
99.13%
0.12
0.00
2.82
99.17%
gSeg (frob.)
0.52
Inf
Inf
90.12%
0.47
88.71%
kerSeg (frob.)
0.26
0.00
5.76
98.35%
0.30
0.00
98.11%
CPDmrlpg
0.00
0.00
0.00
100%
0.00
0.00
0.00
100%
gSeg (nets.)
1.60
Inf
Inf
67.68%
1.58
69.24%
kerSeg (nets.)
0.15
0.00
1.53
99.32%
0.16
0.00
99.31%
gSeg (frob.)
0.23
Inf
Inf
97.71%
0.16
0.04
1.65
99.1755 17%
kerSeg (frob.)
0.35
0.11
3.43
98.37%
0.40
0.02
4.42
97.81
CPDmrdpg
0.19
0.14
95.11%
0.00
0.02
99.98%
gSeg (nets.)
0.98
Inf
Inf
68.93%
0.69
Inf
Inf
80.10%
3
keršeg (nets.)
0.16
0.18
2.06
98.901%
0.17
0.000
3,26
99.16
gSeg (frob.)
0.92
Inf
Inf
66, 78%
0.79
Inf
Inf
72.11%
kerSeg (frob.)
0.82
48.52
5.11
73.55%
0.79
48.82
4.75
73.80%
CPDmripg
0.00
0.02
0.02
99.98%
0.00
0.00
0.00
100%
gSeg (nets.)
5.00
Inf
Inf
0.00%
4.98
Inf
Inf
0.77%
4 kerSeg (nets.)
0.36
0.14
2.65
98, 58.56%
0.34
0.08
2,93
98.47%
gSeg (frob.)
Inf
74.92%
1.86
68.57%
kerSeg (frob.)
0.40
0.05
3.71
98.12%
0.42
0.06
2.93
98.63%
Table 2: The 95% confidence interval coverage (average length) for change points across all scenar- ios.
71
Scenario
Scenario
Scenario
Scenario 4
100
100% (0.003)
100% (0.106)
76.67% (1.528)
100% (0.605)
150
100% (0.001)
100% (0.029)
95.33% (0.653)
100% (0.294)
violated. For gSeg, Frobenius norm (frob.) inputs yield better results than networks (nets.), while kerSeg performs better with networks, benefiting from its high-dimensional kernel-based design. Although both competitors exhibit low Hausdorff distances d(ClC), their higher reverse distances d(C]C) and frequent errors estimating the number of change points suggest they often detect spurious change points.
Table reports the coverage and average lengths of the confidence intervals constructed via the procedure Section 3.1 for node size E {100, 1503. The proposed method generally achieves strong coverage with reasonably narrow intervals. Coverage is lower in Scenario 3, where violations of Model [ and relatively small, layer-specific changes pose greater challenges. The performance improves with larger n as the change magnitudes Ką increase.
4.2 REAL DATA EXPERIMENTS
Our analysis incorporates two real data sets, the worldwide agricultural trade network data set pre- sented here and the U.S. air transport network data in Appendix G.2
The worldwide agricultural trade network data are available from Food and Agricultural Orga- nization of the United Nations (2022). The dataset comprises annual multilayer networks from 1986
to 2020 (T 35), with nodes representing countries and layers representing agricultural products. directed edge within layer indicates the trade relation between two countries of a specific agri- cultural product. We use the top L. = agricultural products by the trade volume and the = 75 most active countries based on import/export volume. Tuning parameters follow the setup described in Section 4.1
Table summarizes the change points detected by the proposed and competing methods for worldwide agricultural trade network data. Notably, the gSeg method fails to detect any change points after 2010, regardless of input type. Meanwhile, the kerSeg method detects change points in 1990 and 1992, which are temporally too close. In contrast, our proposed method (CPDmrdpg) iden- tifies four major change points (1991, 1999. 2005, and 2013) that align well with known geopolitical and policy-related events. Furthermore, confidence intervals, which may be constructed for the pro-

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Published as a conference paper at ICLR 2026
Table 3: Detected change points for the worldwide agricultural trade network data.
Method
Detected change points
CPDmrdpg
1991, 1999, 2005, 2013
gSeg (nets.)
1993, 2002, 2010
kerSeg (nets.)
1990, 1992, 1999, 2005, 2012
gSeg (frob.)
1993, 2002, 2009
kerSeg (frob.)
1990, 1992. 1997. 2003, 2012
Table Detected change point from Algorithm [and 95% confidence intervals via Section 3.1 for the worldwide agricultural trade network data.
Detected change points
Time point
Confidence interval
1991
6
(5.97, 6.03)
1999
(13.98, 14.02)
2005
20
(17.97, 18.05)
2013
28
(25.99, 26.06)
posed method via the procedure from Section level.
3.1 are presented in Table 95% confidence
The 1991 change point aligns with the German reunification and the dissolution of the Soviet Union, both of which triggered major political shifts that significantly affected the trade dynamics. The 1999 change point corresponds to the World Trade Organization's 's (WTO) Third Ministerial Conference, a key moment in debates on globalization, particularly regarding agricultural subsidies and tariff' reductions, with developing nations demanding fairer trade terms. The 2005 change point marks a WTO agreement to eliminate agricultural export subsidies, promoting greater equity in global markets. Finally, the 2013 change point corresponds the adoption of the WTO's Bali Package, the first fully endorsed multilateral agreement, which introduced the Trade Facilitation Agreement and key provisions on food security and tariff quota administration, significantly impacting agricultural trade.
5 CONCLUSION
In this paper, we study offline change point localization and inference in dynamic multilayer net- works setting that, to the best of our knowledge, has not been previously addressed. We propose a two-stage algorithm with consistency guarantees for estimating both the number and locations of change points. We further develop local refinement procedures, derive limiting distributions and introduce a data-driven method for constructing confidence intervals for the true change points.
The current framework assumes temporal independence, but it can be extended to incorporate tem- poral dependence structures (e.g. Padiilla et al. 20022) Cho and Owens 2023); see Appendix B for details on the framework and corresponding adjustments to theoretical analysis.
Several limitations of this work remain open for future research. First, the assumption A ((T) precludes frequent change points. This could be relaxed using alternative selection strategies such as the saroowest-over-thresholl approach (Baranowski et al.) 2019)) instead of the greedy selection in this paper. Second, our inference procedure is limited to vanishing jumps. It would be interesting to explore practical procedures for the non- non-sinishin regime, potentially building bootstrap methods (e.g. Cho and Kirch 2022).
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APPENDIX
All technical details are deferred to the Appendix. Appendix/AJeesablistee the limiting distributions in the non-vanishing regime, while Appendix B discusses the extension to incorporate temporal dependence and Appendix/Cpresente
the extension to allow changes in latent positions. Additional algorithms used in procedures are provided in Appendix [ The proof of Theorem iis given in Appendix E while the proofs of the limiting distribution results, including Theorem 2 in the main text and Theorem B] in Appendix A are presented in Appendix [F] Section 4 are collected in Appendix G
Further details and results for
A LIMITING DISTRIBUTIONS IN THE NON-VANISHING REGIME
Theorem 3. Let (A)) (E
(E(t) (B'(t)
{0. 1 be mutually
independent adjacency tensor sequences generated according to Definition satisfying Model
Assumptions I and Let {ix be defined in (5) with { R obtained from Algorithm 1 using
a threshold satisfying condition stated in Theorem 7
For k E K if Nik Pk: T + co, with Pk > 0 being an absolute constant, then when T we have 77 Tk = (p( and
-rps 2ps Eiert VR. Ex(())
<<0 0,
The Th
Dj
arg min Pa(r), where Pa(r) =
0.
O, r >> O,
rEZ
rpp 2p Σt I (1)),
for E R. Here, the normalized jump tensor Vr is defined in Model MRDDG (Wo)(m))eea))
7 and forany
any k E
[K
1] and
t E Z. (t = A (t) ((m)) with {Ax(t) 1..d.
The proof of Theorem 3 is given in Appendix P
Similar to Theorem Theorem establishes the uniform tightness = - m O1) and further derives the limiting distributions of the refined change point estimators defined in 5 which are associated with two-sided random walk.
B EXTENSION TO TEMPORAL DEPENDENCE
To incorporate temporal dependence, we modify the data-generating process in Definition 2 by introducing exponentially decaying correlations governed by a parameter æ E [0, 1]. Specifically, at time points E {T + +1 adjacency tensors sampled independently: for any 1 1 S
and E []]
(t. ~ Bernoulli (t). where ((.,(()= ((),, For l E [K]. L E [m=+] NM + 1), 1l,ssis S n and [Z]. edges evolve as follows:
At {= Ai.,,
with probability π
Bernoulli (P... (TR + 1)), with probability 1 1-
When π = O, the framework reduces to the independent case. For π > 0, the adjacency tensors exhibit dependence across time, requiring modifications the theoretical analysis.
Adjustments to theoretical analysis. Temporal dependence introduces correlations across time, which invalidate the standard concentration inequalities used our original analysis. To address this, we adapt our results using Lemma 14 from Padilla et al. (022)) which extends Bernstein's 's inequality to account for weak temporal dependence.
Two main components of our theoretical framework require modification:
Proof of Proposition 4 We redefine the event
A(s, t) = {(AA (1), B 2 (+)) [P)(()))))
CologT))
+ (1 /1--m)-log(T)nn Log(T))vV))
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By modifying the proof of Lemma S.4 in Wang et al. (2021) and applying Lemma 14 in Padilla al.
20222. we obtain that P(A(sstte)) I C,T
Proof of Theorem 1 By revising the proof of Theorem 4 in Wang al. 2025 and applying Lemma 14 Padilla et al. 2202)) together with Assumption (z) and (#t) and Lemma in Wang
al. 2025), for any (8, e), we have that
PP
e (0ll s Ci Ci (dmm + nd [m x) log(T) 2 11-7- 1-7
Then by revising the proof Lemma 5 account for temporal dependence via Lemma 14 Padilla al. (2022), and for any O, we establish that
Et
P( Vek
P(t) (1))-A((t),) Por.er (bx)) (bb))) 2E)
(bw))lF)
8k 1m
<2exp { --4 (1- (-
e/ (C3
Combining these modifications, the signal-to-noise ratio (SNR) condition becomes
kV ((A(1- T) 2 CSNR log(T) /nL/22 + d'm + nd + Lmmmx: Under this condition, have that
P K = and iik mhl S Ek. Vk E [K]] 2 21-C7 CT where EK C. log(T))
(1 T)K
Compared the independent stronger by factor of (1 - 1/2 and the localization rate worsens by factor of (1 л) m-- reflecting the impact of the temporal dependence.
(æ
shown in Theorem [ the signal-to-noise ratio is
C EXTENSION TO CHANGES IN LATENT POSITIONS
In this section, we outline an extension that allows latent positions (t) to change across segments. We redefine the D-MRDPG model
(A(t) D-MRDPGs((X())), (Wo(())))))
where both latent positions (() E Ra and layer weight matrices (W(r(1)ee1) C Rdxd are allowed vary over time. Assume that there exist change points
0 0= < <m < ... < <K < <T= T
such that for & [T'-]. 1].
((W)))(WWW) # (X(+ (
and only if E {n Let (() denote the probability tensor at time I. We retain the definition of the k-th jump size,
Kg P( P(m))lF;
where P(() is the probability tensor at time t.
For any 0 < f T, let (t) denote the expected CUSUM-ransformed tensor as in of the main text. To guarantee the Tucker low-rank structure required for TH-PCA, Assumption must be modified accordingly.
Modified assumptions: (t) Mode-1 full-rank condition. Assume that
(P*(()))))) CCa, (P*MM(P=*(()) 2 CT.
rank(M, (p* (()) d,
(Pse(()ss()))
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When (t) is time-invariant, this reduces to Assumption (t).
(i) Mode-3 low-rank condition. Let my rank(Q"((()) Assume
(MMss(P*(()) (t)) O 0 (Ms(P*.(tt)
S Co,
(MM)((()))) 2 Cgap.
This parallels Assumption (ii).
These conditions ensure the necessary low-rankness the expected CUSUM tensors. We acknowl- edge that, compared with the original assumptions, the extended conditions, particularly (i), are less direct or transparent in terms of the explicit structural properties of the network model.
Under the modified assumptions above, and provided the signal-to-noise ratio condition
KV N CSNR log (T)g(T))LLL)) //2 + dd + nd + Lmmmxx:
where Mmma maX +1] rank (M3(P mm )),, the same two-stage algorithm continues to achieve This shows that our methodology naturally extends to
the same localization rate as in Theorem | settings with time-varying latent positions without altering the core statistical guarantees.
D ADDITIONAL ALGORITHMS
We present the tensor heteroskedastic principal component analysis (TH-PCA) algorithm introduced in Han et al. (2022), incorporating an additional truncation step, in Algorithm Its subroutine, the
heteroskedastic principal component analysis (H-PCA) algorithm proposed by Zhang et al. (2022). provided in Algorithm 3
Algorithm Tensor heteroskedastic principal component analysis, TH-PCA(A, (r1, ra), T
INPUT: Tensor A I2I Xps XPs, ranks T1, T2, r3 N+ thresholds TI, T2 for E [3] do
U. H.PCA(M,
.Ta)
See Algorithm for H-PCA and Section for (.(
end for
P
A Ü,Ü x2 02Ü X3 0g0g for (ij. E [PI] x [P2] x [Pa] do
See Section 1.2 for
P.j. <- min {TI, max - P P(ja)
end for OUTPUT: P XPaXps
Algorithm 3 Heteroskedastic principal component analysis, H-PCA(E,r))
INPUT: Matrix Σ E R rank r E N+
Initialise: ((0) + Σ, diag (E)) < 0, + 5 5logfomin (i))mnnnnn)
for L E [0] [T 1] do
Singular value decomposition {(0) = Σn ol(0 (0000) a..00) ... o0(0 20
550) + E (((()) u:00 (vi
((++)) (t), (iag($((+)))
diag (1))
end for
U (u' ,.... u') from top-r left singular vectors of (T)
OUTPUT: E gaxr
E PROOF OF THEOREM 1
The proof of Theorem 1 in Appendix E.1 with all necessary auxiliary results in Appendix E.2
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E.1 PROOF OF THEOREM] |
Proof. We first define the event
nv Log
VA
A K K and bbg m/l E Ak [K1].
where ė Ca lTg(T)
R²
where {by } k= are preliminary change point estimates obtained from Stage in Algorithm by Proposition holds that
Then
(((A] 21 - CoT
and Cr Co, c > O are absolute constants. Since (A (A'O) U (B(()) are independent of {A (A))) U (B()) the distribution of (A'(1) U (B remains unaffected under the conditioning on the event A. All subsequent analysis in this proof is carried out under the event A. Consequently, can derive that
1bg nul E E
Ak E [K)
where the last inequality follows from Assumption and the fact that CSNR is a sufficiently large constant.
Step We first establish that for any l [K]. each working interval (sk. ex) contains exactly one true change point, namely Thk and the two endpoints are well separated.
From we obtain that 7)s E [ha-1: bk+1];
T]s bk 7l Th--1 |M:-1 b
A
A/6
5A/6,
and
bk+1 7/k 7/++1 T} |1+41 bk+1 A
A/6 50/6.
Similarly, we can derive that
min(by - bp- bb+1 - bx3 2 22/3.
As result, the working interval (SK; ex] = (ba-1 + (be - 1)/2, bb+1 (ba+1 - bz)/2],
contains exactly one change point M)- For any E (x), denote m meu.ch rank (aak(Qaxvon 0x (t)) with Qul.ex (t) defined in Then we have that
mi fkick my + m++11 < 2mmma:
(8)
where my = rank ((1x)) with Q(Tx) defined in (4 and Mmax = max my. In addition, we have that
by -bb by 1)2 2 A/3,
and
6k by (bb+1 - bx)/2 A/3. minfek - by. by 8k) 2 A/3.
Therefore,
Step 2. We now show that the population statistics Po (bx) provide a sufficiently strong signal within each working interval (sk, ex).
By Lemma it holds that
[ 1-8t
(cx
222
< t S Te-
x 1x
(1ls 8% 2
Th < t ex.
Tes 2N st (t-58- 5z)
Define the scaling factor
(bs - 3x (ex- bk)
A
ek 8
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Assuming without loss of generality that by mh and using obtain
min by - Sk. Ek bx}
A
2
(10)
2
6
Thus, we have that
pe (bx)
by 8k
(ek n1k)²
2
A
Pk 1/k
K/
16
(ex (ex) b₁)
Ck bk
A 17 b%
2
A
A
K
j Rn = A
(11)
Ek
bn
A/3
where the first inequality follows from
and TO
3² (bg) is independently cy-sub-Gaussian distributed Step Note that each entry the tensor B'
with mean tensor E B' [bx]} Periek (m7) and an absolute constant c, 0. By Theorem 4 and Lemma in Wang et al. (2025), and Assumption (i) and (ii), for any E (s, e), it holds that
p p ( S SGI (dmm + nd + Lmn [mng")og(())
2 21-7-
for some constants C 0 and 3. By {8 derive that
p (1s - (t) p S Cav (d' + nid + Lmx log(T))
p
2 21-7" =
where C2 > 0 a constant, Define the event
B
sup 05 <I<C ST (5L.e2) contains only one change point
ps..cstt) po (t)ls= -
CB / (d + nd + Lm log(T))
(12)
with a constant Cg > 0. By the union bound argument, holds that
P(B} 1-7--1
with constant > 0. By the event B and the triangle equality, we have that
Pesiel (oxieL(be)l p (bx)lb CB (d mmmax nd + Lmmmax) log(T) 2 A& v VA/4, (13)
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where the last inequality follows from large constant. As a consequence,
11
Assumption 2
and the fact that CSNR is sufficiently
p s (bx) Pon.ce
(bx) Pru.en (bx)/ | Poa.cy (bb)lle)
Ponee (bx)
Ponner (bx)
=2-
Pre.ch (bx)lle
IF
(pseice (bx) - poc.en (bx)) paree (bw)ll
-2
(px)) (b))))(bb)ll
Peziek (bx) (|Pekiek (bb))le (psslox(bx)ller
|pseies (bx)ll Pooreen (b:)ll
ppeeee (bx) Pokiek (bs) (/seel/
22
|Pekiek (bs))lF) Per ex (bx)llF
psr.r (bx) || Pas..c (b:) ps 2
Pse. (bx) || (px)))((b))l))
por.ek (bp || + peekee (bx) )
Pokie (bx) Pok.ek (bx)llr)
-2
Pssn.ce (bx) 1p
Psnien (be)llr
2
pst. Pek.ek (bb)llb
22
min ( [|| Pexiee (bx) ||f PoL.ex (bx)llF}
48 O82Oz (d, ie + nd + Lm, log(T)) 21,
22
K A
where the first inequality follows from the reverse triangle inequality, the third inequality follows from the definition the event B, (IT and (13) and the final inequality follows from Assumption and the fact that OSNR a sufficiently large constant. Therefore.
(prnee (bx), P (bx)/1 (ba)ll=) >
| P (ba)lE/2 KK A/V6/96
(14)
where the last inequality follows from
Step 4. Since {A'(1))= is independent of (B())) the distribution of (A'( remain
unaffected under the conditioning on the event By the truncation in the construction of PoL.es (bx)
stated in Algorithm 2 we have that
(ex - (ex-bx)(br - sk) (ek 8k))
p (bx)lox S
Combined with 13 it follows that
(e* s+) per.es (bo) Peree (bx))bbblll
CAAA V
we obtain for any =>0 for some constant C3 > 0. Applying Lemma
P(
P(t) A') Pau.c (bx) (bx)/ Pa (bc))) 2c
Vel 3h 1=
22
52cxp
-ca1 1 1
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where 0 is constant. Choosing C3 (og(T) for large enough constant C3 > 0, and applying Assumption and the fact that CANR is a sufficiently large constant, we finally derive that
P
P(t) (t) PrKee (bx) (bx)/| P (ba)lle) (ba)ll))) 2C
I 27
vex Ck
Sk b1
(15)
where C5 > 3 is a constant. similar argument also demonstrates that
P ( Porees (t) A Su (t), Por.ee (bx) P se. (ba))le) > CsVlog(T) cs, (16)
Step 5. We now consider the univariate time series defined for all t (sk, ex) as
v(() ( (A)))), Pes.cs (bx) Ps (ba))))) (ba)llF)
and
ym (t) A tarey (t), (s) (ps.)>(b))///) (bx)/||P) P (bb)lE))
Conditional on the event B. define the corresponding mean functions
f(t) E(y(t) (t), Pen et (bx) prr.e (bx))l)
and
sa (1) B(y au (1)) (p.n.ce (1), (L), (bx)/ P (bw)ll)).
The function (t) a piecewise constant on (sk, 6x] with a single change point at me-- Using 14 obtain that
(*****(() 2 K& / A/96,
Moreover, from 115.. (16)
and an union bound argument, we have that
P
sup <E>L<I<E
Σ (u(t) f(t))
Vea 8k t=
(5).0)) contains only one change point m
2 Coylog(T)
S 27
and
sup 0 <t<er (seseu) contains only one change point
ee (1) - (s 2 C3y/ooll
< 27
for constant C9 0. Applying Lemma 12 from Wang et al. (017) with À Collo(TT) (t)
follows that the estimated change point îi = arg max sy <t<e point and satisfies for large enough constant Cs > 0.
is an undetected change
log(T)
Th nel < Css
which completes the proof.
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E.2 ADDITIONAL RESULTS
Proposition 4. Let {bu]kzz denote the output of Stage I in Algorithm applied to two indepen- dent adjacency tensor sequences (A(t) (e(t) C. {0.1]=xmz generated according to Definition and satisfying Model [ [and Assumption
K
2 Suppose the threshold is chosen such that
Cr,nnyLoog
3/2 (T) < C,262
(17)
where G,,1, Cr,2 > O are sufficiently large and small absolute constants, respectively.
Then, it holds that
PfR= K and [by, mel S Ah [K]] 2 21
where
nv L (T)
= C. (Tg
x²a
K
and C₂, C. > are absolute constants.
Proof. The proof presented here is minor modification of Theorem in Wang et al. (202 l.. For completeness, we include the full details below.
For <<t<esss T, define the event
A(8, e), (A 5. ())) - P P (())) S Callog(T)
+ log 1/2 (T))TT))
where (t) is defined in (2 and C > 0 is a constant. Due to Lemma S.4 in Wang et al. 2021
holds that P(A(s, L, e)) S C₁ T for some constants Ci > 0 and ei 3. By an union bound argument, it holds that
P(A) P(A)= =P P
U
OSSCt<eST
All subsequent analysis in this proof is carried out under the event A.
We aim demonstrate that, conditioned the event A and assuming that the algorithm has accu- rately detected and localized change points so far, the procedure will also successfully identify any remaining undetected change point, if one exists, and estimate its location within error of غ. To this end, it suffices to consider an arbitrary time interval 0 S 8 < e < T that satisfies
7-- Th
Trtg < E 1144+1
q -1,
and
max { min (1)
s, - } min minn(r+eeri e, 1) S
where q -1 indicates that there is change point contained in (s,e) e) and
ny L 2/2
(T)
VA
C3 log(T)
x2
K
with absolute constant C2 O. By Assumption
2
we have that
E C2A
$ A/64,
(18)
CSNR log 1/2
log (T)
CSNR VnLA/2
where the final inequality follows that CSNR is large enough. Consequently, for any change point Th- must be that either 17 s| S ē or 17k- 2 A-E 2 34/4. This implies that min[ee-na Im sI} < ē, then The corresponds to a change point that has already been detected and estimated within an error of most during the previous induction step. We refer to a change point 7} (s, e) as undetected if
min(nk 8, Mh c) 2 3A/4.
To complete the induction step, suffices to show that SBS( e), T, J) satisfies the following tow properties: (i) It does not detect any change point within (s, e) if all change points in that
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interval have already been detected; and (it) It detects point b in (s, e) such that Iine bl S ē if there exists at least one previously undetected change point 7 in (s, e).
Step 1. Assume that there are no undetected change points within the interval (s,e). Then, for any interval (a', 33'] E J with (a',3" C (s, el, one of the following scenarios must hold: (i) The interval 3²) contains change points; (ti) The interval (a', (8)) contains exactly one change point 7 and min( - a', f - T I 2; (ziz) The interval (a', 3²) contains two change points 7 and 7,,,, and m
3' 1 S É.
We focus on analyzing the scenario (it)), as the other two scenarios are similar and more straight- forward. If scenario (tii) holds, then by (8)), have
غ
S (5' - a
This implies that the interval
(a, + 64 -
644 -a)].
contains change points. Note that pa.. (t) 0 for all t E (a,, B), since there no true change points within (a, 3). Moreover, by the event A,
a. (t), B (1)
(1)) < CAnvVLlog (
Therefore, if the input parameter T satisfies
L 3/2 (T),
Algorithm
I will correctly reject the existence of undetected change points.
Step 2. Now suppose there exists a change point 7) E (s, e) such that
min(1k 8, 7 c} 2 3A/4.
Let az, by and I' be as defined the procedure (BS((s, e), T, J). Denote I' = (a",g"1) By Lemma 8 Madrid Padilla et al. (0022), for any change point vik E (8, e) satisfying min 1k 8, e } 2 3A/4, /4, there exists interval (a', ,'1 C (s, e] containing only one 17s such that
3A/4 a' S 32/16 and Th + 36/16 S B' S T + 32/4.
Since (a, 3] = aa' + (8' - a (8' /64/ we have
7 A3/4 S a A A/8 and
me + + 7k A3/4.
On the event A, it holds that AA 0.8 (m)), B (m))) > | (ma) 11 - Calog(T) log (2
(19)
Furthermore, by Lemma it hold that
pa. (1x))1 (Mk a)(B TTp))
ki.
B -aa a
Then we can derive that
pa.9 (m)) 2 A///16 and | < 3x/A/4
(20)
Combining 19p and constant, we obtain
20 and using Assumption 2 along with fact that CsNR a sufficiently large
a (m)) B (mx) > -
4/32.
By the definition of I", it follows that
Qr- (Āa² 8" (bzz) Ba .8' (bT-)) 2 (nitee (2 /2A/32,
(21)
where
mmex max frse: min(ne- 8,e m)]] 2 31/4).
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Therefore, if the threshold 7 satisfies
7 122/32
Algorithm Tiill consistently detect the existence of any previously undetected change points within the interval.
Step 3. Suppose there exists at least one undetected change point 7/ E (8, such that
min(n: 8, 72k e} 2 3A/4.
Let oz, by and I' be defined according to the procedure SBS((s,e), e), J, and denote I 18") To complete the induction step, it suffices to establish that there exists an undetected change point The E (a",.") satisfying
min a a0, ,3" - -nk} 2 32/4,
(22)
and that
1bg. - mal Sτ. غ
(23)
Step 3.1. Proof of (22 Let
[a", = (a" " + (8" - (64) 3rd - -(3" aa"
(24)
Assume for contradiction that
max <*<8"
|pan.3" (t) (minax) //44
(25)
Then on the event A, we obtain
<8 (t), (e), .8" (8))
max
a <tccs
P 3 [f) + CA log(T)
pp" ,9" (t)lp +
max -
max <t-"
<( tie ( A//64 64 + CA log(T)) max A/8 C log (T)nLL
<(N 2 ////6 164+ + (m )2 2A/128 (N ) P 128
2//32,
where the second inequality follows from (25), the third inequality follows from Assumption 2 and the fact that CSNn is large enough constant. This contradicts the inequality (1). Thus,
p .9" (1) 2 (m )² P//64.
max a <1<8
(26)
Now, observe (i) If (a*, (8*) contains at least two change points, then 8r - a² 2 A. (ii) If it contains exactly one change point nk. but min(rk a", 8" m A/6 then by Lemma 6 we would have
|| pa 00" (0)l =1 =| 7. (p=))) (nh - a" (B* - -7k]
max <I<B
K
1rr a-
[ - 7kk 77 a" } < (I )² 2 /64, contradicting (6). Therefore, it has to be the case that min(T a², ,r m > A/64. Moreover, by 24). it holds that
2 a A/64.
Then, using Assumption 2 and the the fact that OSNR is large enough, we have that
E5 C.A
SNR log (T) CANR VnL log (T())
-1"'=a"'))66
Hence, by a similar argument as in Step 1, no previously detected change point lies in Note that by 21. there is at least one undetected change point in (a*,8*)
Step 3.2. Proof of 23 To this end, we apply Lemma S.5 Wang et 2021).
Deñin
A max
A a ." (t),
B
.99 (f))
|pa .9" [1]
(27)
<t<3
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From 26
Assumption
2 and the fact that OSNA is sufficiently large constant, it follows that
| .8' (0 2 max {Ca log(T)
| 20 (4lF.
max <I<g
max <'<t<""
CA
(28)
where > 0 is small enough constant. By the definition of the event A. we obtain
A logT))
max CLC8
| "0" (t) + log log (TT)/))))) S
max C
P "9" (+)
(29)
where the last inequality follows from (28). Note that 21 27 and 29 verify conditions (2), (3),
(4) of Lemma S.5 in Wang al. (2021) respectively. Therefore, Lemma S.5 in Wang al. (2021)
implies that there exists an undetected change point 7/s within (s, e) such that
C3AA
17 - bz S
and
11p"".." .9 (TH)
2
max StSR"
| 18 (t)lle
(30)
|a , [1)
where Cz, C1 > 0 are constants. Then combining
29 and 30 we can derive that
IT - 6--- CaCAA log(T) mmxxo=ctef pa (1)
(TT
Pa' 3' (t)ll
C4 maxa SiS0"
CoCAlog(T)
A
A Alog 1/2 (T)
(
C1
max <t<0
||Pa-. (t)llp
max gtss" |a' .3" (t)ll
CaCAlog(T)
18/A
64 log
1/2
(T))/LL
R²
VA
log 1/2 1/
(TTyL
<Cylog(T)
K
where second inequality follows form induction step and therefore, the proof.
(26 and C3 > 0 is an constant. This completes the
Lemma 5.
Let {A(t))eT() follow D-MRDPGs in Definition (wr) C lR satisfy Ete wi 1. Then for any e >> 0, it holds that
Let
RaxaxI and
wi (A(1) P(I) V/IVII+) 22) S 2cxxp (-9
E2
P
1+ V
|V maxişişT /ui
where e > is an absolute constant and Vllao = mex Vigal
Proof. By definition of the tensor inner product, have that
T
wi (A(t) - P(t), V/IVIl=)
V we (A(()) - -(P(t)),
We can derive that
T L
ΣΣΣΣ HENA
F m 22 (. =1
and
max JE
|I
V
|
h0
max
[wy]
15t5T
Since
((A()) (P(t))
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mutually independent centered Bernoulli random variables By Bernstein inequality (e.g. Theo- rem 2.8.2 Vershynim 2018). holds with an absolute constant Co > that
T Σ
( (- (-c0 1
w. (A()) P()) v/IVIIF> 25) < <2exp
No maxigIsT [.
which completes the proof.
Lemma 6. Suppose the adjacency tensor sequence (B(t) C [0,1] is generated ac- cording Definition 2 and satisfy Model] 1 For any 0 S 8 <t < T, (te((t) be defined as in If (s,e) e) contains exactly one change point Tee then for any t (s,e)
| POle
(c-s](e-i) (e m7s 2
8 < S Te
(e-2)((-s) (N 8) mk:
U <
Proof. This follows directly the definition of Pa (t)
F PROOFS OF THEOREMS 2 AND 3
Proof. Step 1. Preliminary bounds. We first define the event
K
K and
7h mel S CR. Ak [K]]. where
6₁ C. log (T)
x
Then by Theorem holds that
P[A) 21 CoT
and Cz. Co. co 0 are absolute constants. Since A holds with probability tending to 1 as T' +
we condition the remainder of the proof on A.
From A, Assumption 2 and the fact that CSNR a sufficiently large constant, we have for all k E [K]
that The 1 1]
log (T'
Uhe - mh T---1 176-1
Tik-1
>
C.
- 52/6,
(31)
x2
and
log (
7]++1 7/ 2 71 7J
|7k+1
Ti++1 2 Ce
A/6
52/6.
(32)
x²
Similarly, we can derive that for any & E [K + +]
i 7 24/3.
(33)
As result, the working interval (5x, Ex]
(īj--1 + (Th 7/- 1)/2, T/
(7++1 7/2)/2]
contains exactly one change point 7)--
Next, by Theorem 4 and Lemma 5 in Wang et al. (025 and Assumption (i) and (iiz), for any k E K + 1], we have
10-00p(
(d'ma nh + nd + Lmg, .h log(T')
1.
Mi 7
for some absolute constant C1 0. For any & E [K + by 31 and 32). each interval (7) 77k) contains at most two true change points n 1 and тą- Consequently, we have that
Ma 1.
m 1 m& m&+1 3mmax-
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Thus, it holds that
mmmxx + nd + mmx log
P - 1. l1e C2p
4a E [K 11,
(34)
iis 7k-1
with absolute constant C2 > O.
Step Characterization of bias. From 31 and 32), for any ks E [K], the interval (7 Mh+ 1) may contain one, two or three change points. One example that contains three change points is illustrated in the figure below. We analyze the biases in this case. The analyses for the other scenarios are similar but simpler and therefore omitted.
T/k
T
7]
TJk
16+17
7h+ TjK+11
In the following, we analyze three types of bias terms. Denote or = log(T), then or - oo as T oc. Observe that
P(m)) - p-n-11.0k1] |F
P(mn) 7h-1 - 7h-
P ((
1h - 7k 7 P(TK))
7he - 11
P P(
7Kk 71k-1
17k
77--1
T/k 97--1
7h- 1 71-1
P((T=) -P(m
+
77
P( P(n))lp
1/k 71k=1
7/k -
3Cc log(T)
3C. log(T)
- 2A
Kk-1 +
S oT Kk:
(35)
2Am
where the second inequity follows from the event A and 33 and the last inequality follows from Assumption 2 and the fact that OSNR is large enough constant. Similarly, have that
P - Me+
P()++1)=
+ 7T P(T++)) 7Tk
Mh+1 7h P
Mk+1
7k+11 7K
Mk+1 176+11
|| - P(1++2))
M6+1 7k
3C, log(())
Kk+1 S ar - Kk.
(36)
2
and
P P(1++1) - pd 1:: 1|
P(Tx+1) 76-1 1 Mh-1
Th ne P(
7) mh P P(
P P(
?7k 17k-1
?Tk 7
-1-1 Tik-1
P(()+11) P(nk-11)l
1)k 9k-1
P( P(m))l
176- 7K-1
17s -
3C. log(T")
(K +k) Kx S a No + Kk < CaKRs
(37)
for some constant C₃ > 0.
Step 3. Uniform tightness of kL m/l. In this step, we show that 1g np Op(1). Let r = Tix 77 and without loss of generality, assume r 2 O. Our goal is to establish that
rxe 0,(1)
If TR2 1. the conclusion holds trivially. Thus, for the remainder of the argument, assume
that rx k 2 1. Since î = + r. it follows that
Qs (Tk + r) (*(n)) $ 0.
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Now observe that
2x(m + -2(())=
Σ
||A(1) p p.m-1...l -1)((((1)) P |
I=
Σ
{IA(1) PiL- 1. Mi |A(t) - P pil--1.8e |1
+11
n +r
{|A(()) 00
|F -IA((t) - Pii 2 FF
t=
{[A(t)) PKR- 1.
||A(t) P(n
Σ {IA(0) P ||A(1) - P(m++ 1) (1
{IA(0) P(m)l - P(n++
1 11 + 11I -V--IV+
Therefore, we have that
- IT 1IT + IV < 17| + 177| +17++ I.
(38)
Step 3.1. Order of magnitude of I. We start by analyzing the term
[IA(t) - -i m
- A|())
P - - IIF)
+1
tht
P 1.kk-100
Σ
(A P Pi 1.lk-
FF
-2
(=
t= +1
Net Σ
1 15
p 1. [
Σ
(A(t) P( 1th 1.5h -p 1.56)
t=
1=
mi+r
2
Σ
(P( - puu 1.He, pr 1.
p 1.56)
t=
=1.1 21.2 21.3.
(39)
By 34 we have that
(mmmax + nd + Lmf log(T')
1 1.
me Ip SC2
Ti 7p-11
<3C (d md'maax + nd + Lmm log(T)
S aT kg.
(40)
2A
where the second inequality is by 33 and the last inequality follows from Assumption 2 and the fact that CaNn is a sufficiently large constant. This yields that
mete
- me O, xx))
(41)
t= +1
We now turn to the term 1.2 (39..
By Lemma 5 and 44),, we obtain that
[.2] = O (1'/22
P Th
n Ti |1r) Op(r 1/2 1/2kx))
(42)
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Next, by the Cauchy-Schwann Schwarz inequality, derive that p
[/.3] P()++1)
P 1:8e 1| |
1.7h
pu
1.76
1F -
Op(rar x12))
(43)
where the last inequality follows from and (40)
Combining
(39 and we conclude that
(T) = op(rk + kx))
(44)
Step 3.2. Order of magnitude of IT. We now analyze the term
===
[|A(t)) - p M
Att)
P T [1:e
(= +1
Th
Th +
Pia.8k1 11
(A(t) - -P Pu...a11 -
[=
(= +1
Σ
(A(1) (1)++1), Pic. Pit.Bal1)
t=
(=
2
Σ
(P(n++)) - Pie.ee+ Mi P
t=
[T.1 2IT.2 2II.3.
(45)
By 34
we have that
SC2 (d'mmax + nd + Lmmmax] log(T)
Mx+1
<3C2 (d (m + nd + Lmx log(T) < OeT Kk:
2
(46)
2A
where the second inequality follows from 333 and the last inequality follows from Assumption and the fact that CSNR is a sufficiently large constant. It then follows that
|
Σ
1 || Op(rar* (n)))
(47)
+=
To control IT.2, by Lemma
and 46 obtain that
[II.|] Op (12/2 0,(1122)) P i441 n l1 = 0p( k2))
(48)
Next, by the Cauchy-Schwarz inequality, we derive that
|1.3|
(r|[P((
ñi+ p pii || Op((rap -3/2 x/2222)
(49)
where the last inequality follows from 36 and (46).
Combining
445
47
48
and
49
we conclude that
17 (r(r + 1/2
(50)
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Step 3.3. Order of magnitude of III. We now analyze the term
Th tet
{IIA(t) -P
1 11 - At)) - P(m))) }
=II=
The
PP())) P 1.0k ||
(A P())), P 1. P((m))
(=
PT())) - pds 1.8k 1|
(P( - P(m)); p 1.5h. P(n)))
+r
Σ (A P(m++)) Pic- -P(N))
tm Vike
111.1 - 1 2III.3.
(51)
From 35
obtain that
| = Op(rap ( 2xt)
(52)
Using the Cauchy-Schwand inequality and again (35), we have that
|111.2| S r (r|(((()+1))) - PP)n)))pp p P(mm)le Op(ra7''13 ki))
(53)
To bound 111.3, by Lemma and (35 we get that
|111.3 = Op 1/2 m P(n))ll) Op 21/2 1/2kk))
(54)
Combining 511.
52p
53 and (44. we conclude that
| = op (TRă + 1/2 1/2 KR).
(55)
Step 3.4. Order of magnitude of IV. Consider the term
th IV = Σ
{|A(()) - P.c.ee F1
1 F
||(()) - P(
thtr
Σ
P(1++))
Pu 111 -2
Σ
(A P(m++) 1), Pic.ee P( 1)
tm
=IV.1 2/V.2.
(56)
By 36 we derive that
[VV1] Op(rTT" 2x
(57)
By Lemma and 36 we have that
[V:2 = Op (,2/2 P()++1) - Pii |1r) = a2 11xk)))
(58)
Combining (56). 57 and (58). we conclude that
IV = (r + 1/2 1/2kk))
(59)
Step 3.5. Order of magnitude of V. We now analyze the final term
+r
==
Σ
||(t))
P(m)) ||((t) P P(n++1)l
t1
tatt
Σ
[[ - P(7++1) 2
Σ
(A(t)
P(1+1),, P(T)) - P()--1))
1=
(=
=rk?
2
2V.1
(60)
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Using Lemma 5 obtain that
| = Op
(1 P(R++)))l))
Opp(21/2xx)) xxx))
(61)
Step 3.6: Combining 338
550
55
599.
600
and
61
we have for all rK 2 that
TH (1()).
Step 4. Limiting Distributions. For any E (5k, εx), define
Eo Σ w=t+1
(1)
Σ
|A(u) - I
|A(u)
(()++))ll
0=
Note that the term V defined in (38 satisfies
in rr 2x (1k)),
and hence by 38
44
55
59 and Tx = =1,(1), have that
[n< 2x (m))
{ (Ox (n r) (nx)) - 1 [L [TTT] [VV] 3 0.
Therefore, by Slutsky's theorem, it suffices to derive the limiting distributions of @n + r) (mk) as T co. We consider the two scenarios for lig-
Non-vanishing scenario. Suppose Ph. T + with 0 being an absolute constant. For O, we have that
n Σ
(m +r) - -e((m)
{1A(()
(1)++1)))) |I(()) P(m))]
t= +1+1
nik
P( P( 1)l
Mi +1+1
2 Σ (A(t)
P(nk)
(=
2, --p 2pk
Σ ( Ex(
(62)
1mr+1
with F defined in Model 1 and for any is [K 1] and Z, (t( Ax P(n)) with
{A+ (t) ii.d. MRDPG (Ton(n))e)
For r > 0, we have that when + xc.
tet
Ōx(n + 3r)-Qan -Qx(m)=
Σ
{[A(() - P(m))l - |A(t) - P()++1)
Netr
Σ
[ -
1=
nk+
+2
(A(1) P( P(1++)) - P(n)))
t= +1
2 Prp
2 + 2pk
L (t))
(63)
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By Slutsky's ' theorem and the argmin continuous mapping theorem (see e.g. Theorem 3.2.2 in Well- ner et al. 2013), we obtain
7hk 2 arg min (x.
7h
which completes the proof of part Theorem
3
Vanishing scenario. Let m = Ki noting that m - oc as + co. For O, we have that
2* + rm) (mk)
trmm
Σ
[|A(t) - P(m))ll |A(t) P(
1=
Th +r
The tum
Σ
[ P(T++1) +2
Σ
(A(1) P( P(1++1) P(n)))
(=
2 +-m
Σ (A(I) - P(T++1), VR)-
=r
By the functional central limit theorem, we have that when T - cc.
Th +m
(A(t) - P()++1), ux) 2 ok.N+1Bi(r),
[=
where B1 (r) is a standard Brownian motion and for any k [K] and K' [k,k 13, of Var (( k. Ey (1))). Consequently, as T
0c
2x + +m) [x(7))) 2 r 20k
Similarly, for r < O, we have that when T oc
Če(m + rm) - 2x (TR:) 2 - 20 2oR,
where (B2 (r) is a standard Brownian motion. Applying Slutsky's 's theorem and the argmin continuous mapping theorem (see e.g. Theorem 3.2.2 in Wellner al. 2013). we conclude that
m)) 2, arg min (r.)).
which completes the proof of Theorem 2
G ADDITIONAL DETAILS AND RESULTS IN SECTION
All experiments were run on CPU with 16GB RAM. For each synthetic scenario with node size 100, number of layers L 4 and time span T 200, the compute time about 10 hours to localize the change points and to construct the confidence intervals over 100 Monte Carlo trials. For each real data experiment, the computation time is approximately 15 minutes to perform change point localization and confidence interval construction.
G.1 ADDITIONAL RESULTS IN SECTION 4.
Sensitivity analysis: We examine the sensitivity of Algorithm to the threshold constant C7,1- Guided by Theorem I the threshold is set as 7 Cr. nonnnnLL Lo 2 The constant Cr, is calibrated by evaluating the false positive rate under an MSBM with no change points (four equally sized communities; see Section 4.1 We find that C, 0.1 yields a false detection rate of about 1%, demonstrating effective control. Smaller values (c,,1 E 0.05, 0.08) increased false detections, while
larger values (c,,1 0.12, 0.15, 0.20) reduced detection power.
30

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Table 5: Means of evaluation metries for dynamic networks simulated from Scenario varying Cr,1-
d(Clc)
d(cic) + C(G.
7
Cr,1
0.25
0.00
0.00
0,00
100%
0.20
0.00
0.00
0.00
100%
0.15
0.00
0.00
0.00
100%
50
0.12
0.00
0.00
0.00
100%
0.10
0.01
0.00
0.42
99.86%
0.08
0.25
0.00
6.68
97.80%
0.05
5.18
0.00
52.86
67.50%
0.25
0.00
0.00
0.00
100%
0.20
0.00
0.00
0.00
100%
0.15
0.00
0.00
0.00
100%
100
0.12
0.00
0.00
0.00
100%
0.10
0.00
0.00
0.00
100%
0.08
0.15
0.00
4.98
98.54%
0.05
5.02
0.00
53.84
67.56%
Table Mcans of evaluation metries for dynamic networks simulated from Scenario 2, varying Cr,1-
71
Cr,1
d(Cic)
d(cic) 4
C(G. g') 1
0.25
0.00
0.00
0.00
100%
0.20
0.00
0.00
0.00
100%
0.15
0.00
0.00
0.00
100%
50
0.12
0.00
0.00
0.00
100%
0.10
0.00
0.00
0.00
100%
0.08
0.02
0.00
0.64
99.68%
0.05
3.79
0.00
28.46
75.43%
0.25
0.00
0.00
0.00
100%
0.20
0.00
0.00
0.00
100%
0.15
0.00
0.00
0.00
100%
100
0.12
0.00
0.00
0.00
100%
0.10
0.00
0.00
0.00
100%
0.08
0.05
0.00
1.14
99.38%
0.05
3.53
0.00
28.60
76.50%
Tables 8
report results for Scenarios 1-4 with Cr,1 E {0.05, 0.08, 0.10, 0.12, 0.15, 0.20, 0.25). These results demonstrate that our method is relatively robust against the choices of C,,1-
We also assess sensitivity to the input ranks Table 9 We find that the method remains robust across these choices.
E (10, 15,20) 20} with results shown in
Frequent change points: To assess performance under increasingly frequent changes, we conduct simulations with varying numbers of change points K {2, 12). The results are summarized in Table 10 As shown in Table although localization accuracy decreases slightly K increases, all change points are consistently detected, demonstrating the robustness of the method even in high-frequency settings.
To evaluate performance under randomly located structural changes, we conduct simulations in which change point locations randomly sampled in each iteration rather than equally spaced. The number of change points is set to two for Scenario and five for Scenario 2. The results summarized in Table II We conclude that even with randomly located change points, our method demonstrates good and robust performance, outperforming the competitors in most settings.
Random change points:
31

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Table 7: Means of evaluation metries for dynamic networks simulated from Scenario 3, varying Cr,1-
72
Cr,1
K
d(C)c) +
d(cic) +
C 1
0.25
1.00
50.00
0.00
75.00%
0.20
0.96
48.00
0.00
76.00%
0.15
0.64
32.00
0.00
84.00%
50
0.12
0.39
19.58
0.08
90.17%
0.10
0.19
9.64
0.14
95.11%
0.08
0.09
4.30
0.50
97.61%
0.05
4.27
0.36
32.54
71.55%
0.25
0.43
21.50
0.00
89.25%
0.20
0.15
7.52
0.02
96.23%
0.15
0.00
0.02
0.02
99.98%
100
0.12
0.00
0.02
0.02
99.98%
0.10
0.00
0.02
0.02
99.98%
0.08
0.06
0.02
1.08
99.57%
0.05
4.04
0.02
32.76
73.62%
Table 8: Means of evaluation metrics for dynamic networks simulated from Scenario 4, varying Cr
72
Cr,1
K
d(cic) +
d(Cic) 4
C(G. 1
0.25
2.67
83.20
0.00
62.47%
0.20
1.19
28.40
0.00
85.63%
0.15
0.13
2.60
0.00
98.67%
50
0.12
0.01
0.22
0.02
99.88%
0.10
0.00
0.02
0.02
99.98%
0.08
0.00
0.02
0.02
99.98%
0.05
0.75
0.02
11.94
93.36%
0.25
0.01
0.20
0.00
99.90%
0.20
0.00
0.00
0.00
100%
0.15
0.00
0.00
0.00
100%
100
0.12
0.00
0.00
0.00
100%
0.10
0.00
0.00
0.00
100%
0.08
0.00
0.00
0.00
100%
0.05
0.89
0.00
12.46
92.55%
Temporal dependence: We further evaluate robustness under temporal dependence by modifying Scenario 1. Instead of sampling (t) ~ Bernoulli(P independently across time, we generate temporally dependent edges follows:
Bemooulli ((1 ( (t,a((t 1))= Pay,(t 1)),
(t,,((t) 1,
(t + 1)
Bernuulii (P.,,, + 1)(1 -))
(t.j.(t) = 0;
with π (0.005, 0.01, 0.02) controlling dependence strength. Results in Table 12 show that our method maintains strong performance when temporal dependence is weak (л S 0.0I) and degrades gradually as dependence increases.
Comparisons with other competitors: The method of Wang et al. 2025 denoted CPDonline, designed for online change point detection in dynamic multilayer networks. The method of Li al. (202)). denoted AutoCPD, is deep learning based and typically requires myriad labeled data indicating whether a time window contains a change point to train a classifier, which is not available in our fully unsupervised setting.
Nevertheless, we conduct simulations comparing our method with CPDonline and AutoCPD. For AutoCPD, trained the method using simulated labeled data and then applied it testing dataset. As shown in Table our method (CPDmrdpg) outperforms CPDonline and AutoCPD across nearly all evaluation metrics for both small and moderate-sized networks.
32

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Table 9: Means of evaluation metrics for dynamic networks simulated from Scenarios 1 and 3 with 50, L = T3 = L, varying input ranks 11
12 r.
Scenario
r
K
K
d(C'c) +
d(cic) 4
C( +
10
0.00
0.00
0.00
100%
15
0.01
0.00
0.42
99.86%
20
0.00
0.00
0.00
100%
10
0.15
7.80
0.30
95.96%
3
15
0.19
9.64
0.14
95.11%
20
0.05
2.70
0.20
98.55%
Table 10: Means of evaluation metrics for dynamic networks simulated from Scenario with n 100 and L 8, varying K.
K
K
K|
d(cic) 4 d(cic)
C(G,
2
0.00
0.00
0.00
100%
0.00
1.00
1.00
96.08%
12
0.00
1.00
1.00
94.19%
G.2 ADDITIONAL DETAILS AND RESULTS IN SECTION
This section provides a detailed analysis of the U.S. air transportation network data, evaluates the performance of competing methods (introduced in Section 4.1, and presents the constructed confi- dence intervals using the procedure in Section 3.1
The U.S. air transportation network data consist of monthly data from January 2015 to June 2022 (T = 90) and available from Bureau of Transportation Statisties (2022). Each node corresponds to an airport and each layer represents a commercial airline. A directed edge a given layer indi- cates direct flight operated by a specific commercial airline between two airports. We choose the
L = 4 airlines with the highest flight volumes and the n = 50 airports with the most departures and arrivals.
Table 14 reports the detected change points for all methods. Although the kerSeg method using networks as input demonstrates good performance in the simulation study, it detects an excessive number of change points in this real data experiment, making the results unreliable and raising concerns about false positives. Similarly, the kerSeg method that uses layer-wise Frobenius norms as input has detected change points that are too close, yielding clusters of change points that could potentially be grouped together. On the contrary, the gSeg method that uses the Frobenius norms as input detects too few change points, while the gSeg method using networks input has detected too many change points. The proposed CPDmrdpg method (Algorithm I identifies five change points that align well with known disruptions and policy changes in the aviation sector, discussed below.
The change point in December 2015 coincides with increased regulatory scrutiny over airline consol- idation, following concerns raised by the American Antitrust Institute about reduced market com- petition after series of mergers. The June 2017 change point aligns with the proposal of the Aviation Innovation, Reform and Reauthorization Act, which advocated for privatizing air traffic control and influenced route planning among carriers. Moreover, the February 2019 change point follows the U.S. government shutdown (December 2018- January 2019), which caused Transporta- tion Security Administration staffing shortages and significant operational disruptions, prompting stabilization efforts the months that followed. Lastly, the most significant structural disruptions emerged in February 2020 and February 2021, aligning with the initial shock and continued fallout of the COVID-19 pandemic, which triggered widespread flight cancellations, demand collapse and structural reconfiguration in the aviation industry.
Table 15 reports the detected change point from Algorithm and the 95% confidence intervals constructed using the procedure in Section 3.1
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Table 11: Means of evaluation metrics for networks simulated from Scenarios 1 and 2 under domly located change points.
n
Scenario Method
K K
d(ci)) L
d(cic)
C(),
CPDmrdpg
0.05
2.58
0.77
98.84%
gSeg (nets.)
1.40
Inf
Inf
31.82%
kerSeg (nets.)
0.25
Inf
Inf 94.53%
gSeg (frob.)
0.60
Inf
Inf
83.10%
kerSeg (frob.)
0.35
Inf
Inf 93.93%
50
CPDmrdpg
0.59
7,48
0.94
95.87%
gSeg (nets.)
1.90
35.55
5.65
82.96%
2
kerSeg (nets.)
0.25
3,05
1.25
98.23%
gSeg (frob.)
0.50
Inf
93.15%
kerSeg (frob.)
0.60
0.45
10.20
96.43%
CPDmrdpg
0.05
2.57
1,41
98.54%
gSeg (nets.)
1.40
Inf
Inf
29.26%
kerSeg (nets.)
0.20
Inf
Inf
94.25%
gSeg (frob.)
0.65
Inf
Inf
78.30%
kerSeg (frob.)
0.40
Inf
Inf 96.68%
100
CPDmrdpg
0.43
5.03
0.95
96.80%
gSeg (nets.)
1.75
Inf
Inf
80.47%
kerSeg (nets.)
0.40
2.15
1.55
98.81%
gSeg (frob.)
0.35
0.90
2.80
98.06%
kerSeg (frob.)
0.30
0.25
4.80
97.97%
Table 12: Means of evaluation metrics for dynamic networks simulated from Scenario with 50, varying levels of temporal dependence π.
π
K K 4
d(CiC)
d(CIC)
C(G g') t
0.000
0.01
0.00
0.43
99.86%
0.005
0.05
0.00
0.20
99.90%
0.010
0.60
0.00
15.00
93.90%
0.020
5.50
0.00
57.00
62.60%
Table Means of evaluation metrics for dynamic networks simulated from Scenario comparing CPDmrdpg with competing methods.
Method
K K 4
d(C)c) 4
d(CiC)
C(g.g") 1
CPDmrdps
0.01
0.00
0.42
99.86%
50
CPDonline
0.00
3.00
3.00
95.13%
AutoCPD
0.00
0.69
0.69
99.12%
CPDmrdpg
0.00
0.00
0.00
100%
100
CPDonline
0.00
2.00
2.00
97.05%
AutoCPD
0.97
0.89
16.17
88.29%
Table 14: Detected change points for the U.S. air transportation network data.
Method
Detected change points
CPDmrdpg
2015-12, 2017-06, 2019-02, 2020-02, 2021-02
gSeg (nets.)
2015-11, 2016-10 10, 2017-09, 2018-09, 2019-09, 2020-10, 2021-08
kerSeg (nets.)
2015-11,201610 2016-10, 2017-05, 2017-09, 2018-05, 2018-10
2019-03, 2019-09, 2020-03, 2020-10, 2021-03, 2021-09
gSeg (frob.)
2015-11, 2020-01, 2021-03
kerSeg (frob.)
2015-11, 2017-10, 2020-01, 2021-03, 2021-05 105, 2021-09, 2022-01
34

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Table 15: Detected change point from Algorithm and 95% confidence intervals via Section 3.1 for the U.S. air transportation network data.
Detected change points
Time point
Confidence interval
2015-12
12
(11.55, 12.11)
2017-06
30
(28.79, 30.98)
2019-02
50
(49.67, 53.22)
2020-02
62
(59.66, 60.36)
2021-02
(73.58, 71.27)
35


Reviews:
[{"review_id": "0raw8QtqbU", "reviewer": "Reviewer_8CQ5", "summary": "This paper proposes a two-stage algorithm for offline change point localization in dynamic multilayer random dot product graphs. Stage I uses seeded binary segmentation with CUSUM statistics; Stage II refines estimates via tensor heteroskedastic PCA. The authors establish consistency for change point detection/localization, derive limiting distributions under vanishing and non-vanishing jump regimes, and provide a data-driven confidence interval procedure. Experiments on synthetic and real networks demonstrate strong performance.", "questions": "Please refer to the question in the Weakness section.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The theoretical contributions are novel, deriving limiting distributions for change point estimators in network data. The localization rates match minimax optimality for single-layer networks while extending to multilayer settings.\n\n2. The paper provides consistency results, limiting distributions, a data-driven confidence interval procedure, and extensive robustness checks.", "weaknesses": "While online change point detection in multilayer networks has been recently studied (Wang et al., 2025), the paper does not clearly articulate what specific technical challenges arise in the offline setting or why such an extension from online methods is non-trivial.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "W6j65Od552", "reviewer": "Reviewer_yKYp", "summary": "The paper proposes and analyzes a new moethod for offline change point localization and inference in dynamic multilayer networks (following the D-MRDPGs model). They prove consistency of their estimator, and derive limiting distributions of the refined estimators in general regimes. They provide methods for computing confidence intervals, and demonstrate the favourable performance of their estimator on both synthetic and real data.", "questions": "- Middle of page 3: “We allow all model parameters … to diverge with T”. Please clarify this, how do they relate to T exactly?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The paper is original in that they consider a model that has not been studied before. This model is well motivated from the literature and appears in practice. To the best of their (and my) knowledge, this is the first result of its kind in the context of dynamic network data. \n\nThe paper is well written, they state the scalings very clearly which is important in the area of change points. Their theory is sound, and they use standard techniques/sub-methods from the change point literature such as CUSUM statistics. They also acknowledge potential improvements in the conclusion section which are sound. \n\nThe contribution is significant to some extent, as the experiments indicate an experimental improvement over previous models that seems to come from the fact that they are looking at a more specific problem than competing methods (some competing methods are nonparametric). However, I must mention that the techniques are not in themselves novel (at least at first glance), but it would be good if the authors could comment on this.", "weaknesses": "From my understanding, competing methods only work change point detection in single layer dynamic networks, or they are inherently nonparametric methods. Can these be adapted to the multilayer setting? Can you comment on how fair this comparison is? It is not clear to me at this point. \n\nAlso, implications from the experimental results could be made more clear, for example, by including figures in the main body of the text. You state that “across all scenarios, our method achieves best performance” but that does not seem to be true, as if I dig into Table 3 in Appendix F.1, it seems that in the n=50 case you do not clearly outperform kerSeg (nets.). Could you comment on this, and at least make it clear in the paper that you do not outperform in all cases, and maybe some intuition as to why? Showing some of these results graphically also would help for clarity. \n\nThe comparison with other methods on real world data should be made more clear (in Appendix F.2 for example), and brought into the main paper. This is very important, and can help better clarify the benefit of your method, which is currently not so clear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aWHQomYAsP", "reviewer": "Reviewer_USG5", "summary": "This paper develops a two-stage CUSUM pipeline for change-point detection in dynamic multilayer random dot product graphs. Stage-1 performs seeded/binary segmentation with a tensor CUSUM; Stage-2 applies a low-rank TH-PCA/matched-filter refinement that exploits shared latent positions across layers, yielding a $\\sqrt{L}$ SNR gain and $O(\\kappa^{-2})$ localization rate under large minimal spacing and global layer-weight changes. Proofs establish consistency and limit laws; simulations and small real data are provided.", "questions": "1. Comparison with $\\ell_0$ approaches. Please include a discussion comparing against spacing-free $\\ell_0$ methods. Proper references should be cited and compared; I list some, but I think more can likely be found.\n\n2. I think including a vectorized version of multiplayer networks and then using binary segmentation is worth being included as a baseline to see how much the improvement in stage 2 is. \n\n3. To demonstrate robustness beyond the current (near-)even spacing and global-weight-jump setting, please add stress tests that randomly change times i.i.d. from a distribution. \n\n\nThe paper has very strong assumptions, and the model is not very applicable given those assumptions. However, the paper is technically solid, and the TH-PCA refinement plus distributional results are valuable. I would not oppose acceptance if proper questions are addressed or at least discussed.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper is clearly written; the model, assumptions, and the two-stage algorithm are easy to follow.\n\nClean multilayer insight: the Stage-2 projection/matched filter makes precise how low rank and layer pooling provide $\\sqrt{L}$ SNR gains.\n\nTechnical development is careful (ensuring consistency and limiting distribution for the refined estimator); a standardized pipeline (screen then refine) is appealing.", "weaknesses": "1. $\\textbf{Scope (extension vs.~novelty):}$ The contribution appears closely aligned with prior multilayer RDPG pipelines [1] that (i) \\emph{screen} for candidate changes via CUSUM/binary segmentation–type statistics and then (ii) refine via a low-rank projection step (and the target is similar). In the current draft, the primary difference seems to be a regime shift (online $\\rightarrow$ offline) while keeping the same modeling assumptions (multilayer RDPG/GRDPG) and the same screen–then–refine structure and target. As written, this reads more like an extension to the offline setting than a fully new methodological framework.\n\n\n2. $\\textbf{Narrow regime.}$ The methodology focuses on time-invariant latent positions with only changes in global layer weights. This means that the communities of nodes remain consistent over time; in this scenario, the change points appear to be narrowly defined, as they cannot disrupt the community structure of the networks. Many practical settings involve node-localized changes alongside shifts in community structure, where the latent positions $X(t)$ should also be able to vary (See [2] for an example). The current scope does not account for these regimes. \n\n\n3. $\\textbf{Spacing dependence.}$ The paper assumes a strong gap at the order $T$--with a gap at the order of $T$ and the number of change points is $O(1)$, spacing-free $\\ell_0$ approach can already achieve $\\textbf{exact selection}$ and the same $O(\\kappa^{-2})$ localization approach without a minimal spacing condition. In contrast, using binary segmentation requires setting up the minimal spacing window, which is not necessary and could be misleading. For example. The paper effectively must generate evenly spaced change points. If, instead, change times are drawn i.i.d.\\ from a distribution—e.g.,\n$\n\\eta_1,\\ldots,\\eta_K \\sim \\mathrm{Unif} \\{ 2,\\ldots,T \\},\n$\nthe minimal spacing\n$\n\\Delta_{\\min}:=\\min_{1\\le k\\le K+1}\\bigl(\\eta_k-\\eta_{k-1}\\bigr)\n$\ntypically collapses. This destroys the large-spacing assumption required by a CUSUM screen and makes any fixed spacing window ill-posed. By contrast, spacing-free $\\(\\ell_0\\)$ approaches remain optimal/consistent in this setting and do not require enforcing even spacing.\n\n\nFor the $\\ell_0$ approach to estimation of dynamic (possible multilayer) networks, I refer to [2], but I believe more literature can be found.\n\nFor comparison between CUSUM and $\\ell_0$ approach, I refer to [3]. See Section 4, 'However, compared to the $l_0$-penalization methods, WBS is computationally more expensive and involves more tuning parameters.''.\n\n4. Simulations use (nearly) evenly spaced changes, reinforcing the strong-spacing assumption; real-data experiments also use very small $n,T,L$, This leaves the reader with the impression that the method hinges on large spacing and may not scale or adapt to realistic, short-gap or node–localized scenarios.\n\n[1] Wang, Fan, et al. \"Multilayer random dot product graphs: Estimation and online change point detection.\" Journal of the Royal Statistical Society Series B: Statistical Methodology (2025): qkaf051.\n\n[2] Zhao, P., Bhattacharya, A., Pati, D., & Mallick, B. K. (2022). Factorized fusion shrinkage for dynamic relational data. arXiv preprint arXiv:2210.00091.\n\n[3] ''Wang, Daren, Yi Yu, and Alessandro Rinaldo. \"Univariate mean change point detection: Penalization, CUSUM and optimality.\" Electronic Journal of Statistics 14 (2020): 1917-1961.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)