Change Point Detection on A Separable Model for Dynamic Networks
Authors: Yik Lun Kei
Hangjian Li
Yanzhen Chen
OSCAR HERNAN MADRID PADILLA
Venue: Accepted by TMLR ()

Abstract: This paper studies the unsupervised change point detection problem in time series of networks using the Separable Temporal Exponential-family Random Graph Model (STERGM). Inherently, dynamic network patterns are complex due to dyadic and temporal dependence, and change points detection can identify the discrepancies in the underlying data generating processes to facilitate downstream analysis. In particular, the STERGM that utilizes network statistics and nodal attributes to represent the structural patterns is a flexible and parsimonious model to fit dynamic networks. We propose a new estimator derived from the Alternating Direction Method of Multipliers (ADMM) procedure and Group Fused Lasso (GFL) regularization to simultaneously detect multiple time points where the parameters of a time-heterogeneous STERGM have shifted. Experiments on both simulated and real data show good performance of the proposed framework, and an R package CPDstergm is developed to implement the method.

Published in Transactions on Machine Learning Research (08/2025)
Change Point Detection on A Separable Model for Dynamic Networks
Yik Lun Kei* Deportment of Statistics University of California, Santa Cruz
ykei@uusc..dd
Hangjian Li
Cangiian.&@hwalmmee
Walmart Global Tech
Yanzhen Chen
imnyunzhen@usll.l
Deportmend of Information Systems, Business Statistics and Operations Monagement Hong Kong University of Science and Technology
Oscar Hernan Madrid Padilla
oeco.madridd@stat.uooo ucla. edu
Department of Statistics and Data Science University of California, Los Angeles
Reviewed on OpenReview: https: // openreview. net/ permmf id= JFkbaHF
Abstract
This paper studies the unsupervised change point detection problem in time series of net- works using the Separable Temporal Exponential-family Random Graph Model (STERGM). Inherently, dynamic network patterns are complex due to dyadic and temporal dependence, and change points detection can identify the discrepancies the underlying data generat- ing processes to facilitate downstream analysis. In particular, the STERGM that utilizes network statistics and nodal attributes to represent the structural patterns is flexible and parsimonious model to fit dynamic networks. We propose Alternating Direction Method of Multiplicrs (ADMM) procedure and Group Fused Lasso (GFL) regularization to simultaneously detect multiple lime points where the parameters of a hime-bhetroggenouus STERGM have shifted. Experiments on both simulated and real data show good performance of the proposed framework, and an R package CPDstergm is developed to implement the method.
estimator derived from the
1 Introduction
Networks are often used to describe relational phenomena that are not limited merely to the attributes of individuals as tabular data. In investigation of the transmission of COVID-19, Fritz et al. 2222]) used networks to represent human mobility and forecast disease incidents. The analysis of physical connections, beyond the health status of individuals, permits policymakers to implement preventive measures effectively and allocate healthcare resources efficiently. Yet relations by nature progress time, and dynamic relational phenomena are occasionally aggregated into a static network for analysis. To this end, temporal models for dynamic networks are in high demand to study the evolution of relational phenomena.
In recent decades, plethora of temporal models has been proposed for dynamic networks analysis. Snijders (2001), b, Suijders (2005), and Snijders et al. (2010) developed Stochastic Actor-Oriented Model, which is driven by the actor's perspective to make or withdraw ties to other actors in network. Нoff et al. (0022)
"Equal Contribution

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Published in Transactions on Machine Learning Research (08/2025)
2005 Sewell & Chen 2015): and Sewell & Chen (2016) presented latent space models, by assuming the edges between actors are more likely when they are closer in the latent Fuclidean space. Matias & Miele| 2017 : Ludkin et al. (2018). and Pensky (2019) investigated the dynamic Stochastic Block Model (SBM), and Jiang et al. (2020) developed an autoregressive SBM to characterize the evolution of communities, Kolar et al. (2010) locused on recovering the latent tirie-nryying graph structures of Markov Random Fields from scrial observations of nodal attributes. Furthermore, the Exponential-family Random Graph Model (ERGM) that uses local forces shape global structures (Hunter al. 2008b) is promising model for networks with dependent ties. Hanneke et al. (2010) defined a Temporal ERGM (TERGM), by conditioning on previous networks in the network statistics of an ERGM. Desmarais & Cranmer (201bb) proposed a bootstrap approach to maximize the eseudo-likelihodd of the TERGM and assess uncertainty. In general, nelwork evolution concerns the rale at which edges form and dissolve. Demonstrated in Krivitsky & Handcock 20114), these two factors can be mutually interfering, making the dynamic models used in the literature difficult to interpret. Posing that the underlying reasons that result in dyad formation are different from those that result in dyad dissolution, [Krivitsky & Handoock (2014) proposed Separable Temporal ERGM (STERGM) to dissect the entanglement with two conditionally independent models.
Sarkar & Moore
In time series analysis, change point detection plays a central role in identifying the discrepancies in the underlying data generating processes over time. In reality, network evolution is usually time- netereeetoog..ooss.. Without taking the structural changes across dynamic networks into consideration, learning from the time series may not be meaningful, by confounding the network effects before and after a change occurs. As relational phenomena are studied in numerous domains, it is practical for researchers to first localize the change points, and then analyze the network effects within stable segments, rather than overlooking the time points where the network patterns have substantially changed.
There has also been an increasing interest in studying the unsupervised change point detection problem for dynamic networks. Wang et al. (2013) focused on the Stochastic Block Model time series, and Wang
ct al. (021)) studicd a sequence of inhomogencous Bernoulli networks. Larroca ct al.
(2021). Marenco et al.
(2022), ,, and Madrid Padilla et al. (2022) considered a sequence of Random Dot Product Graphs that are both
dyadic and temporal dependent. Methodologically, Chen & Zhang (2015) and Chu & Chen (2019) developed graph-based approach to delincate the distributional differences before and after change point, and Chen
2019 utilized the nearest neighbor information to detect the changes in an online framework. Zhao et al.
2019
proposed two-step approach that consists of an initial graphon estimation followed by a screening
algorithm, Song & Chen (2024) exploited the features in high dimensions via a bernel-based method, and
2020a) employed embodding methods to detect both anomalous graphs and anomalous vertices. Chen et al.
Chen ct al.
(2024] and |Athreya el al. (2024) developed a model for network time series based on a latent
position process, using spectral estimates of the Euclidean mirror to detect first-order change points. Zhang
et al. (024) combined Variational Graph Auto-Encoder and Gaussian Mixture Model, and Kei et al. (2025)
focused on graph representation learning for change point detection in dynamic networks. Moreover, Liu
ct al. 20188 introduced an eigenvector-bseed method to reveal the change and persistence in the gene communities for a developing brain. [ybee & Atchade (2018) focused on a Gaussian Graphical Model to detect the change points in the covariance structure of the Standard and Poor's ' 500. [Onduu et al. (2021) proposed a factorized binary search method to understand brain connectivity from the functional Magnetic Resonance Imaging time scries data.
Allowing for interpretable network statistics to determine the structural changes for the detection, we make the following contributions in the proposed framework:
To detect multiple change points from a sequence of networks, we fit a time-hettroogencous STERGM while penalizing the sum of Euclidean norms of the sequential parameter differences. Essentially, we impose a Group Fused Lasso (GFL) regularization on the model parameters, smoothing out minor variation and highlighting significant. structural changes. An Alternating Direction Method of Multipliers (ADMM) procedure is derived to solve the resulting optimization problem.
We exploit the practicality of STERGM, which manages dyad formation and dissolution scparately, to capture the structural changes in network evolution realistically. The flexibility of STERGM and the extensive selection of network statistics also boost the power of the proposed method. Moreover,
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demonstrate the capability of including nodal attributes to detect change points, and an R. package CPDstergm is developed to implement the proposed method.
We simulate dynamic networks to imitate realistic social interactions, and our method can achieve greater accuracy the networks that are both dyadic and temporal dependent. Furthermore, punctually detect the winter and spring vacations with the MIT cellphone data (Eagle & Pentland 2006), and we detect three significant financial events during the 2008 worldwide economic crisis from the stock market data analyzed by James & Matteson (2015)
The rest of the paper is organized follows. In Section 2 we review the STERGM for dynamic networks. In Scction g present the likelihood-based objective function with Group Fused Lasso regularization, and we derive an Allernating Direction Method of Multipliers to solve the optimization problem. In Section 4 we discuss change points localization after parameter learning, along with model selection. In Section implement our method on simulated and real data. In Section 0 we conclude our work with a discussion on the limitation and potential future developments.
we
2
STERGM Change Point Model
2.1
Eseudo-likelihone of ERGM
For a node set N = (1.2.... ,n]. n): for all pairs (i,j) E Y N. The network y has dyad Yi E 13 to indicate the absence or presence of relation between node and node j. The relations in network can be either directed undirected, where undirected network has Yı Уyı for all (i.j)= E Y.
we use a network y E y= {0, 1 to represent the potential relations
The probabilistic formulation of an Exponential-family Random Graph Model (ERGM) is
P(y: 0) = cxp(e - (0)
(1)
where g with : y , is vector of network statistics; E IR is a vector of parameters; exp[v>(0) = Eyey exp(0" g3)] the normalizing constant. The network statistics g may also depend on the nodal attributes x. For notational simplicity, we omit the dependence of g on æ.
With a surrogate in Besag (9774;
Strauss 1keda 19900; Robins al. (2007) Van Duijn et al. (2009
Desmarais & Cranmer 200 2b] Hummel et al. (2012), the pseudo- Iseudo-likelhhood of ERGM, product of likelihoon
for each dyad Vij conditional on the rest of the network y-17, is
PL( = II (uully-1fi0) = II h(0 (g) Vii [ [[- h(e (00)g)g(())) (U))
(2)
(FJ)
where h(z) 11/(11eepp((1))) E (0, 1) is the sigmoid function with P(y. = 1y .y: 0) h(O Agn (3)). The change statistics (y R2 denote the change in g when Yiy changes from 0 to 1, while rest of the network y-ij remains the same. Then the logarithm of the pseudo-likelihood is given by
10 ERGM
Σ (LJEY
(0 Ag (v)) log { exp(O (gg
which is helpful in ERGM parameter estimation. Moreover, the Hessian matrix of FRRM is
H(0) = Σ h(0 Agu(()) [1- h(@ )g((t))) [A.,(
Since h(0 (y) E (0, and Ag, (g) RPxp positive semi-definite, the Hessian H()) = VØ ((0) is also positive semi-definite. other words, the negative logarithm of the pscudo-likclhhodd is convex with respect RP. Next, we introduce the Separable Temporal ERGM (STERGM) used in change point detection model.
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2.2 Sheudo-likeliood of STERGM
For network time series, network evolution concerns (1) incidence: how often new ties are formed, and (2) duration: how long old ties last since they were formed. Pointed out by Donnat & Holmes (2018), Goyal & De Gruttola (2020), and Jiang et al.] (2020), modcling snapshots of networks often gives limited information about the transitions between consecutive networks. To address this concern, Krivitsky & Handcock (2014)
designed two intermediate networks, formation network and dissolution network, to reflect the incidence and duration. In particular, the incidence can be measured by dyad formation, and the duration can be traced by dyad dissolution. Many applications real-world data support the separable mechanism for dynamic networks (Broekel & Bednarz 2018: Uppala. & Handcock 2020 Ando et al. 2025
Let y' E = {0, 1 be a network observed at a discrete time point t. The formation network y+ E y is obtained by attaching the edges that formed at time to y² and y+ = {y E y y 2 y The dissolution network y . y- obtained by deleting the edges that dissolved at time from y and y- (y 3" y S y We can also use the notation from Kei al. (2023) to specify the respective formation and dissolution networks between time I - 1 and time as
Vij" + max 1-1 v(t) and Vig" min(y!) vig)
for all (i,5) V. In summary, y t+ and y yo incorporate the dependence on y' through construction, and they can be considered as two latent networks recovered from both y²- and y to emphasize the network transition from time t t- to time t.
Posing that the underlying factors that result. in edge formation are different from those that result in edge dissolution, Krivitsky Handcock (2014) proposed the Separable Temporal ERGM (STERGM) to dissect the evolution belween consecutive networks. Assuming y conditionally independent of y given y² the time-heterogeneous STERGM for y conditional on g'-1 is
II P(u'l'' 0') II P 0th) P(y 1o-"ly" "10"
(3)
with the respective formation and dissolution models:
P(y + 1 0+.) = exp @e+ g+ (3 t 1 11 + yy ))
P(y " 0) = exp [ g yy y 1) v (0 y ")]
The parameter 0'= (0+1 0-*) E RP is a concatenation of 0+ E RRi and 0_ E RR such that P1 + Notably, the normalizing constant in the formation model exp|x>*(B (0+.t y² }] is a sum over all possible networks in y++,,, and the normalizing constant in the dissolution model exply (0-+,="'')) is a sum over all possible networks in y- Since measuring these normalizing constants computationally intractable when the number of nodes n is large (Hunter & Handcock 2006). many parameter estimation methods exploit MCMC sampling (Geyer & Thompson 1992 Krivitsky 2017] Bayesian inference (Caino & Friel 2011 Thiemichen et al. 2016) to circumvent the intractability of the normalizing constants.
However, for change point detection with a time-heterogencous model, these parameter estimation meth- ods remain computationally intensive, making them prohibitive for relatively long sequences of networks. Hence, extending from the pseudo-likelihood ERGM in (2), the pseudo-likelihood of a time-heterogeneous STERGM in (3) is given by
h@t Ag ((t) 1 [[ ii-i 2g
II PL t=2
II II 2=2 (K
[n(00.. ..g. ()) v.
[1- 1 1(0 AgT (gr 1
where h(x)= 1/(14 exp(-x)) =x(1)= E (0, 1) is the sigmoid function. Since th and y .f inherit the dependence -1 by construction, we use the implicit dynamic terms, g (y+.t) with gf y+ RR and g (y
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with g -/ R²² discussed in Krivitsky & Handcock (2014). The change statistics Ag.j (+) E R denote the change in gyy when Vij F. changes from to 1, while rest of the y* remains the same; the change statistics Ag. (+)))))))))))))r) E _ denote the change in g (y (y))) when Vij" changes from 0 1, while rest of the y yー remains the same. Similar to (2), the pseudo- likelihood of STERGM is a product of conditional likelihood for each dyad in the respective formation and dissolution networks, given the rest of the networks
and the previous network (Besag) 1975 Strauss & Tkoda 1990 Cranmer & Desmarais 2011 Schmid &
Hunter 2023). Moreover, the logarithm of the pseudo-likelihodd of the time-hetcrogencous STERGM is
T
1(0) = Σ Σ
Uij" [ott (g +t]] - log {1 + exp [0t . (t+} +
1=2 (K,))
(4)
yij" ' 10- -... (y 4)] log (1+ exp 10- Ag ()}}}
where 0 =
E RTXP with T -
Empirically, for change point detection, the pseudo-likelhood of STERGM is preferable to the true likelihood for the following three reasons. First, the dyadic dependency in y and y mitigated by conditioning on the previous network y'- and by separately modeling through the formation and dissolution processes. Specifically, conditional on yij 1, the Vij" +. myx vyy)) can only be 1; while conditional on y 0, the Viy" min(y'y 1-1 y can only be 0. This design explicitly restricts the states of dyads by partitioning network evolution into formation and dissolution processes, thereby reducing the dyadic dependence within y+ and y Hence, conditioning on the previous network which already captures the structural depen- dencies, the pseudo-likelihone of STERGM becomes a reasonable surrogate. Second, the primary objective in change point detection is localize substantial structural changes over time, rather than to recover the cocfficient estimates for network effect interpretation. In the former case, the pscudo-likclhhodd of STERGM remains adequate, as large parameter shifts can still be reliably identified, even if the estimates are subject to mild bias by using common approximation to the true likelihood. Third, we adopt the logarithm of the pseudo-likelihodd to particularly avoid using MCMC sampling or Bayesian inference, which is compu- tationally challenging for the optimization problem defined in Section 3 Instead, the psendo-lkeelhhod of STERGM improves the scalability of the eslimation procedure, and the sigmoid function that involves the pre-computed change statistics permits efficient calculation of the gradients and Hessians for iterative param- eter updates. In summary, the pseudo-likelihood of STERGM provides both computational feasibility and effectiveness to facilitate change point detection in dynamic networks, advantage that the true likelihood cannot offer at scale.
Now we consider the change points to be detected in terms of the parameters in STERGM. Let [Be) K++ C [T be a collection of ordered change points with 2 = Bo < B1 < .... < BB < BK+1 = such that
0 Pk e Bi+l
0 Bh P 11 k 0, O....K.
0 BB + BLI1 k= ................................ K 1, and g BBx+ Bxx
Our goal is to recover the collection { k== from a sequence of observed networks TT' where the number of change points K is also unknown. Intuitively, the consecutive parameters 0³ and 0++1 are similar when change occurs, but one more components in B R can be different from 0B, RR after change happens. For this setting, we present our method in the next section.
3 STERGM with Group Fused Lasso
3.1
Optimization Problem
Inspired by Vert & Bleakley (010 and Bleakley & Verl 2011 we propose the following estimator for our change point detection problem:
| - 02.112
θ = arg min -1(0) + +
(5)
d
0
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where ((0) is formulated by 4 The term >> 0 is tuning parameter for the Group Fused Lasso penalty, and the term d E I+4- is position dependent weight such that d; VT/ t)] for E [1, 1]. Intuitively, the inverse of d; assigns greater weight at the time point that is far from the beginning and the end of a time span, as the end points are usually not of interest for change point detection.
The Group Fused Lasso penalty expressed as the sum of Fuclidean norms encourages sparsity of the param- differences, while enforcing multiple components in 0++1, - Oz across j I....,p to change at the same group i. This is grouping effect that cannot be achieved with the & penalty of the differences. Along with the user-specified network statistics in STERGM, the sequential parameter differences learned from the observed networks with (5) can reflect the magnitude of structural changes over time. By penalizing the sum of sequential differences between the STERGM parameters, the proposed framework focuses on capturing significant structural changes while smoothing out minor variations.
Figure gives an overview of the proposed framework. The shaded circles on the top denote the sequence of observed networks as time passes from left to right. The dashed circles in the middle denote the sequences of formation networks t and dissolution networks y recovered from the observed networks. Note that each observed network is utilized multiple times to extract useful information that emphasizes the transition between consecutive time steps. We learn the parameters denoted by the dotted circles at the bottom, while monitoring the sequential parameter differences.
Sequence of T observed networks
y²
1
m
y+3
y
10000
0
100
0
6
Sequence of T - learned parameters
Figure 1: An illustration of change point model with STERGM.
To solve the problem in we first introduce slack variable z E x and rewrite the original problem constrained optimization problem:
Zi 1,+ 2i, 112
i arg min -1(0) +
d,
(6)
subject to = z.
Let XTTX be the scaled dual variable. The augmented Lagrangian can be expressed as
C. (0, = -1(0)
2 Zi,
+
a - z + ull
uu]
(7)
d,
2
1=1
where R2 is another penalty parameter for the augmentation term.

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Fused Lasso regularization has been widely used for one-dimensional change point detection problems Leevy-
ledue & Harchaoui 2007; Rojas & Wahlberg 2014: Lin et al. 2017]. Following Bleakley & Vert (201 we
make the change of variables (7, E RIX x R 1xxp to formulate the augmeettd Lagrangian in 7 as a
Group Lasso regression problem (uuan & Lin 2006 Friedman et al. 2010
Alaíz et al. 2013 where
21, and B., Zi+1, 21,
d: 11,
(8)
Reversely, the matrix E RTxP also be collected by
= 1,,7 X
where X E RTx(r-l) is a designed matrix with X = dj for i > j and 0 otherwise. Plugging y and B into
we have
8,) A
Q Σ B1, - 1,,17 XB ull / 11lll
(9)
Thus we derive the following Alternating Direction Method of Multipliers (ADMM) procedure to solve
6
/ ((++1) =arg min 1(0) + e 2a)) + 1/
(10)
(a+1) 1) g(att) arg min A
1e,.112
a
( 1,17 - x + uu())
(11)
7.8
2
i=1
u (0+1) ((++1)
(a+11) + (a)),
(12)
where a denotes the current ADMM iteration. Once the update 11
is completed within an ADMM iteration,
we collect z(a+1) = (1+1) + X x3(++)) until the next decomposition of z. We recursively implement the
three updates until a convergence criterion is satisfied. By adapting the idea from Boyd et al. (0011) we
have the following result for the proposed ADMM procedure:
Proposition 1. Denote the respective primal and dual residuals the ath ADMM ileration as
(a) primal
1
g(u)
2 and (a) 'dnat
(a) ij
10-1))3
2
xxP
T
1=
Assume the updates (() "primal 0 and dual residual dual 0 00.
10 and 11 attain minimum at each ADMM iteration.
Then the primal residual
The proof is provided in Appendix A Next, we discuss the updates 10 and
11 in detail.
3.2 Updating
In this section, derive the Newton-Raphson method for learning θ in the update (10p. We choose to the Newton-Raphson method because it is more efficient than gradient descent: the Newlon-Raphson method utilizes second-order information to adaptively determine the step size, leading to quadratic convergence and more stable updates (Galántai 2000). Specifically, to implement the Newton-Raphson method in a compact
form, we vectorize E T as 0 vec,p R and we construct the following matrices:
t +4
and H
A
The matrices t. E Expi and A E Exps abbreviate the respective change statistics Ag and Ag,,(y that ordered by the dyads. The dimension of A is thus 2F xp, where the quantity F. n

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is due to vectorization, and the double in the number of rows is due to the separability of STERGM. In practice, the matrix H E BRTP that consists of the change statistics for i t= ....., 7 is calculated before the implementation of ADMM.
For the Hessian matrix, we also need to calculate पř h(H θ) E R where h(z) = 1/(1 + exp(-2)) is the element-wise sigmoid function with 1 fx(x) = h1xx((1 h(x)). Furthermore, we construct the following matrices:
W2
W the
W
W-1 and W-
W
where W+ = diag(u
)) (1- 1041/E with Poj h(0t.t Ag (yt+)) (0, 1). The matrix
W E R ExE defined similarly except for notational difference.
The Newton-Raphson method to iteratively update the parameter 0 E R for the proposed framework is implemented follows:
0+1 -. (-
- +(()))
(13)
where e denotes the current Newton-Raphse iteration. The derivations provided in Appendix B The diagonal matrix W with diagonal entries between 0 and 1, along with quadratic form of the matrix H, shows that the matrix H W is positive semi-definite. The identity matrix Lpp inherited from the augmentation term 10- in [10 ensures the Hessian matrix H WH+alm is not only invertible but also positive definite. Thus the objective function (10p strongly convex with respect the parameter 0 and unique global minimum is guaranteed
exist. Once the Newton-Raphson method concluded within an ADMM iteration, we fold the updated vector 0 back into matrix as ((a+1) (00 RRx before implementing the update in [1 which is discussed next.
3.3 Updating y and
In this section, derive the update in 11 which is equivalent to solving Group Lasso problem. We decompose the matrix to work with y and 3 instead, and the objective function is convex with respect to these parameters. With ADMM, the updates on and 33 do not require the network data and the change statistios, but the updates primarily rely on the 0 learned from the update 110
By adapting the derivation from Vert & Bleakley 2010 and Bleakley Vert 2011). the matrix f ((r-1)xp can be updated in block coordinate descent manner. Specifically, the block coordinate descent mothod update Br, for each block i 1,,,..... .,....T- 1 is implemented follows:
1 Br. ax a.
(1 (1- ||sill2 (1-1e21a)
A 11A111) 8,
(14)
where (-)_ = max(.,0) 0) and
8, = a... (glaal)))) + (a) -1,17 - x.,
The derivations are provided in Appendix and the convergence of the procedure is monitored by the Karush-Kuhn- Tuccer (KKT) conditions:
Pr. ax. a. + (a))) 1,,17 Xx3) =
vB,. * 0,
11.1112
| (0(++1) (0) 1,,17 x13
vB,. 0.
Subsequently, for any 3 e 1)x,, the minimum in E I 1 is achieved at
7= (1/r)11,r (0(++1) u X
Once the update concluded within an ADMM iteration, we collect 1,,17 + X33 and proceed update the scaled dual variable E xR with 12
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4 Model Selection and Change Point Localization
In this section, we discuss additional details for change point detection with STERGM. The proposed method network statistics including nodal altributes, and the estimator is consistent under certain assumptions. Moreover, use Bayesian information criterion to perform model selection, and we provide data-driven threshold for change point localization.
4.1 Network Statistics and Nodal Attributes
As a probability distribution over dynamic networks, STERGM allows us to generate different networks that share similar structural patterns with the observed networks, by using a carefully designed MCMC sampling algorithm (Besag. 2001 Suijders 2002: Krivitsky 2017). Hence, a dynamic network modeling problem with STERGM, network statistics are often chosen to signify the underlying process producing the observed networks or to capture important network effects interpreting for research question.
In the change point detection problem with STERGM, network statistics are chosen to determine the types of structural changes that are searched for by the researchers. The R library ergm (Handeock et al., 2022) provides an extensive list of network statistics that boost the power of the proposed method. Since the underlying reasons that result in edge formation are usually different from those that result in edge dissolu- tion, the choices of network statistics in the formation model can be different from those in the dissolution model. Moreover, we permit the inclusion of nodal attributes in network statistics, a capability that many change point detection methods for dynamic networks do not provide. For an in-depth discussion of network statistics in an ERGM framework, see Handcock et al. (2003), Hunter & Handeock (2006), (nnjddrss et al.)
(2066) Hunter et al. (2008a), Morris et al. 20088 Robins et al. (2009), and Blackburn & Handcock (2022)
4.2 Error Bound under Structured Sparsity
For our change point detection framework with Group Fused Lasso regularization, provide the following estimation error bounds by adapting the idea from Negahban et al. 2001
Proposition 2. Denote 0 R the true parameter and suppose that ||0* S M/2 for some M > Let Ô E R be the minimizer of the objective function in Define the set of true change points as
5 subject to the constraint ||0||cc I M/2.
S= fiE (1.....7--11 0 01, * 0
Suppose the loss function L(0) = -1(0) satisfies the Restricted Strong Converity condition:
L(0 + A) > L( (VL( A) + 4
A 2
2
for all perturbations A E XP that satisfy the structured sparsity condition:
A A+1.- A
1 Ai+,. A
(15)
di
d.
$
iES
for some constants & > 0 and norm of the grudient salisfies
> 0. Also, assume that with probability at least 1 8, the restricted dual
IV A
where the restricted dual norm ||. is laken with respect to the Total Variation (TV) norm ||-1|
VL(0))
sup A 11
(VL(0"), A) and 1 = Σ
A A
A
1m1 d
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Then if the estimation error Å == θ 0- E RR also satisfies the structured sparsity condition in follows with probability at least 1 - å that. the mean squared error is bounded as
15 it
1 10 1- - TP TP
6Λ (1+a) k
Σd dj FES
2AM
max
The proof is provided in Appendix D Specifically, the Restricted Strong Convexity assumption enstres that L(0) exhibits sufficient curvature when the estimation error has most of its total variation aligned with the true change points. The restricted dual norm condition on VL(@0) controls the influence of noise, preventing stochastic fluctuations from suggesting spurious jumps in the estimated parameters. Together, these conditions facilitate a bound that reflects the estimator's sensitivity to the signal structure and noise level. Moreover, when a K, M 1, O, and
x2 E d +2
- 0,
tp
SE
obtain that our estimator is consistent for estimating l in terms of the mean squared error:
110 1 0
tp
While previous works have developed error bounds for Total Variation and Fused Lasso estimators (Rojas & Wahlberg 2014 Hütter & Rigollet, 2016 Lin et al. 2017), we focus on the Group Fused Lasso regularization applied to the parameters of STERGM for dynamic networks.
4.3 Model Selection
In practice, to determine the optimal set of change points over multiple STERGMs learned with different tuning parameter X, we can use Bayesian information criterion (BIC) to perform model selection. Consider the STERGM with learned i and fixed X, we have
BIC A) (0) + log( x p x Oeg().A).
(16)
For list of X, we choose the set of change points obtained from the STERGM with the lowest BIC value. Different from the number of nodes the network size Nuet is (2) for an undirected network and 2 x (*) for a directed network. general, for a dyadic dependent network, the effective network size often smaller than Neet and it may be difficult to quantify the effective size (Hunter et al. 2008a).. In node clustering problem for a static network, Handcock
al.] (2007) used the number of observed edges to quantify the effective network size. In this work, we Nuct to consider greater value the network size, since the procedure is to select model with the lowest BIC value. Furthermore, the term Seg(O, A) in (16) gives the &+1 that are learned with a particular À value. In other
number of segments between change points [ words, Seg( A) K 1, where K is the mumber of detected change points.
4.4 Data-driven Threshold
Intuitively, the location of a change point is the time step where the parameter of STERGM at time / differs from that at time t t-1. 1. To this end, can calculate the parameter difference between consecutive time points in I E BTXP as
A = 1 - - Vi [1,--]1
and declare a change point when a parameter difference greater than a threshold.
Though rescarchers can choose an arbitrary threshold for 2 based on the sensitivity of the detection, this work provide data-driven threshold with the following procedures. First standardize the parameter
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differences 2
AC: Δ0, median(A0) (A))
Vi [1,7-1]
(17)
sd(A0)
The Aς E in (17) can be considered as the change magnitude in networks over time. Then the threshold based on the parameters learned from the data is constructed as
Cthe mcan AC) Z x sd()
(18)
where Z is the (1-a)% quantile of the standard Normal distribution. We declare change point B when ACB 6thr- The data-driven threshold in 18) is intuitive, the standardized parameter differences at the change points are greater than those values in between the change points, derived from the Group Fused Lasso penalty. When tracing in a plot over time, the values AC can exhibit the magnitude of structural changes, in terms of the network statistics specified in the STERGM.
5 Simulated and Real Data Experiments
In this section, we evaluate the proposed method on simulated and real data. For real data where ground truth is unknown, we align the detected change points with real world events for interpretation. For simulated data where ground truth is known, we use the following metrics to compare the performance of the proposed and competitor methods. The first metric is the absolute error |K K| where K and K are the numbers of detected and true change points, respectively. The second metric is the one-sided Hausdorff distance:
d((cc)= max min |2 cl, cEC aEc
where è and C are the respective sets of detected and true change points. We also report the metric d(C When é = 0, we define d(C)C) = oo and d(C)C) = ac. The third metric described in van den Burg Williams 2020 is the coverage of a partition g by another partition g", defined as
1 - C(G,9')
AANA' AUA'
IAI- AEg
max A'EG'
with A,A' S [1. The and g' are collections of intorvals between consecutive change points for the respective true and detected change points. These three metrics are chosen to reflect a progression in eval- uation difficulty: the absolute error assesses the number of detected change points, the one-sided Hausdorff distance captures the largest deviation in change point location, and the coverage metric measures alignment over the full time span, requiring a comprehensive match between the true and detected segments.
5.1 Simulation Study
We simulate dynamic networks from three particular models to imitate realistic patterns. First, we use the Stochastic Block Model (SBM) to attain that participants with similar attributes tend to form communities, and we impose a time-dependent mechanism in the network generation process. Second, we simulate dynamic networks from STERGM, which separately takes into account how relations form and dissolve over time, as their underlying reasons are usually different. Third, we utilize the Random Dot Product Graph Model (RDPGM), where edge formation is driven by the similarity in latent positions between nodes, and we allow these latent positions to evolve over time.
For each specification, we let the time span T 100 and the number of nodes {50, 100, 200). The K = 3 truc change points are located at t = {26, 51, 76). and the K + 1 = 4 intervals in the partition G are Ai = f 25}, A = (26...., 50), Aa = (51, 75), and A4 = {76.... 1003. In each specification, we report the means and standard deviations over 15 Monte Carlo trials for different evaluation metrics.
Specifically, to detect change points with the proposed method, we initialize the penalty parameter = 10,
and we let the tuning parameter 10" with ò e 2, 3,4]. For each X, we run A 200 iterations of
ADMM and the stopping criterion in (66 uses étll 10-7 Within each ADMM iteration, we run C 20
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iterations of the Newton-Raphson method, and D 20 iterations for the Group Lasso update. The stopping criteria for the Newion-Raphson method is || 0-0c-1 - 0.112 10-3. To construct the data-driven threshold in (18, we use the 0.9 quantile of the standard Normal distribution. Additional details about the model specifications are provided in Appendix E Throughout, the network statistics are calculated directly from the library ergm (Handcock el al. 2022). and the formulations are provided in Appendix
We compare our proposed method against five competing methods: gSeg (Chen & Zhang 2015 1, kerSeg (Song & Chen 2024). CPDrdpg (Madrid Padilla et al.] 2022), CPDnbs (Wang et al. 2021 and CPDker
(Madrid Padilla et a.. 2021 The gSeg method utilizes graph-based based scan statistic and the kerSeg method employs a kernel-based scan statistic to assess distributional differences between two segments before and after a candidate change point. The CPDrdpg method identifies change points by estimating the latent positions of nodes using a Random Dot Product Graph Model (RDPGM) and by constructing a nonpara- metric CUSUM statistic that accounts for temporal dependence. The CPDnbs method intrgrates sample splitting with wild binary segmentation (WBS) and detects change points by maximizing the inner product between two CUSUM statistics derived from the split samples. The CPDker method detects change points by using kernel density estimation approach to identify distributional shifts in multivariate data sequences. In particular, the gSeg and kerSeg methods are available in the respective R libraries gSeg (Chen et al. 2020b) and kerSeg (Song & Chen 2022). The CPDrdpg, CPDnbs, and CPDker methods are available in the R library changepoints (Xu et al. 2022
For gSeg, we use the minimum spanning tree to construct the similarity graph, and we use the original edge-count scan statistic to test the mull hypothesis that there is no change point within a time span. The significance level is set to 0.05. For kerSeg, we use the kernel-based scan statistic fGKCP, and we set the significance level networks (nets.) and network statistics (stats.) as two types of input data for comparison. For CPDrdpg, we let the number of intervals for wild binary segmentation (WBS) be W = 50, and we let the mumber of Icading singular values of an adjacency matrix in the scaled PCA algorithm be d = 5 to fit a RDPGM. For CPDnbs, we let the number of intervals for WBS be W = 15 and we set the threshold for detection to the order of n nog" (T)) as suggested by Wang et al. (2021). For CPDker, we let the number of intervals for WBS be W = 30 and
0.001. Since gSeg and kerSeg are general methods for change point detection,
use
set the bandwidth to the order of (log(T)/T))* for the Gaussian kernel as suggested by Madrid Padilla et al. (2021), where p is the number of network statistics. Throughout, we use these chosen settings, since they produce higher coverage metries C(G, g') for the competitors across different scenarios on average. Changing the above settings can improve their performance on some specifications, while severely jcopardizing their performance on other specifications.
Scenario 1: Stochastic Block Model (SBM)
In this scenario, we use Stochastic Block Model (SBM) to generate dynamic networks. As in Madrid Padilla ct al. 2022), we construct two probability matrices P, Q [0, 1]mx= and they are defined
Psj = 0.5, i E E and Qú = 0.45, i E B,, 1 [3], 0.3, otherwise,
0.2, otherwise,
where B1, B3 are evenly sized clusters that form a partition of [I....,n). We then construct a sequence of matrices E. for t I...., such that
Paj. te tE Us,
E by
Qij,
AU
Lastly, the networks are generated with p E {0.0, 0.5, 0.9} as time-dependent mechanism. For any p and t == ,,...,T T 1, we let y ~ Bernoulli E and
Bernoulli (p(nllippp((11) (p(1- E1 + E.+1)
y = 1, y 0.
+1 yi
Bernoulli
E
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When p O, the probability to draw an edge for dyad (2,5) at time t + remains the same. This imposes time-independent condition for sequence of generated networks. On the contrary, when p 0, the probability to draw an edge for dyad (i,j) becomes greater at time ++ when there exists an edge at time t, and the probability bocomes smaller when there does not cxist an edge at time
100
100
100
100
100
100
34 67
100
34
100
34
67
100
Figure 2: Examples of adjacency matrices generated from SBM with p = 0.5 and n 100. In the first row, 25, 50, 75 respectively, In the second row,
from left to right, each plot corresponds to the network from left to right, each plot corresponds to the network at t 26, 51, 76 respectively (the change points). Tn each display, red dot indicates one and zero otherwise.
Figure 2 exhibits examples of generated networks at particular time points. Visually, Scenario produces important pattern in these networks. Hence,
adjacency matrices with block structures, and mutuality to detect the change points with our method, we two network statistics, edge count and mutuality, in both formation and dissolution models. For gSeg and kerSeg, besides dynamic networks {y' } also use the two network statistics {g(3')) " as another specification. The CPDrdpg and CPDnbs methods directly use the networks as input data. Tables and 3 display the means and standard deviations of evaluation metrics for different specifications.
As expected, the CPDrdpg, CPDnbs, and kerSeg methods can achieve good performance in terms of the covering metric C(G, g') when p O, since the time-independent setting aligns with the assumptions of these competitor methods. However, their performances the networks in the sequence are time-dependent, the gSeg, kerSeg, and CPDnbs methods can effectively detect the true change points, as the one-sided Hausdorff distances d(C(C) are small. Yet the reversed one- sided Hausdorff distance d(C(C) and the absolute error j K| suggest that they tend to detect excessive number of change points as the sequences of networks become noisier under the time-dependent condition. Morcover, the kerSeg method can achieve good performance when the temporal dependency moderate with p = 0.5. The CPDrdpg method remains robust when the temporal dependency is strong with p = 0.9, and the performance improves the number of nodes increases. Regardless of the temporal dependence, our CPDstergm method, on average, achieves smaller absolute error, smaller one-sided Hausdorff distances. and greater coverage of interval partitions, outperforming the competitor methods.
worsened when p O. In particular, when
Another aspect worth mentioning is the usage of the network statistics in two competitor methods. The performance of gSeg and kerSeg methods, in terms of the covering metric C(g. g'), improves significantly when we change the input data from networks network statistics, which demonstrates the potential of using network level summary statistics to represent. the enormous amount of individual relations.
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Table Means (standard deviations) of evaluation metrics for dynamic networks simulated from the Stochas- tic Block Model with 0.0. The best coverage metric bolded.
Method
K
d(C(c) 4
d(c()) 4
C(G. g')
CPDstergm
0.2 (0.6)
0.8 (0.4)
1.7 (2.7)
95.99%
CPDrdpg
(0.8)
(1.2)
(3.6)
89.32%
CPDubs
0.1 (0.3)
3.3 (5.8)
1.8 (0.8)
92.17%
CPDker
0.0 (0.0)
17.3 (7.0)
10.7 (18)
58.67%
0.0
50
gSeg (nets.)
2.8 (0.4)
Inf (na)
Inf (na)
9.08%
kerSeg (nets.)
0.0 (0.0)
0.0 (0.0)
0.0 (0.0)
100%
gSeg (stats.)
(0.4)
Inf (na)
Inf (na)
43.68%
kerSeg (stats.)
0.1 (0.3)
0.1 (0.3)
0.3 (0.8)
99.67%
CPDstergm
0.7 (1.3)
0.8 (0.4)
5.0 (6.4)
91.34%
CPDrdpg
0.3 (0.6)
1.3 (0.6)
2.3 (2.3)
92.33%
CPDubs
0.1 (0.4)
3.3 (5.7)
2.5 (2.6)
90.46%
CPDker
(0.3)
21.3 (10.6)
10.5 (2.7)
57.92%
0.0
100
gSeg (uets.)
2.9 (0.3)
Inf (na)
Inf (na)
3.20%
kerSeg (nets.)
0.0 (0.0)
0.0 (0.0)
0.0 (0.0)
100%
gSeg (stats.)
(0.6)
Inf (na)
(na)
45.55%
kerSeg (stats.)
0.0 (0.0)
0.0 (0.0)
0.0 (0.0)
100%
CPDstergm
(0.4)
(0.4)
(3.9)
95.33%
CPDrdpg
0.1 (0.3)
1.0 (0.0)
1.5 (1.8)
92.85%
CPDubs
0.1 (0.3)
3.3 (5.7)
1.9 (0.4)
91.40%
CPDker
0.1 (U.3)
29.2 (10.5)
11.7 (2.4)
48.40%
0.0
200
gSeg (nets.)
2.9 (0.4)
Inf (na)
Inf (na)
6.01%
kerSeg (nets.)
(0.0)
(0.0)
(0.0)
100%
gSeg (stats.)
1.9 (0.6)
Inf (na)
Inf (na)
42.28%
kerSeg (stats.)
0.0 (0.0)
0.0 (0.0)
0.0 (0.0)
100%
Table Means (standard deviations) of evaluation metrics for dynamic networks simulated from the Stochas- tic Block Model with = 0.5. The best coverage metric bolded.
Method
jK K
d(cic) 4
d(c))) 4
C(G.g") f
CPDstergm
0.0 (0.0)
1.0 (0.0)
1.0 (0.0)
98.04%
CPDrdpg
1.3 (1.7)
2.5 (1.4)
7.5 (5.9)
81.47%
CPDubs
1.6 (0.6)
3.3 (3.2)
11. (1.1)
73.54%
CPDker
(0.4)
(9.9)
10.7 (2.3)
56.80%
0.5
50
gSeg (uets.)
12.9 (1.8)
0.0 (0.0)
19.3 (0.7)
27.20%
kerSeg (nets.)
6.3 (1.4)
0.0 (0.0)
16.5 (2.6)
45.87%
gSeg (stats.)
(1.1)
42.7 (20.2)
(7.8)
50.92%
kerSeg (stats.)
0.7 (0.8)
0.0 (0.0)
5.3 (7.1)
95.13%
CPDstergm
(0.0)
(0.0)
(0.0)
98.04%
CPDrdpg
0.3 (0.6)
1.4 (0.5)
2.8 (3.1)
91.07%
CPDnbs
1.8 (U.7)
2.9 (2.9)
12.2 (1.1)
72.17%
CPDker
(0.3)
(8.1
10.3 (2.9)
58.33%
0.5
100
gSeg (nets.)
12.5 (1.1)
0.0 (0.0)
19.3 (0.7)
27.60%
kerSeg (nets.)
6.1 (1.0)
0.0 (0.0)
15.0 (2.1)
46.40%
gSeg (stats.)
(U.7)
(na)
(na)
53.18%
kerSeg (stats.)
0.9 (0.6)
0.0 (0.0)
8.1 (7.3)
93.67%
CPDstergm
(0.0)
(0.0)
(0.0)
98.04%
CPDrdpg
0.0 (0.0)
1.0 (0.0)
1.0 (0.0)
93.32%
CPDubs
1.8 (0.7)
3.2 (3.6)
11.8 (0.9)
71.43%
CPDker
0.1 (0.3)
26.7 (12.7)
11.9 (2.2)
53.18%
0.5
200
gSeg (nets.)
12.2 (0.6)
0.0 (0.0)
19.1 (0.5)
27.87%
kerSeg (nets.)
4.5 (0.7)
(0.0)
13.8 (1.7)
52.00%
gSeg (stats.)
166 (0.7)
Inf (na)
Inf (na)
54.36%
kerSeg (stats.)
0.5 (0.6)
0.0 (0.0)
4.4 (6.1)
96.33%
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Table Means (standard deviations) of evaluation metrics for dynamic networks simulated from the Stochas- tic Block Model with 0.9. The best coverage metric bolded.
n
Method
RK_ Kl
d(C)C) 4
d(ci)) 4
C(G,9) (GG.G) f
CPDstergm
0.0 (0.0)
1.0 (0.0)
1.0 (0.0)
98.04%
CPDrdpg
(1.4)
2.5 (1.1)
8.4 (5.7)
83.13%
CPDubs
1.3 (0.6)
2.7 (2.6)
10.7 (2.6)
76.38%
CPDker
0.0 (0.0)
16.9 (6.8)
10.5 (2.2)
60.43%
0.9
50
gSeg (nets.)
12.4 (1.1)
0.0 (0.0)
19.1 (0.5)
27.67%
kerSeg (nets.)
11.1 (0.7)
0.0 (0.0)
18.9 (0.6)
31.53%
gSeg (stats.)
(3.8)
(14.8)
16.7 (4.8)
57.38%
kerSeg (stats.)
4.1 (1.7)
0.0 (0.0)
15.1 (3.4)
72.67%
CPDstergm
0.0 (0.0)
1.0 (0.0)
1.0 (0.0)
98.04%
CPDrdpg
0.1 (0.3)
1.6 (0.8)
1.9 (1.6)
92.14%
CPDubs
1.4
2.5 (2.1)
10.4 (2.5)
76.44%
CPDker
(0.6)
21.1 (14.7)
(3.1)
60.52%
0.9
100
gSeg (uets.)
12.4 (0.9)
0.0 (0.0)
19.0 (0.0)
27.67%
kerSeg (nets.)
11.9 (0.3)
0.0 (0.0)
19.0 (0.0)
28.27%
gSeg (stats.)
(2.3)
(9.0)
18.7 (3.0)
62.81%
kerSeg (stats.)
3.7 (1.2)
0.0 (0.0)
17.1 (3.5)
73.27%
CPDstergm
(0.0)
(0.0)
(0.0)
98.04%
CPDrdpg
0.0 (0.0)
1.0 (0.0)
1.0 (0.0)
93.19%
CPDubs
1.3
2.5 (2.1)
10.3 (2.5)
77.38%
CPDker
0.1 (U.3)
22.5 (12.5)
11.1 (2.5)
54.72%
0.9
200
gSeg (nets.)
12.3 (1.0)
0.0 (0.0)
19.1 (0.3)
27.73%
kerSeg (nets.)
12.0 (2.00
(0.0)
18.9 (0.3)
28.00%
gSeg (stats.)
7.4 (2.4)
0.9 (1.8)
17.9 (3.8)
57.61%
kerSeg (stats.)
4.8 (1.7)
0.0 (0.0)
16.7 (1.9)
66.87%
Scenario 2: Separable Temporal ERGM
In this scenario, we employ time-homogeneons STERGMs (Krivitsky & Handcock 2014b between change points to generate sequences of dynamic networks, using the R package tergm (Krivitsky & Handcock 2022). For the following three specifications, we gradually increase the complexity of the network patterns, by adding more network statistics in the data generating process. First we use two network statistics, edge count and mutuality, in both formation and dissolution models to let p = The parameters are
-1, -2, tE A U
1, -1, A U4.
Next, we include the number of triangles in both formation and dissolution models to let = 6. The parameters are
tEA 115, 1, -1, 2, 1, 1.5, te AuA
-2, -2 2,,
g+. 0
Finally, we include the homophily for gender, an attribute assigned to each node, in both formation and dissolution models to let p = The parameters are
-2, 2, -2, -1, -,,
LEA,
t 0
-1.5, 1, -1, 1, 2, 1, 1.5, 2, A2
The nodal attributes, E (Female, Male) for e [n], are fixed across time / in the generation process.
Figure exxiibis examples generated networks at particular time points. Specifically, Scenario 2 produces adjacency matrices that are sparse, which is often the case in reality. For comparison, detect change points with our method, we use the network statistics that generate the networks in both formation and dissolution models. For gSeg and kerSeg methods, besides using the networks specification, also
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use the same network statistics that generate the simulated networks another specification. The CPDrdpg and CPDrbs methods directly use the networks as input data. Tables standard deviations of evaluation metries for different specifications.
and D display the means and
200
100
100
50
100
50
100
50
100
Figure 3: Examples adjacency matrices generated from STERGM with p = 6 and n = 100. In the first row, from left to right, each plot corresponds the network 25, 50, respoctively. In the second row, from left to right, each plot corresponds to the network al / 26, 51, 76 respectively (the change points). In each display, red dot indicates and zero otherwise.
For p= 4, the performance of the kerSeg method in terms of the covering metric C(g, improves signifi- cantly when we substitute the input data from networks to network statistics. The CPDrdpg and CPDnbs methods also achieve good performance when the dyadic dependency is weak. However, for p 6, the competitor methods tend to detect excessive number of change points, when the simulated networks are highly dyadic dependent due to the inclusion of the transitivity. Using network statistics as input can no longer improve the performance of gSeg and kerSeg, in contrast to Scenario where the dyadic dependency is relatively mild. Nevertheless, our CPDstergm method, which dissccts the network evolution using formation and dissolution processes, can achieve a good performance when the simulated networks are both temporal and dyadic dependent. Lastly, for p = 8, the performance of CPDrdpg, CPDnbs, and CPDker methods does not deteriorate, and that of the CPDrdpg method improves the number of nodes increases. Notably, our method permits the inclusion of nodal attributes to facilitate change point detection, feature that many existing hods for dynamic graphs do not offer. On average, our method produces smaller absolute error, smaller one-sided Hausdorff distances, and greater coverage of interval partitions, outperforming the competitor mothods at different levels of dyadic dependency.
Scenario 3: Random Dot Product Graph Model (RDPGM)
In this scenario, we simulate dynamic networks using the Random Dot Product Graph Model Young & Scheinerman, 2007: Athreya el al. 2018). At time point / 1, we generate two latent positions X Z E शि for each node [n] from a multivariate Normal distribution as X z (11,, Ia). For subsequent time points, the latent positions evolve according to the following autoregressive process:
x p + p)ep z pZ; + p)ct t= L.
where oi NN Throughout, we set p 0.9 to induce the temporal dependence, and we normalize the resulting latent. iositions each time point. Next, to incorporate structural patterns in the dynamic
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Table Means (standard deviations) of evaluation metrics for dynamic networks simulated from the STERGM with p =44 The best coverage metric bolded.
P
n
Method
K
d(C))) 4
d(ci)) 4
C(G.9))
CPDstergm
0.0 (0.0)
0.0 (0.0)
0.0 (0.0)
100%
CPDrdpg
(0.0)
(0.7)
(0.7)
92.48%
CPDubs
0.1 (0.3)
3.7 (5.7)
2.3 (1.0)
87.67%
CPDker
0.3 (0.5)
19.7 (13.1)
10.7 (2.3)
61.61%
4
50
gSeg (nets.)
1.6 (1.0)
23.1
13.1 (8.3)
45.62%
kerSeg (nets.)
2.6 (0.7)
0.0 (0.0)
13.5 (4.3)
80.53%
gSeg (stats.)
(0.4)
48.1 (13.4)
(7.0)
50.26%
kerSeg (stats.)
0.0 (0.0)
0.0 (0.0)
0.0 (0.0)
100%
CPDstergm
0.0 (0.0)
0.0 (0.0)
0.0 (0.0)
100%
CPDrdpg
0.1 (0.3)
1.0 (0.0)
1.3 (1.3)
92.86%
CPDubs
0.1 (0.3)
3.5 (5.7)
2.0 (0.0)
88.13%
CPDker
(0.5)
17.7 (6.5)
(3.4)
60.44%
100
gSeg (uets.)
1.5 (1.2)
20.5 (6.5)
15.5 (7.3)
45.21%
kerSeg (nets.)
2.7 (0.5)
0.0 (0.0)
16.2 (2.9)
76.87%
gSeg (stats.)
(0.5)
(na)
Inf (na)
40.42%
kerSeg (stats.)
0.1 (0.4)
0.2 (0.4)
0.6 (1.1)
99.21%
CPDstergm
(0.0)
(0.5)
(0.5)
99.48%
CPDrdpg
0.0 (0.0)
1.0 (0.0)
1.0 (0.0)
93.19%
CPDubs
0.1 (0.3)
4.1 (6.1)
2.7 (2.6)
87.08%
CPDker
0.1 (U.3)
26.5 (11.2)
11.7 (2.3)
53.58%
200
gSeg (nets.)
1.5 (1.4)
22.7 (6.3)
15.6 (7.3)
46.82%
kerSeg (nets.)
2.5 (0.6)
(0.0)
15.3 (3.0)
76.60%
gSeg (stats.)
2.0 (0.9)
Inf (na)
Inf (na)
37.35%
kerSeg (stats.)
0.1 (0.3)
0.0 (0.0)
1.0 (2.6)
99.33%
Table Means (standard deviations) of evaluation metrics for dynamic networks simulated from the STERGM with p = The best coverage metric bolded.
Method
jK K
d(cic) 4
d(cic) 4
C(g.g") f
CPDstergm
0.0 (0.0)
1.1 (0.3)
1.1 (0.3)
94.20%
CPDrdpg
1.2 (1.7)
6.0 (5.5)
8.0 (4.9)
75.61%
CPDubs
1.5 (0.5)
4.3 (2.3)
11.3 (1.0)
75.64%
CPDeer
(0.0)
17.9 (7.9)
11.1 (2.6)
54.90%
6
50
gSeg (uets.)
13.1 (1.2)
0.0 (0.0)
19.4 (1.1)
27.47%
kerSeg (nets.)
10.1 (1.0)
1.5
18.5 (1.5)
35.82%
gSeg (stats.)
15.3 (1.2)
(0.6)
20.6 (0.6)
25.18%
kerSeg (stats.)
9.7 (1.2)
3.3 (1.3)
19.4 (1.8)
35.28%
CPDstergm
(0.0)
(0.4)
(0.4)
93.95%
CPDrdpg
0.9 (1.0)
4.9 (1.7)
8.6 (3.8)
77.59%
CPDnbs
1.4 (U.5)
4.9 (2.2)
10.9 (1.0)
73.84%
CPDker
(0.0)
29.9 (9.4)
10.7 (2.2)
50.48%
100
gSeg (nets.)
12.0 (0.0)
0.0 (0.0)
19.0 (0.0)
28.00%
kerSeg (nets.)
10.2 (0.9)
0.5 (0.5)
18.1 (1.1)
36.13%
gSeg (stats.)
14.9 (3.2)
2.8 (4.9)
20.5 (0.6)
25.09%
kerSeg (stats.)
8.1 (1.2)
5.2 (1.6)
17.9 (1.8)
38.32%
CPDstergm
(0.3)
(0.0)
2.1 (4.4)
97.06%
CPDrdpg
0.3 (0.6)
2.4 (0.8)
333 (2.3)
91.08%
CPDubs
1.4 (0.5)
4.3 (1.9)
11.9 (0.5)
75.58%
CPDker
0.0 (0.0)
35.2 (1.3)
10.7 (1.7)
48.77%
200
gSeg (nets.)
12.0 (0.0)
0.0 (0.0)
19.0 (0.0)
28.00%
kerSeg (nets.)
10.0 (0.7)
(0.4)
18.1 (0.9)
36.17%
gSeg (stats.)
5.5 (6.6)
28.6 (22.2)
20.4 (0.9)
31.1%
kerSeg (stats.)
8.7 (1.0)
3.4 (0.8)
18.9 (0.4)
42.86%
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Published in Transactions on Machine Learning Research (08/2025)
Table Means (standard deviations) of evaluation metrics for dynamic networks simulated from the STERGM with p =8 The best coverage metric bolded.
P
n
Method
- Kl
d(C)c) 4
d(ci)) 4
C(g.g)) f
CPDstergm
0.4 (0.6)
3.7 (5.5)
5.1 (3.8)
86.32%
CPDrdpg
(1.1)
9.4 (7.1)
11.3 (5.4)
70.72%
CPDubs
1.3 (0.6)
4.9 (4.2)
11.8 (0.6)
75.57%
CPDker
0.1 (0.3)
20.5 (8.7)
12.6 (2.1)
51.15%
8
50
gSeg (nets.)
13.5 (1.2)
0.0 (0.0)
19.7 (1.2)
27.07%
kerSeg (nets.)
10.4 (1.5)
1.5 (1.2)
19.1 (1.8)
36.58%
gSeg (stats.)
14.6 (2.2)
(1.7
19.9 (1.2)
27.10%
kerSeg (stats.)
9.3 (1.3)
4.8 (1.9)
18.5 (1.8)
36.23%
CPDstergm
0.0 (0.0)
1.9 (1.2)
1.9 (1.2)
92.65%
CPDrdpg
0.8 (1.3)
7.1 (8.1)
9.0 (3.6)
73.13%
CPDubs
1.3 (0.5)
5.1 (3.2)
12.0 (0.0)
75.51%
CPDker
(0.3)
29.8 (9.5)
12.0 (1.8)
49.71%
100
gSeg (uets.)
12.0 (0.0)
0.0 (0.0)
19.0 (0.0)
28.00%
kerSeg (nets.)
9.6 (1.2)
1.3 (1.3)
18.0 (1.1)
37.31% %
gSeg (stats.)
12.9 (1.3)
(2.4)
19.9 (0.7)
28.56%
kerSeg (stats.)
8.7 (1.4)
5.9 (2.2)
18.4 (1.7)
35.78%
CPDstergm
(0.0)
(0.0)
(0.0)
94.45%
CPDrdpg
0.7 (1.0)
4.20
5.7 (2.2)
85.59%
CPDubs
(0.5)
(3.1)
11.1 11.1
73.38%
CPDker
0.1 (U.3)
33.3 (5.9)
10.5 (2.0)
49.89%
200
gSeg (nets.)
12.0 (0.0)
0.0 (0.0)
19.0 (0.0)
28.00%
kerSeg (nets.)
9.1 (0.6)
(0.0)
16.9 (1.0)
38.13%
gSeg (stats.)
14.1 (0.9)
1.1 (0.8)
19.6 (0.5)
26.15%
kerSeg (stats.)
8.8 (1.0)
2.9 (0.4)
17.6 (0.5)
38.60%
networks, we generate two weight matrices U, V E Bdddd where each entry sampled independently as
Ur, Uniform(0,1/11) 1/16), Vr Uniform(1/11.
Vr,se [L....,d}.
and the weight matrices remain fixed within the corresponding time segments between consecutive change points. Lastly, at each time point t, the adjacency matrix y E ( {0,1] 1| generated as
Bernoulii U UZ)) te AuA.
y/
Bernoulli [ VZJ,, t
The latent dimension d is varied across simulation settings with (10, 15, 20).
Figure presents examples of generated networks at specific time points. In particular, Scenario produces adjacency matrices that sparse, and no specific network pattern be noticed. To dctect change points with the proposed method, use two network statistics, edge count and mutuality, in both formation and dissolution models, since the networks are generated based on the similarity in latent positions between nodes. For gSeg and kerSeg methods, besides dynamic networks, we also use these two network statistics as another specification. The CPDrdpg and CPDnbs methods directly use the networks as input data. Tables and g display the means and standard deviations of evaluation metrics for different specifications.
The weight matrices, generated from the Uniform distributions with small support, induce sparsity in the simulated networks, posing challenges for both the proposed and competitor methods. Although most of the methods are able detect the true change points, they also tend to identify additional change points that deviate substantially from the ground truth, as reflected by high absolute errors and large ono-sided Hausdorff distances. Moreover, as the networks are generated from latent positions that differ by nodes, we observe a degradation the performance gSeg and CPDnbs when the latent dimension d grows, partly due to the incrcased complexity of the underlying network structures. In contrast, the performance of kerSeg, which uses network statistics as input, and that of CPDrdpg, which aligns closely with the data generating mechanism, improve when the latent dimension increases. Similarly, the proposed CPDstergm
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Published in Transactions on Machine Learning Research (08/2025)
100
100
100
1
34
34
34
100
67
100
100
100
100
67
34
67
100
34
67
100
34
100
Figure 4: Examples adjacency matrices generated from RDPGM with 15 and n = 100. In the first row, from left to right, each plot corresponds the network 25, 50, respoctively. In the second row, from left to right, each plot corresponds to the network al / 26, 51, 76 respectively (the change points). In each display, red dot indicates and zero otherwise.
method demonstrates robustness to different latent dimensions and the performance improves with larger network sizes. On the other hand, the gSeg and kerSeg methods show substantial gains when applied to network statistics instead of dynamic networks. The performance of CPDker remains consistent across varied latent dimensions. The proposed CPDstergm method, which also utilizes network statisties, outperforms the competitor methods, indicating its adaptability to sparse and temporal dependent networks.
5.1.1 Time Comparison
We compare the computation times of different methods across varying node sizes, and the results are provided in Table IO. We focus on three representative configurations where the change point detection performance is comparable across methods, as shown in Section 5.1 The reported computation times in seconds reflect the total runtime over three simulated network time series. Tt is worth noting that some competing methods exhibit shorter runtime when they fail detect the correct change points and terminate early. Although the proposed method requires more computation time, achieves better performance on average compared the competitors.
5.2 MIT Cellphone Data
The Massachusetts Institute of Technology (MTT) cellphone data (Fagle Pentland 2006) consists of human interactions via cellphone activity, among 96 participants for a duration of 232 days. The data were taken from 2004-09-15 2005-05-04, which covers the winter and spring vacations in the MIT academic calendar. For participants i and j, connected edge yřy indicates that they had made at least one phone call on day t, and yi = 0 otherwise.
As the data portrays human interactions, we use the number of edges, isolates, and triangles to represent the occurrence of connections, the sparsity of social networks, and the transitive association of friendship, respectively. For gSeg, and kerSeg methods, we use these three network statistics ( as input data, since they produce more informative results than using the dynamic networks fy' (-') The CPDrdpg and CPDubs methods directly the dynamic networks, and CPDker method uses network statistics.
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Published in Transactions on Machine Learning Research (08/2025)
Means (standard deviations) of evaluation metrics for dynamic networks simulated from the RDPGM with 10. The best coverage metric is bolded.
Table
n
Method
Kl
d(Cic) 4
d(cic) 4
C(g.9") f
CPDstergm
18 (1.1)
1.1 (0.9)
13.1 (5.5)
81.61%
CPDrdpg
(1.1)
(5.0)
18.3 (2.2)
72.82%
CPDubs
1.0 (0.5)
5.2 (3.4)
10.2 (3.3)
76.71%
CPDker
0.1 (0.3)
20.0 (8.3)
9.7 (3.1)
64.49%
10
50
gSeg (nets.)
3.0 (0.0)
Inf (na)
Inf (na)
0.00%
kerSeg (nets.)
4.2 (1.4)
3.3 (4.4)
16.0 (3.1)
61.06%
gSeg (stats.)
(1.3)
16.9 (17.3)
19.7 (1.1)
64.26%
kerSeg (stats.)
2.5 (1.5)
6.0 (5.5)
19.5 (2.0)
73.06%
CPDstergm
0.4 (0.5)
0.9 (0.4)
6.5 (7.5)
93.34%
CPDrdpg
1.9 (1.0)
6.6 (4.6)
17.7 (2.3)
71.83%
CPDubs
1.0 (0.8)
5.3 (4.1)
9.5 (4.1)
77.52%
CPDker
(0.5)
(9.4)
(2.3)
57.39%
10
100
gSeg (uets.)
2.9 (0.4)
Inf (na)
Inf (na)
5.87%
kerSeg (nets.)
7.1 (1.1)
1.5 (3.8)
17.8 (2.0)
46.90%
gSeg (stats.)
(2.2)
(10.9)
20.4 (1.0)
60.88%
kerSeg (stats.)
4.9 (1.5)
0.8 (1.7)
19.3 (2.3)
68.43%
CPDstergm
(0.0)
(0.0)
(0.0)
96.36%
CPDrdpg
2.3 (0.7)
5.6 (2.5)
18.7 (2.2)
71.59%
CPDubs
1.5 (0.7)
6.5 (3.4)
11.4 (2.7)
70.36%
CPDker
0.0 (U.0)
24.4 (8.5)
10.4 (1-7)
57.14%
10
200
gSeg (nets.)
2.7 (0.6)
Inf (na)
Inf (na)
13.66%
kerSeg (nets.)
8.1 (1.8)
(0.4)
18.5 (1.4)
43.41%
gSeg (stats.)
4.9 (1.3)
7.3 (3.9)
20.1 (0.3)
62.78%
kerSeg (stats.)
5.4 (1.3)
2.2 (3.9)
20.1 (1.5)
61.71%
Table Means (standard deviations) of evaluation metrics for dynamic networks simulated from the RDPGM with 15. The best coverage metric is bolded.
Method
KK K
d(Cic) 4
d(c)c) 4
C(g.g") f
CPDstergm
1.7 (1.0)
1.9 (3.9)
14.7 (2.7)
80.38%
CPDrdpg
2.4 (0.6)
1.6 (0.6)
16.9 (1.9)
75.90%
CPDubs
1.5 (0.7)
6.0 (4.1)
11.3 (1.7)
70.88%
CPDeer
(0.4)
18.6 (10.0)
(2.5)
66.67%
15
50
gSeg (uets.)
3.0 (0.0)
Inf (na)
Inf (na)
0.00%
kerSeg (nets.)
5.3 (1.4)
3.3 (4.5)
18.7 (1.8)
56.01 %
gSeg (stats.)
(1.8)
25.8 (27.7)
19.9 (0.5)
54.73%
kerSeg (stats.)
3.2 (1.1)
2.7 (5.2)
20.3 (2.1)
76.76%
CPDstergm
(0.6)
(0.0)
11.1 (6.5)
88.32%
CPDrdpg
1.1 (0.8)
4.0 (5.6)
15.5 (1.8)
79.18%
CPDmbs
1.5 (0.8)
7.1 (3.6)
10.8 (2.0)
68.99%
CPDker
(0.3)
22.3 (7.5)
10.3 (2.5)
60.73%
15
100
gSeg (nets.)
3.0 (0.0)
Inf (na)
Inf (na)
0.00%
kerSeg (nets.)
6.5 (1.6)
2.7 (5.7)
17.9 (2.0)
47.57%
gSeg (stats.)
(2.2)
10.4 (15.8)
20.0 (1.2)
63.45%
kerSeg (stats.)
4.3 (1.3)
0.1 (0.3)
20.6 (2.4)
74.23%
CPDstergm
(0.3)
(0.0)
2.1 (4.4)
95.65%
CPDrdpg
1.7 (0.9)
4.3 (4.7)
15.6 (1.4)
76.17%
CPDubs
1.9 (0.7)
7.9 (2.8)
11.8 (0.6)
66.32%
CPDker
0.1 (0.3)
24.2 (7.5)
10.3 (1.4)
56.99%
15
200
gSeg (nets.)
2.8 (0.4)
Inf (na)
Inf (na)
8.91%
kerSeg (nets.)
(1.9)
(5.3)
18.8 (1.7)
41.86%
gSeg (stats.)
4.2 (2.0)
9.1 (16.5)
18.9 (0.7)
61.83%
kerSeg (stats.)
4.5 (1.2)
0.2 (0.6)
20.0 (2.6)
72.11%
20

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Published in Transactions on Machine Learning Research (08/2025)
Table Means (standard deviations) of evaluation metrics for dynamic networks simulated from the RDPGM with 20. The best coverage metric is bolded.
n
Method
Kl
d(C))) 4
d(c(c) 4
C(g.9")
CPDstergm
1.1 (0.7)
5.6 (8.4)
15.9 (2.1)
79.76%
CPDrdpg
(1.0)
(0.8)
16.8 (1.3)
73.88%
CPDubs
1.5 (0.8)
8.4
11.8 (0.6)
66.97%
CPDker
0.3 (0.5)
16.2 (10.0)
7.5 (3.7)
71.19%
20
50
gSeg (nets.)
2.9 (0.3)
Inf (na)
Inf (na)
3.19%
kerSeg (nets.)
5.3 (1.4)
2.8 (4.5)
16.7 (2.1)
58.03%
gSeg (stats.)
3.1 (1.8)
21.8 (30.1)
19.9 (0.5)
58.70%
kerSeg (stats.)
3.0 (1.2)
0.4 (1.1)
17.3 (3.1)
79.34%
CPDstergm
1.2 (0.7)
0.9 (0.3)
15.7 (4.4)
87.37%
CPDrdpg
1.6 (0.9)
3.0 (4.9)
16.3 (1.0)
78.92%
CPDubs
1.7 (0.8)
8.3 (3.0)
11.6 (1.6
66.22%
CPDker
(0.0)
19.3 (8.9)
(2.0)
64.06%
20
100
gSeg (nets.)
2.9 (0.3)
Inf (na)
Inf (na)
6.18%
kerSeg (nets.)
7.5 (1.3)
1.0 (2.6)
18.3 (1.6)
44.46%
gSeg (stats.)
(2.8)
26.4 (25.4)
19.7 (0.6)
50.98%
kerSeg (stats.)
3.1 (1.1)
0.0 (0.0)
18.2 (3.1)
81.13%
CPDstergm
(0.5)
(0.0)
11.3 (6.9)
89.83%
CPDrdpg
1.8 (0.4)
1.2 (0.6)
16.9 (1.0)
80.81%
CPDubs
1.7 (0.7)
8.3 (3.5)
11.8 (0.4)
67.29%
CPDker
0.2 (U.4)
24.7 (9.2)
10.9 (2.0)
54.76%
20
200
gSeg (nets.)
2.8 (0.4)
Inf (na)
Inf (na)
9.54%
kerSeg (nets.)
(0.7)
(0.0)
18.3 (2.0)
39.80%
gSeg (stats.)
3.9 (2.3)
16.8 (21.0)
20.1 (1.3)
54.09%
kerSeg (stats.)
4.1 (0.9)
0.0 (0.0)
17.0 (3.0)
73.33%
Table 10: Time comparison in seconds across different methods and node sizes.
SBM (p 0)
STERGM (p
RDPGM (d = 20)
Method & Node
50
100
50
100
50
100
CPDstergm
15.033
39.833
17.085
39.293
29.607
40.065
CPDrdpg
2.899
6.289
2.961
6.271
3.352
6.690
CPDnbs
0.308
1.124
0.310
1.078
0.338
1.436
CPDker
9.622
12.062
12. 105
12.692
7.749
7.418
gSeg (nets.)
0.112
0.238
0.183
0.509
0,088
0.221
kerSeg (nets.)
3.971
4.303
4.707
4.922
5.957
7.877
gSeg (stats.)
2.698
7.244
2.321
6.071
1.276
1.960
kerSeg (stats.)
6.290
10.893
6.207
10.066
6.161
7.119
In addition, the proposed CPDstergm method supports the usage of node level covariates, define nodal attribute a; E (High-Activity, Low-Activity) for each node [n], based on its cumulative activity over time. Specifically, we calculate the total degree of each node across the time span, and classify nodes as 'High-Activity' their cumulative degree exceeds the median, and 'Low-Activity" otherwise. The four network statistics, edges, isolates, triangles. and homophily for activeness, are used both formation and dissolution models of our method. Figure 5 displays the change magnitude AÇ of Equation (17) and the detected change points from the proposed and competilor methods. Moreover, Table []] provides list of potential nearby events that align with the detected change points, and Figure D displays raster plot of active edges over time, as visual validation to the detected change points.
The two shaded areas in Figure correspond to the winter and spring vacations in the MIT 2004-2005 academic calendar, and our method can punctually detect the pattern change in the contact behaviors. The competitor methods can also detect the beginning of the winter vacation, but their results on the spring vacation are deviated, Visually, the substantial reduction of interactions in Region I and the subtle shifts in Region II of Figure p support the change points detected by the proposed method, aligning with the
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Published in Transactions on Machine Learning Research (08/2025)
g5eg
kerSeg
CPDrdpg
CPDnbs
CPkee
10
Change magnitude
2004-10-41
2004 2004-100-24
2004-11-16
2004 2004-12-47
2005 03-244
Figure 5: Visualization of the change magnitude AỨ and the detected change points from our method for the MIT cellphone data. The detected change points from the competitor methods (blue bars) are also displayed. The two shaded areas correspond to the winter and spring vacations in the MIT 2004-2005 academic calendar. The data-driven threshold (red horizontal line) is calculated by (88 with Z0.975-
largest spike for the winter vacation (2004-12-17) and the smaller spike for the spring vacation (2005-03-24) in Figure 5 Furthermore, we detect a few spikes in the middle of October 2004, which correspond to the annual sponsor meeting that happened on 2004-10-21. About two-thirds of the participants have prepared and attended the anmual sponsor meeting, and the majority of their time has contributed to achieve project goals throughout the week (Fagle & Pentland 2006). The drastic shifts Region III of Figure also reveal the frequent changes in the network patterns during the sponsor meeting week.
Table 11: Potential nearby events that align with the detected change points (CP) of our method.
Detected CP
Potential nearby events
2004-10-13
Preparation for the sponsor meeting
2004-10-24
2004-10-21 (Sponsor meeting)
2004-11-16
2004-11-17 (Last day to cancel subjects)
2004-12-17
2004-12-18 2005-01-02 (Winter vacation)
2005-03-24
2005-03-21 to 2005-03-25
(Spring vacation)
5.3 Stock Market Data
The stock market data consists of the weekly log returns of 29 stocks included in the Dow Jones Industrial Average (DJIA) index, and it is available in the R package ecp (James & Matteson 2015). We consider the data from 2007-01-01 to 2010-01-04, which covers the 2008 worldwide economic crisis. Morcover, we focus on using the negative correlations among stock returns to detect the systematic anomalies in the financial market. First, we use a sliding window of width to calculate the correlation matrices of the weekly log returns. We then truncate the correlation matrices by setting those entries which have negative values as 1, and the remaining as O. In the 158 constructed networks, a connected edge yi indicates the log returns of stocks and j negatively correlated over the four-week period that ends at week t.
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Published in Transactions on Machine Learning Research (08/2025)
J. Q%. i 1 it we p W
Win of 1550 w pop int
ineeeii o ip
TICgt A Cil a 206 1. hoe
W 3
ammo wes The bhale The auan
"oote hhe 1025
Wiih Afterr
Active: edges
144
122
11
H 25 MTH 1aA
7 11-55445
1
2F
M M
50
100
150
200
Time points
Figure 6: Raster plot of the MIT cellphone data. Each black tile represents an active edge between a pair of nodes at a given time point. The red vertical lines indicate change points detected by the proposed method, and the blue shaded regions highlight notable changes in network interaction patterns.
As network statistic, the number of triangles signifies the volatility of the stock market, since the three stocks are mutually negatively correlated. In other words, the more triangles in a network, the more op- posite movements among the stock returns, suggesting a large fluctuation in the financial market. On the contrary, when the number of triangles low, the majority of the stock returns either increase or de- crease at the same time, suggesting a stable trend in the market. In addition, we define nodal attribute æ; (Hedging- Prone, Forket-Follooingg for each stock i E [n]. based on cumulative degree over time, Specifically, stocks with total degree above the median are labeled as 'Hedging-Pronee reflecting defensive behavior relative to the market, while the remaining stocks are labeled "Market- indicating synchronized movement with the broader market. For the proposed method, we use three network statistics, edges, triangles, and homophily for risk orientation, in both formation and dissolution models. For the competitor methods, we use dynamic networks as input data, since they produce more reasonable results than using the network statistics. For CPDker, we use network statistics as inpul. Figure 7 displays the change magnitude AC of Equation (17 and the detected change points from the proposed and competitor methods. Table 12 provides list of potential ncarby events that align with the detected change points, and Figure displays raster plot of active edges over time, visual validation to the detected change points.
As expected, the stock market is highly volatile between 2007 and 2009, The competitors have detected excessive number of change points, aligning with the smaller spikes in AÇ of Figure Those change points could be detected by our method, if we manually lower the threshold to adjust the sensitivity. Since the networks are constructed using sliding window, a detected change point indicates the pattern change occurs amid the four-week time horizon. As supporting evidence to the three detected change points in Table [2] the New Century Financial Corporation Brothers one of the largest investment banks. Their bankruptcies caused by the collapse of the mortgage industry severely fueled the worldwide financial crisis, which also led the Dow Jones Industrial Average to the boltom. In Figure S Region shows a substantial increase in edge density following the collapse of New Century Financial Corporation, indicating heightened volatility in the stock market. As fewer edges suggest stable trend, Region II captures the market downturn following the bankruptey of Lehman Brothers. Finally, Region III shows a rebound in market activity, as edge density rises after the Dow Jones Industrial Average reaches its lowest point, suggesting the beginning of a recovery phase.
the largest U.S. subprime mortgage lender in 2007, and the Lehman
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g5eg
kerSeg
CPDrdpg
Dnbs
CPDker
Change magnitude
2
0
2007 2007-04-23 -04-23
2018- 1008-10-06
2019 -44-00
Figure 7: Visualization of the change magnitude AC and the detected change points from our method for the stock market data. The detected change points from the competitor methods (blue bars) are also displayed. The dala-driven threshold (red horizontal line) is calculated by (18) with Z0.975
Table 12: Potential nearby events that align with the detected change points (CP) of our method.
Detected CP
Potential nearby events
2007-04-23
2007-04-02 (New Century Financial Corporation filed for bankruptey)
2008- 10-06
1008-09-15 (Lehman Brothers filed for bankruptcy)
2009-04-20
2009-03-09
(Dow Jones Industrial Average bottomed)
6 Discussion
In this work, we study the change point detection problem in time series of networks, which serves as a prerequisite for dynamic network analysis. Essentially, we fit a time-heterogeneous STERGM while penalizing the sum of Euelidean norms of the parameter differences between consecutive time steps. The objective function with Group Fused Lasso penalty is solved via Alternating Direction Method of Multipliers, and we adopt the pseudo-likelihood of STERGM to expedite parameter estimation.
The STERGM (Krivitsky & Handcock 2014) used in our method is a flexible model to fit dynamic networks with both dyadic and temporal dependence. manages dyad formation and dissolution separately, as the underlying reasons that induce the two processes are usually different. reality. Furthermore, the ERGM suite (Handcock al., 2022) provides an extensive list of network statistics to capture the structural changes, and we develop an R package CPDstergm to implement the proposed method.
Several improvements to our change point detection framework are possible. Relational phenomena by nature have degrees of strength, and dichotomizing valued networks into binary networks may introduce biases for analysis (Thomas & Blitzstein, 2011). To this end, we can extend the STERGM with a valued ERGM (Krivitsky] 2012; Desmarais & Cranmer 2012a Caimo & Gollini 2020 to facilitate change point detection in dynamic valued networks. Morcover, the number of participants and their attributes are subject change over time. It is necessary for change point detection method adjust the network sizes as in Krivitsky et al. (2011), and to adapt the time-evolving nodal attributes by incorporating the Exponential-
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Active odiess
50
100
150
Time points
Figure 8: Raster plot of the stock market data. Each black tile represents an active edge between a pair of nodes at a given time point. The red vertical lines indicate change points detected by the proposed method, and the blue shaded regions highlight notable changes network interaction patterns.
family Random Network Model (ERNM) as in Fellows & Handcock (2012) and Fellows & Handcock (2013) In addition, the proposed framework can be extended to detect change points in dynamic multilayer networks (Sohn Park 2017 Wang el al.] 2025 where different lypes of interactions for a fixed set of nodes are observed simultancously. Lastly, recent advances contrastive learning (Ermshaus al. 2023 Puchkin & Shachrbakova, 2023) present novel and promising direction for developing adaptive and data-driven approaches to change point detection in dynamic networks.
Code and Data Availability
The R package CPDstergm is available at http::://gituub..cmmalaleeeeee/s duce the results in the manuscript using the R package is available at https://github. com/ anlenkeenkei/ CPDstergm / The MIT cellphone data is available in the R. package CPDstergm, and the stock market data is available in the R package ecp.
The code to repro-
Acknowledgments
We thank Mark S. Handcock for helpful comments on this work. Also, we are extremely grateful for the constructive comments provided by the anonymous reviewers and Action Editors. Yik Lun Kei and Oscar Hernan Madrid Padilla were partially funded by NSF DMS-2015489.
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A ADMM Convergence
In this section, we provide the proof for Proposition
Consider the constrained optimization problem
2i 1,- -Z,,-12
arg min -1(0) + A
d;
im1
subject to 0 - z = 0.
The Lagrangian can be expressed as and the augmented Lagrangian be expressed as
=) )(()) + trlpp -]]]
(19)
p) 9(z) tr(p (0- x)]
2
0 x1?
(20)
2
where
2 22.112
f(0) - 20, g(z)
2
d
and p E TRT the Lagrange multiplier.
Let (0", p²) be the optimal point of the Lagrangian L(O, z, p) in Equation
19
Also, let
n S(0") + y2²²)
where h is the current ADMM iteration. Denote pi+1 = g zh+ E RTXP as the primal residual in the k+1 matrix form. Since C(0k+1, , p²) = ++1 + tr((p") T , and L(, ,,,,p"'' p²) s 1(0 2+1 pp)), we
have
p² S tt + tr((p") Txktl]] p" - p b+1 s trl(p") (T ktll.
(21)
Note that the dual update of ADMM procedure for the optimization problem in 20 is p p² ar
p² pk11 ap Since g++1 arg ming (0, z p²), the optimal condition gives
E 0f( + pt a 2²).
(22)
Substituting the p in side of (22 we have
22 with p+1 art+1 and then adding and subtracting azh+1 to the right hand
0 1 + (p - har a10h-1 - z²t) + azz+1 k+l x+1
k+1
0 0f()++)) + r+1 kr + afttt gh+1 + (zxt2 x²)) htl + (2 211)
0 00 pt Ap + ap
0 E + pt + aa(xht1 - =²).
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This implies that @h+1 minimizes the objective function
f(0) + tr (p
h+ + - 0)
and
f(@h+1) ) + tr (p + a(z k+1 - 211) T gh+!|| - f(0)) tr((p + -(xxtt1=2"))
-0-1-=2)) =²))² 00)
(23)
Similarly, since x =+1 arg min, (0kt) 2, p²)), the optimal condition gives
0 0g(2 pt a(B - 2111).
(24)
Substituting the p in 24 with k+1 arttl have
0 E 0g(2*-1) (p - arktl) ar 0y(2*~1, p k-1
This implies that *+1 minimizes the objective function
g(2) tr((ph+1)T
and
g(2ht1) tr((p ++1 S g(2*) - tr((ptt1))=* 2*).
(25)
Adding the two Inequalities
23 and 25
together, we have
[(0+t)) + g(2 - -(00) g(=²) S (tr((p + -= T (0r - 1h+ = (2^+1 - =²*)]
p p² s tr tr[ tt1 + -(z
- )² T (0- 1h+1)] - tr((p (2 2²)]_
(26)
Separating and grouping the terms on the right hand side
26 we have
k+1 p" s (T)
T(0 (0r oh+1)] tr tr(a(= )²) (0- pbtii] + tr((p+t1) + (Til))((2)) (2 - 2*)]
s tr((p+t)] (0r x - -0 2]+] + tr trfa(z*t1 - - (0- - pbtti]]
s (tr(( T 1 + tr tr|a(2 k+l T o
(27)
where 2" and g++1 2b41 x+| Moreover, note that
x+l gh+1 =+1 (0) 2²)
x+1 = (0 0²) - (2 - =²)
- k+1 = (0' 1 1 + (2kt1 - 2²)
(0" 0 k-h (zk+1 =²).
Substituting the term (0- - 0k+1) in Inequality 27 we have
xx k+1 1 p¹ S (r ++1) TT nt1 + tr (r(a(2* k+1 2²) ()) kkt (2kt1_ -
5 str * kt1] - tr(a(2= k+1 - 2 ) [dtt]] - tr(a(2=++ ht -x -
(28)
Then adding Inequalities
21 and
28 we have
s tr((p")T T x ++1 11 tr (p ++1 I x - tr [ k+1 - - 2 [ tr a(2 - T2) x²)]
0 < ur((p" p2+2) T x1 irfa(z k+1 - 2²) TT [T - -r([a(z>)) - T (2 _ =2)]]
0 (p TT) p + 21r(a(= [T)T,,tt) + 2 21r(a(= 2 (2 2*)]
(29)
where 29 followed we multiply the inequality by two and change the sign both sides.
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Now we consider the expansion of the first term on the right hand side in Inequality 29 ph+1 with the dual update p*+1 = pt + anttl, we have
Substituting the
20rl(ptt1= - -p-) Totty
(30)
= 2ull(p + tar - -p") T ]
2tr((p" -p")T [ + 2t 2tr(a(1 [(p+t))]
= 2url(0" - -p") , + +all 1/ + alll
(31)
Since g++1_ p² + ark-l, we also have ph+1= (p p²). Substituting the r ph+1 in the first two terms of 31 and expanding the matrix multiplications, the expression in (30) proceeds as
21rl(p ++1 - pp-) T p
1 10/2(p - p²) potti-phe + a t -p)) ( - )²1 + +all
1
a
1 -r(((ttt)) ITT pptt1= - -2(0')T pa+1 + +2( pt - (p¹)) T p²1 all 112 113
1 -ul(ott /pptt1 - 2(p")) p patl + 2('))) Tp - ( ii((((ottt) (1) 1x+2_ 1+41-((p)) -2(p') p r+ + (0')T T pi] - [(o²)] pt - 2(02) "² + (0') 0']} + a
T pt + +p¹²) pr - (pp') p²] + +all
=
a
1
a
1 a (1p -011 -1pp" p0)))) + 11 11..
(32)
Next we consider the expansion of the second and third terms on the right hand side in Inequality an additional term a||r=+11/? By adding and subtracting a [|* - 2 ||2, we have
29 with
allr*++1|3 |1 + 2tr (a2**t - 2²) p + 2tr [a - =x*)T (**t1 - ==))]
(33)
=all*++1|0 || atr [2(2 - = ) T b+1 + 2tr [a(z (= - 2/17 (2 =²)] + all [|| - 2113 allz =xx1/ 112
= =a(I 112- + tr [=(2 k+1 - =x")Tx*+1] 'I + || k+ - x² *0122 allz - x* =1 + 2tr [a(x (+1 - -x (0T(x*1)))) (z/t1 =))]
= -all k+1 + h+1 - x² )1? -all x+1 - = II? + 2tr 2tr[a(( k+1 T x+1 - z²)]
(34)
Since x x+1 2" = (=++1_ - )²) + (z* - 2)) we substitute the 21+1 x" in the third term of 34 so that
33
proceeds as
allr + 2tx [a(2 k+1
(t1 - : [t + 2tr 2tr k+l - =t) (x - x²)]
=allntt + (2 - > -allzht1 - =²11 1/ + 2 a( 2+2 - 2²) |1 1(= - =² + (2* - ==')]
=a|lrtt] + (z/+1 - 2² >²11 allz - $1 + atr (I - -²² (1(( =² + +2(x - =''}]
-
=a|ppbt + (z - 2* 10 + atr (z h+ - 2/) [[ - + 2(2²* -
- 2²)1}
(35)
Since x+1 x² = (2tt1 =²) (zx x²), we sequentially substitute the x+1 - 26 in 35
that 33
procceds as
allr 112 + 2tr [a(= 24) T [-t1] + 2tr [0(2++1 24) ( (2*+1 =²]]]
= allpbtt
+ 1=1 - x² )1? + atr
(** - )²) T [[*
k+1 - =²²)= - -(** - x²) + +22 - xx x11}}
= =allntt1 + (2 - -21)1 + atr
(x*+1 - 2 [T((2 k+1 - 2²) + +(2 - -2')]}
-allnbtl + (- - =1 + atr
[[2/101] - =²) - -x* - =="]] (2 - =²)) + (z" - z1)}}
allntt1 + (2/ - )| + ||| - 21112- || xx"| [
(36)
33

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Published in Transactions on Machine Learning Research (08/2025)
Combining the results from 32 and 36 to substitute the corresponding terms on the right hand side of
Inequality 29 we have
2trl(p R+1 p²) k+ x + 2tr str[a(=*t) ++1 - ) T + 2tr (=
h+1 - x1) [xx xx)] s 0
(1pp k+1 p² - -11p* - -p"ll²) + a ptt + (2 - = + a ||| k+1 - =01 - || - -x" =1:21 =
1 1lptt1 p² 1 + aalz I
1pt p=11²- al1 =" = l + ael allnttz + (2 - = 0. (37)
a
Define
1
v*
11 - p'1l + a|12** - ==1 2
a
Then Inequality (37 can be expressed as
vk+ v < -alln + +2 2²) )|²
$ aal k+1 1/? + 2tr &+1 1) (x k+1 )² + || - x² (2:))
s -all a||p all all - = - 22tr ( [(x/+( - x*)]
(38)
Recall that 2+1 minimizes g(2) tr((ph+1)) 2) and 2 minimizes g(z) - tr +r((p"))=2)
Tz]. Then
g( - tr((p*+1) TT
z = S g(= +T(( =*1
and
g(z²) tr((et)) T x11 S g(*h+1)== - (=((pt)T*ktt)) 2t11
Adding the above two inequalities, we have
tr((p')) 2 + tr((p ( [-r]]-(((+tt]]]) 24+11- - tr((p²)) 22* S 0 tr((p") tr((p")((x* - =2")| + tr((p (1 (x* - =2++111 < 0
-trl(p /pi T T(*ht1 - 2*1] < 0.
Recall p*+1_ p² ark+1, Then
atr((r+tl) (2++1 =²²)] s
Back to Inequality 388
can see that
Fk+1 - yk <
-a|lz - 2
also hold. Since V and
vk+1 <V² - -a--aaalle (1|n+t11 + - 111)²²)
(39)
we know that V is bounded by below decreasing sequence.
By iterating
39
we have
(1 1 P. || **) 0
which implies the primal residual || k co. Similarly, for
|| = 1 - 1| + 0 and dual residual ||
2/ 1/ - 0 as
1
0h 2
and d
19
primal
T xXP
p
T x
we have p.kk primall
+ O and y r
0 x co. This concludes the proof for Proposition
34

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Published in Transactions on Machine Learning Research (08/2025)
B Newton-Raphson Method for Updating 0
In this section, we derive the gradient and Hessian for the Newton-Raphen Raphson method to update A. The first-order derivative of (()) with respect to 0+. E RP,, the parameter in the formation model at particular time point t, is
exp Agg - e Ag Agij(yttt))
Vol. 1(0) = E
Uij Aggj(ytt))
(8,))X
Σ (u 11 Aot((t"
(1.))E
E (0, 1). The - 1 is the element-wise sigmoid function. Likewise, the first-order derivative of I(0) with respect to f-~ E R22 the parameter in the dissolution model at particular time point t, is similar except for notational difference.
where Mij h(0+.h. (u -0
Denote the objective function in Equation
10 as (0). To update the parameters 0 E RTXP in compact R R The matrices Brxp and R also vectorized
form, we first vectorize it as 0 vec+p(0)
(0)
as z = vzCrp(z) E 1 and u (w) E RTPx1, With the constructed matrices H E R2rExTp, the gradient of C.(0)) with respect 0 is
V ()) - -H (3) +a z(0) + t())
where = h(H Ø) E (0,1)
Furthermore, the second order derivative of ((0) with respect 0+ 12d1 is
V 1(0) = Σ -uty(1 (
(
(KJ)Y
and the second order derivative of ((0) with respect to θ-it E TR is similar except for notational difference. Thus, with the constructed matrices H E E2TExTP and W E (0.1/4)=B the Hessian of (o) with respect to 0 E R is
C H + al,p
where Ipp is the identity matrix. By using the Newton-Raphson method, the Ø E RTpx1 is updated as
O WH + Q
7 (-
z(a) tt)))
where e denotes the current Newton-Raphn Raphson itcration. Note that both W and jł are also calculated based on 0.. The constructed formation and dissolution network data is vectorized in the form of y E (0, (0,1)2rExL to align with the dyad order of the matrix H E 2=xxx
C Group Lasso for Updating 3
In this section, we provide the derivation to update 3, which equivalent to solving a Group Lasso problem (Yuan & Lin 2006). Denote the objective function in [1] La(y,8). When Br.. # 0, the first-order derivative of (r. (9) with respect Bi,, is
C. (T.8) = A B
11. (x + (a) 1,,17 - X., - X. x_,..B...))
08.
where X., E xx
1 is the ith column of matrix X E Rix(r-1) and Bi, E RIxp the ith row of matrix B E (r)xx) Setting the gradient to O, we have
-1
B (XX.XX.. +
1B,
8,
(40)
35

---

Published in Transactions on Machine Learning Research (08/2025)
where
8; ==ox[[((0(at) +u(a)
,,17 X. x1_-BB1,,))
Taking the Euclidean norm of 40 on both sides and rearrange the terms, have
||,,.112 = (a.X. (1|s:]12 A).
Plugging 1/,,.112 into 40 the solution of Bi, is
Be. ax.,X.,s (1 (1- ||:|||2 8i-
d
When Br, = 0, the subgradient v of || needs to satisfy ||0||2 S 1. Since
O Au a.. + v(()) 1,117 X. -..-BBi..h
we have e = 1- 8; and we obtain the condition that fi. becomes o if ||s:|l2 A. Therefore, to update Br. for each = ,,...,T 1, we can iteratively apply the following equation:
1 ax, X., (1-
B
|s:112/
where (+) max(:,0). The matrix X E (x-(r-1) is constructed from the position dependent weight d E R
D Error Bound under Structured Sparsity
In this section, we provide the proof for Proposition Denote 0 E RT as the true parameter and suppose that ||0" < M/2 for some M > Let Ô E RTXP be the minimizer of the objective function in (5) subject the constraint |10| M/2. Since â minimizes the penalized objoctive, we have
||0 02.112 < L((0))
10 07.112
L(O)+ +
1=1 d.
1=1 d
1
07. 07.112 10
-0,-112
L( L(0') <
(41)
d;
1=1
where L(O) -0)).
Define the estimation error as Â = 0 0" E RTTP, Using the triangle inequality I|a bll2 2 ||b||2 ||a||2
with a = Â A. and 0 - 0 such that a + b 6,41.- d.., we obtain:
0 1 0..112 1 2 10 - - - A i+..- i
À H1. Á,, 12.12 > |0 01.112 110_1,- 02.
(42)
for = 1,.., , TT- Applying Inequality
422 to the right- hand side of 41 we obtain
H+, A
L( - L()))
(43)
d
Now, we define the set of true change points as
$ S= (i (1 (1....,7
1} 01. # #
36

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Published in Transactions on Machine Learning Research (08/2025)
Suppose the loss function L(0) satisfies the Restricted Strong Convexity condition:
I(O"" + Δ) T(00) + (VI(0"). A
(44)
2
for all perturbations A RTXP that satisty the structured sparsity condition:
Σ
1+1, A.,112 12
1+.
1
(45)
d;
d;
igs
iS
for some constants & > 0 and a >
Assume the estimation error Å - 0"
RTXD satisfies
45
Then from Inequality
we have
A.,1
i s+1, A.,112
T((O) T((0)) S
Σ
d:
d
iES
i
P 6, l|2
L(θ) L(0)) S
+1,
(46)
d,
iES
where 466
follows by the condition in 45
Morcover, since (VL(0"), A) 2 -I(VL(0") А) and 0"
+ Â 0, it follows from the Restricted Strong
Convexity condition in 44 that:
T(O"- + Á) T(0") 2 2 Á 7 + (VI(0"), A)
L( - L(
A 2 (V A
Å H1,
A..
x- (1 a) Σ
A ( A
(47)
d;
where 47
follows by (46
Next, we denote the Total Variation (TV) norm II, for a matrix A RTxp, as
A A 20
A TV =
d:
where [d.) the position dependent weights. Denote constraint as
= |||| S 1, |||| 00 S 11]. Then we define a restricted dual norm ||, for a matrix B E RTXp, as
BI. = sup (B, A). AEC
Let
C = max | rv. |
20
and
Á A.
c
Note that
1
1 C
Å IIV
-
-- 1,
1
no
=11
A
Iox
---
37

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Published in Transactions on Machine Learning Research (08/2025)
Á C and
(VL(O"), 3) I sup (VL(0"), A) = [V
Âec
Furthermore, we have
(VL(0"), A = c- (VV(( I c- |V IlI-
|VT(0)) )|- max ( TV, 100
$ |V )ll. max [Tv, M]
(48)
where || Å co < 1001 + 1101 I M/2 + M/2 = M.
We now assume that
V - X
holds with probability at least 1 - 6. Then from 48
obtain
KVL(O"), A) S max Â |TV, M}
1+1 Ai.llz M
max
d;
^ 1+1 Av,1l2
|| - A. 1l2 M
+ Σ
max
d,
YES
iEs
Á., MM
1+1,
max
(1 + a)
(49)
d:
iES
A 1,,
Ac,112
$ 2
(1+a)
+ M
(50)
d,
where 49
follows by the condition in
45
and 50
follows by max(a, b) S + for a.
Substituting 50
into
47
we have
k 2
A (1+a) a)
1, A + +M 5
Â i+1,
2r.112
2
2
d;
dj
iES
iES
1+1. Anill2 M x- (1+a)
Å M+1,
Ai,ll2
|
(1+ a)
d
iES
3
Å A 1,-
A
X(1 + aa)
Σ
M.
(51)
F
2
Note that Eiesll A4t1 -
Then with Sauchy-Schward inequality, we have
AHLI. Acrlk d
Σ d
<
V
iES
iES
ies
d
|||| rr-
(52)
NES
38

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Published in Transactions on Machine Learning Research (08/2025)
Substituting 52
into 61
we have
k:
A(1
Σ iES
d ||
X + M
2
2
A F - 33(1++aa
Σ d 3
ف
|
+
(53)
iES
Note that for a. a, > 0, we have a a b S max(2a, 26). Then from
53
we have
k
< max (6((+a))
Σ
d 2
|
2AM
iES
This implies that either
1|| S
Edi² -|Å|1r
(54)
YES
or
A 2AM
(55)
In the first case with 554 we have
6Λ k
Å ||
<
(1 a0)
Σ d,²
iES
In the second case with 55 have
2AM
| 1| S
k
Combining the two cases, we have
2
6Λ
2AM
| || S max
(1 (1+a)
d
k
k
iES
Thus,
2
6Λ (1+
2AM
0
10
< max
Σ d
Tp
iES
This concludes the proof for Proposition
E Practical Guidelines
As in Boyd et al. (2011), we also update the penalty parameter for the augmentation term to improve convergence and to reduce reliance on its initial choice. Specifically, after the completion of an ADMM iteration, we calculate the respective primal and dual residuals:
P
(a) primal
2/10 and rdual
(a)
(0-1)/3
21
TXP
TxP 1=1
==
at the ath ADMM iteration. following schedule:
We update the penalty parameter and the scaled dual variable a with the
39

---

Published in Transactions on Machine Learning Research (08/2025)
ala+1)= 2a("),
u (a+1)
(a) 1 ua(a) if primal 10 x ddual 2
aa+t)) 1 a0(0)
u(at1) = 2u (a)
(a) dual 10 x primal
2
Moreover, since STERGM a probability distribution for the dynamic networks, in this work we stop ADMM learning until
1(01a+1)) - -((0(a)) ((B(a)) S
Ctol
(56)
where étol is tolerance for the stopping criteria. By convention, also implement two post- post-processing steps to finalize the detected change points {Ba) When the spacing between consecutive change points is less than threshold Bx- BR-1 < Sspe, we keep the detected change point with greater AC value to avoid clusters of nearby change points. Furthermore, interest, we discard the change point B₃ that is less than threshold send and greater than T dend- In Section 5 we set ōapc = 5, and we set dend = 5 and dend = 10 for the simulated and real data experiments, respectively.
the endpoints of time span usually not of
The algorithm to solve ( via ADMM is presented in Algorithu [ The complexity an iteration for the Newton-Raphsee method is O(7²p²) and that for the block coordinate descent method is O 1)p). In general, the complexity of Algorithm 1 is at least of order O(A[Cr2p² + Dr(r 1)pl), where A, C, and D are the respective numbers of iterations for ADMM, Newton-Raphson, and Group Lasso.
Algorithm Group Fused Lasso STERGM
1: Input: initialized parameters ((2) (1), (()), (1),, luning parameter X, penalty parameter a, number of iterations for ADMM, Newton-Raphsoo and Group Lasso A, C, D, vectorized network data y. network change statistics H
2: for == 1, A do
Ø vec (0()), ( (ec-p(1,,17000 + X(3()),, (t) vectp(u(1))
for c= C do
Let ac+1 be updated according to 13 end for
g(a+1) voc ( (0+11)
7:
Set 5 =7(() and B g(a) for d = -1.... D do
Let hdlt1 be updated according to 14 for i = ,,,
10:
gdt1= = (1/T))1,, -00.rr(((0))))))) (0(a+1) + (a) gd+1)
11:
12:
end for
((+11) =7d+1, g(++)) gd+1
13:
z(4+1) 1,,17(a+1) + X (1)(++1)
14:
15: (a+1)- ((a+1) z(++1) + (a)
16: end for
17: x ((a+1)
18: Output: learned parameters ²
To understand the role of two specific components in our framework, we conduct an ablation study on (1) the adaptive update of the penalty parameter R4 for the augmentation term, and (2) the use of position dependent weights d E R in the Group Fused Lasso penalty. Under the same settings as in Section 5.1 Table 13 reports the performance across different scenarios. While enabling eit adaptive a or weighted d alone yields good performance, enabling both simultaneously produces similar results. Given that the differences in performance across configurations are modest, is recommended to enable both components for the proposed method. This choice provides robust and automated approach that leverages the benefits of adaptive optimization and time-aware regularization, without requiring further determination.
40

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Published in Transactions on Machine Learning Research (08/2025)
Table 13: Means (standard deviations) of evaluation metries for dynamic networks across different scenario with n 100, varying the use of position dependent weights and adaptive penalty updates.
Scenario
Weighted d Adaplive
|k K
d(cic) +
d(clC)
C(G.
X
0.7 (0.6)
0.9 (0.3)
7.7 (6.3)
91.45%
SBM
0.7 (1.3)
0.8 (0.4)
5.0 (6.4)
91.34%
(o = 0.0)
0.7 (1.3)
0.8 (0.4)
5.0 (6.4)
91.34%
X
0.0 (0.0)
1.3 (0.6)
1.3 (0.6)
93.71%
STERGM
0.0 (0.0)
1.1 (0.3)
(0.3
93.95%
(p=6) 6)
/
0.0 (0.0)
1.1 (0.4)
1.1 (0.4)
93.95%
X
0.2 (0.4)
0.9 (0.3)
3.9 (6.3)
94.78%
RDPGM
0.2 (0.4)
0.9 (0.3)
4.1 (6.8)
95.37%
(d = 10)
0.4 (0.5)
0.9 (0.0)
6.5 (7.5)
93.34%
F Network Statistics in Experiments
In this section, we provide the formulations of the network statistics used the simulation and real data experiments. The network statistics of interest are chosen from the extensive list ergm (Handcock et al. 2022), an library for network analysis. Tables 14 displays the formulations of network statistics used in the respective formation and dissolution models of our method for 2,...,T and the formulations of network statistics used in the competitor methods for t =, The formulations are referred to directed networks, and those for undirected networks are similar.
Table 14: Network statistics and formulations.
Network Statistics
Formulation
Edge Count
Σ kj Vig
Mutuality
iej Vij
g+ (tt+
Triangles
is Yos y 4.1 +.t Eijek Vij Yí y
Vik
+.t tht
Homophily
kj Yig 1(x; = =x,)
Isolates
[(degin (gy (t,, = 0 A degout (y txt,2)
Edge Count
Vij"
Mutuality
iej Mij yj
Triangles
ijk Yes Yjk Vik + Eijek Vie""
" Uzk yki
Homophily
Vig" 1(x; =x,)
Isolates
I (degin (y-";)) 0 degour (y Vig
Edge Count
Mutuality
k Di
g(v")
Triangles
ijk Viyu + Eijek yiyy
Homophily
Vig 1(x; = xx))
Isolates
E, (deg.. (y', = degout (y', = 0)
41


Reviews:
[{"review_id": "RUf9ykZUR8", "reviewer": "Reviewer_XU3V", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n\n- Change point detection needs substantially more attention as a critical task for analytics\n- Solid and sound solution for the particular problem\n- Results support the overall claims and shows the strengths of the solution\n\nWeaknesses\n\n- Tuning of parameters for the proposed method and baselines requires more details. It's unclear how fair is the comparison currently when multiple parameters require tuning\n\n- Only two baselines are used. Several of the datasets used are time-series based and hence baselines for that community could serve as potential baselines. For example [a] and relevant works cited there.\n\nErmshaus, Arik, Patrick Schäfer, and Ulf Leser. \"ClaSP: parameter-free time series segmentation.\" Data Mining and Knowledge Discovery 37.3 (2023): 1262-1300.\n\n- Runtime performance is missing, the method seems computationally heavy\n\n- An ablation study can help understand strengths of sub-components of the solution", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a novel framework for unsupervised change point detection in time series of dynamic networks by leveraging a time-heterogeneous Separable Temporal Exponential-family Random Graph Model (STERGM). To effectively capture shifts in the underlying data-generating processes, the proposed method fits a sequence of STERGM models while enforcing smoothness via a Group Fused Lasso (GFL) penalty on the parameter differences across time. The resulting optimization problem is efficiently solved using the Alternating Direction Method of Multipliers (ADMM), and parameter estimation is expedited through the use of the STERGM pseudo-likelihood. This approach addresses the challenges posed by temporal and dyadic dependencies in evolving networks and allows for the detection of multiple structural changes.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Check weaknesses. Especially new baselines needed and details on fair parameter tuning.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "9y6lqBmbL9", "reviewer": "Reviewer_pZTb", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Although the reading flow is not particularly a strength, the paper is well-written in general. \nThe model formulation and the optimization seem to be correct.\n\nTo me, the main weaknesses are related to the focus of the paper, in the following sense.\nAs far as I understand, the model (all the Section 2) is not new. Or is there a contribution in this section that I'm missing?\nOn the other hand, the use of fused lasso for change point detection is also not new. Moreover, the optimization (formulation and resolution through ADMM with its subproblems) is essentially the same as in the paper by Bleakley and Vert. \nTherefore, the contribution of the paper is to use these two tools toghether. Which is a valid contribution to me.\nHowever, if this is the case (I might be wrong), I would've expected a more profound experimental section.\nThe quality of the experimental section is ok for a completely new method with new ideas, where one wants to show that it makes sense in specific scenarios.\nIn this case, I expect the experimental section to be the core of the paper.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this paper, the authors propose a method for change point detection of graphs, specifically when the graphs stem from a Separable Temporal Exponential-family Random Graph Model. The method is based on the estimation of the parameters for each graph, exploiting the fused lasso to impose continuity in the parameters for graphs belonging to the same model. The change points are then determined with a simple threshold in the norm of the difference for continuous parameters. The method is compared with two other R packages, both on simulated and real data. The code is provided along the paper.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- I would've liked a better explanation of the exponential model and its true and surrogate likelihoods.\n \n - The experimental section needs a lot of work to gain the protagonism that I expect in this kind of papers. In particular, the code of several graph CPD methods cited in the paper is available, and therefore, they should be included in the comparison. I would also like to see how this method performs with graphs stemming from other dynamic network models.\n \n - The description of the methods (in particular, the methods with which the authors compare) is quite poor.\n \n - There's no time comparison or mention of the complexity of any of the methods.", "broader_impact_concerns": "No concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "rxOhGDcetZ", "reviewer": "Reviewer_CGpB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths:\n- The problem itself is interesting\n- The simulation experiments are fairly convincing about the merits compared to competitors.\n- The writing is fairly clear:\n    * It is well-structured\n    * The methodology is explained well. Both the experimental setups and the algorithm itself were fairly easy to follow.\n- The analysis of the ADMM method is nice to have.\n\n# Weaknesses\n- The applications to real data are difficult to interpret. \n    * The authors do a reasonable job discussing events that coincide with the detected changes. However, the algorithms are detecting changes that don't always line up with the explanations.\n    * Ideally, however, there would be some way to display the data in a way that would make it clearer what was causing the changes to occur:\n        - Maybe display some snapshots of the adjacency matrices near the change points\n        - Alternatively, something like a raster plot might help (where each pair is its own row)\n        - Additionally, since the matrices for the stock data were computed after some preprocessing, it might be useful to alternatively plot the raw data and line it up with the detected change points. \n- Propositions 2 and 3 are stated awkwardly. \n    * For Proposition 2, it would probably be better to say something like \"A step of Newton-Raphson can be implemented by...\"\n    * For Proposition 3, it would probably be better to say something like \"Block coordinate descent can be implemented by...\"", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper describes a methodology for detecting changes in the parameters generating data for dynamic networks. The method is based on solving an optimization problem, constructed from a Separable Temporal Exponential-family Random Graph Model and a group norm regularization. A specialized ADMM method for the problem is formulated and analyzed. The method is tested on simulated and real-world data, and compared with existing approaches to network data change detection. On simulation data, at least, it appears to give improved performance over competitors, likely due to the ability to model richer temporal evolution patterns and handle more complex random behaviors.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "# Critical for Securing Recommendation\nDo a bit of work to improve the explanation of the performance on real data. Plots, as suggested above, would really help. It would be really helpful to understand what was happening with all of the methods.\n\n# Helpful to Strengthen the Work\nImprove the wording in Propositions 2 and 3.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)