$\ell_1$ Latent Distance based Continuous-time Graph Representation
Authors: Zhao-Rong Lai
Zheng-Sen Zhou
Liangda Fang
Yongsen Zheng
Ziliang Chen
Venue: ICLR 2026 Poster (2026)
Keywords: $\ell_1$ distance
graph representation
sequential survival process
ultra-low-dimensional embedding
Primary area: learning on graphs and other geometries & topologies

Abstract: Continuous-time graph representation (CTGR) is a widely-used methodology in machine learning, physics, bioinformatics, and social networks. The sequential survival process in a latent space with the squared $\ell_2$ distance is an important ultra-low-dimensional embedding for CTGR. However, the squared $\ell_2$ distance violates the triangle inequality, which may cause distortion of the relative node positions in the latent space and thus deteriorates in social, contact, and collaboration networks. Reverting to the $\ell_2$ distance is infeasible because the corresponding integral computation is intractable. To solve these problems, we propose a theoretically-sound $\ell_1$ latent distance based continuous-time graph representation ($\ell_1$LD-CTGR). It facilitates a true latent metric space for the sequential survival process. Moreover, the integral of the hazard function is found to be a closed-form piece-wise exponential integral, which well fits the ultra-low-dimensional embedding. To handle the non-differentiable $\ell_1$ norm, we successfully find a descent direction of the hazard function to replace the gradient, enabling mainstream learning architectures to learn the parameters. Extensive experiments using both synthetic and real-world data show the competitive performance of $\ell_1$LD-CTGR.

Published as a conference paper at ICLR 2026
1 LATENT DISTANCE BASED CONTINUOUS- -TMMO
TIME GRAPH REPRESENTATION
Zhao-Rong Lai', Zheng-Sen Zhou Liangda Fang', Yongsen Zheng Ziliang Chen Guangdong Key Laboratory of Data Security and Privacy Preserving. College of Cyber Security, Jiman University *Department of Computer Science,
College of Information Science and Technology, Jinan University
*College of Computing and Data Science, Nanyang Technological University
*Digital Trust Centre, Nanyang Technological University
RResearch Institute of Multiple Agents and Embodied Intelligence, Peng Cheng Laboratory
Laizhrê . en, sunchou@stu200 jnu. edu. en, fanglde jnu. edu. en yongsen. zhengentu.edu. sg, c. c.ziliang@yahoo.
ABSTRACT
Continuous-time graph representation (CTGR) is a widely-used methodology in machine learning, physics, bioinformaties, and social networks. The sequential survival process in a latent space with the squared & distance is an important altra-low-dimensional embedding for CTGR. However, the squared l2 distance violates the triangle incquality. which may cause distortion of the relative node positions the latent space and thus deteriorates in social, contact, and collabora- tion networks. Reverting to the & distance is infeasible because the correspond- ing integral computation intractable. To solve these problems, we propose a theoretically-sound & latent distance based continuous-time graph representation (D-CCR), facilitates a true latent metric space for the sequential survival process. Moreover, the integral of the hazard function is found to be closed- form piece-wise exponential integral, which well fits the -iaranlowwdim embedding. To handle the -on-differentiable l norm, successfully find a de- scent direction of the hazard function to replace the gradient, enabling mainstream learning architectures to learn the parameters. Extensive experiments using both synthetic and real-world data show the competitive performance of & L.D.CG...
INTRODUCTION
Graph (or network) representations widely used across various fields, ranging from natural sci- ences to social sciences, including areas such as particle interactions, biotic activities, and social networking. One of the key tasks in graph representations is predict the properties of connections and nodes within a graph (Hamilton, 2020; Zhang et al., 2020). Early methods mainly focus on static networks, employing approaches like random walks (Grover & Leskovee, 2016), matrix fac- torization (Qiu et al., 2019) and Graph Neural Networks (GNNs, (Wu et al., 2021)). More recently, these techniques have been extended accommodate more complex types of networks, including signed networks (Nakis et al., 2023) and hierarchical structures (Nakis et al., 2024a).
Compared with static networks, dynamic networks can better reveal intrinsic patterns and interac- tions between nodes over time (Çelikkanat et al., 2024). At first, researchers focuses on discrete- time networks, using probabilistic models (Heaukulani & Ghahramani, 2013), multiscale structures (Herlau et al., 2013), and hyperbolic space (Yang et al., 2021). More recently, continuous- networks have emerged, modeled by methods like random walks (Zuo et al., 2018; Nguyen et al., 2018), Poisson processes (Fan et al., 2021; Çelikkanat et al., 2022), Hawkes processes (Hawkes, 1971; Arastuie et al., 2020; Huang et al., 2022), and sequential survival processes (Çelikkanat et al., 2024). However, Poisson and Hawkes processes treat network connections as instantaneous events, largely ignoring the persistence of edges in the likelihood function. To address this limitation, the
"Corresooniing author.

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Graph Sequential Survival Process (GRASSP (Çelikkanat et al., 2024)) is proposed. which dynami- cally captures both the existence and absence of connections. Since survival analysis is effective at modeling the duration leading up to the failure or termination of an entity, it is also suited for mod- eling the lifespan of connections between nodes, which is particularly effective in social, contact, and collaboration networks.
The distance between two nodes in the latent space plays a crucial role in determining their likeli- hood of being connected. Nodes are more likely connect when they are closer in the latent space, while they are more likely to disconnect when farther apart. This property can be captured using seent-dittacee-basee hazard function within the sequential survival process (Çelikkanat et al., 2024). A common choice for latent distance is the squared Eg distance, which is rooted in the gen- eral real-world Euclidean geometry. It well fits the altra-low-dimensiona embedding (Çelikkanat et al., 2024; Nakis et al., 2024b; Huang et al., 2024a), which serves as dimensionality reduction scheme, improves the conciseness of graph representation learning, and saves computational cost (Hamilton, 2020; Xu, 2021; Barros et al., 2021). Moreover, the integral of the corresponding hazard function is the well-studied Gaussian integral, commonly used in probability and statistics.
However, the squared lg distance violates the triangle inequality (see Eq. 8), thus it is not valid distance for the latent metric space. The corresponding hazard function may distort the relative node positions in the latent space, and thus deteriorates in social, contact, and collaboration networks. For example, to determine whether A has a closer relationship with B than C in this latent space, the triangle inequality should not be violated. This problem cannot be fixed by reverting the l2 distance (without a square), or using other lp distance with p > 1 and * 2, because the integral computation of the corresponding hazard function is intractable. Some recent works indicate that the l distance can serve as an alternative of the & distance when the latter becomes ineffective in certain scenarios (Lai & Wang, 2024; Çelikkanat et al., 2024; Wang et al., 2025). It motivates us to use the I distance to facilitate a true latent metric space for the sequential survival process. We find that the integral of the hazard function with & distance a closed-form piece-wise exponential integral, which well fits the setra-low-dimensinn
embedding.
In this paper, we propose a theoretically-sound & latent distance based continuous- graph rep- resentation (e, LD-CTGR) Our main contributions can be summarized as follows:
1. We verify that the average & distance during an event time can be bounded by two values that are determined by the connected and disconnected states and the survival function, which is similar to the bounded property of the squared lg distance case.
We derive the integral of the hazard function for LLDDCTGR, which is a closed-form piece-wise exponential integral, markedly different from the Gaussian integral of the squared & distance case.
3. We propose an efficient integral computing algorithm for the ultra-low- diwediinniiinaal embedding of LD..G..
4. The & distance, the corresponding hazard function, and the survival function all non- differentiable. Hence we find a descent direction to replace the gradient, allowing mainstream learning architectures learn the parameters of the graph.
2 RELATED WORKS AND UNSOLVED PROBLEMS
We introduce the preliminary framework based on (Çelikkanat et al., 2024), then raise some prob- lems of the currently-used squared & distance.
2.1 CONTINUOUS-TIMA GRAPH REPRESENTATION
Denote 9 == (V, E) as graph, where V == (1, N) represents the vertex set, and E := Un represents the edge set. Here, E., : (i,j) E v² denotes that nodes and j are connected by an edge. In the continuous-time interval scenario, the time interval [7] := [0, T') must also be considered, Correspondingly, a connection between nodes i and may last from tę E [T] to tk-1 E (T). Therefore, the edge set should be extended to the following set of tuples E := {(i.), (x+1) E 22 x [7]2 14 < [x+1). where (i, lk, 141) and (i, 4, 4441) represent two distinct connections if [tx, (1+1) n [b, f141) = U.
An event time e. is defined as the time when an edge gets connected or disconnected, with eo = 0 always being an event time. If a node pair undergoes M events en = 0 e < e2 <... < eap-1 < T, then there M consecutive intervals {[em; cm+1) M each with different states. The state
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function 12 [T] + S with S == {1, -1] indicates whether edge (i, is connected (s 1) or disconnected (s = -1] at time t.
A survival function describes how long object exist or function until it fails dies:
S(t) : > = exp (- (-f," (T))
(1)
where R 10, : + R denotes the hazard function for this survival process, and T represents the lifetime random variable. Thus, the "surviving" and "dying" states of connection can be characterized as part of a sequential survival process. Let 80 E S be the initial state and (t) be a counting process that records the number of events up to time The probability function of M (t) equaling m is given by
pek GG
A(SK, T) dr
(tM(()) m} EE k= exp ( M(kk T)
dp.
(2)
where : ((11,2,,.. bm) [T]m : & t < l2 < ... < bm < T) denotes the integration domain, and ()s,, represents the hazard rate state Sy and time The integration variables (t1, 12.t2,.. tm) depend on the event times (c1,62.-* 62.-.. .Cm). According to the fundamental theorem
of calculus, the density function for IP (t) m] (e1, c.. , is:
A(5K, Ex) k=1 exp (Je. (Sk)7)
P(M())=m)(11 (e1, 6m)
II
Next, we develop suitable hazard function X(s, t) control the connections between nodes. This function is embedded in a latent metric space (X, dx), where dx denotes the distance function. The position of node i time is represented by a point (t) X. To model the connection based on the distance between nodes, (i () can be defined
(s, t) == (0x (t), ry (t)),,
(4)
where 2i* : : + R+ a continuous function. Then the log- oilliiklihood function of the entire graph given by:
Eigl
ck f >es(Su, dr -ay-1
log ((]
Σ
log dys()
(5)
(6.))V
where Ω represents the set of hyperparameters. Recent attention focuses the ultra-low- dimensional embedding where the latent space X = R has a very low dimensionality D, ideally D 2 (Çelikkanat et al., 2024; Nakis et al., 2024b; Huang al., 2024a). It reduces dimensionality directly, improves the conciseness of graph representation learning, and saves computational cost (Hamilton, 2020; Xu, 2021: Barros al., 2021).
2.2 PROBLEMS OF SQUARED & DISTANCE
In GRASSP (Çelikkanat et al., 2024), the following specific form for (s, t) proposed:
(s 1) = exp (3(s) + + (()):(() (()-:(())7)) (0)1))
(6)
where the parameter ((s) captures global connectedness (Krivitsky et al., 2009; Çelikkanat et al.,
2022), and the squared &a distance |1 has a simple computational form:
||r - ([]]]]])) = 2r.
(7)
This advantage makes GRASSP efficient and effective in the ultra-low- low-dimensiooal embedding = However, || 1/ violates the triangle inequality, thus not valid distance for the latent space. Specifically,
|r: - 1 [|: rell3 ||re rr,12. Fr.
(8)
simple counterexample is that (r: TerTeT(r (r+ -r,) 0. Due to this problem, (sy f) in (6) cannot determine whether node i is closer to node than node j, which may cause distortion in the latent metric space (X, 0x) and affect the performance social, contact, and collaboration networks. To better see this distortion,
can divide the interval [-1, 1] into Q segments and consider set of nodes ry (-1+ + 2g E RR. 1, ..., Q. Then the indirect distance between

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To and rq is
Σ |r - - 1 q=1
Q
2 -2)==
- 0 as Q
+ oo.
Q
It indicates that adding infinite nodes between ro and r can shrink their indirect distance to nearly zero, which is counter-intuitive and unconventional. This may affect the local geometrical structure and relative node positions in the latent space.
One may argue that this problem can be fixed by reverting to the & distance without a square. However, using the fz distance makes it intractable and inaccurate to compute the integral Jee (kk T) dr in the log-likelihood (5). Following the same variable change technique as (16), the core computation for the Eg distance case is:
exp (s|lArry + Auizt(12) dt.
(10)
There is no closed-form solution to this integral, thus it should be computed via numerical methods. This causes two main drawbacks: 1) The interval [0, e,, Ei] should be divided into a sufficiently large number H of subintervals with length At. then O(D) is required conduct the summation and l2 norm in each of the subintervals. Last, (H + 1) exponents are evaluated to compute the entire integral (the complexity of exponential operation the total computational complexity for the &a distance integral is O(HD), which can be hundreds or even thousands of (D) for both GRASSP and &LD-CTGR (see the last paragraph of Section 3.4 and Table A1). The numerical integral involves an approximation error compared to the exact integral, which can be amplified by composite function relationships. According to Simpson's 's Rule with an even H, the numerical integral for (10) is
be tens of that of multiplication). Thus
Al
H
It, =
exp (s11A 4.
exp Ami + Any (hat)ll)))
3
k=1. is odd
H
+ 2
exp exp Amy + Anr(hAAt)la) + exp exp As) + An,((HAt)]2)) (11)
h is even
yields approximation error of O((At)*). For an integration interval [0, 1]. /l 1000
leads to At = 10-3 and an approximation eтror of O(10-12). However, even such an approx- imating accuracy cannot bring about good performance our task (see Table A4). In contrast, Je (5k, r) dr using the squared & distance is well-studied exact Gaussian integral, as ex- plained in the following subsection.
2.3 GAUSSIAN INTEGRAL INDUCED BY SQUARED l DISTANCE
To model the continuous time movement of a node, a piece-wise linear approximation with B bins can be used:
[t/n]]
(+(() x + A
+ mod(t, ( (1/An]++))
(12)
b=1
(0) where x; represents the initial position, AB == T/ denotes the bin width, and v (.)) is the velocity for the 6-th bin. The integral of (s, t) in (6) over an interval (et, ew) can be computed as:
exp (3(s) + sl|r:((1)) (0)))))) de
2| exp (B(s)+ -
spł)) (((s))))) e=((((e.))- ((y(s eff(((ee))
T(1) = |Aviill2 + Pijs
Amy := r.)))) (e)),
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Aviy := vz(ei) 0y(e)). Pij := Ax, |
(13)
where erf and erfi denote the error function and the imaginary eтrоr function, respectively, for the Gaussian distribution. Both functions are well-studied Gaussian integrals that can conveniently computed by sophisticated numerical methods. This is main reason why the squared & distance preferred to the &2 distance. As for other ep distance with p > and p + 2, the corresponding integral is still intractable.
3 METHODOLOGY
We propose the following I distance based hazard function:
(14) where || ||1 represents the & norm. It drops the square of the distance and facilitates a true latent metric space (X, 0x) with dx := || ||a, whose integral is tractable and found to be closed-form piece-wise exponential integral. By this means,
(s, exp (3(() s||r:(t) - (t)ll),
solve the problems of latent space distortion and integral computation in Section 2.2. To handle (14), we need to reform the entire sequential survival process and overcome several challenges: 1. The signs of different dimensions for (t) - (t)) may change asynchronously within the interval [er, e,), which com- plicates the integral computation Je (s, f) dí. 2. An efficient integral computing scheme should be developed for the wal-loow-dmeesiinee embedding to save the computational cost. 3. Aiy(s, t) is non-differentiable with respect to (w...t.) т; or Tjs which could hinder the learning of model parameters.
3.1 BOUNDEDNESS OF AVERAGE & DISTANCE
important property in the sequential survival process is that the connected and disconnected states of a node pair, together with the survival function determine the lower and upper bounds of the corresponding distance (Çelikkanat et al., 2024). We verify that this property also holds in the l distance case, presented by the following proposition.
Proposition 1. Let and ck+1 be two consecutive event times for a node pair (i.5). Then the average l distance of this node pair in the interval [ek, Ck+1) with survival function Siy(.) and state E 11, can be bounded by:
cx+1
B(-1) < ct+1 ck
[|r((t) - rt(())1 dt < B(1),
B(s):=s(log (log(S:) (ex)) loo)))-oo((Sp)g log(S: (cx+ 1))) (og(e))1oo((++1-e=)))
(15)
The proof is given in Appendix A.1. This proposition indicates that the average & distance of a node pair (i, j) during time span [ck,Cl+1 Ck+1) is controllable by the state in practical sense. The left incquality indicates that the average ε distance should be no less than B(-1) in the disconnected state, while the right inequality indicates that the average &, distance should be no larger than B(1) in the connected state. In other words, nodes and j cannot get too close to (or too far from) each other to maintain a disconnected (or connected) state. This aligns with the general understanding of nature and society.
3.2 INTEGRAL COMPUTATION
The integral of the hazard function 1 (s) dt is key component in the maximum log- likelihood estimation (MLE) as given in (5). To compute this, we employ piece- linear ap- proximation (12) similar to the E2 distance version in (13).
Asy(s, dt= dt= [ exp (3(() de
1mm exp (3(8) + sI| + Au - de
= exp
exp As|lAr + Augttli dt,
(16)
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(et()) and Av (c: vz(ee) vy(e)) represent the instantaneous relative position and velocity of the node pair (i.j). These quantities are recorded at each key time ey and used to compute the integral up Cu. Since exp (3()) is constant w.r.t. the integral over t, it can he put outside this integral. Then the key work lies in computing the latter part integral of (16). The main result is summarized as the following theorem, while the detailed computation and proof are provided in Appendix A.2.
where Az,((e) r.(e))
Theorem 1.
exp (3(s)) I I...0c--e: exp (B(s)) + Ei- Iu.teett +1 [Ts,0))
if (iM > Eu - OF Zij.(Z) <0
tt)(s,t) dt= Jer where zit(s) and 2ij,()) denote the zero points defined in (34) and (35). (et,)),(o+1)) Ly,0 and Iis,())e are integral pieces defined in (37), (38), (40), and (41), respectively.
else (17)
Theorem indicates that 1 Aey(s, t) dt is a closed-form pise exponential integral in the l distance case, which is very different from the Gaussian integral (13) in the & distance case. be computed successively and conveniently from g 7 1 with slight updates of some intermediate variables in (44). We demonstrate the application of Theorem with simple example Appendix A.2.
Moreover, (ej.te)((++1
The ULTRA-LLOWDDIENNSONN EMBEDDING
3.3
embedding has been catching attention recently (Çelikkanat et al., 2024; Nakis et al., 2024b; Huang et al., 2024a), since it reduces dimensionality by its nature, improves the conciseness of graph representation learning, and saves computational cost (Hamilton, 2020; Xu, 2021; Barros al., 2021). As for LD-CTGR, we find that there are at most 2 zero points in the durr-llo-diimnnsione embedding with D = 2, which are compatible with the two limits e& and ek+1 of integration. By this means, we can develop an efficient integral computing algorithm in tensor-patee cassoo-paaallelieed form.
Theorem 2. the ultra-low-dimesionn dimensional embedding with = 2. the integrals can be computed in a tensor-paralllllzeed way: i.j.k
Sch sekt1 t))
[r Xig ts)
= cxp (I, +
(18)
iji
where Iı I2, Is R
tensor parall-praleeeized integral pieces defined in (58), (59), and (60), respectively. @ denotes the element-wise multplicaaiio.. B(x) E R Rl.x1 and E (1,-1) 1 The dimensionality of the above tensors is the
the total number of observations L L:= (i. k.
The proof is given in Appendix A.3, which also demonstrates the entire computing process for I,, I2, and I3- By this means, & LDDCTGR not only well fits the ultra-low-ding -idelloowddieesiinalll embedding like GRASSP, but also facilitates true latent metric space.
3.4 DESCENT DIRECTION AND SUBGRADIENT COMPUTATION
There are 3 primary sets learnable parameters in the proposed LD-CTGR model: the initial positions (0) E N-D the velocities for the piece-wise linear approximation bins E N.B..D. and the global connectedness parameters [3(s 1), (s 1)] E R2.. Each of the N nodes is represented as point in a D-dimensional latent space, resulting in a total N parameters for (0)) Additionally, with B bins per node and different velocities for each bin, there are N B parameters for v in total (see Eq. 12).
The learning process involves minimizing the negative log-likelihood logr(G]() defined in (5) w.r.t. (0), v, and ((s). This typically achieved through gradient descent. Specifically, we need to compute the following partial derivatives:
2 [raut \sg 1)
i log Aig ex) dΦ
1
aAt cx)
pek-1
dt,
Aey (yK
UΦ
Jec
dΦ
(19)

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where 20)) v. B(s), respectively. In both terms,
@Ag(ed) 34
must exist. While (5,(s.))
does
03
exist, (A,,(s.) and (),, may not. Since this subsection focuses on the primary parameters, we 8e temporarily omit the variables (s, t) of Xij for clarity.
0m("))
Mainstream learning architectures, such Pytorch', rely backpropagation, which essentially applies the chain rule for gradient computation. Combining (12) and (14), we have
ahi ahy Ôr, Qbg I Dhy axg DNg Ôr; aby aiy. -.
Diy (20)
ax: (0) Dr:
a (0) δr:
dr: au: (b) òr: i (b) Ör:
dr:
where I denotes the identity matrix with its dimensionality inferred from the context. In the squared
&g distance form (6), the gradient BAu exists, making backpropagation valid. However, in the &
ör;
distance form (14), 8ho does not exist due the nondiifferentiabee nature of the & norm, making ör. backpropagation invalid.
To address this issue, we need to find descent direction of Xy w.r.t. T3 to replace the negative gradient @A4 which leads to the concept of subgradient. Dr.
Definition 1 (The Fréchet Subdifferential). For proper lower semicontinuous function h R RU {+o)), the subgradient set at point dom h is defined as follows:
W lim inf (3) h(r) w
0h(r):=
(21)
4-
y
v/r
In brief, subgradient w (if it exists) represents ascent direction of h, possessing similar proper- ties to a gradient (Lin et al., 2024). In our task, is well-known that sign(r) e 0|rll1- Therefore, directly set 0|[r|| == sign(r). By slightly abusing the use of chain rule, we further obtain
the following ascent direction:
exp (3(s) + s s||r: ryy(0ll) (t))) (t)((l))
= eexp (3(s) + s (1) - (01l)) 8 sign(r,(() - (())
(22)
Theorem 3. Let (ri((Ts) be defined in (22), and ler := {d D: T =rja} be the index set of dimensions where nodes and j have the same coordinate values.
1. IfC =D, =D, then Xag is differentiable at T; and (rss(rs) (ri)(ra) = 0.
2. IfC _ D, then (ra)((ra) is descent direction of Xig al
3. IfC = 0, C + 0 and 1, then (x:z(r;) is subgradient of Xij at T.
The proof is given in Appendix A.4. This theorem indicates that in all cases, -ig (r:) in (22) char- acterizes either a stationary point or a descent direction. Particularly in Item 3, AAy (r:) becomes subgradient. However, when 1, @rig(r:) is not subgradient demonstrated in Case 4 of the proof in Appendix A.4, because
exp(- Ed (y ri.al) 1+ Eage lya-ra,d ri.dl
z 0
(23)
V (ya-ridd)
in (72). To see this, consider counterexample with D = (1, C {1} and ya V - for d = 1,2. Then (23) becomes
exp( - -2c + o(-2c) +e 1
of 2 <
(24)
V2e
/2c
V2
as 6 + 0. This violates the inequality in (21), which means that ax (r:) not a subgradient. Nevertheless, -r);;(r;) is still descent direction of Xij at in this case.
To summarize this subsection, we can directly compute D%g by (22) and check whether (r, (which is equivalent to C D). If so, then @ri((r)) = VAg (r:) 0. If not,
httts://pytorcch.rr//
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then (T.)) is a descent direction of Aty r,, and a line search method can be applied to reduce Asg. In this way, dAz (T:) is compatible with mainstream learning architectures like Pytorch for the proposed & LDDCCGG..
As for computational complexity, the negative log-likelihoode of & 2,LD-CTGR is nonconvex, which requires (1/(2) iterations to achieve convergence tolerance of O. In each iteration, e LD- CTGR has a computational complexity of O(L M), where L and M denote the sample size and the number of trainable parameters, respectively. The closed-form integral computation of & LD- CTGR requires only (1)) complexity for the ditr--low-dimensiona embedding with D = 2. Hence the entire computational complexity of LLD-CTGR&L is O(L - .M/e), which is the same as that of GRASSP. Table A1 also shows that both GRASSP and 4LD-CTGR have the same order of actual runtime. As for 'D > 2. GRASSP requires (D) for one integral computation in (13), based on the D-dimensional squared l2 distance computation. LLDDCTGR needs to implement sorting of D zero points, which requires O log, (D)) complexity. Then it needs to compute at most (D + 1) one-dimensional closed-form integral (with complexity (()), based on (43) and (44). Hence the overall complexity of &LD-CTGR for one integral computation is O log, (D) =
Ö(D), which is the same asymptotic complexity as that of GRASSP. Moreover, the comparison operation is generally faster than multiplication, thus the gap between & LDDCCGR and GRASSP is small when D is small. Additional runtime experiments regarding higher dimensionalities D = 10 and D 100 are provided in Table A2, which shows that & LD-CTG is also efficient in these cases.
4 EXPERIMENTAL RESULTS
Our experimental setup extends the standard benchmark for CTGR methods outlined in (Çelikkanat et al., 2024) to include more data sets and competitors. It covers broad range of networks with varying sizes or characteristics. Detailed experimental settings are provided in Appendix 1. Eleven synthetic and real-world benchmark data sets with diverse characteristics ation: Synthetic-a- (Çelikkanat et al., 2022), Synthetic-3, Contacts, (Génois et al., 2015), HyperTeet (Isella et al., 2011), Infectious (Isella al., 2011), Facebook (Viswanath et al., 2009), NeurIPS (Globerson et al., 2004), US Legis. (Poursafaei et al., 2022), Can. Parl. (Poursafaei et al., 2022), Wikipedia (Poursafaei et al., 2022), and Reddit (Poursafaei al., 2022), detailed in Appendix A.5.2. NeurIPS has about 5 thousand nodes and 1.2 million edges, while Facebook has about 82 thou- sand nodes and 200 million edges. Both data sets contain large-scall real-world networks test the scalability of different competitors. Eight state-of-the art methods are taken into comparisons: Nede2 (Grover Leskovee, 2016), CTDNE (Nguyen et al., 2018), HTNE (Zuo et al., 2018), PIVEM (Çelikkanat et al., 2022). TCL (Wang et al., 2021; Yu et al., 2023), GraphMixer (Cong et al., 2023; Yu et al., 2023). DyGFormer (Yu et al., 2023), and GRASSP (Çelikkanat et al., 2024). All these competitors have their own default settings (e.g., the embedding dimensionality), which all adopted in our experiments. Details are provided in Appendix A.5.3. The code available at https: / / github...b.... com/ aaazhrr/iiil CTGR.
used for evalu-
The assessment is conducted in three main aspects: in-sample, out-of-sample, and across-sample evaluations. The Adam optimizer (Kingma & Ba, 2015) with learning rate of 0.1 is used in model learning. Hyperparameters are optimized based on the best performance on the validation set.
Network Reconstruction (in-sample). This evaluation focuses on reconstructing both the connec- tion and non-connection periods within the training set, thus serving as an in-sample assessment. Its purpose evaluate how well method captures the temporal structural changes occurring within the network. As shown in Table AS, L.D.CTTGR achieves competitive performance, particularly on Synthetic-a, HyperText, and Reddit. This indicates that L,LDCTG effective in real-world network reconstruction tasks. DyGFormer ranks the second place in this scenario, which shows its advantage performing in-sample tasks as a Transformer-based (Vaswani et al., 2017) model.
Network Completion (out-of-sample). In this evaluation, each method generalizes the learned connection structure from the training set to the test set, making it a fully out-of-sample assessment. The goal is to assess how well the method can complete the entire network with only partial in- formation. Table 1 shows that &LDD-CTR. outperforms other methods in all but two situations. Moreover, the challenging real-world data sets Facebook and NeurIPS, l LDCCTGR maintains robust performance. Hence & LD-CTGR has a strong generalization capability in continuous -atti network completion tasks.
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Published as a conference paper at ICLR 2026
Table 1: Performance of different methods for network completion (out-of-sample) across diverse
data sets STD).
Data Set
Vec
CTDNE
ITTNE
PIVEM
TCL
GraphMixer DyGFommer GRASSP LD-CTGR
Synthetic-a
ROC
0,459
0.489
0.542
0.578
0.602
0.619
0.680
0.559
0.817
#0.009
#0.021
=0.019
#0.030
#0.020
=0.016
#0.021
#0.024
=0.030
PR
0.471
0.493
0.574
0.562
0.590
0.586
0.607
0.327
0.813
+0.013
+0.019
+0.020
+0.027
+0.016
#0.011
+0.020
#0,021
+0.031
Synthetic-88
ROC
0.636
0,507
0.377
0.542
0.550
0.364
0.604
0.612
20.007
+0.009
=0.009
+0.007
#0.011
20.043
20.018
#0.018
#0.013
PR
0.694
0.569
0.578
0.566
0.556
0.563
0.649
0.540
0.641
+0,007
#0,011 1
#0.004
+0.009
+0,010
#0.043
#0.022
+0.024
#0.018
ROC
0.517
0.489
0.461
0.557
0.610
0.62T
0.667
0.670
0.680
Contacts
+0.021
+0.029
#0.025
+0.009
#0.001
#0.002
#0,016
#0.016
+0.017
PR
0.526
0.553
0.509
0.579
0.602
0.687
0.711
0.714
+0.019
+0.031
#0.023
+0.017
+0.003
#0.001
#0,021
#0,025
#0.028
ROC
0.570
0.498
0.613
0.554
0.641
0.658
0.660
0.619
0.671
HyperText
#0.011
+0.015
#0.014
+0.015
+0.016
=0.001
+0.017
#0.011
#0.012
PR
0.595
0.554
0.651
0.571
0.645
0.652
0.660
0.591
0.672
+0.013
+0.017
#0.008
+0.008
+0.001
+0.001
+0.018
+0.024
40.015
ROC
0.681
0,534
0.578
0.728
0.726
0.728
Infectious
+0.004
+0.009
#0.018
+0,003
#0.000
=0.001
+0,019
#0.029
#0.017
PR
0.632
0.585
0.611
0.592
0.731
0.723
0.744
0.711
0.779
+0,011
+0.008
#0.016
+0.004
+0,001
#0.003
+0.018
+0.028
+0.017
ROC:
0.329
0,340
0.463
0,482
0.533
0.571
0.569
0,500
0.572
Facebook
+0.002
+0.005
#0.003
+0.002
+0.002
#0.004
+0,010
+0,000
+0.004
0.572
0.501
0.608
0.624
0.500
+0.004
+0.005
#0.003
+0.003
#0.001
=0.002
+0.009
#0,000
20.004
NeurlPS
ROC
0.355
0.455
0.222
0.469
0.503
0.467
0.496
0.360
0.533
#0.002
#0.018
=0.026
#0.014
#0,000
=0.001
+0.022
=0.031
#0.022
PR
0.355
0.435
0,289
0.468
0.504
0.536
0.530
0.468
0.559
+0.002
+0.022
#0.028
+0.027
+0.000
=0.002
#0.018
+0.026
10.019
US Legis.
ROC
0.393
0.490
0.492
0,510
0.749
0.770
0.727
0.656
0.776
+0.003
#0.009
20.014
#0.010
#0.006
=0.015
#0.016
#0.013
#0.013
PR
0.486
0.534
0.542
0.529
0.684
0.707
0.699
0.587
0.725
+0.004
#0.014
=0.016
#0.011
+0,005
=0.013
#0.017
=0.015
#0.012
ROC
0.675
0.309
0.473
0.529
0.734
0.801
0.807
0.678
0,810
Can. Parl.
+0.003
+0.010
#0.011
+0.012
+0.008
=0.014
#0,010
+0.009
#0.009
PR
0.616
0.568
0.538
0.545
0.692
0.739
0.874
0.709
0.761
20.004
#0.013
=0.016
#0.010
#0.002
20.012
#0.011
#0.008
20,010
ROC
0.345
0.488
0.520
0.615
0.958
0.969
0.972
0.874
0.982
Wikipodia
+0.008
#0.010
=0,009
#0.012
#0.005
=0.007
+0.009
=0.006
+0.004
PR
0.384
0.512
0.555
0.615
0.941
0.971
0.960
0.907
0.983
+0.007
+0.012
=0.010
+0.009
#0.006
=0.008
#0.010
#0.007
#0.005
Reddit
ROC
0.445
0.576
0.522
0.698
0.964
0,963
0.977
0.918
0.988
20.010
20.012
20.011
20.013
20.008
20.008
20.006
#0.009
20.005
PR 0.471
0.531
0.580
0.632
0.965
0.963
0.980
0.910
0.983
#0.011
#0.013
#0.012
+0.014
+0,008
20.008
+0.006
+0.010
+0.005
Future Connection Prediction (across-samplee. This evaluation involves using the sequential sur- vival processes learned from the training to predict future connection states in the prediction set, representing an across-sample assessment. As shown in Table 2, E LD-CTGR outperforms other competitors most situations, including the challenging real-world data sets Facebook and NeurIPS. Additionally, the performance of & LDDCTG is more robust than that of the other meth- ods across different situations. Hence L,DDCTTGH tasks, even when the prediction period extends beyond the training set.
also effective future connection prediction
To summarize, LDDCCTG is effective and robust across network reconstruction, completion, and prediction tasks, and shows good generalization ability in the latter two tasks.
5 CONCLUSION AND DISCUSSION
The sequential survival process in latent space with the squared 62 distance is a widely-used method for continuous-time time graph representation. However, the squared lg distance is not a valid distance for the latent metric space, because it violates the triangle inequality. The corresponding hazard function building on the squared l, distance may distort the relative node positions in the latent space and thus deteriorates in social, contact, and collaboration networks. Neither can the l, distance with p > 1 be

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Published as a conference paper at ICLR 2026
Table 2:
Performance of different methods for network prediction (across-sample) across diverse
data sets (TD)
Data Set
Vec
CTDNE
IINE
PIVEM
TCL
GraphMixer DyGFurmer GRASSP (,LD-CTGR
ROC
0.486
0.511
0.575
0.588
0.420
0,446
0.427
0.875
0.922
Syntheric-a
=0.003
#0.019
#0,016
#0.014
+0.015
#0.066
#0.010
=0,020
=0,018
PR
0.491
0.495
0.614
0.502
0.454
0,510
0,562
0.819
0.890
#0.012
+0.019
+0,020
+0.017
+0.011
#0.014
+0.012
20.019
#0,019
Syntheric-8
ROC
0.314
0.491
0.393
0.588
0.456
0.363
0.353
0.861
0.864
=0.003
#0.012
#0.006
#0.006
#0.008
#0.056
#0.009
#0.014
#0.014
PR
0.578
0.555
0,639
0.598
0.503
0.465
0.521
0.829
0.831
=0.007
+0.018
#0,005
#0,006
+0.009
#0.035
+0.011
#0,014
#0,016
RCC:
0.738
0.509
0.604
0.493
0.681
0.676
0.702
0.763
0.767
Contacts
#0.009
+0.016
+0.003
#0.011
+0.013
#0.004
+0.012
10.016
#0.018
PR
0.687
0.565
0.601
0,497
0.691
0.692
0.690
0.714
0.721
20.015
#0.017
+0.004
+0,010
+0,003
+0.001
+0.014
#0.020
#0.018
ROC
0.552
0.491
0.501
0.516
0.513
0.525
0.568
0.607
0.568
HyperText
#0.003
+0.011
+0.019
#0,006
+0,005
#0.001
+0.011
#0,007
+0.005
PR
0.518
0.552
0.502
0,516
0,525
0.537
0.556
0.569
0.576
+0.011
+0.005
+0.018
+0.004
+0.008
+0.004
+0.012
+0.009
+0.009
ROC
0.869
0.508
0.730
0.517
0.867
0.859
0.887
0.898
0.901
Infectious
=0.002
#0.006
#0.017
#0.008
+0.003
+0.003
#0.006
#0.015
20.016
PR
0.875
0.555
0.771
0.602
0.866
0.852
0.859
0.861
0.888
+0.007
+0.014
+0,013
+0,009
+0.007
+0.005
+0.007
#0,017
+0.016
RCC:
0.489
0.503
0.468
0.483
0.493
0.472
0.502
0.491
0.528
Facebook
#0.002
+0.005
+0.003
+0.002
+0.001
+0.004
+0.008
#0.006
20.004
0.513
0.517
0.462
0,491
0.512
0.517
0,520
0.498
0.535
20.006
#0.005
#0.009
#0.003
+0,001
#0.002
#0.009
#0.006
20.003
NeurlPS
ROC
0.445
0.504
0.510
0.507
0.5
0.5
0.502
0.761
0.778
20.004
#0.009
#0.018
#0,014
+0,000
#0,000
#0.015
=0,010
=0,011
PR
0.470
0.569
0.517
0.505
0.5
0.5
0.502
0.675
0.723
20.004
#0.011
+0.022
#0.012
+0,000
+0.000
+0.014
#0.019
10.013
US Legis.
ROC
0.475
0.466
0.490
0,463
0.482
0.469
0.742
0.565
0.754
20.003
#0.011
#0.017
#0.012
10.011
20.015
#0.010
#0.012
20.014
PR
0.496
0,513
0.593
0.481
0.505
0.505
0.706
0.537
0.711
=0.004
+0.013
#0.020
#0,012
+0.008
#0.013
#0.011
20,018
+0,012
ROC
0.675
0.309
0.473
0.529
0.734
0.801
0,862
0.678
0.810
Can. Parl.
=0.003
#0.010
#0.011
#0.012
+0.008
#0.014
#0.009
=0.009
10.009
PR
0.616
0.568
0,538
0,545
0.692
0.739
0,856
0.709
0.761
20.004
20.013
20.016
#0.010
+0.002
20.012
#0.010
20.008
20.010
ROC
0.393
0.415
0.445
0.502
0.823
0.841
0.729
0.655
0.845
Wikipedia
=0.010
#0.012
#0.011
+0.009
+0.007
20.006
#0.010
20,008
20,007
PR
0.398
0.374
0.425
0.489
0.860
0.884
0.758
0.654
0.836
=0.011
#0.013
#0.012
=0.010
+0.008
#0.007
+0.009
=0.008
10.007
Reddit
ROC
0.473
0.503
0,524
0,517
0.840
0.829
0.863
0.768
0.882
20.011
20.012
20.013
20.014
20.009
20.010
20.011
20.011
20,008
PR 0.454
0.460
0.490
0.486
0.877
0.852
0.904
0.751
0.895
=0.010
#0.012
#0.013
#0.014
+0.009
+0.010
10.011
20.011
+0.008
used, because the corresponding integral computation is intractable. To address these problems, we develop a theoretically-sound l latent distance based continuous-time graph representation (f, LD- CTGR). Specifically, we verify that the average & distance during an event time enjoys a similar bounded property that of the squared & distance case. We also derive the integral of the hazard function for the & distance case, which results in a closed-form piece-wise exponential integral. This differs markedly from the squared &g distance case, where the corresponding integral is the well-known Gaussian integral. We further develop an efficient integral computing algorithm for the ultra-low- wow-ddenniiinaall embedding. Finally, although the l distance, hazard function, and survival function are non-differentiable, we successfully find a descent direction for the hazard function to replace the gradient. This allows us to leverage mainstream learning architectures such as Pytorch to optimize the model parameters.
Extensive experiments conducted on eleven synthetic and wollwwrld benchmark data sets show that C, LD-CTGR achieves good performance across network reconstruction, completion, and prediction tasks, outperforming eight tfat--ofthhe-a art methods in most cases. Moreover, E LD-CTGR not only achieves the same computational efficiency GRASSP, but also facilitates true latent metrie space. Future works may focus on exploring additional types of valid metrics to further improve the geometric diversity of the latent space.
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Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
This work is supported in part by the National Natural Science Foundation of China (grant numbers: 62541606, 62176103, 62276114, and 62206 10)), in part by the National Research Foundation, Singapore and Infocomm Media Development Authority under its Trust Tech Funding Initiative, and in part by The Major Key Project of PCL (No. PCL2025A02 and No. PCL2024A04). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore and Infocomm Media Development Authority.
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Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S. Yu. A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems, 32(1):4-24, 2021.
Mengjia Xu. Understanding graph embedding methods and their applications. SIAM Review, 63(4): 825- 853, 2021.
Menglin Yang, Min Zhou, Marcus Kalander, Zengfeng Huang, and Irwin King. Discrete-time tem- poral network embedding via implicit hierarchical learning in hyperbolic space. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pp. 1975-1985, New York, NY, USA, 2021. Association for Computing Machinery.
Le Yu, Leilei Sun, Bowen Du, and Weifeng L.v. Towards better dynamic graph learning: New architecture and unified library. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (eds.), Advances in Neural Information Processing Systems, volume 36, pp. 67686- 67700. Curran Associates, Inc., 2023.
Daokun Zhang, Jie Yin, Xingquan Zhu, and Chengqi Zhang. Network representation learning: A survey, IEEE Transactions on Big Data, 6(1):3-28, 2020.
Yuan Zuo, Guannan Liu, Hao Lin, Jia Guo, Xiaoqian Hu, and Junjie Wu. Embedding temporal network via neighborhood formation. KDD '18, pp. 2857-2866, New York, NY, USA, 2018. Association for Computing Machinery. ISBN 778145035550.
Abdulkadir Çelikkanat, Nikolaos Nakis, and Morten Mørup. Piccewise-velocity model for learning continuous-time dynamic node representations. In The First Learning Graphs Conference, 2022.
Abdulkadir Çelikkanat, Nikolaos Nakis, and Morten Mørup. Continuous-time graph representation with sequential survival process. Proceedings of the AAAI Conference on Artificial Intelligence, 38(10) Mar. 2024.
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A APPENDIX
A.1 PROOF OF PROPOSITION 1
To prove this proposition, we need the following Jensen's 's inequality.
Lemma (Jensen's inequality). Let f (a, bj - R+ be non-negative Lebesgue-integrable func- tion, and yp : + IR be convex ietesgue-integrble. function. Then:
f (f((t) dt > 1 a
b (--a f(t)dt)
1
s()
(25)
Proof. We begin by examining the integral 0411-6, value of (s. f) in the interval [ete, 1.)1). It can control the average & distance by using convexity and Jensen's 's inequality.
Sea. Xey dt. It can be seen the average
rett1
exp (B(s) + (()r::(() ((1ll) de
ck+l 6k Jeh.
=exp(3(s))
exp (8)(r:(() (()l) de
Cx+1 - Ck Jec
exp(((()) exp
[[r(()) rg (f) dt).
(26)
Ck+1 ek
The above incquality follows from Lemma with f(t) == (t) ry (t) ppxx) == exp(s x), 6g and b Cx+1- Note that (xx) is convex w.r.t. for both 1 and -1. Taking the logarithm of both sides of (26) yields:
log (f () - log(Ee41 - -ex)
pe11 (1r((t- (r)( dt.
B(s) + Ek+1
(27)
On the other hand, it follows from the definition the survival function (1) that
Cht1
(s d
(28)
Inserting (28) into (27) and rearranging terms yield:
- (f))) dt
ek+1 cE
< log (log(S:)(ex))
B3()) -og(ee+1e)+))))
(29)
By setting 1 and - in (29), we have
1
[I((t) (r(t)-r((()ll::))
CR41 Ek
< log (log(Siz
log( (o)()(((((((()))) - -10))=4)))-(())-oo((((e-))-))))
(30)
-1
reu+1
[|r((1) (1))) di
Ek ce < log (log(S (ex)) log(Si -3)))))33(111
ck+1
(31)
Multiplying both sides of (31) by s = -1, then the upper and lower bounds (15) can be derived.
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A.2 PROOF OF THEOREM 1
Proof. We focus on the latter part of (16):
81
exp (s| Amiy + Aviyt(li) dt
Arij. + Avasial)
exp
dl,
(32)
d=1
where Arij,d denotes the d-th dimension of Az; and D is the dimensionality of the latent space.
The key technique is handle the D absolute operators simultaneously within the integral interval [0, eu ei). First, since Arzj. Ausj.al = Arij. - Auij,tl, we can adjust the sign of A Abjjd ensure that is non-negative:
Aijja Argja; Arij. Arija
if Auija O.
(33)
Next, we find the zero point Zij,d for the equation Arijd Anjj.at = 0:
Arrid 2ijdi== Aussid
if Auij, #
2.1,,,....2
(34)
2ijdd== O, ATi..t Amij,al
if Auyyd = 0
When Awij. = O, we can set Zij,d O and assign AAzija to Arijd Then we proceed with the following steps. We sort all the zero points the ascending order:
(1)(1) 21).(2) E... (.)(
(35)
Accordingly, denote (d) and A Aeg,(d) having the same sorting order (d) that of (ij. Then the sum in (32) over the interval (c) (t)((+))) becomes
Ariy. + Avij,tt
(rii(d)) + Aess,ay
d=1
AAiggntt Ao
Σ
(-Ar
d=o+1
Aotj.(d) Σ
Σ Amij,(d)
Σ Ar d=c-1 1
d=
Vis,te + Xss.()):
(36)
where Visto) and Nayte) denote the linear coefficient of and the constant term on the interval [2),(e) (t)()+))), respectively. Therefore, if Vij,(e) * O, the part of the integral (32) over the interval [Zi.((); (e)()+)) be computed by
D Ariya + Auigall dt
Isg,(cl(e+1) =
exp
Fsite)
d=1
exp (s (Vinco) + Xis.te)) de
Fai(e)
exp (sXij.(e)
exp (sVig,(ett Sunee1
sViy,ie)
exp (sX)))o) (exp (sVis.c) - exp (sVo..e)))))))) (37)
sVij.(c)
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If Vij(e) 0, then
Iosle)oo+1) = f (s (Vencest + Xis(o))) dt
=exp (*X (2i5(C+11) 2..(())
(38)
Next, we need to further investigate the case of (1) > 0 or (D) < Cu et, where the lower or upper limit of the integral goes beyond the smallest or largest zero point. In both cases, we can extend the definitions of Vis,te) and Xij,e)) c 0 and c =
V1, == Σ
Au (1 X1,(0)
Σ d=1
Ar b,d))
d=1
VD)
Σ Avij.dd) d=1
X1,(D) == Atij(d)
(39)
d=1
Let (g).(e) be the smallest zero point that is greater than 0, and 2ij(a) the largest zero point that is less than (ew - et). We compute the following two integrals:
Fisi(e)
I=,00
exp (s (Vi + Xi)((-n)) dt
exp(sXu,(c-1)) (exp (sVisle- -1) -Viile-1)
if Viy(e-1) +
(40)
exp (sXi). =t)((
if V Vile-1 =
exp (s (Vin Xi)(e))) de
(exp (e) - -1)) - exp (>V (sVi,.))...)))
if V # if Vi1(o)
exp (s))i,())) (ea el 2))
(41)
If (c) > or (c).(c) < 0, then the integrand will not change sign in the interval [0,c, - et).
In either case, we denote (c) == (c - 1) or (ट) = (z). respectively. Then the integral (32) has the following simplest form:
Iij,0.e--jj-
exp (s (Vi,cett + Xis,te))) dt
exp(xx...tee) (exp (sV - -1) (Vij,ca) exp (sX.j.ta)) (en - ei)
if Vy,(a # if Vi) 0
(42)
In other cases, the integral (32) can be computed by:
Aryja + Suiiaat) dt
exp
IIj.0,00
+
Iis10)oo11 + Iiple)oe-aa
(43)
with (e)))) Iij,0.e) and e.. defined in (37), (38), (40), and (41). Summarizing all the analyses of this proof, Theorem indicates that the integral of the hazard function based on the L distance is closed-form piece-wise exponential integral.
BE-1I IT (c+1) in (17), it is unnecessary to compute each Vyk( or Xis(e) individually as per (36). Instead, we can exploit the following relationships to compute Leyle)((+11 successively from e = g 2
Note: When computing Σe=1
Vole
xxll +2Amg.et11-
(44)
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A demonstration of Theorem 1: Let the hazard function
(s, t) = exp (It 0.1| + It - 0.2))
(45)
with 0 and eu 1. Its plot is shown in Figure Al. The zero points 2jjkel 0.1 and 2ij.(z) 0.2. Therefore, the entire integral should be divided into parts and computed using the second line of (17):
()
exp - 0.2)) dt
0.1
exp (0.3-21) dt+ / 0.2
exp(0.1) de+ I exp (2r-0.3)
J
oxp (0.3-21) 10 +0.1 (x1( cx.1x((0 +
2 exp (2t- 0.3) 1022
2
=0.5(exp(1.7) + exp(0.3)) - 0.9 exp(()))
22.4172
(46)
Xy (s exp (|t-0.1| |t-0.2])
(s,t)
0
0.2
0.4
0.6
0.8
t
Figure A1: A simple example of (s, t) in the interval [0, 1].
A.3 PROOF OF THEOREM 2
Proof. When D the integral (32) becomes
Che
exp (s(IAr + Aug, + |Arg, + Aug dt.
(47)
Then we examine the core part in the exponent:
f(t) = Am.ja + Ang + Amzj + A0sz2t).
(48)
In CTGR, the probability of Autjd is essentially zero, especially when each node pair has relative movement at any dimension and any time. Hence with slight abuse of definition, we
Ansid for d 1, and sort uzja such that Uzj(1)) & assume Auy,d # 0, 2. Let uij.d Auss.
(2).(2)- Then (48) becomes
f(1) Amg,ollug,00 0 AA 4/
(49)
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As the argument t falls into different intervals, ft) also falls into 3 different segments:
fi( = - (eign(Aet..1)) Az + sign (2,,,12)
(1) + (1
if (2)
(2()) (sigu( (1),,2))) (2)).(2) Sign(Arij... A
(50)
+ +()
Allj,())) if Uij((2) < t 2ij.(1)
[s(() (sign(Autj, (2) sign(An Arij.(1))
+(AA0,,(2) + Amss,wy)
if - (1).11
where the sign function is defined as follows:
if if = 0 if x
sign R + R, sign(r) =
(51)
Note that it can operate on tensor in element-wise way. Next, need to break the integration interval [0, Ek+1 ex) into parts according to the 3 segments in (50). Moreover, we should also consider the relative positions of the break points (1) (2) and the integration limits O, (ek+1 - -e). To tackle this issue, we define the following two break points:
S = min (0. (00,-44,,000 -1,,(2),,, - cx)
= min max {0, -1,,11)) cx).
(52)
Then the integrals regarding the segments of ftt are:
I / exp(sf((1)) de
<<oop(-s(signAAuu <ongl eexp( (op(-=(((AA),all (mr),)))))
(53)
I2 L exp(sf2(1)) de
5
=(5 () (xpps((sig(((()treee)) Azire) - -
+ eexp(s( ((4 - (Ang)))))
(54)
I3 =
exp(ssa(())
=(E++1-E Ae Aai
+
+ (A
(55)
The entire integral is
/ Ee+z
=xp (I + +I 12++2 Tz).
(56)
To see how {Se t) can be computed in tensor-parallelized way, first indicate that all the operators from (48) (56) are tensor-parallel. These operators include sign, exp, the absolute value operator, element-wise division, and others. Therefore, we can stack each variable w.r.t all the observations (one line for each observation) into single tensor as follows:
ij
An(a = (ew)) Ae(a) = (Amgngyeaa)es d=
= js(i.j := (:=l<(i,,,ea))uuu
::= ((i.j.
e [ex (ix ek+1 (e (k))s.
(57)
These all L tensors if there are L observations. Accordingly, the 3 stacked (tensor- parallelized) integrals for (53), and (55) can be computed by
I = 3 exp( (iggn(Am)) Am sign(( Amp))
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+ exp( (( + A
(58)
22 hagp://ooaaaaiaaoolooooo/o/a////
+ exxp(s @ (C - @ 2 - Am()))
(59)
Σ3 -cx+11 expl
+ expp( @ (ex+1 - c₁ 2) o (( + (Am()))))
(60)
where @ denotes the element-wise multiplication. Finally, the entire stacked integral can be com- puted by (18).
A.4 PROOF OF THEOREM 3
Proof. We omit the variable i in the expression of Xij for simplicity, which does affect the proof. We start from the simplest case.
Case 1: c= =D. In this case, Aty (rx) exp (3(s)). Hence (r,) (ra) O.
Case 2: C - D. We need to prove that there exists some > 0 such that Aıg 60Ac,(r:) <
Xi (r:). Taking a subtraction yields
>i - (r.); (ry(T:)
=exp (3(s) + s (t: (t)(())))
(exp (()))))))))))))))))))))))))))))) (rt) -1 -1].
(61)
Hence it suffices to prove that the second multiplier is negative. Denote (r.),(r.)) as the d-th dimension of aAi (ra).. For all d C, ri.d Tjid and thus (Asg,((r)) O. For all d C, it can be seen that (A.g,((r)) * O. Then we can set
8 = min Iiid rj,al
(62)
dec 2 (
This is uniform step size such that sign - 80Aij,a We further set
(r;) - -Tj,d) = sign(ri,d - aaa) for all d & C.
8= 5/ exp (3(s) + (|)) (t(() O.
(63)
then
exp (s(1r; -,),
Σ (I(ria- (((raa--s--igurrr.c. signsss.ggrrrrr.....a. -maaa
exp
ddc
Js sign(r.. - r. (aaaa)((raaa -(ra-13aa))
= exp
dgc
sign(rid- sigu(riaa -1
exp
ddc
8 Σ 32 (sign(rs..
exp
dgc
expl (osp--)DD(C)-11.
(64)
where [D)C] denotes the cardinality of the index set DIC. The last equality holds because and (rg.a From (64) back to (61), it can be seen that Aiy(r: -50;;(rr)) < Xisrr)) and -0x, (r:) is a descent direction of Xij at r:.
1. Since C S D, have (DiC and thus exp(-5) D D 0.
Case 3: C = 0. We need to verify that the following inequality holds for any + rr:
>ig(y) Ars(rs) > (O)ig(r:)) rz).
(65)
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By substituting the expressions of Ai (y), Aig (r:). and 0)y (r:), and rearranging terms, the inequal- ity (65) can be transformed into
exp (s (s((lyr)) ((r=-v;laa) -1
ss-sign(r: T (y
sigin(() ((y-r;)) (r::T,)
(66)
Since 0, we define
minded [i,ddTy.dd > 0.
=
(67)
2
Then when y - r, and
/lyrr:l1 < e, we have sign(y-rr) sign(r;-rs) Thus
s-sign(r,-rj) ((y ((g-r)))(rrr)))
=s(sign( Aysign(y-1)) y r, (y-r) - sign((::-r) (r:-r;)) (
= - ||, - r1la))
(68)
Let & [1y-ry 1,, and define
g(t) == exp(xf) 85 -
(69)
Then g gE) s(exp(sE) 1). Regardless of whether : = 1 or = 1,
1, we have g'()) > 0 for & >
and g (E) for Hence g(E) 2 g(0) 0. Substituting == ly || [r: ry 111 we have
exp 1101by-r,11111r----
-1 > <1yy-r ry 1 ||r.
(70)
for any + Ty. It verifies that 0A.g (r;) is subgradient of Xıj at by Definition 1. Case 4: + and 1. Directly computing the quotient in (21) gives
exp(s( 1 (1varra -ra.d r., Σ dgc
lya-rr (aar),))))--
> We rarad /
Σ (ya Tva
dEC
dgc
(71)
Let y Fr, then sign(ya Tj.d) sign(ri,d rj.d) for any d C. Combined with ri,d = Fjdd for
any d E C. (71) becomes
D
exp(s(
Σ
(lya ri,a)) +
Σ
Dd rr.al)) - 8(
Σ
1ya
riaal) /
Σ
yy ri.d)²
dgc
dEc
dec
D
D
exp(s(
Bd (yd Σ
1Bas (iaal /1
Σ
Ud (za)2.)))2
(72)
d=1
dgc
When
1, we have
exp
yd
r(al)
>
Va ri.al +
Σ
lya
rs,al + 1.
(73)
d=1
dec
The first inequality follows from the same properties of g(E) defined in (69) with E == Σb 1ya
T.al. Therefore,
D
D Tr.))
exp
Bd (u--raal)-4
Σ 1y rial
/
Ua
(74)
d=
Hence (ri((re) is subgradient of Xij r; by Definition 1.
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A.5 EXPERIMENTAL DETAILS
A.5.1 EXPERIMENTAL SETTING
The entire network is divided into two parts. Events within the last 10% of the timeline form the second part, which is used as the prediction set. Meanwhile, 20% of node pairs from the first part are randomly selected and then equally split into validation and test sets. The training set consists of the residual network that excludes any connections from the nodes the validation and test sets. If node pair does not have any connected periods during the training time but is selected for the prediction set, is excluded from the entire network.
The next step is to generate labeled data for the connection prediction tasks. This involves dividing the timeline of each sampled node pair into segments based on their state values. For each segment, random time f is selected to create sample interval [t st +el, where 0.017 and 'I' represents the total length of the timeline. a sample interval includes different states, it is considered as an indeterminate state and discarded.
The resulting samples are then categorized into "hard" and "simple" sets. The "hard" set contains samples from node pairs that have experienced least connected and one disconnected periods throughout the entire timeline (Huang et al., 2024b). In contrast, the "simple" set includes node pairs with only one state, making their state prediction relatively straightforward. Samples in the prediction set are categorized based on their connection status during the training time, following the setting in (Poursafaci al., 2022).
For each training, validation, test, or prediction set, let h be the size of the "hard" set, and define = min 1000, connected set size, unconnected set size).
(75)
Then each class (connected or disconnected) in the training, validation, test, or prediction set con- tains h/2 samples from the "hard" set, and (k: h/2) samples from the remaining "hard" and "sim- plc" sets. Performance evaluated using AUC-ROC and AUC-PR scores, which assess true or false positive rates and precision-recall characteristics, respectively. All the experiments are repeated for 10 times to produce the mean and standard deviation results SSD))):STD).
A.5.2 DATA SETS
The network is treated undirected, with the finest available temporal granularity, such mea- surements in seconds or milliseconds, used for the input timestamps. Additionally, the data sets are adapted fit different competitors ensure fair comparisons. For instance, dynamic networks transformed into static weighted or unweighted networks by aggregating connections over the entire timeline for static competitors. Furthermore, since competitors cannot handle non-connection events, these events are excluded from the evaluation.
We employ 11 synthetic and real-world benchmark data sets with diverse characteristics for our experiments, detailed below.
Synthetic-a. This data set is generated using the sequential survival process outlined in Section 2, incorporating both connection and non-connection event times. The initial node positions and velocities are sampled using a procedure similar that in (Çelikkanat et al., 2022). The global con- nectedness parameters are set as (3(1) 1 and 3(1) = -1. Velocities are sampled from mul- tivariate Gaussian distribution vec(e) NO, E) where Σ == diag (a II, . BLB, aplp) set as 0 ;= 3, ON == 10 oB := 10-6 and op == 1. Visualized key frames are shown in Figure A2, illustrating that the ε distance case leads to more between-group and longer-distance connections compared with the squared Eg distance case.
N.--B-D. B--D These hyperparameters
Synthetic-8. The entire timeline is divided into bins of equal length, with nodes randomly clus- tered into 10 groups. The probability of node pair connection within the same group is set 0.8, while that of connections between different groups is set as 10 These connections persist until the next bin event occurs.
Contacts (Génois et al., 2015). This data set records interactions among individuals in an office building over span of days in 2013.
HyperText (Isella et al., 2011). It captures face-to-face interactions tracked by radio badges of participants during 2.5-day conference.
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Infectious (Isella et al., 2011). This data set includes another interaction network during an event in Dublin. the Contacts, HyperText, and Infectious data sets, each timestamp represents a 20-second connection. Multiple connections within two-minute window treated single connection duration.
Facebook (Viswanath et al., 2009). This is friendship network that indicates whether user is on the friend list of another user. It is a large-scale network with about 82 thousand nodes and 200 million edges.
NeurIPS (Globerson et al., 2004). The original version tracks collaborations among authors for NeurIPS conference papers from 1989 to 2001. It is later extended cover papers from 1988 to 2003 (Globerson et al., 2007). It is a large-scale network with about 5 thousand nodes and 1.2 million edges.
US Legis. (Poursafaci et al., 2022). It is a senate co-sponsorship graph that records social interac- tions between legislators from the US Senate. The edge weights indicate the number of times that two congress persons have co-sponsored a bill in a given congress.
Can. Parl. (Poursafaci et al., 2022). is a dynamic political network that records the interactions between Canadian Members of Parliaments (MPs) from 2006 to 2019. Each node represents one MP for an electoral district, and each edge formed if both of the corresponding two MPs voted yes on a bill. The edge weights indicate the number of times that one MP voted yes for another MP in year.
Wikipedia (Poursafaei et al., 2022). It is co-editing network on Wikipedia pages over one month. Editors and wiki pages are nodes, while one edge represents that given user edits a page at specific timestamp. The task is to predict with which wiki page user will interact given time.
Reddit (Poursafaci et al., 2022). It is an interaction network for users and subreddits from 2008 to 2010. Both users and subreddits are nodes, while each edge indicates that user posted on subreddit at a given time. The task to learn the interaction frequency towards the subreddits of a user the next week.
A.5.3 COMPARED METHODS
To assess the performance of the proposed LDDDCTG method, we compare it against 8 leading methods with static, dynamic, and continuous-time graph representations. These methods serve as strong baselines for evaluating the effectiveness of our approach. We adopt all the default settings of these methods in the experiments.
Nede2Vec (Grover & Leskovec, 2016). Node2Vec introduces flexible concept for defining the neighborhood of node and designs a biased random walk algorithm to efficiently explore various neighborhoods. Its default embedding dimensionality is D 128.
CTDNE (Nguyen et al., 2018). Continuous Time Dynamic Network Embedding (CTDNE) extends random walk methods to dynamic networks, which capture and integrate temporal information into network embeddings. Its default embedding dimensionality is D 128.
(Zuo et al., 2018). Hawkes Temporal Network Embedding (HTNE) adopts a Hawkes pro- cess based approach to capture the influence of historical neighbors on the current neighbors into temporal network embeddings. Its default embedding dimensionality is D 128.
HTNE
PIVEM (Çelikkanat et al., 2022). Piece-wise Interpolation for Temporal Network Embedding (PIVEM) uses piece-wise linear interpolations approximate temporal node evolution. Its project link? indicates that the default embedding dimensionality is = 10. It uses B 100 bins in the timeline segmentation.
TCL (Wang et al., 2021; Yu et al., 2023). It applies contrastive learning to representation learning dynamic graphs, which builds graph-topology-aware transformer. Its default embedding dimensionality is D = 322.
hhttps://abcee wtw.aaatoiiikkkak.g.g.k htt.::/ieeeeittiiiiiiioiooooerre http://gttttbbit///r//st///////e vem/
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GraphMixer (Cong et al., 2023; Yu et al., 2023). It consists of three components: link-decoder that treats temporal links, a node-encoder that summarizes node information, and a link classifier that performs link prediction. Its default embedding dimensionality is D 322.
DyGFormer (Yu et al., 2023). is a Transformer-based architecture that only needs to learn from the historical first-hop interactions of nodes a neighbor co-occurrence encoding scheme and patching technique.
GRASSP (Çelikkanat et al., 2024). It employs the squared & latent distance based sequential sur- vival process to model continuous-time graph evolution. It uses the altra-low-dimensinn embed- ding with D 2 and the number of bins B 100 that of PIVEM.
& LLD.CTGR (ours). also uses the ultra-lo--dimeniinn number bins B 100 that of PIVEM or GRASSP, be consistent.
embedding with 'D = 2, and the same
A.5.4 RUNTIME COMPARISON BETWEEN GRASSP AND &LLD-CTGR (TABLE A1)
Table A1: Average runtime (in seconds) per epoch of GRASSP, &.LD-CTGR and the & distance version (H- 1000) with ditra-low-dimensionne
= 2 on different data sets (mean + STD). The training criterion of GR GRASSP is followed, which is three-stage 300-epoch procedure. Results are conducted on device with an Intel(R) Xcon(R) Gold 6330 CPU, 1 TB RAM, and cight NVIDIA A100 GPUs. Both GRASSP and & LDDCCT have the same order of actual runtime.
Data Set
GRASSP
4LD-CTGR
& Distance
Synthetic-a
1.93E 4 + 8.97E - 6
1.94 + 9.44E 6
2.29E - + 1.13E 3
Synthetic-B
1.64E 4 7.15/
1.66/ + 7.81E 6
1.94E
9.34E
Contacts
3.21E 33 3 1.60E
3.09E 3 = 1.45E
4
3,80E -1 + 1.70E 2
HyperText
6.18E -3 3 + 2.98Æ 4
6.22E -3 + 2.90E 4
7.10E -1 + 3.53E 2
Infectious
6.15E
3.00E
6.15E = 2.79E
7,26E
3.36E
Facebook
4.49 = 2.18E -1
4.55 + 2.33E -
531 1 28.4
NeurIPS
2.74F 1 + 1.41
2.77 + 1.37E
33.8 1.56
US Legis.
1.86E - 2 9.39E
1.76E -2 = 9.10E 4
2.14 - 1.0SE -1
Can. Parl.
2.49F - + 1.20E
3
2.4A -2 + 1.31
2.93
+ 1.50E
A.5.5 ROBUSTNESS OF LD-CTGR TO BIN NUMBER AND EMBEDDING DIMENSIONALITY
We examine the robustness of & LD-CTGR bin number (default B 100) and embedding dimen- sionality (default D 2). We first fix 'D = 2 and let B change in [92, 108), and then fix B 100 and let change (2, 6). The corresponding results on the Contacts data set are shown in Tables A3 and A4, respectively. They indicate that LDD.CTGR is robust to small changes of these hyperparame- ters. Morcover, we directly compare the performance of LDDCC,,,, GRASSP, and lg Distance on the Contacts and Infectious data sets in Table A4. Results show that & LD-CTGR achieves the best performance in most cases, while & Distance generally underperforms & LDDCTGR and GRASSP to certain extent.
A.5.6
NETWORK RECONSTRUCTION PERFORMANCE (TABLE A5)
A.5.7
CONTINUOUS-TIME VISUALIZATION (FIGURE A2)
Following (Çelikkanat et al., 2024), we visualize the evolutions of the é latent distance and the squared & distance based Synthetic-a networks in Figure A2.
23

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Published as a conference paper at ICLR 2026
Table A2: Average runtime (in seconds) per epoch of GRASSP and &DDDCTGR with higher mensionalities different data sets (mean + STD).
Data Set
GRASSP
2,LD-CTGR
D 10
Synthetic-a
1.98 + 9.20 6
1.99 9.67 6
Synthetic-$
1.68E - -4 = 7.33E
6
1.69E -4 + 8.00E
6
Contacts
3.29E -3 = 1.64E
3.16E - 1.50E
HyperTText
6.33E -3 + 3.05E
4
6.36E -3 2.99E -4
Infectious
6.32F -3 + 3.07E
6.31E 2.86E
Facebook
4.61 + 2.23F -1
4.66 + 2.39F -1
NeurIPS
2.81E - = 1.45E 2
2.85-11- 1.40E -
US Legis.
1.90E - 2 = 9.69 4
1.80E - = 9.34E
Can. Parl.
2.55E 2 + 1.23E 3
2.49E 2 + 1.34E 3
D = 100
Synthetic-a
2.21E 4 + 1.02E 5
2.43E 4 + 1.20E 5
Synthetic-B
1.88E + 8.20E
2.08E + 1.01E 5
Contacts
3.72E -3 + 1.82E
4
3.92E -3 33 1.86E
HyperText
7.12E -3 -3 = 3.44E
7.93E -3 3.73E
4
Infectious
7.02E -3 = 3.44E
4
7.83E -3 3.60E
Facebook
5.09 2.47F -1
5.88
3.05F -
NeurlPS
3.14F -1 + 1.62F 2
3.54F - + 1.71
US Legis.
2.18E -2 + 1.07E
2.23E -2 + 1.19E
Can. Parl.
2.85 = 1.37E
3.09
+ 1.70E
Table A3: Performance of & LDDDCTGRR with respect to different bin numbers on Contact (mean of 10 repetitions).
Completion ROC
Reconstruction ROC
Prediction ROC
Bin. Num.
PR
PR
PR
92
0.660
0.714
0.636
0.646
0.772
0.726
94
0.655
0.709
0.641
0.651
0.767
0.721
96
0.645
0.694
0.661
0.656
0.757
0.711
0.650
0.699
0.676
0.666
0.762
0.716
100
0.680
0.724
0.676
0.676
0.767
0.721
102
0.635
0.689
0.636
0.641
0.752
0.706
104
0.640
0,694
0.631
0.636
0.757
0.711
106
0.650
0.704
0.636
0.641
0.762
0.716
108
0.645
0.699
0.636
0.641
0.757
0.716
24

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Published as a conference paper at ICLR 2026
Table A4:
Performance of LLDD-TTG,, GRASSP, and the l distance version (H 1000) with respect to different dimensionalities on Contact and Infectious (mean of 10 repetitions).
Contact
Infectious Reconstruction PR
Completion
Reconstruction ROC PR
Prediction ROC
Completion ROC
Prediction ROC PR
Dim. Size
ROC
PR
PR
PR
ROC
LDLD-TGGR
Mcan 0.680
0.724 0.676
0.676
0.767
0,721
0.756 0.779 0.861
0.832
0.901
0.888
STD 0,017
0.028 0.016
0.021
0.018
0.018 0.017 0.017 0.021
0.019
0.016
0.016
Mean
0.664
0.704 0.657
0.680
0.737
0.699 0.753 0.765 0.846
0.817
0.885 0.883
STD 0.016 0.009 0.024
0.020
0.016
0.023
0.013 0.018 0.018
0.020
0.020 0.016
Mean
0.655 0.686 0.638
0.668
0.733 0.682
0,745 0.767 0.858
0.820
0.888 0.875
STD 0.019 0,018 0.021
0.022
0.023
0,016
0.019 0.015 0.015
0.019
0.024 0.016
5
Mean 0.647 0.696 0.636
0,654
0.744
0.704
0.736 0.748 0.859
0.811
0.882 0.876
STD 0,020 0,018 0.023
0.015
0.019
0.021
0.022
0.024 0.019
0.014
0.016 0.015
Mean
0.633 0.690 0.674
0.671
0.777
0.735
0.735
0.737 0.869
0.830
0.879 0.886
STD 0.028 0.016 0.019
0.018
0.012
0.017
0.013 0.022 0.016
0.019
0.017
0.020
GRASSP
Mean 0.670 0.714 0.589
0.634
0.763
0.714
0.728 0.711 0.738
0.708
0.898 0.861
0.016 0.025 0.013
0.019
0.016
0.020 0.029 0.028 0.018
0.016
0.015 0.017
Mean
0.659 0,704 0.605
0.632
0.734
0.703
0.718 0.703 0.729
0.698
0.873 0.885
STD 0,022 0,025 0.023
0,018
0.021
0,015
0.024 0.013 0.014
0.015
0.004 0.027
Mean
0.645 0.689 0.593
0.639
0.739
0.679
0.713 0.706 0.734
0.713
0.885 0.877
STD 0.021 0.015 0.027
0.010
0.019
0.018
0.021 0.021 0.015
0.019
0.019 0.023
Mean 0.656 0.689 0.568
0.612
0.736
0.689 0.702 0.686 0.732
0.693
0.876 0.851
STD 0.025 0.025 0.015
0.020
0.013
0.014
0.019 0.027 0.015
0.021
0.018 0.025
Mean
0.652 0.685 0.562
0.626
0.744
0. 705 0.719 0.702 0.729
0.700
0.887 0.862
STD 0.014 0,020 0.022
0.025
0.027
0.020 0.012 0.016 0.014
0.029
0.016 0.023
Eg Distance
2 Mcan 0.605 0.625 0.614
0.606
0.692
0.623 0.688 0.691
0.721
0.694
0.834
0.811
STD 0.038 0,034 0.035
0.029
0.022
0.024 0.021 0.013 0.017
0.021
0.008
0.012
Mean 0.594 0.639 0.552
0.617
0.644
0.593 0.678 0.677 0.716
0.688
0.809 0.803
STD 0.028 0.027 0.027
0.023
0.023
0.027
0.023 0.019 0.023
0.018
0.019 0.013
Mean 0.599 0.589 0.559
0.574
0.655
0.607 0.676 0.673 0.719
0.701
0.814 0.804
STD 0.026 0.023 0.034
0.026
0.020
0.026
0.019 0.021 0.024
0.017
0.017 0.023
Mean 0.567 0,626 0.544
0.569
0.624
0.623
0.677 0.675 0.719
0.698
0.807 0.803
STD 0.025 0,026 0.022
0.020
0.026
0.023
0.024
0.014 0.021
0.014
0.019 0.020
Mean
0.530 0.609 0.632
0.587
0.678
0.647
0.673
0.666 0.720
0.709
0.818 0.810
0.030 0.022 0.029
0.023
0.022
0.019
0.013
0.018
0.027
0.029
0.014 0.017
25

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Published as a conference paper at ICLR 2026
Table A5: Performance of different methods for network reconstruction (in-sample) across diverse
data sets STC).
Data Set
Node2 Vec
CTDNE
IITNE
PIVEM
TCL
GraphMixer DyGFurmer GRASSP 4,LDCTTGR
ROC
0.524
0.550
0.535
0.567
0.281
0.300
0.436
0.577
0.793
Synthetic-a
=0.013
#0.016
#0.021
20.018
+0.042
#0.038
=0.020
#0,028
=0,038
PR
0,538
0.555
0.602
0.642
0.400
0.412
0.446
0.509
0.708
#0.023
#0.027
+0.019
#0.016
+0.009
+0.027
#0.022
=0.029
#0,039
Syntheric-9
ROC
0.541
0.493
0,535
0.531
0.528
0.448
0,536
0,843
0.610
20.006
#0.008
#0.008
#0.006
#0.061
20.006
#0.018
#0.015
#0.021
PR
0.545
0.557
0.591
0.536
0.621
0.573
0.616
0.756
0.554
+0.004
+0.006
#0,006
=0,006
+0.054
#0.016
+0.020
#0,011
#0.017
ROC
0.674
0.508
0.335
0.339
0.394
U.539
0.396
0.589
0.676
Contacts
=0.011
#0.021
#0.011
#0.012
+0.007
#0.012
#0.016
#0.013
#0,016
0.657
0.570
0.563
0.567
0.616
0.614
0.642
0.634
0.676
20.016
20.019
#0.013
#0.009
20.016
20.013
#0.019
20,019
20,021
ROC
0.589
0.486
0.619
0.560
0.602
0.662
0.68T
0.607
0,694
HyperText
=0.006
+0.014
#0.011
#0,004
+0.010
#0.006
#0.014
20.006
#0.011
PR
0.569
0.542
0.624
0,372
0.627
0.663
0.640
0.580
0.691
+0.008
+0.013
+0.007
+0,004
+0.007
+0.003
+0.015
+0.009
#0,014
Infectious
0,851
0.613
0.839
0.738
20.003
#0.009
#0.011
#0.005
#0.001
20.001
#0.012
20.018
20,021
PR
0.742
0.566
0.819
0.630
0.812
0.801
0.848
0.708
0.832
+0.008
+0.011
+0,009
#0,007
+0.004
#0.001
+0.013
#0,016
+0.019
ROC
0.306
0.473
0:445
0.482
0,510
0.500
0.502
0.500
0.612
Facebook
#0.002
+0.005
+0.003
+0,002
+0.002
+0.009
+0,010
#0.000
20.004
0.515
0.481
0.520
0.530
0,540
0.588
20.004
20.005
#0.003
#0,003
+0.001
20.006
#0.011
20.000
+0.004
NeurlPS
ROC.
0.433
0.489
0.431
0.510
0.633
0.634
0.683
0.548
0.528
+0.004
+0.011
+0.011
+0.009
+0.001
+0.001
#0.020
#0.018
+0.009
PR
0,476
0.541
0.448
0.525
0.580
0.578
0.613
0.506
0,501
#0.004
#0.015
+0.008
+0.008
+0.004
+0.006
20.018
#0.025
+0.008
ROC
0.493
0.490
0.525
0.750
0.866
0.767
US Legis.
20.003
#0.011
=0.017
#0.012
+0.005
+0.014
#0.016
#0.012
20.014
PR
0.510
0.524
0.576
0.561
0.515
0.700
0.856
0.588
0.712
#0.004
+0.013
+0.020
#0.012
+0.004
+0.008
+0.017
#0.018
+0.012
ROC 0.701
0.479
0.583
0.308
0.687
0.761
0.906
0.593
0.826
Can. Parl.
20.004
+0.009
10.011
+0.009
+0.004
+0.009
10.010
#0.013
10.004
0.649
0.542
0.643
0.634
0.743
0.911
0.614
0.793
20,004
20,009
#0.015
#0.014
+0.005
+0.005
#0.011
20.008
20.003
Wikipedia
ROC
0.312
0.480
0,430
0.620
0.852
0.868
0.760
0.705
0.870
#0.010
+0.011
#0.012
+0.009
+0.006
+0.007
#0.011
+0.008
#0.005
PR
0,321
0.453
0.471
0.680
0.890
0.908
0.805
0.668
0.838
+0.009
#0.012
+0.013
#0.010
+0.007
20.008
#0.012
=0.009
10.006
ROC
0.467
0.501
0.653
0.750
0.766
0.790
0.707
0.801
Reddit
20.012
20.013
20.014
20.016
40.010
20.010
#0.012
#0.011
20.009
PR
0.456
0.465
0.530
0.681
0.769
0.767
0.801
0.689
0.803
10.011
+0.012
#0.013
20.014
+0.010
+0.010
+0.012
10.011
20.009
26

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Published as a conference paper at ICLR 2026
4
(a) E = 150
(b) f = 200
(C) t = 250
(d) 300
(c) 350
t 400
4
(g) = 150
(h) L = 200
(i) = 250
(j) 1- 300
(k) f 350
()) - 400
Figure A2: Continuous-time graph representation via the latent space Synthetic-a. (a)~(f): the
squared & distance case. (g)~(l): the & distance case.
27


Reviews:
[{"review_id": "dO6q3iVt99", "reviewer": "Reviewer_syKN", "summary": "This paper studies continuous-time graph representation via sequential survival processes and asks how to embed evolving networks in an ultra–low-dimensional latent space without distorting geometry. The main difficulty is that the widely used squared latent distance violates the triangle inequality, which can warp relative node positions. To overcome this, the paper proposes ℓ₁LD-CTGR, which replaces the latent distance with ℓ₁ while preserving tractable learning. Specifically, it (i) proves boundedness of the average ℓ₁ distance across event intervals, (ii) derives a closed-form, piece-wise exponential integral for the hazard, (iii) designs an efficient tensor-parallel algorithm tailored to D=2 embeddings, and (iv) introduces a descent-direction/subgradient scheme that makes the non-differentiable ℓ₁ objective trainable in mainstream autodiff frameworks with the same order of complexity as GRASSP. Theoretical results establish a true latent metric space and closed-form integrals compatible with ultra–low-dimensional modeling. Empirically, across 11 synthetic and real-world datasets and three evaluation regimes (reconstruction, completion, prediction), ℓ₁LD-CTGR attains competitive or superior performance to eight strong baselines, with robust gains on large, challenging networks.", "questions": "1.In Table 1, DyGFormer outperforms the proposed method on some datasets, and the manuscript does not provide an explanation or discussion of this phenomenon.\n\n2.In Appendix Table 5.4, the proposed method appears slower than existing baselines on several datasets. Does this suggest the claimed efficiency gains only emerge with longer training time, or that further optimization is still needed?\n\n3.In Table A4, the proposed method underperforms several strong baselines on multiple datasets. Please explain the underlying causes and provide analysis to clarify this discrepancy.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.The paper clearly shows that the squared ℓ2 distance violates the triangle inequality and can distort latent-space geometry. It also argues that the resulting integrals are less tractable, giving the work clear and well-founded motivation.\n\n2.The paper presents a generally sound theoretical development supported by a careful empirical evaluation.", "weaknesses": "The paper claims “competitive performance” across multiple synthetic and real datasets. However, in several tables, the proposed method either trails strong baselines or offers only marginal gains. The manuscript lacks a systematic analysis (e.g., error breakdowns, dataset-specific failure modes) to reconcile these discrepancies, which weakens the empirical claim.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Bi47T53jyP", "reviewer": "Reviewer_XmKP", "summary": "The paper introduces ℓ1-LD-CTGR, a continuous-time graph representation that uses an ℓ1 latent distance to form a true metric space, overcoming distortions caused by squared ℓ2 distances and enabling tractable hazard function computation. Experiments show that this approach achieves competitive performance on both synthetic and real-world graph datasets.", "questions": "No more questions.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "(1) This paper presents a solid theoretical motivation and analysis for adopting the L1 distance over the traditional L2 distance, offering a complete and self-contained formulation.\n\n(2) The experimental evaluation is comprehensive, covering multiple benchmarks with sufficient baselines and datasets to convincingly demonstrate the effectiveness of the proposed approach. The reported results clearly show that the proposed method outperforms prior baselines.", "weaknesses": "Although the paper is well-structured and self-contained, its overall contribution appears limited. The main novelty lies in replacing the L2 loss function with an L1 formulation and adjusting the corresponding learning algorithm and optimization strategy. Moreover, the work is confined to temporal network learning, which limits its broader impact on the wider research community.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ghccnOxjKV", "reviewer": "Reviewer_hFhi", "summary": "This paper addresses a fundamental theoretical flaw in existing Continuous-Time Graph Representation models, such as GRASSP, which rely on the squared $l_2$ distance in the latent space. The authors correctly point out that $\\|\\cdot\\|_2^2$ violates the triangle inequality, leading to distorted metric properties. They propose $l_1\\text{LD-CTGR}$, which replaces the squared $l_2$ distance with the theoretically sound $l_1$ distance ($\\|\\cdot\\|_1$). The core technical contribution is the derivation of the continuous-time integral of the $l_1$ hazard function as a closed-form piece-wise exponential integral and finding a descent direction to handle the non-differentiability of the $l_1$ norm. Experiments show competitive performance and identical computational complexity compared to the $l_2^2$-based baseline.", "questions": "1.\tFor dimensions $D > 2$, how does the computational complexity of the tensor-parallelized integral (Theorem 2, Eq. 15) scale with $D$? Specifically, what is the complexity when $D=10$ or $D=100$, and is $l_1\\text{LD-CTGR}$ still competitive with GRASSP in these higher dimensions?\n\n2.\tThe core motivation is correcting the metric distortion. Can the authors provide a metric (e.g., a latent space uniformity measure, or the preservation of long-range topological fidelity) that *directly* quantifies the benefit of using a valid metric space ($l_1$ vs. $l_2^2$), thus supporting the strong theoretical claim empirically?\n\n3.\tIn Case 4 (disconnected state $s=-1$ and $C \\ne \\emptyset$), the computed direction is not a subgradient. Given that maintaining the disconnected state is essential for modeling complex networks, what are the observed empirical consequences (e.g., convergence speed) of using a descent direction that is not guaranteed to be a subgradient?\n\n4.\tSince you successfully solved the non-trivial $l_1$ integral, please elaborate on the specific mathematical properties of the $l_2$ distance integral ($\\int \\exp(\\dots + s\\|r_i - r_j\\|_2) dt$) that make it intractable, contrasting it with the solvable nature of the $l_2^2$ integral (Gaussian) and your $l_1$ integral (piece-wise exponential).\n\n5.\tProposition 1 confirms that the bounded property of the distance holds for $l_1$ distance as well (L199-204). Beyond verification, how does this property actively guide or constrain the optimization process during training, and is the derived bound $\\mathcal{B}(s)$ (Eq. 12) used anywhere in the final loss function or regularization?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tThe motivation is excellent: correcting the use of the squared $l_2$ distance ($\\|\\cdot\\|_2^2$) which violates the triangle inequality (L059), ensuring the latent space is a valid metric space using the $l_1$ distance.\n\n2.\tThe derivation of the closed-form piece-wise exponential integral for the $l_1$ hazard function (Theorem 1) is a non-trivial technical accomplishment, successfully tackling the intractability concern associated with non-$l_2^2$ distance metrics (L234-236).\n\n3.\tThe use of descent direction (subgradient computation) to handle the non-differentiable $l_1$ norm (Theorem 3, L309-317) is a sound approach that ensures compatibility with mainstream optimization frameworks like PyTorch (L282, L333-336).", "weaknesses": "1.\tWhile $l_1$ distance is theoretically sound, the paper does not convincingly demonstrate that resolving the metric space flaw translates into significant empirical gains. $l_1\\text{LD-CTGR}$ is competitive but rarely dominates its flawed counterpart, GRASSP (e.g., in Table 2, GRASSP often achieves higher scores).\n\n2.\tThe efficiency gains from the piece-wise exponential integral seem heavily reliant on the assumption of $D=2$ (L249, L342). The paper should explicitly discuss how the complexity scales with increasing dimensions $D$ (where $D$ determines the maximum number of zero points $z_{ij, d}$), and whether the method remains feasible for $D>6$ (L1197).\n\n3.\tTheorem 3 correctly identifies that the derived term $-\\partial\\lambda_{ij}(\\mathbf{r}_i)$ is *not* a subgradient when $s=-1$ and $C \\ne \\emptyset$ (Case 4, L997-1005). Although the authors state it is still a descent direction, the potential impact of using a non-subgradient in the optimization step for the important \"disconnected\" state ($s=-1$) must be discussed.\n\n4.\tThe paper argues reverting to $l_2$ distance is intractable due to integral complexity (L065). Given the technical achievement of solving the $l_1$ integral, a more explicit comparison showing *why* the $l_2$ distance integral remains intractable (e.g., complexity of integration beyond $D=2$) would strengthen the necessity of adopting $l_1$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ahVfnjbdbB", "reviewer": "Reviewer_Y6yo", "summary": "This work expands upon the recent GRASSP framework for the modeling of continuous time dynamics using the sequential survival process and piecewise velocity modeling procedure developed in (Celikkanat et al., 2024), Specifically, the paper here makes the following contribution:\n\n1)\tAdvance the modeling to a true metric embedding model based on the l1 distance as opposed to the squared Euclidean distance used due to computational convenience in (Celikkanat et al., 2024).\n\n2)\tDemonstrate that this procedure despite added complexity of inference in practice have same scaling properties with a subgradient procedure proposed well optimizing the L1 distance while preserving other essential properties of GRASSP.\n\n3)\tExtensively evaluate the procedure and contrast the framework to more approaches than previously considered finding that the L1 distance based framework has enhanced performance when compared to the previous squared L2 distance embedding framework of GRASSP – pointing towards metric properties enhancing performance.", "questions": "Experimentally the superiority of the L1 distance metric over squared Euclidean distance is shown. The method compared against have been run in their default setting. It is unclear if the proposed method have been more extensively tuned in hyperparameters in terms of learning rates etc. Only used dimensionality is discussed. Please clarify how learning rate etc. were tuned for the methods.\n\nCould the L2-metric approach using a truly Euclidean metric as opposed to squared Euclidean distance representation be feasibly implemented using numeric integration as opposed to analytical expressions using the squared distance? – and how would such numeric integration based metric approach compare to the presented L1 approach?\nThis would better position the paper as it is unclear in the current experimentation what the influence of having a metric vs. non-metric embedding implies (this would be comparing squared L2 to conventional L2-norm (metric) - which could have been addressed by accurate numeric integration potentially on smaller toy problems) and what is the effect of L2-norm (metric) vs. L1-norm (metric) presently invoked. These different norms will by themselves substantially influence the results and here the L1-norm may also have benefits over the L2-norm which cannot currently be assessed. This would strengthen the paper to address and help further position the contribution and understanding the merits of the L1-norm.\n\nGiven the above I consider this a borderline paper but would be willing to increase my score.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "•\tThe paper is very well written and clear in its presentation.\n\n•\tThe contribution is sound and valid and includes a detailed experimental comparison to existing methodologies.\n\n•\tThe inference procedure is non-trivial and ensures metric properties in latent distance based continuous time link persistent networks.\n\nOriginality: The paper expands upon the GRASSP framework to make the embeddings truly metric by use of the L1 norm as opposed to relying on the squared Euclidean distance. The experimentation closely follows the GRASSP framework but is extensive. The contribution is sufficiently original. \n\nQuality: The developed methodology is non-trivial and the paper well written and the experimentation extensive. It would however improve the paper to compare the L2-norm metric version with the proposed L1-norm metric version. It is unclear what comes from change of metric and what is the result of non-metric squaring the L2-norm. The metric L2-norm would be less computational attractive but could be implemented by use of numeric integration instead of relying on analytical expressions of the integral representations.\n\nClarity: The paper is clear and well presented.\n\nSignificance: The approach appears useful and superior to using the squared L2 distance with results well supporting the utility of the proposed approach.", "weaknesses": "•\tThe novelty of the paper is somewhat limited but valid expanding the prior work on GRASSP to encompass true metric embeddings as opposed to relying on computational convenient squared L2 distances.\n\n•\tIt is unclear how difficult in practice using the standard Euclidean distance would be. I.e., the integral rather than being analytically solved could here be solved by numeric integration which would likely not be very heavy in computation and potentially work well in practice. This would strengthen the paper to consider.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Rk8iffUz6g", "reviewer": "Reviewer_51D8", "summary": "This paper proposes a latent position model for continuous-time using a hazard function based on $\\ell_1$ distance between latent positions motivated by issues regarding the triangle inequality from previous methods using squared $\\ell_2$ distance. The model is benchmarked against continuous-time graph representation models and other common graph representation techniques against commonly used synthetic and real dynamic networks with impressive results in incredibly low embedding dimension.", "questions": "Some datasets have worse performance when you move from in-sample to out-of-sample to across-sample assessment, for example, synthetic-$\\alpha$. In general, I would expect the in-sample to be the easiest assessment as it is looking within training data, but it occasionally underperforms compared to the other assessments. Can the authors provide some intuition as to why this phenomenon occurs?\n\nDoes using the $\\ell_1$LD-CTGR technique with dimension $D > 2$ lead to any improvement in performance? If not, this would further motivate the argument for using the computational efficient $D = 2$ embedding.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 4, "strengths": "The technical work in this paper is strongly motivated and the argument for using $\\ell_1$ distance for social networks, highlighting issues with using the squared $\\ell^2$ distance used in GRASSP. While the computations are different in nature to GRASSP, the new approach has the same complexity cost as the previous approach, usurping the technique for incredibly low dimensional embeddings ($D = 2$).", "weaknesses": "The experimental results lack explanation. For instance, it is not explained how comparison continuous-time methods based on instantaneous edge models (CTDNE/HTNE) are applied to the persistent edge datasets. These methods are trying to model a different problem, so it not surprising they underperform. However, $\\ell_1$LD-CTGR often outperforms GRASSP which is a like-minded comparison.\n\nMore information than what is given in Appendix A.5.1 about in-sample, out-of-sample and across-sample evaluations is needed. I understand the intended aims of the three approaches, but I don't see how it relates to the language of 'simple' and 'hard' sets. More details are needed to make this a repeatable experiment.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)