Diffusion Model for Graph Inverse Problems: Towards Effective Source Localization on Complex Networks
Authors: Xin Yan
Hui Fang
Qiang He
Venue: NeurIPS 2023 poster (2023)
Keywords: Diffusion Model
Graph Inverse Problems
Source Localization
Information Diffusion
TL;DR: This paper addresses the inverse problem of information diffusion over complex networks by proposing a diffusion model that enables us to reason about diffusion sources and paths given the observed diffusion graph.

Abstract: Information diffusion problems, such as the spread of epidemics or rumors, are widespread in society. The inverse problems of graph diffusion, which involve locating the sources and identifying the paths of diffusion based on currently observed diffusion graphs, are crucial to controlling the spread of information. The problem of localizing the source of diffusion is highly ill-posed, presenting a major obstacle in accurately assessing the uncertainty involved. Besides, while comprehending how information diffuses through a graph is crucial, there is a scarcity of research on reconstructing the paths of information propagation. To tackle these challenges, we propose a probabilistic model called DDMSL (Discrete Diffusion Model for Source Localization). Our approach is based on the natural diffusion process of information propagation over complex networks, which can be formulated using a message-passing function. First, we model the forward diffusion of information using Markov chains. Then, we design a reversible residual network to construct a denoising-diffusion model in discrete space for both source localization and reconstruction of information diffusion paths. We provide rigorous theoretical guarantees for DDMSL and demonstrate its effectiveness through extensive experiments on five real-world datasets.

Diffusion Model for Graph Inverse Problems: Towards Effective Source Localization on Complex Networks
Xin Yan 'Hu Fang?* Qiang Hel
CCollege of Medicine and Biological Information Engineering, Northeastern University, China 2RIIS & SIME, Shanghai University of Finance and Economics, China yanx inwuttlok....oo.oooooooooooooo.. com, fang. hui@mail shufe. edu. en, heg.hnggaii@@aiill..............
Abstract
Information diffusion problems, such the spread of epidemics rumors, are widespread in society. The inverse problems of graph diffusion, which involve locating the sources and identifying the paths of diffusion based on currently observed diffusion graphs, are crucial to controlling the spread of information. The problem of localizing the source of diffusion is highly ill-posed, presenting major obstacle in accurately assessing the uncertainty involved. Besides, while comprchending how information diffuses through a graph crucial, there is a scarcity of research tackle these challenges, we propose a probabilistic model called DDMSL (Discrete Diffusion Model for Source Localization). Our approach is based on the natural diffusion process information propagation over complex networks, which can be formulated using message-passingg function. First, we model the forward diffusion information using Markov chains. Then, we design a reversible residual network to construct denoising-diffusion model in discrete space for both source localization and reconstruction of information diffusion paths. We provide rigorous theoretical guarantees for DDMSL and demonstrate its effectiveness through extensive experiments on five real-world datasets.
reconstructing the paths of information propagation. To
Introduction
Information diffusion is a pervasive phenomenon in our daily lives. Data scientists have conducted ex- tensive research on information diffusion along the direction of entropy increase, such maximizing the influence of nodes social networks [15] and developing control policies to curb the scale of epidemics in disease transmission networks [53] However, merely comprehending the mechanism of forward diffusion is insufficient. When destructive information spreads across a network, can cause huge damage the entire system. For example, the rapid spread of rumors in social networks cause harm to society (54), and the spread of computer viruses on the Internet can paralyze a system consisting of millions of users [I]. Additionally, pandemics such as SARS and COVID-19 human interaction networks pose serious challenges to human health and social functioning (31). Therefore, accurately identifying the sources of transmission and cutting off their possible transmission paths in time is crucial. This can help limit the spread of negative information and maintain the stability of the network.
Common information dissemination models like SIR (Susceptible-Infeed -nteeeeeeeeedd-ccoveeedd)) and SI Seacepttile-Infeeteee [19] subject uncertainties during the process of information diffu- sion. As illustrated Figure T the diffusion graph of information should follow a non-explicit distribution, implying that a single diffusion source may correspond to multiple different diffusion
"Equal contribution. CCoresponding author.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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graphs and a diffusion graph may have multiple different diffusion sources. Therefore, the inverse problem of information diffusion is underdetermined.
Susceptibles
Intextivee
Rezoward
P
Q
Q
DK 1K..11) [4)
p (g)
plx, (2g)
1-T
pb (2)
p(, (x)
Figure 1: The SIR diffusion process of information over complex networks. Distinct sources may produce identical infection graphs (IT), and initiating forward diffusion from 2T may lead to different diffusion outcomes. The distribution of sources and forward diffusion can be represented by probability models p-1 and P. respectively.
Previous source localization research has primarily relied on manually formulating rules to filter source nodes, such as using Jordan centrality (24 [6] locate multiple source nodes SI model. or unbiased intermediary centrality to identify source nodes in SIR model [4]]. Besides, IVGD 4 exploited fixed point theorem for source localization of IC model [1]. However, these algorithms impose relatively striet requirements on information diffusion patterns, which could hardly be satisfied in real-world applications. To address this issue, some algorithms [38] 36 have been proposed using graph neural networks (GNN) [35] to learn source nodes under different diffusion models. Unfortunately, most deep learning-based algorithms ignore the underdeterminacy mation propagation and attempt to establish direct mapping between observed data and source nodes. As a result, inference results are often deterministic and fail to quantify the uncertainty of source localization. In addition to source localization, it is equally important to recover the possible propagation paths of information. For example, when an infectious disease breaks out, is necessary to promptly trace the virus transmission trajectory and corresponding close contacts. However, in reality, this can only be achieved by analyzing the genetic evolution tree of the virus strain [4], which is both time-consuming and
infor-
In summary, there is a current dearth of methods that can simultaneously locate information sources and restore information propagation paths.
Inspired by diffusion phenomena, the diffusion denoising model [16] has garnered significant achieve- ments in the realm of image generation. It effectively restores the original image within noise distribution through gradual refinement. This process bears resemblance to the challenges encountered in information reconstruction and diffusion, demanding our attention. Throughout the propagation process, the source information continually undergoes blurring, presenting an exceedingly arduous task of restoring the original source node. In this scenario, employing diffusion model for the reverse recovery of the entire information dissemination process proves highly appropriate. However, using diffusion models to solve inverse problems in graphs presents several challenges: (1) Infor- mation diffusion occurs on non-Euclidean space graphs, making learning discrete non-Euclidess data difficult; (2) Diffusion over graphs is governed by information propagation rules that operate in non-Euclidee spaces, making it challenging to establish both the forward process and reverse inference process of information diffusion within the network; and (3) Most existing algorithms only work for specific propagation patterns, suffering from the problem of model generalization.
To address the aforementioned challenges, we propose new framework, DDMSI DaessSSDDE Discrete Diffusion Model for Source Localization). DDMSL excels at source localization and reconstructing the evolu- tion of diffusion, showing promising results across different propagation patterns. Our contributions can be summarized as follows:
"The code for DDMSL is available at httttp:s http://hhhhcc.cc///e///eeeee////

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We model the information diffusion process using Markov chains and demonstrate that at each moment, the state transition matrix converges, enabling us reconstruct the information diffusion paths from any time point.
We design residual network block based on graph convolutional networks to approximate the information diffusion process and provide a theoretical proof for the model's reversibility.
We propose an end-to-end framework being capable of both reconstructing the evolution of information diffusion and source localization. To the best of our knowledge, this is the first study employ denoising diffusion models for solving graph inverse problems. Extensive experiments on five real-world datasets demonstrate its effectiveness.
2 Related work
Diffusion models for inverse problems. The diffusion denoising model [16] 29 is currently one of the best available deep generative models, which has been extensively applied in image-related inverse problems 33
Previous studies have extended DP3Ms [2] to discrete spaces and used this work as foundation for developing multiple generative diffusion models for graph data [14 42). Additionally, Stevens et al. [39 and Chung et al. [7 demonstrated that the diffusion model can solve nonlinear inverse problems. These works collectively demonstrate the effectiveness of diffusion models across different domains in addressing inverse problems.
23 34
Localization of information sources the graph. To facilitate the localization of infection sources, various algorithms have been proposed [25] [52 44 9 Early algorithms focused on screening source
nodes by employing feature engineering. For instance, Luo et al. (25) identified the diffusion source by the centrality of the diffusion subgraphs, while Prakash et al. (30] and Zhu et al. [52) respectively developed NETSLEUTH and OJC algorithms based on the minimum description length of nodes and the shortest path. Wang et al. [44 designed series of label propagation algorithms based on the aggregation process of information diffusion, on which Dong et al. [9] further improved using GCN [20]. Recently the proposed reversible perception algorithm IVGD based on graph diffusion 43 has been applied to the IC model. However, these algorithms generally have strict limitations on diffusion patterns or network structures. DDMIX X was the first algorithm to use generative model for reconstructing the dissemination paths of information. leverages VAE (variational autoencoder) to learn the state of nodes across all time steps; however, as the number of time steps increases, the solution space of the dissemination path becomes significantly more complex, leading to low accuracy in source localization. Ling et al. later introduced SLVAE [22], which is also based on VAE and can efficiently localize the source node but cannot directly reconstruct the propagation paths. Current generative model-basee approaches for source localization cannot simultaneously handle the problems of source localization and reconstructing diffusion paths.
3 Methodology
3.1 Problem definition
We consider an undirected graph G = (V, E) where V == (=² ,.... "=............. is the node set and E sm is the edge set. Let S = {25 x x } denote the set of initial diffusion source nodes at t = 0. The source nodes undergo diffusion on the graph for I' time steps, governed by the diffusion pattern g()), and the set of node states at time step / denoted by X, = (*/,... a with xt E (0, 1) (c.g.. if g()) the SIR model, then M E {SS, 1, RJ. where S, 1, denote susceptible, infected and recovered state, respectively). If the node E S, then z 1. otherwise z = O. We define the research problem as finding the intermediate state X (Xo. X,....,Xy that maximizes the likelihood function X argmarx P ((Xx(XX,)
given the observed XT.
3.2 Information diffusion in discrete spaces
Busceptible-Infeetee Infected-Recoveeede (SIR) and Susceptible-Infeeted (SI) models 19 are commonly used to model diffusion phenomena in nature, such as the spread of epidemics [51 and rumors [50]. In this paper, we will demonstrate our approach using the SIR model. The SIR model categorizes node states into susceptible (S), infected (I), and recovered (R) states. The S state represents individuals who are
3

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susceptible to infection, while the I state represents those who have been infected, and the R state indicates recovery from the infected state. The transition between these three states is inreversible. We assume that all nodes on the graph are homogeneous and follow the same transition process for the different states of the SIR mode
( = - (1-8(0)))())) (--
(I)
P ((r R æ 1) - 7t(
And, the SIR diffusion process can represented using a state transfer matrix:
1
()
(7(()
0 1 1- (t)) (t(())
(2)
0
where Qi denotes the state transfer matrix of node at moment t, and [Qi] uv denotes the probability of transferring from state u to state [ (t) and (gg(t) denote the infection rate and recovery rate at moment I, respectively. Let (t). (t). (t)) to be the probabilities of node i being in three
states 5, I, and R at time t, then (t) and (t) can be calculated by the following equation:
(1(1) Pf + 1) -f(()(1-ff(())
Pg 1)
(t)
Ps(t)
7k(1) Ph(t + 1) (t)
Pi())
Theorem 3.1 Given graph G and the infected seed set S, it can be determined that the state transfer matrix Q for the STR diffusion process on G converges at all times.
Sketch of Proof. Given G and S the initial state of information diffusion can be determined. Based on the SIR propagation rule, an iterative equation can be constructed for the state distribution of the nodes at any given time. Finally, Q can calculated according to Equations 2 andd
Referring to Equations and we can readily demonstrate the convergence of the state transition matrix Qj every moment, rendering the discrete diffusion model practicable solution 2 The proof is shown in Theoren 3.1
3.3 Discrete diffusion model for source localization
3.3.1 Forward process
We represent the state of node i by one-hot vector æii E IR 1x1MM and the state distribution of node i at time is written as:
(x) a{{1) 2/a"" Cat ( -00==11-20 xi-1
Information diffusion is Markov process that allows for inference of the node state at any given moment based the initial state:
q [
21) =
Σ
II
(xt xq_1) = Qix Cat (æ):p- Q:)
(5)
A=1
3.3.2 Reverse process
Reconstructing the information diffusion process requires backward inference of the forward process, which can be achieved through Bayesian formulation
(4Q2"") 2 (xt(Qt11)
( xi mn)) ~ Cal x xi-1iP-
(6)
x Qi21" T
A denotes the graph's adjacency matrix, and (t) represents whether neighbor is in infected state at 'In SIR model, u, u (1,2, 3), representing three states: S, R, respectively.
5 Pllease refer to Appendix [ for the formula derivation of forward and backward processes.

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In the absence of knowledge about zi the posterior distribution q (xi-1 z 25) becomes in- tractable compute directly. As result, we must approximate this distribution using other methods. Continuing with Bayes' theorem, we can express the posterior distribution as follows:
Enj 9 1, x/)
q (w) mi) q (at-1 m!)
E. (agllet) (ari-1 a0j an)
(7)
q (x)))
We utilize neural network model (nm)) learn about pe (xt xx) and estimate (at xx)- The detailed design of mas will be elaborated later in Section 3.4
(2)-11 az) po (w) ax))
E a aai-no (ab (ablec) (a:-11 x m6)
q (x) x
ES a
(o) 1 2 (1)
()) 2 l2f)
(8)
pe
q [x) xx))
Similar to [2]. we predict zt and then use this prediction to learn the distribution of q (æ) æ)). This approach offers several benefits. First, it leverages prior knowledge by exploiting the fact that at t there are only two states: susceptible and infected. This makes mng easier to train. Second, locating =o is key target of our task. Finally, predicting ag can help constrain errors the information reconstruction process (Refer to Appendix(C)
Lib -B (at))
DKL. la (2r at) 1p (arr) +
LT
E (al DKkL t=2
æ xt) [a (wt
llpo (x)_1 xx)]]
(milaa)))
E8
[log pe (ag xi)]
Le
Lo
Equation presents the variational lower bound loss function of the denoising diffusion model where D DL represents relative entropy. Furthermore, as highlighted in our earlier discussion, it is crucial for us to supervise zá directly. As result, we arrive the simplified variational lower bound loss simple)) which is denoted as:
[6
lssimpte
1 N
(Tib 18 (o(t)a)) [-
log Po (aré a)
(10)
3.3.3 Supervision of information propagation rules
Our reconstructed information diffusion must adhere to the propagation rules of model g(-). During early training, the predictions of mn for x/_1 may not be accurate enough, which could lead to violations of the propagation rules. For example, using the SIR model, if the state of node at time t classified as I, then the state of é time 1 can only be either or I. However, the prediction result of nng may erroneously output state R, resulting in the failure calculate g (xf_1 ot, =t).
To prevent violations of the propagation rules, we can take two measures: (1) cases where the inference result of mng violates the propagation rule, g( (it_//lt... .x)) should be set to 01,0, and |
(2) To supervise the training of nng, the propagation rule loss function Leonstrain = Leonstraini + Leonstroin2 is introduced. The detailed derivations of (1) and (2) are shown in Appendix C
Liconstrain1 Relu (X,-1 (A + 1)x,)
Leomstrain2 max 0,
x 1- _
2
VX 2 X
(1I)
In order to evaluate Equation we must first sample X,-1 from the discrete probability distribution
computed in Equation 8 To ensure gradient preservation during the sampling process, we employ the G'ubbllSoff Soft--r technique [77] X,-1 ~ Cuumbel-Softrin (q(x; :. =))). Equation 11
demonstrates that each node's contribution to the information diffusion process is non-negative. Tn
summary, the constrained loss function is Leanstrain= Irconstrainl Leemstrain2 and the loss function of DDMSI. is:
Loss Loimple + Leonstrain
(12)
5

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3.4 Design of the inference model
In this section, we discuss the design of the mng in detail. Let
X
Xo = ( a ... 20 and YT [ah. 2 ................................ ,,.,,,,,yy The diffusion of information
FC
[0%
Residual Block
Residual Block
F
complex net- works can be modeled using the following processes [6 43):
Residual Block
Residual'Block x
Sigmoid
f", F
Dense
Xo
(13)
Within this framework, Sw and serve as the feature vector construc- tion and feature propagation func- tions, respectively. The relationship between YT and Xo be expressed as: YT P(Xo) (Xo)) Our research objective focuses on the inverse problem of the SIR or SI diffusion model, which entails large number of stochastic processes. Thus, when F = RD, solving P-1 can be extremely challenging. In the following subsections, we will explore how to construct suitable functions fw and F facilitate the calculation of P-1.
Figure 2: Multi-layer reversible residual network.
3.4.1 Relationship between GCN and SIR diffusion models
According to [37 diffusion models such SIR are analogous to message passing neural networks (MPNNs [1), where each node's state is only updated based on states its neighboring nodes. However, the operational architecture must be designed to enable each node to aggregate both its own features and those of its neighbors, and update its state via nonlinear activation as shown in Equation 26 Therefore,
propose using MPNNs to remodel the SIR diffusion process. Denoting (t) P(æ Ω2) as the probability that node is in state E {S, RR at time in the SIR diffusion process. It can be observed that this process is structurally equivalent to:
Ejenty M. (M(P)()) U. (M(P)((t) (( ((P(()) = =o (WnPh()) bn)
+1)
Ph
(14)
where Mr is the aggregate function and U. represents the node status update function. (()) is nonlinear activation function. Eig represents the edge between nodes i and j. Additionally, since 10 (4)) already contains nonlinear transformations, more complex forms of transformation are not necessary and U, be defined Ue(a, b) b. To up, Equation 14 can be simplified as:
Pt + = +
Σ M(P)())
(15)
(EN())
We can achieve this using residual block composed of graph convolutional networks, which allows us to easily fit the SIR model:
h SN(U+;) with U E RCXM
g(h) = (D A22 1/2 h + bb)
(16)
(I+1) (1 + [SN BN
h
Among them, U is a linear transformation, h denotes the representation of the (-th layer of the network, D is the degree matrix, SN(() and BN(() represent spectral normalization and batch normalization, respectively. af.) and a.(.) are nonlinear activation function In fact, Sw and F in Equation 13 are the SN (U(.) and residual blocks in Equation 16
We set o() and (.(+) )) to Mish and LeakyReLa respectively.
' { (o(), where fo(E) o (SN (BN (E))).

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3.4.2 Reversibility of residual blocks
We now discuss the invertibility of Equation 13 as reconstructed from Equation 16 Equation 13 can be written as:
{
SN (U(Xo)) = for (Xo) = É (SN (BN (o(E))) = + fo(E) = =F(E) Yrr-
(17)
Lemma 3.1 Denoting the Lipschitz constants of fro and fo to be Li and Lo, then Lo < and To
Sketch of Proof. Since fu and BN (g(E)) are spectrally normalized, their Lipschitz coefficients are less than 1 [28]. Additionally, since the a--) is Mish(-), it can be determined that the Lipschitz constant of their composite function (i.e., fo) is less than 1.
Theorem 3.2 If Lu < and La < 1, then YT = (Xfw((o)) is reversible.
{
X = _ xo- (Xo
Sketch of Proof. YT = (Xo)) can be written as
Since
f= Y Yr(5
fw 1 and fo < 1. constructing the iteration according to the Banach fixed point theorem YT F Ff (Xo)) is invertible.
In practical applications, we typically use multiple residual blocks to form residual network. This approach is employed to improve the receptive field of the GCN and alleviate the problem of node smoothing caused by multi-layer graph convolution. The multi-layer reversible residual network that we designed depicted Figure [ where DP represents dropout. If the dropout rate r and the i-th layer residual block is F,, then we have Fi(E) = (P (E))
Theorem 3.3 If dropout-rate = 0.5, La < 1, and Lo < 1, the residual network Yr
(FioF2o... oF (f (Xo)) is reversible.
Sketch of Proof. Denoting this multilayer residual network as F, the upper bound of Lp is the product of the Lipschitz constants of each function [13. Additionally, with dropout rate of r, the Lipschitz constants of the functions will be limited to (1 r) times their original values. Using these two conclusions, we can calculate the upper bound of the Lipschitz constant of a multilayer reversible network with LF < 1 when = 0.5.
4 Experiments
4.1 Datasets and evaluation metrics
The diffusion of information occurs in a broad range of network types, and DDMSL was evaluated on five distinct realistic datasets: Karate 48 Jazz [12. Cora-ML (27), Power Grid 45). and PGP [5). The detailed parameters of these networks are provided in the Appendix F.2 Following previous research [33 9 222, we conducted SIR and SI diffusion simulations on each
Datasets.
dataset by randomly selecting 10% of the nodes as source nodes at the initial moment, and stopping the simulation when approximately 50% of the nodes were infected (For (For PGP dataset, simulation stopped at 30% infection rate). We randomly divided each generated dataset into a training set, validation set and test set in the ratio of 8 1.
Evaluation Metrics. Our objective consists of two components: source localization and information diffusion paths recovery. The source localization task is binary classification task, and thus evaluated using four metrics: Precision (PR), which denotes the proportion of nodes predicted as sources that are true sources; Recall (RE), which represents the proportion of actual source nodes that are correctly predicted; F1 score, the harmonic mean of PR and RE; and ROC-AUC (AUC), quantifying the model's ability to classify accurately. To evaluate the performance of the recovering information diffusion
paths, we adopted the Mean Squared Error (MSE) eиrоr in g MSE = NT ΣT-1 X, - X
where X₂ is the ground truth. This metric solely computed for infected nodes, with nodes in the recovered and susceptible states being marked as O.
'While some previous studies 9 commenced source localization when the infection rate reached 30%, set it to 50% to consider more complex scenarios.
7

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4.2 Baseline algorithms and experimental settings
Six algorithms for source location were selected to compare with DDMSL: DDMIX
8
employs a
Variational Autocncoder (VAE) to reconstruct the outbreak propagation process; LPSI information to learn source detection through label propagation; GCNSI Ω uses graph convolu- tional network to learn manually formulated node features, enabling the discrimination of multiple source nodes; OJC [52] locates the source node by minimising the eccentricity of the infection; NETSLEUTH (30 filters multi-source nodes based on minimum description distance, but works solely on the SI diffusion model: and SLVAE [22] deploys generative models to learn the distribution of source nodes. Appendix [T.] provides the experimental settings and tuned hyperparameters for
aggregates
2014
these algorithms, while implementation details on DDMSL are shown in Appendix
4.3 Experimental results
while comparing DDMSL with other algorithms on different diffusion models, we have focused on the widely used SIR and SI models, which are effective in capturing the dynamics of various information diffusion processes found in nature. It is noteworthy that DDMSL's flexibility and robustness enable it to handle real-world scenarios where the diffusion process may undergo significant variations by simply designing the corresponding state transfer matrice 9
Performance comparison (mean of five rounds) of source localization under the SIR model. The best results are highlighted in bold, while the underlined indicates the best of the five baselines. Statistical significance from paired t-test depicts: ***, **, for p-value < 0.01, 0.05, 0. respectively.
Table I:
Karate
Jazz
Cora MI
Power Grid
PGP
Methods
F1
AUC
PR
F
AUC
DDMSL
0.708
0.736
0.722
0.853
0.817
0.881
0.848
0.930
0.894
0.867
0.880
0.928
0.833
0.879 0.855 0.930 0.856
0.903
0.879
0.943
GCNSI
0.275
0.410
0.329
0.671
0.301
0.363
0.330
0.641
0.247
0.273
0.260
0.591
0.165
0,182
0.173
0.54
0.543
0.549
0.748
0:211
0.393
0.274
0.646
0.400
0.098
0.026
0.048
0.509
0.193
0.012
0.022
0.503
0.518
0.437
0.696
SUVAR
0.582
0.400
0.464
0.696
0.750
0.576
0.651
0.77%
0.815
0.721
0.765
0.852
0.908
0.719
0 0.803
0.8030.8560
0.17
0.721
0.766
0.851
OIC
0.178
0.265
0,213
0.594
0.147
0.161
0.154
0.535
0.114
0.114
0.114
0.508
0.109
0,109
0.109
0,505
0.128
0.128
0.128
0.516
DDMIX
0.289
0.258
0.215
0.238
0.273
0.204
0.250
0.287
0.346
0.182
Improve.
28.
55.750
22.5%
8.97
52.9%5
19.6%0
9..7%
20.25%0
15.0%
8.9%
-8.3%
22.2%
6.6%
5.796
4..9%
25.3%
14.3%%
10.5%
St Significunce
Table 2: Performance comparison (mean of five rounds) of source localization under the SI model. The best results are highlighted in bold, while the underlined indicates the best of the six baselines. Statistical significance from paired t-test depicts: ***, **, for p- v-lluee < 0.01, 0.05, 0. respectively.
Karate
Jazz
Cora M
Power Grid
PGP
Methods
PR
RE
AUC
PR
RE
ACC
PR
RE
F1
AUC
PR
FI AUC
PR
RE
F1 AUC
DDMSL
6.706
0.980
0.798
0.972
0.782
0.853
0.513
0.914
0.750
0.368
0.345
00.941
0.763
0.960
01852 0.966
0.754
0.887
0.815
0.925
GCNSI
0.357
0.156
0.401
0.687
0.366
0.426
0.394
0.676
0.321
0.354
0.337
0.6370.636
0.636
0.335
0.315
0.330
0.487
0.370
0.421
0.665
LPSI
0.114
0.494
0.291
0.592
0.343
0.306
0.453
0.284
0.349
0.623
SLVAE
0.591
0.477
0.503
0.733
0.888
0.579
0.691
0.785
0.841
0.728
0.856
0.815
0.780
0.800
0.844
0.633
0.723
0.810
OJC
0.267
0.395
0.318
0.663
0.120
0.127
0.123
0.517
0.125
0.125
0.525
0.178
0.178
0.118.0.511
0.541
0.118
0.118
0.118
0.510
DDMIX
0.253
0.377
0.363
0.654
0.244
0.133
0.172
0.212
0.220
0.222
0.221
0.247
0.345
0.235
0.3440
0.189
0.186
0.187
0.207
NeiSleuth
233
0.217
0.229
0.211
0.562
0,200
0.210
0.558
Improve.
19.4%
105.35
58,
555
32.55
11.9%
47.9%
17.6%
16.
16.35
6.00
24.8%
N.3%
9.9%
-6.3%
23.8%
7.0%
9.88%
10.05%
44..5%
12.7%
14.5%
Signlicano
Source Location. Table I shows the performance of DDMSL in SIR diffusion mode, where it outperforms all baseline algorithms regarding almost all metries. Notably, DDMSL leads the best of the baseline algorithms by approximately 25% and 14% on Fl score and AUC, respectively. Meanwhile, Table 2 shows similar results in the SI diffusion mode. While SLVAE has higher precision than DDMSL, DDMSIL surpasses SLVAE by approximately 21 % and 17% in terms of F1 score and AUC, indicating that DDMSI. more accurate identifying source nodes. Moreover, the performance of DDMSL varies across datasets and is mainly influenced by the network's 's typologies (Refer to Table and the scale of information diffusion. The performance of DDMSL in solving the inverse problem generally has negative correlation with the average clustering coefficient and density of the network. The larger these two parameters are, the more complex the scenario of information diffusion is, and the accuracy of DDMSL in detecting source points decreases, which aligns with our intuition.
Pleaase refer to Appendix E for measuring the state transition matrices for different propagation models.
8

---

0253
PGP
PGP
Q12
0112
11151
0.201
PowerGrid
PowerGad
0.123
0.116
0.137
0.177
Cora
Cora ml
0:111
0.115
0.175
0..12
Jazz
Jaz
a.116
K1116
0.111
DDMIX DDMSL
10009
DDMIX
Karaee
Karate
0.057
J.074
DDMSL
0.00
0.06
0.10
Q115
0.20 MSE
0.25
0.30
0.35
0.00
0.05
0.10
0.15
0.20
0.25
0.50
0.35
(a) MSE reconstruction error SIR mode.
(b) MSE reconstruction error in SI mode.
Figure Comparisons of DDMSL and DDMIX on reconstructed information diffusion processes.
Reconstructing information diffusion paths. Both DDMSL and DDMIX can reconstruct the evolution of information diffusion, but DDMIX can only recover the states of susceptible nodes (S) and infected nodes (7), whereas DDMSL. can reconstruct nodes of all states. To facilitate fair comparison, we calculate MSE loss for node reconstruction states S and as shown in Figure 3 DDMSL consistently outperforms DDMIX on all datasets, especially in SI diffusion mode where DDMSL's average reconstruction loss is 69% lower than DDMIX's. Unlike DDMIX, which recovers the entire graph state, DDMSL allows for fine-grained node-level reconstruction.
Additional experiments. DDMSL represents source detection algorithm relying on diffusion models, necessitating an assessment of its aptitude for generalization. Concurrently, we assessed its computational complexity while also incorporating experiments utilizing two real diffusion datasets. We have documented the results of these experiments in Appendix G
Visualization. To further showcase the efficacy of DDMSL in solving the inverse problem of information diffusion on complex networks, we designed a visualization that demonstrates source localization and the propagation evolution process. Due space constraints, we have presented the visualization in Appendix H
4.4 Ablation study
Table 3: The results of the ablation study.
Jazz
Power
PGP
Methods
PR
RE
F1
AUC
PR
RE
AUC
RE
F1
AUC
PR
RE
F1
AUC
PR
RE
F1
AUC
DDMSL
0.708
0.736
0.722
0.853
0.81
0.881
0.848
0.330
0.894
0.867
0.850
0.92%
0.833
0.879
0.855
0.930
0.356
0.903
0.879
0.943
(DMSL
0.264
0.910
0.379
0.827
0.254
0.683
0.367
0.741
0.277
0.626
0.384
0.722
0.459
0.605
0.522
0.706
0.366
0.857
0.51
0.845
DDMSLbb
0.655
0.850
0.718
0.860
0.648
0.822
0.723
0.889
0.834
0.856
0.845
0.915
0.818
0.875
0.845
0.919
0.811
0.888
0.848
0.922
DDMSL
0.359
0.185
0.236
0.592
0.931
0.111
0.1943
0.556
0.817
0.043
0.08
0.521
0.998
0.2756
0.432
0.638
0.997
0.656
0. 0.792
0.823
To evaluate the effectiveness of each component, we conducted ablation experiments on DDMSL in SIR diffusion mode. One essential compo- nent of DDMSL is the reversible residual net- work. We removed this component and replaced each reversible residual block with a GCN net- work, denoted DDMSL(a). Additionally, DDMSL is also supervised by Leonstarin on the propagation rules. The model without the propagation rule supervision module is denoted as DDMSL(b). Finally, to assess the impact of GCN (Graph Convolutional Network) in re- versible residual blocks, DDMSL(C) represents the model after removing the GCN modules from residual blocks. We ran the three vari- ants on five datasets and compared the results,
0.35
DDMSI
DDMSL(a)
0.30
DDMSL()
DDMSL(e)
0.
0.
0.00
0.00
Jazz
Figure 4: MSE emor in ablations.

---

which are shown in Table and Figure 4 As can be seen in table 3 we have these observations: (1) removing the residual module (aDMSL(()) leads significant degradation in model performance. with the Fl score and AUC decreasing by 49% and 17%, respectively. Although the GCN network is also a form of MPNNs, we noticed that the output of the inference network composed of GCNs were very similar, indicating node feature oversmoothing, and further highlighting the effectiveness of the reversible residual network module; (2) Constraint is also effective, which contributes to 5% and 2% performance improvement in terms of Fl score and AUC, respectively. Similar results can be obtained in Figure + regarding the task of reconstructing the information diffusion processes; and (3) upon the drop of the GCNs from the reversible residual network (DDMSL(c)), noteworthy deterioration model performance was observed. manifesting as a substantial decrease in F1 scores and AUC by 59% and 32% respectively. This compellingly signifies the indispensable role of GCN in acquiring the diffusion mode of the SIR model through the process of learning. Similar results can be obtained in Figure 4 regarding the task of reconstructing the information diffusion processes.
5 Conclusions
In this paper, we introduced reversible residual network block based on the relationship between diffusion phenomena and message passing neural networks, while ensuring the reversibility of the network by limiting its Lipshitz coefficient. Using this, we constructed a discrete denoising diffusion model (DDMSL) which can locate the source of graph diffusion and restore the diffusion paths. Extensive experiments on five real datasets have demonstrated the effectiveness of DDMSL and its constituent modules. Our work offers insights into how calculate the distribution of solutions to graph diffusion inverse problems based on the information propagation laws on complex networks.
Solving the inverse problem of graph diffusion plays crucial role many social operations, including controlling the spread of infectious diseases, rumors, and computer viruses. It provides valuable insights on enhancing source detection performance and fills the gap in methods for recovering diffusion evolution. However, our work has some limitations, For instance, DDMSL requires the prior knowledge about propagation models, such as infection rate and recovery rate. Although we can infer these parameters from existing observation data [3 21 it limits the application of DDMSL in situations with insufficient observations of propagation conditions. Our future research will focus on reducing the dependence of DDMSL on prior conditions.
6
Acknowledgement
This work was supported in part by the National Key R&D Program of China under Grant No. 2021 the National Natural Science Foundation of China (Grant No. 62202089, 72192832 and U22A2004), the General project of Liaoning Provincial Department of Education (Grant No. LJKZ0005), the Shanghai Rising-Star Program (Grant No. E3QA1403100 the Natural Science Foundation of Shanghai (Grant No. 217R 1421900), and the Fundamental Research Funds for the Central Universities (Grant No. N2319004),
10

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A
Broader impacts
Overall, our work offers valuable insights into how limit the spread of malicious information. For example, by tracing the spread of infectious diseases, we can identify potential contacts of infected individuals, effectively containing the outbreak. However, it is important to consider the potential negative implications of this approach for society, such as compromising the privacy of individuals living with infectious diseases like HIV. Addressing these concerns should be important.
B Derivation of forward and backward processes
Derivation of forward processes. Extending Equation 4 by means of the Markov property:
a (a) 2
E
II
g (a
at-1)
11:-11 k=1
Σ
o Qia T
Σ
2 0iat ... ...,/-
(18)
Σ x " aai1 Qiot"
Σ
x ef
IQ' ... IQ I IQ T
a/ 2 12' ~ Cat (at:p Qi)
Where I is the identity matrix.
Derivation of backward processes. The detailed derivation Equation 6 is as follows:
(x) x;/ xτ)) q (2}_1 xx)
q (x)__ 1/. x/))
[x) x6)
y (#t ax/_)) 1) (w)_1 xx))
(ar) ax)))
æ/ 2// T 20 12 T
mn T m[_1") 7)
(n1 1 T
(19)
x 2iot"
(
+ (xf (in))
(2i_1 T
2 Qixt""
Cat xi-1ip= x2/22"
(z))Qi
Qiat""
15

---

From the Bayesian formula, it follows that:
Enj 2 (zt-1) 26 20)
q (2)-1 oi z/) = (a/-1 1 x²)
2 (2+))
Ems 1 (1ot-1
oot, mn)) a (2ng an() a (at))
(a:t)
(20)
Σ (ai-1
c m²)) (ar xi))
E (abbie)) (a a xn)
Fitting this distribution using neural network:
q (oi_1 x!) ~ po (x/_1 | æ
Egg upo (eggle))
( (wi-1 æi, 2n))
(x) x)-1, (x) x%))
= E (at
q {x) x3))
a (ar) a (ar mg))
po(glln))
9 (a) mx)
(21)
Eg (q (x)
x
و 1
2
(i)
po
(1)
xx
q [x]
x/)
9 (af x
(5)
(1)
00
q
[x:
xx))
C Propagation rule constraint of information diffusion reconstruction
To investigate the circumstances under which the propagation rules may be violated, let's revisit Equation 6 Note that the denominator serves as the normalization term, while the numerator is composed of two key terms rule.
(x(Qi") and (i)) that are crucial in preserving the propagation
The three rows of Qi correspond to the distribution of q<))|=]_ 2/)) when æi is in one of three states: S, I, and R. Similarly, the three rows of Qi represent the distribution of q(=i-1l=t) when zj is in one of three states: S, 7, and R. Let @ and @ be denoted by of and de respectively,
then of
7,, dी 0.
I1
(1 - -B)
I1 3 + 1 - x8) I (1 7
Li p + di Σ IE (1 7h Thrt
Q
0
(22)
0
x/)) = (at(("") c Qin)) are When 2; and sn are in different states, the results of |mt,
shown in Table 4 The table reveals that ((af_ılzf, = = [0, 0, when the propagation rule is violated. Since we are predicting z/g there are only two possible states for z/: S (Susceptible) and I (Infected), with 2/ R (Recovered) being excluded. In such instances. q( |=t =n) can be set to [1, 0, 0] to resolve the issue. Furthermore, =t R and zj = S, then s/- = would violate the propagation rules. However, as shown in the 9th row of Table 4 the probability of æt_1 = much smaller than that of - and -i-1 = R, and such situations will not lead to training failure. Hence, there is no need for any specific handling of this scenario.
To further minimize propagation rule violations during the training process, we incorporate supervi- sion of the propagation rule. Specifically, when using this supervision function, nodes that have state of R are set to / to enforce the propagation rule.
Leonstrain1 Relu (X,-1- (A + 1)Xo)
(23)
16

---

Table The distribution of the unnormalized q xb) in different cases.
g
Tiy mb
S
R
3k)
S
0
0
qi
[
98
0
7k
0
0
0 -
S
d:
(1
(()
9i
R
0
j=
7h 7644
(1
%
0
where (A + 1) represents the total number of infected nodes in a given node's first-order neigh- borhood. Equation 23 penalizes Xo if the node infected and there are no other infected nodes within its first-order neighborhood. To maintain the stability of inferred Xo generating X we apply the same constraint the process. Specifically, we utilize the monotonicity regularization of information diffusion from [22]. If the source set X is superset of x!" then the generated Xt-1 resulting from their diffusion needs to satisfy the following equation.
max (0.
x(2i|l xx 2 x
Leonstrain2
(24)
D Proofs of lemmas and theorems
D.1 The proof of theorem 3.1
Proof Set the initial infection seed as: S {2r =000= ,....10 At the initial moment, the infection status distribution of node is:
Pi=Sm. (0) Pi= (0)
P pips (0) 1, P S (0)
(25)
(0)
At time t, the infection status distribution of node is:
1- 1- (t()) (t - (
(f( = Pj(t - 1)(1 - -| -
(26)
(t( = Pj(t-
where is the adjacency matrix and j is the neighbor node i. When dealing with a static graph G. A is fixed, allowing for determination of the state distribution any time based on the initial node state. Specifically, if both graph G and the seed node set S are known, it becomes possible to calculate the state distribution of each node any given time using Equation 26 Utilizing Equations
2 andd 3 Q can also be determined.
D.2 Proof of lemma 3.1
Proof The graph convolution layer with batch normalization BN (g(E) can be abbreviated as GCow.. In our approach, apply spectral normalization to both the linear transformation U and convolutional layers GConv. a result, the weight parameters of both the networks J and oCow possess 1 1-Lipschitz continuity after spectral normalization, described in (28).
17

---

Let GConv R" + R", 21,22 R Note that the nonlinear activation functtioo o(-) is set to Mish(?), which obviousty possesses L-pischhitz continuiry.
[ - fo( (|o(GConu(z)) (((Coo/))))
$ [GConv(a)) GCCone(za))
(27)
|| - x2l1p
where Ill1pp represents the p-norm (p = or p oo). Therefore, the Lipschitz constant of fo is less
than 1.
D.3 Proof of theorem 3.2
Proof To prove that Yr F(fw((oo) is reversible, it is necessary ensure that F and f reversible (43]
{ = tu = + 1 {= YT fo(() + fo(E) = YT
{ X = Xo - sw (Xo)
=
(28)
We construct the following f
{
x - X3 fw (Xb)). x = gh+l =YT f ) =Yr
limg-roo X = Xo +1 limms 5h =
(29)
By Lemma 2 we can ensure that Tow < 1 and Lo < 1. Moreover, the Lipschitz constant of (X)
Inw which is less than 1. Thus, when the number of iterations h is sufficiently large, it follows that the transformation YT F (Xo)) is reversible according to the Banach fixed point theorem 4
(x
D.4 Proof of theorem 3.3
Proof Specifically, Theorem 3.2 proves that (FioF2o.... o...Fo....FF....... is invertible when = 1. There- fore, we are currently examining whether (F of (E) retains its reversibility for 1. To make the notation simpler, we denote F...o....FFF...... as F,
(E)
カ- + (E) + Σ lm (Ri(t))
F, = FF (.. = DDP
o....... o...o DP
(30)
The application of the dropout function DP will limit the Lipschitz constant of any function f to 1 - times its original value: LDP(S) (1 Ly Additionally, have L II}==1 (LF,) s
- (1-r) Ly,,) [3]. Therefore, the Lipschitz constant F is expressed as:
LF, $ r1
1 Li
Σ Lsw Lp,
1=1
n-1
< (1 r)
1+ Ls
[11 (1 Lsn))*
(31)
11 [1-r)(1 4y,.)l 1- (1--)(111 + Ly.))
= (1 r)"
1 L Li
Note that when L < 1 and n 1, we only need to set r = 1/2 to ensure that LE, < 1, hereby guaranteeing that F is reversible.
E DDMSL implementation details
The DDMSL. approach has been previously explained, and we will now provide further details on the implementation DDMSL. The linear transform U refers fully connected layer, and Figure 2 the dense layer is composed of two fully connected layers that undergo spectral normalization. The
18

---

final output of the meg is represented by an N 1 matrix, indicating the probability that each node is an infected state at We designate nodes with an infection probability higher than a certain threshold as infected nodes.
Given complete information diffusion instance X (Xo. Xr) where X, { 2y }
sample E (I,..., T to be included in the neural network mng based on the probability distribution ptt) Σ t' and use the sine-cosine position encoding [41] to embed The training and inference processes shown in Algorithm| and Algorithm respectively. Additionally, the variable 7' remains consistent with the information diffusion step size.
Algorithm 1: Training
Input: Xo. G, Xt, threshold
repeat
Qt < Equation 2 and Equation 3
3 ((11-1 + Calculate Equation using X, and Q:
1 ~~ , T)), P(L)= -1-1
x mmo(9,X,,()
// Using data from mng infers the source node X--1 which is then used to reconstruct the diffusion graph at t 1.
X
+r-1ixt [X - 1= =
X [x
8 Q + Equation
2 and Equation 3
// Qi is generated by X
Po Po(x2-1/21) + Calculate Equation using Xt, xa-1 and Q
10 X,-1 f- Gumbel - So ffmaz (T)-1())))
Take gradient descent step on:
12 Vo (Issimmpe + L.comstrain)
Algorithm 2: Infering
Input: X,, 9, Empty set X, threshold a Output: X
for t starts from T to 1 do Ama(9. X,.
x
X
1 X
> a
X
-t-1 [ | =
Q
Equation and Equation
g
Po Po + Calculate Equation using X,, x and Q₂
X,-1 Gumbel - Softmar( (Po (
X(1]
X,-1
end
we have Q (Q..... Qi where Q can be calculated using Equations 2 and Additionally, the (t). (t). and (t) Equation can be computed using various methods. Monte Carlo simulations provide the most accurate results, but require at least 10 simulations to be sufficiently precise, leading to a high time complexity. An alternative approach is to utilize a neural network model [10 47 to learn P3 (1), (L),, (t). which significantly reduces the training time of the model. When applied the SI model, DDMSI. utilizes the state transfer matrix by:
In Algorithm
(t) (t)
Q
(32)
where X represents the predicted source node using X,, while Q2 is generated using the same method as above. Ultimately, Po (o(x=-1/zz) is calculated using xt-1 X,, and x We obtain X,-1 1 by sampling from Po(x4-1|22) and label nodes as 0(5), 1(T), or 2( Rt) based on their state. Algorithm proceeds in similar manner to the process described above.
19

---

Additional algorithms and dataset parameters
E1
Hyperparameters of the algorithms
The hyperparameters for each algorithm have been set according to the values shown in Table 5 Any parameter that is not stated default is common both SI and SIR models. In the updated version of SLVAE, the original three-layer MLP network encoder was replaced with a three-layer GCN network, resulting in improved performance. For hyperparameters and implementation details of other algorithms, please refer to the corresponding original papers. DDMSL and deep learning comparison
Table 5: Hyperparameter settings of different algorithms.
Algurilhmm Ha per-pammeeeee
karate
Jazz
cura ml
power gridl
FGF
Search space
Description
learning
Decline Interval
Leaming Rote
(1200,1500]
1200,)000)
1500, 1200)
[500, 12001
100,500,800,
Detemmneed by the 1,088 curves the tmining and validation sess.
decrcases by 0.97
Leurning rale
DDMSL
Number
[moii 3.max 9.step 1]
residual blocks
Drupout rate
0.5
0.5
0.5
0.5
0.5
Determined by Theorem 33
SIR model
0.5
0.4
0.45
(minn00.3 m.ooa:0... patep
Threshold
model
m 7.7.a=pp... Determined by the LOSS surves of the unining and validation ses. monn0.33mma-0 aat.a-:..aateeeiiiie
Epuch
2000
1600
1600
1600
1600
0.5
0.55
Threshold
Learning cate
10
2 x 10
10
3 10
(2 10"
10
10
SLVAE
GUN-based eneader
(61.1282566 [61.2332566641122226500 (61.126.256]
128.255) (1w.)2222556662226666666688
61.128,256)
[64.126.256]
The hadden dum encrder
MIP-hased
1256,128.1]
(256,128.1) (256.128.1)
1256,128.11
[256,128.1]
125.128.111112255..1
decoder parameters
of the decoder
Epoch
200
200
200
200
200
0.3
Theshhldd
Learning rile
10
2 10
10
x 10-3
10
12
10
10
Epoch
100
100
100
100
100
algorithms both running Windows 10 systems and trained using 4090 graphics card. The code for DDMSL is already open source, please refer to:
com Ashenone2/DDMSI.
F.2
Additional details of the datasets
The description of the data sets used for the experiments and their statistics are shown as below:
Karate [8] includes a network of interrelationships between 34 members of the Karate club, comprising 34 nodes and 78 edges. The Karate dataset is real dataset, widely employed in complex network community discovery research.
Jazz 12 The Jazz dataset a network dataset that captures the collaborative relationships between jazz musicians. It comprises 198 nodes and 2,742 edges, and has been extensively used in research on complex network community discovery, node importance metrics, and other related studies.
Cora-ML [ Cora-ML is citation network dataset containing papers from the field of machine learning. Nodes represent papers and edges represent citation relationships between papers.
The Power Grid dataset is network dataset describing the topology of the Northeastern US power grid, containing 4,941 nodes and 6,594 edges.
Power Grid 45
algorithm for secure information exchange, consisting of 10,680 nodes and 24,316 edges.
PGP 5 : It is User network of the
Table 6: Statistics the five datasets.
Datasets
#Nodes
#Edges
#Avg(degree)
#Average clustering coefficient #Density
#Diameter
Karate
34
78
2.29
0.57
0.14
5
Jazz
198
2,742
27.70
0.62
0.14
6
Cora ml
2,810
7,981
5.68
0.28
0.002
17
Power Grid
4,941
6,594
1.33
0.08
0.005
46
PGP
10,680
24,316
4.55
0.27
0.0004
24
20

---

G Additional experiments
Experiments on Real Diffussin Datasets. In order to gauge the efficacy of DDMSL on real- world propagation datasets, we opted for the Twitter [6] and Douban [ datasets, encompassing 12,627 nodes with 309,631 edges, and 23,123 nodes with 348,280 edges, respectively. The detailed performance metrics can be found in Table
Table 7: Additional experiments real diffusion datasets.
Twitter
Douban
Methods
PR
RE
AUC
PR
RE
AUC
DDMSL
0.445
0.286
0.313
0.625
0.484
0.324
0.381
0.622
SLVAE
0.310
0.317
0.253
0.578
0.412
0.140
0.209
0.547
Generalization Performance. We conducted extensive tests datasets of varying scales the generalization performance of both DDMSL and SLVAE algorithms. The comparative results are presented in Table revealing that DDMSL demonstrates commendable generalization performance across a majority of scenarios.
Table 8: Additional generalization experiments: Test results on different network topologies after one training real network, where the original performance represents the test performance of DDMSL on real networks.
Training
Cora
Power
POP
Twiller
Douban
Network
PR
RE
AUC
PR
RE
F]
Original
DDMSL
0.790
0.903
0.845
0.34
U.8S2
0.966
0.754
0.887
0.3
0.92S
0,445 0.256 0.313
0.484
0.324
0.381
0.62
performance
SLVAE
0.721
0.763
0.833 0.948
0.908
0.719
0.803 0,856
0.731 0.731 0.766
01.17
OLSS
0.310 0.317 0.253 05330.37
1.412
0.140
0.209
0.547
Small World
DDMSL
0.752
0.826
0.866
0.499
0.841
0..75
0.290
0.618
0.301
SLVAE
0.324
0.676
0.781
0.781 0.626
0122660.33
0.335
0.666
0.982
0.339
U.769
0.388 0.211 0.227
0.572
0.125
1.166 0.541
DDMSL
0.722 0.584
0.645
00.79
0.687
0.463
0.773
0.99
0.632
0.73 0.439 0.239 O. 0.319/0
0.624
11.20
0.307 0.614
0.336
BA Tree
SLVAE 0.961
HDMSL0.443
0.832
0.577
0.609
0.721
0.757
0.866
0.939
0.772
0.
3999
0.560
0.32 0.881
0.993
0.54
0.545
0.6.2
0.766
0.774
0.252
0.3377377
0.285
0.323
0.193 0.517
0.612
0.412
0.312
0.289
0.180
0.314 0.623
0.525
BA Dense
SLVAE
DDMSL 0.74
0.062 0.61 0.611
0.683
0.633
0.713
0.829
0.788
0.731
0.651
0:35
0.44| 0.550
0.72
0.99
0.
0.33
0.617
0.
0.723 0.286 (0.405
0.305
0.285
0.313
0.622
0.557
(0.374
0.125
0.132
0. 0.553
0.501
Time Complexity. Lastly, we conducted a comparative evaluation of the time complexity between DDMSL and baseline algorithms across diverse datasets, revealing the outcomes illustrated in Table Owing to the gradual inference of diffusion state for each time step, the time complexity of DDMSL. tends be substantial. However, optimization through parallel computing can effectively mitigate this disparity.
Table 9: Additional Time complexity experiment.
Test time
Cora-MI
Power-Grid
PGP
DDMSL
15.84s
22.19s
22.72s
SLVAE
4.77s
7.28s
11.29s
DDMIX
9.34s
15.7s
23.26s
GCNSI
1.4s
1.4
15.41s
LPSI
2m14s
Im51s
21m37s
OJC
6mlls
50m17s
2h41m52s
NetSleuth
2m40s
4m39s
21m24s
Training Time
Cora-MI
Power-Grid
PGP
DDMSL
10m36s
16m57s
32m03s
SLVAE
18s
39s
1m10s
DDMIX
16m5s
24m17s
37m53s
GCNSI
3m53
20m6s
34m26s
21

---

H
Visualization
H.1
Visualization of reconstructing diffusion paths
To conserve space, we displayed the actual node states and corresponding prediction results every 20% of the time. Figures Showcss the results, where the blue nodes denote susceptible ones, the red nodes denote infected nodes, and the green nodes denote recovered nodes. The findings indicate that DDMSL accurately restored the node states at different times. In contrast, DDMIX could only restore infected nodes, revealing that DDMSL far surpasses DDMIX in its expression capability.
FTT/
1-T1/
2
Prediction
Figure 5: DDMSL reconstructs SIR diffusion on Karate.
H.2
Visual comparison source localization
Due to spatial limitations, we only presented the source localization results of DDMSL and benchmark algorithms on the Karate and Jazz datasets. The results are depicted in Figures I0 and [T] The baseline algorithm's 's performance is unsatisfactory, as evidenced by significant positioning errors in source nodes. On the contrary, DDMSL outperforms other benchmark algorithms in accurately identifying source nodes. Moreover, even when misidentifying source nodes, DDMSL locates them near the actual nodes.
22

---

F
F41/5
F-21/
Ground Truth
Predittion
Figure DDMSL, reconstructs SIR diffusion on Jazz.
FT
3T/S
2
Figure DDMSL reconstructs SIR diffusion Coral ml.
23

---

-T/
F3T/
F
Prediction
Figure 8: DDMSL reconstructs SIR diffusion Power grid.
H
HT/5
-3T/5
2
1-0
Ground Truth
Prediction
Figure 9: DDMSL reconstructs SIR diffusion on PGP.
24

---

(a) DDMSL
(b) SLVAE
(c) DDmix
(d) LPSI
(e) GCNSI
(I) OJC
(g) Ground Truth
Figure 10: Visualization comparisons of source localization on Karate.
a DDMSL
(b) SLVAE
(c) DDmix
(d) LPSI
(e) GCNSI
(I) OJC
(g) Ground Truth
Figure 11: Visualization comparisons of source localization on Jazz.
25


Reviews:
[{"review_id": "eUSWvmc991", "reviewer": "Reviewer_NqvF", "summary": "This paper addresses the problem of source identification in a stochastic\nnetwork diffusion process, having observed the set of nodes that are\ninfected at time T. It considers SIR and SI models. The authors propose a\nneural network based solution called DDMSL. The authors approach is based\non a reaction-diffusion process and formulate it using a message-passing\nfunction and use a reversible residual network for source identification\nand reconstruction of the cascade. DDMSL is compared with several\nbaselines and an ablation study is conducted to evaluate the significance of\nthe reversible residual network and propagation rule supervision module.", "questions": "Several definitions of notation are missing. Diffusion model is not\ndefined. These are mentioned in the Weakness section.\n\nIn Algorithm 1 line 4, t is sampled from a distribution where the\nprobability of t increases as its value increases. This is not explained\nanywhere. Why is this sampling necessary?", "limitations": "No negative societal impact.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "The authors use a novel reversible residual network construction for the\nsource identification problem.\n\nThe paper has a strong experiments component considering multiple networks\nand several baselines.\n\nDDMSL shows much superior performance compared to the baselines.", "weaknesses": "The presentation could be greatly improved. Many notation and crucial\nconcepts are missing. The derivations are very dense and lack intuitive\nexplanations. So, it is hard to understand the results. Details follow.\n\nDiffusion models are not clearly defined: There are several types of SIR\nand SI models in the literature. The authors fail to describe their model\nclearly. Is it a discrete-time SIR process, i.e., an infected node infects\nits neighbors with probability equal to the edge weight at each time step\nuntil it recovers, which again is determined by some probability? A second\ntype of process is a Gillespie process which is a discrete-event process\nwhich involves rates as the authors have used. Also, the authors seem to\nuse a single transmission rate per node. This is important to clarify as\nthe baselines considered in the paper use their own definitions of SIR\nmodels. Is it consistent with the definition of the SIR model in this\npaper? If not, then the comparison wouldn't be fair.\n\nReaction-diffusion process is not defined. In section 3.4.1, the authors\nseem to relate GCN with reaction-diffusion models. This is not clear at all\nas the concept is not introduced.\n\nTheorem 3.1: What is the purpose of this theorem? Further, its proof is not\npresented, even in the appendix. Also, it is not referred to anywhere.\nEquation (3) is not used anywhere as well. It is not clear why the rates\ncan be expressed in that manner. Also, even though these are mentioned as\nrates, beta and gamma are used as probabilities.\n\nSeveral symbols are not defined: nn_\\theta, KL, D_KL, Cat(), Q_k^i. It is\nvery difficult to follow the derivations. They are dense and lack\nexplanations.\n\nAblation: In DDMSL(a), the reverse residual block is removed, but replaced\nby a GCN network. Typically, in an ablation study, the component is removed\nand the performance evaluated. It is possible that the GCN network\ndeteriorates the performance.\n\nThe Figures 6 to 11 in the Appendix are very difficult to read and infer.\nBesides, these are just few instances where DDMSL does well. Table 2\nalready quantifies the performance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "   Thank you very much for your comments and suggestions. We have provided some explanations and clarification regarding your concerns as follows:\n\n(1) “Many notation and crucial concepts are missing”.\n\n   Response: Thank you very much for your comments and suggestions.  The neural network model used in the reverse inference process was described in section 3.3.2 on page 4 by ${nn_{\\theta}}$. \nWe have only used $Q_t^i$ instead of $Q_k^i$ in our manuscript, and specific explanations of $Q_t^i$ are provided on page 4, line 121. As for $KL, D_ {KL}, Cat()$, these symbols are from the Loss function of the Diffusion model, which have been used in many diffusion models widely, such as DDIM[3]. We deeply apologize for any confusion caused to readers. We will provide a more detailed introduction to these symbols in subsequent revised version.\n\nAs for the lack of the concept of reaction diffusion process, in our paper, reaction diffusion process refers to individual mediated diffusion processes such as SIR (see equation 1), SI, SIS, etc. We will revise and add this section in subsequent paper revisions. Thank you for your suggestion.\n\n(2) “Diffusion models are not clearly defined”.\n\nResponse: We use discrete SI and SIR models, please refer to equation 1 on page 3 for detailed definitions. During the experiment, all baseline algorithms used consistent SIR and SI models and ensured that the experimental results were generated based on the same datasets.\nIn fact, the transition probability of each node in the SIR process is necessarily different, just as Lokhov et al. [1] analyzed the SIR process using message passing algorithms, and we inferred the state distribution of any node at any time step. Each node follows the pre-defined process throughout the entire diffusion process: an affected node affects its neighbors with probability equal to the edge weight at each time step until it recovers. However, the infection probability and recovery probability of each node during this period depend on the state of the node and its neighbors, as SIR is a Markov process.\n\n(3)“The proof of Theorem 3.1 and the role of  Equation 3”.\n\nResponse: The purpose of Theorem 3.1 is to ensure that the state distribution of nodes at each time step is convergent, which is a necessary condition for the discrete Diffusion model to work [2]. Due to space limitations, we have provided a detailed explanation of its proof and function in the appendix. Please refer to section C.1 on page 16 for details. Therefore, our proof of Theorem 3.1 is not missing. And, we will add a description of Theorem 3.1 in the main text.\n\nEquation (3) is not that what you said has not been used anywhere. If you read the paper carefully, the role of Equation (3) is explained in lines 121-124 on page 4, that is, the parameters used to calculate the State-transition matrix of Equation (2). Meanwhile, the generation of Equation (3) is also very intuitive. For instance, the probability of node $i$ being infected at time $t+1$ is the sum of the two: the probability of $i$ being uninfected and subsequently infected at time t, and the probability of $i$ being infected at time $t$ but not transitioning to a recovery state afterwards. We will add a detailed explanation of this section in the appendix.\n\n(4) “Ablation Study“.\n\nResponse: Thanks for the great suggestion. We have added ablation experiments on the GCN module, and the experimental results are shown in Table 1 and Figure 1 in the attached PDF, where DDMSL (c) represents the model after removing the GCN module.\n\n(5) “The Figures 5 to 11 in the Appendix“.\n\nResponse: Figures 5 to 9 are visualizations of DDMSL in the process of reconstructing information diffusion, while Figures 10 to 11 are visualizations of source localization tasks. Each graph has thousands or even tens of thousands of nodes, and it is impossible to display the reconstruction process of each dataset in such a small space. Therefore, we choose to use vector graphs to clearly see the state of each node and even each edge when zoomed in, but this will result in a significant memory footprint. \n\nWe believe that visualization of reconstruction diffusion is necessary, and the source localization results can already be clearly displayed in Table 2. Therefore, we have reduced the visualization of source localization to prevent further expansion of text memory usage.\n\n(6) “Reason for sampling t in Algorithm 1 line 4“\n\nResponse: The probability of t being sampled is indeed positively correlated with  $t$. Generally, $t$ is uniformly sampled in the Diffusion model of other fields. But we found in the experiment that, compared to the uniform sampling, our setting can significantly reduce the training time and bring some performance improvement. Sorry for missing the explanation on t sampling, and we will supplement it in subsequent modifications. \n\n(7) “Limitations“.\n\nResponse: We believe that the negative impact of our research on society is limited. It is possible that some people's privacy may be exposed in some use scenarios, e.g., AIDS patients who do not want to disclose their diseases. In summary, we will also add this section in the appendix.\n\n[1] Lokhov, Andrey Y., Mézard, Marc, and Zdeborová, Lenka. \"Dynamic message-passing equations for models with unidirectional dynamics.\" (2014).\n\n[2] Austin, Jacob, et al. \"Structured Denoising Diffusion Models in Discrete State-Spaces.\" (2021).\n\n[3]Song, Jiaming ,  C. Meng , and  S. Ermon . \"Denoising Diffusion Implicit Models.\" (2020)."}, {"review_id": "ClEh9xmJw7", "reviewer": "Reviewer_wXsK", "summary": "Information diffusion is common in various domains, such as social networks, the internet, and disease propagation. Obtaining the diffusion paths and localizing the source based on the final diffusion node states is beneficial for researchers to identify key transmission pathways during information dissemination and facilitate control over the propagation. This article proposes a framework for information diffusion called Discrete Diffusion Model for Source Localization (DDMSL) using the Susceptible-Infected-Recovered (SIR) model. The framework utilizes a discrete Markov Chain to characterize information diffusion and recover diffusion paths based on observed results while achieving source localization. For the neural network module in the framework, the authors design an inference model based on a residual network and rigorously prove its reversibility to justify the rationality of the model design. The experiments are performed on five real-world networks, where propagation data is generated using SIR and SI models. The experiment results demonstrate that, compared to the baseline, DDMSL performs better in source localization and diffusion path recovery across all datasets.", "questions": "My most concerning questions are mostly mentioned in the *Weaknesses* part, which are:\n\n1. The reason why the proposed method utilizes the denoising diffusion model for this task.\n2. The performance of the proposed method on real-world propagation datasets is not provided.\n3. In IVGD(Wang et al.), the invertibility of the network is proven so that fixed-point iterations can be performed. I am concerned about how the reversibility of residual blocks, strictly proven, contributes to the prediction of $X_0$.(In Figure 2, the input is $Y_T$, while in the text of sec.3.4, the input is $X_0$.)\n4. Moreover, I would like to know whether the model has been tested on different graph topologies under a single training. Because in most application scenarios, the trained model is asked to generalize on the changing graphs. I wonder whether the results in the paper are from the model trained and tested on the same static network topology(dataset).", "limitations": "The authors have illustrated the main limitation that the model requires prior knowledge about propagation models. \n\nThe provided source code cannot be run directly.", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Interesting idea. This paper proposes a method to establish a correlation between the information propagation process and the denoising diffusion model, which is worth further research.\n- Rigorous theoretical guarantees. This paper presents reliable theoretical proofs to explain the designs of the model, which are also proven effective in the experiment stage.\n- Well performance on different networks(datasets). The experiment results demonstrate the consistent performance across different networks and the improvement compared to the baselines.", "weaknesses": "- This paper needs to clarify why the diffusion model is being used for this task. Only the achievements of the diffusion model in the image inverse problems are mentioned. Personally, it seems that there is a significant difference between the inverse problems of images and the task addressed in this paper.\n- In my view, the method proposed is not highly correlated to the so-called denoising diffusion model, as the initial input is not a particular noise, and the diffusion steps $T$ are set equal to the steps of information propagation.\n- The paper does not show the model's generalization ability on real-world propagation datasets (only SIR or SI model as presented in the paper, no real-world datasets like *Digg* and *Memetracker*).\n- Compared to previous works, the proposed method requires fine-grained diffusion history information, which may largely limit its application scenarios.\n- Compared to previous works that do not explicitly simulate every diffusion step, such as SL-VAE, this method seems much more inefficient. And no efficiency study is provided.\n\n## After Rebuttal\nThe authors' response has addressed most of my concerns. I suggest authors discuss the following points in the revised versions.\n1. How to decide the number of diffusion/denoising steps in the diffusion model, the same as the real-world forward diffusion process on networks? What if this number is extremely large?\n\n2. Please provide generalization performance on other real-world large networks, and comparison with baseline methods.\n\n3. Requiring fine-grained diffusion history information is still a large limitation. Can this forward diffusion history be inferred with parameter estimation?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable feedback. In response to your input, we have conducted experiments utilizing real-world diffusion datasets and DDMSL generalization experiments. The forthcoming section will provide comprehensive responses to each of your inquiries.\n\n(1)\"Why the proposed method utilizes the denoising diffusion model for this task?\"\n\nResponse: Please refer to our global response.\n\n(2)\"Relationship between the proposed method and the denoising diffusion model.\"\n\nResponse: As mentioned in global response, our approach utilizes a discrete diffusion denoising model that is extensively employed in graph data generation tasks [2,3]. Unlike Gaussian distributed noise sets, this model deals with noisy data exhibiting a stable distribution resulting from the initial data's interaction with the state transition matrix. As illustrated in the paper, the SIR, SI, and other models can be effectively represented using state transition matrices. Since the diffusion noise sequence data is naturally generated, there is no need to introduce additional noise for constructing the noise sequence.\n\nFurthermore, Theorem 3.1 provides assurance regarding the convergence of node states at every moment, ensuring the model's functionality at any given time. Consequently, we set the time step to T in our experimentation.\n\n(3) \"Showing the model's generalization ability on real-world propagation datasets.\"\n\nResponse: Thank you for your suggestion. We tested DDMSL using real-world information dissemination datasets. However, the Digg dataset lacked propagation information flows due to recent updates. The Memetracker dataset's significant growth posed processing challenges within limited time. Thus, we used alternative datasets: Twitter [4] (12,627 nodes, 309,631 edges) and Douban [4] (23,123 nodes, 348,280 edges). These datasets underwent SLVAE benchmark algorithm processing, followed by source localization testing. Results can be found in Table 4 in the attached PDF.\n\n(4) \"How the reversibility of residual blocks, strictly proven, contributes to the prediction of $X_0$ ?\"\n\nResponse: We will first explain the relationship between Section 3.4 and Figure 2, and then briefly introduce the process of DDSML and the roles of each module.\n\na. The relationship between Section 3.4 and Figure 2.\n\nFigure 2 illustrates a reversible residual network used in DDMSL. This network functions similarly to Unet or Transformer in an image diffusion model (DDIM). Please refer to Appendix D for detailed training and inference processes. The network takes observed infection information ($Y_T$) as input and outputs the potential source node ($X'_0$) that might have caused $Y_T$.\n\nMoreover, Section 3.4 primarily focuses on constructing a neural network model capable of fitting the SIR inverse diffusion process, which is introduced in Section 3.4 on page 5. Equation 13 demonstrates that assuming $X_0$ represents the source nodes, we utilize $Y_T=P(X_0)$ to represent the generation of $Y_T$ from $X_0$ using the SIR or SI process. Naturally, $X_0$ is unknown while $Y_T$ is known, and inferring $X_0$  from $Y_T$ can be denoted as $X_0=P^{-1}(Y_T)$.\n\nLastly, we devised a residual block based on GCN, built upon the equivalence between SIR/SI diffusion and message passing processes, and assembled these blocks to form a reversible network. Subsequently, we verified that the network is indeed reversible. Our ultimate objective is to design the residual network depicted in Fig2, denoted as $P^{-1}$, ensuring its capability to output $X_0$ after training with $Y_T$ as input.\n\nb. How the reversibility of residual blocks, strictly proven, contributes to the prediction of $X_0$?\n\nNow let's delve into this matter further. The group of residual blocks forms a network that improves the receptive field of GCN and addresses issues like node oversmoothing. This network corresponds to fitting $P^{-1}$. We input the observed infection graph $Y_T$ into the residual network, calculate the loss between the output and the actual $X_0$, and update the network's parameters. Through iterations, the residual network gradually approximates the true $P^{-1}$. Rigorous theoretical proofs assure that residual networks can produce $X_0$ as an output.\n\n(5) \"Whether the model has been tested on different graph topologies under a single training?\"\n\nResponse: Our experiments entail separate training and testing processes for each graph structure, following the prevailing methodology in deep learning algorithms for source localization. \n\nWe wholeheartedly concur with your viewpoint regarding the significance of evaluating model generalization ability. To evaluate generalization, we conducted additional experiments using three diverse network datasets of varying scales. Each trained model was assessed on different network structures, revealing the DDMSL model's remarkable generalization performance. Detailed outcomes are provided in Table 2 of the attached PDF.\n\n(6) \"Limitations\"\n\nResponse: As elucidated in our paper, DDMSL does possess limitations that necessitate prior knowledge of partial diffusion models. However, these limitations are not insurmountable, given the availability of alternative approaches to aid in parameter estimation for information diffusion models (see Sec 5). In addition, We have made updates to the code.\n\n[1] Song, Jiaming ,  C. Meng , and  S. Ermon . \"Denoising Diffusion Implicit Models.\" (2020).\n\n[2] Vignac, Clement, et al. \"Digress: Discrete denoising diffusion for graph generation.\" arXiv preprint arXiv:2209.14734 (2022).\n\n[3] Haefeli, Kilian Konstantin, et al. \"Diffusion models for graphs benefit from discrete state spaces.\" arXiv preprint arXiv:2210.01549 (2022).\n\n[4] Z. Cao, K. Han and J. Zhu, \"Information Diffusion Prediction via Dynamic Graph Neural Networks,\" 2021 IEEE 24th International Conference on Computer Supported Cooperative Work in Design (CSCWD), Dalian, China, 2021, pp. 1099-1104, doi: 10.1109/CSCWD49262.2021.9437653."}, {"review_id": "AVuhQyaRV8", "reviewer": "Reviewer_W7Kc", "summary": "This paper proposes a discrete denoising diffusion model for source localization called DDMSL. DDMSL can simultaneously locate information sources and restore information propagation paths. Experiments results on real-world datasets demonstrate DDMSL’s effectiveness.", "questions": "1. The phrase “graph inverse problem” or “inverse problem” sounds peculiar, perhaps the diffusion inverse problem or the spread inverse problem would be more comprehensible.\n2. The formulation of the proposed denoising diffusion model is not clear. In Eqs. (4) and (5), why is $x^i_t$ included in $q(x^i_t | x^i_{t-1})$ and $q(x^i_t | x^i_0)$? If we already know $x^i_t$, is it necessary to have $x^i_{t-1} or x^i_0$ to calculate $x^i_t$?\n3. How to obtain the distribution $Cat(x^i_t; p)$?\n4. What are reaction diffusion models? Maybe they should be included in the Section Related Work?\n", "limitations": "1. The proposed model requires simulation of SIR from t_0 to t_T to obtain the training data, whereas diffusion models can be trained at any step by sampling Gaussian noise at an indicated time step. It seems that the model proposed in this paper needs more training time and is difficult to apply to large-scale graphs, such as OGB datasets. It is also necessary to compare time complexity with baseline methods.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. This is the first study to solve information spread problems based on denoising diffusion model.\n2. Extensive experiments on real-world datasets demonstrate the effectiveness of the proposed method.\n", "weaknesses": "1. The motivation of employing denoising diffusion model is not clear. Why is this generative model used for source localization problem? Compared to other source localization models, what are the advantages of the denoising diffusion model?\n2. I do not agree that the proposed method is a denoising diffusion model. The final distribution of the forward process and the initial distribution of the reverse process are both random because they are simulated by the SIR model. It is more like a sequence model trained by the SIR data.\n3. The motivation behind the model design is not clear. For example, why are SN and BN employed in Eq. 16?\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you sincerely for your valuable suggestion. Taking into account your feedback, we have incorporated a comprehensive time complexity evaluation for DDMSL and will ensure that all your questions are answered thoroughly.\n\n(1) \"What are the motivations and advantages of using diffusion models to handle inverse problems?\"\n\nPlease refer to our global response.\n\n(2) \"The relationship between DDMSL and denoising diffusion model.\"\n\nAs expounded in global response, we posit that the SIR model's role in the information diffusion process can be perceived as noise, progressively obfuscating the information contained within $X_0$. It becomes challenging to disentangle the inherent noise of the original $X_0$ data from the final input data utilized by DDMSL. In contrast to conventional denoising diffusion models like DDIM, which handle a sequence of artificially added Gaussian noise, our model processes a sequence of SIR data wherein the noise originates naturally. This fundamental distinction sets apart the noise characteristics encountered in SIR sequence data from those artificially introduced in DDIM.\n\n(3) \"The motivation behind the model design is not clear. \"\n\nThe rationale behind the design of $nn_{\\theta}$ stems from our belief that the propagation process of SIR resembles the message transmission process depicted in equation 14, thus enabling simplification to the form presented in equation 15. We will furnish a specific design case (termed as Equation 16) based on equation 15. Employing SN(·) and BN(·), we aim to constrain the Lipschitz coefficients of the model, as discussed in the proof of Lemma 3.1 (page 6, line 209), with additional detailed proof provided in the appendix. We sincerely apologize for any inconvenience caused to the readers. To enhance clarity, we will augment this section with more comprehensive explanations in the next revised version.\n\n(4) The use of the phrase “graph inverse problem”\n\nDiffusion occurs in various spatial and entity relationships. Our research focuses on graph-based analysis, hence the name is set to align with our investigation. We highly appreciate your input and will diligently take your opinion into consideration. \n\n(5) \"The formulation of the proposed denoising diffusion model is not clear. \"\n\nEquations 4 and 5 are crucial for modeling the information diffusion process in DDMSL. Eq.4 captures the relationship between node states at adjacent time steps, while Eq. 5 establishes the connection between $X_0$ and node states at any given time using a Markov chain derived from Eq. 4.\n\nTo calculate our target $q(X_0 | X_t)$, we need to obtain $q(X_t | X_0)$ and establish the relationship between $X_t$ and $X_0$. However, we can only observe $X_t$ in reality, which corresponds to the infection data. Therefore, we must infer $X_{t-1}$ from $X_t$ iteratively until we obtain the inferred result of $X_0$.\n\nVarious methods exist for inferring $X_{t-1}$ from $X_t$. For example, DDIM predicts the noise between $X_t$ and $X_{t-1}$ and subtracts it from $X_t$ to obtain $X_{t-1}$. In our case, we use a discrete denoising diffusion model (D3PM), which utilizes a neural network model to predict an intermediate $X_0^{'}$ (different from $X_0$) based on $X_t$. Then, $X_0^{'}$ is substituted into equation 8 to derive $X_{t-1}$. Appendix D details the DDMSL model's process.\n\n(6)\"How to obtain the distribution $Cat(x_t^{i};p)$?\"\n\nLet's illustrate the calculation process of Equation (4) with an example. At time $t=s$, node A has a state represented by $x_s=[1,0,0]$. The state transition matrix $Q_{s+1}$ at time $t=s+1$, computed using equations 2 and 3, is assumed to be $Q_{s+1}=[0.4,0.6,0; 0,0.7,0.3; 0,0,1]$. Thus, the state distribution of node A at time $t=s+1$ is $x_{s+1} = x_sQ_{s+1} = [0.4,0.6,0]$. \nThis implies that node A is in state S  at time $t=s$, and the probability of it being in state S at time $t=s+1$ is:\n\n $q(x_{s+1}=S|x_s=S)=x_sQ_{s+1}x_{s+1}^T=[0.4,0.6,0][1,0,0]^T=0.4$. \n\nSimilarly, the probability of node A being in state I is 0.6.\n\n(7)\"What are reaction diffusion models?\"\n\nOur sincere apologies for any lack of clarity in explaining the reaction diffusion models. In the paper, \"reaction diffusion models\" refer to models like SIR, SI, etc., where nodes undergo reactions with neighboring nodes to update their states and influence others. We will enhance the introduction and related work sections to clarify these concepts. \n\n(8)\"Limitations\"\n\nDDMSL uses a residual graph convolutional network as its backbone. For a graph with N nodes and E edges, the complexity of the network during inference from time t=T to t=0 is roughly $O(KT|E|)$(K is the number of GCN layers).\n\nDuring  experiments, we found that the generation of $Q_t$ has the most significant impact on time complexity. $Q_t$ is derived by performing m SIR Monte Carlo simulations to calculate $Q_t$, resulting in a time complexity of approximately $O(mTN^2)$(m is the number of Monte Carlo simulations). This calculation can be time-consuming, especially for large-scale graphs.\n\nTo reduce inference time in DDMSL, we employ the following strategies:\n\n1) To optimize the computation of $Q_t$ using Monte Carlo simulation, we support code optimization for parallel computation. Our code enables accelerated computation through multi-threading. By utilizing CUDA operators, we can reduce inference time by 20-30 times (GPU version still undergoing testing). We also provide a comparison of the training and testing times of DDMSL using 12 threads alongside other benchmark algorithms (Due to time constraints, we used default settings for baseline algorithms) in Table 3, where DDMSL demonstrates acceptable inference speed. Please refer to the attached PDF.\n\n2) An alternative approach involves leveraging existing deep learning models (see Appendix D) to fit $Q_t$, requiring only a portion of the generated data for accurate predictions. This substitution can roughly reduce testing and training times by one-third.\n\n\n"}, {"review_id": "A61Y0iIpSG", "reviewer": "Reviewer_c2eK", "summary": "This work discusses the challenges associated with tracing the origin and path of information diffusion in complex networks, such as those involved in epidemics or rumors. To address these, the authors propose a probabilistic model, DDMSL (Discrete Diffusion Model for Source Localization), which utilizes Markov chains to model information diffusion and a reversible residual network to localize the source and reconstruct the paths of information diffusion, with its efficacy supported by extensive tests on five real-world datasets.", "questions": "1. The motivation for recovering the diffusion path is weak. I feel like the source localization only needs to predict the correct set of seed nodes in order to identify the culprit, why would it be necessary to recover the diffusion paths with the already given diffusion status/observation? I am still having a hard to convince myself with the provided motivation.\n2. The authors claim \"existing inference results are often deterministic and fail to quantify the uncertainty of source localization\", but it seems like the proposed method's results are also deterministic from Figure 2. Does the proposed method output a range of possible source nodes with [S, I, R] labels?\n3. In terms of the experiment section, the biggest problem is the efficiency issue of the proposed framework. With the step-wise diffusion model to recover the overall diffusion path, the runtime of the proposed method and the complexity analysis should be demonstrated. ", "limitations": "I am not seeing any potential negative societal impact.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The general topic is interesting, and the paper studies the problem of source localization under two information propagation patterns: SI and SIR. The proposed methods are sound with theoretical guarantee. The overall writing is easy to follow, but the use of notations are sometimes less formal. The idea of simultaneously handling the problems of source localization and reconstructing diffusion paths seems to be interesting, but the motivation is less promising. The experiments are strong, and the proposed method is better than most of existing baselines. ", "weaknesses": "There are three main questions of this work. Please refer to the questions section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "\nThank you for your valuable suggestion. We have incorporated the time complexity analysis of the DDMSL algorithm and will provide comprehensive answers to all your inquiries.\n\n(1)\"The motivation for recovering the diffusion path\"\n\nThank you for your question. In our perspective, reconstructing communication paths plays a pivotal role in controlling information dissemination. This process involves speculating on the distribution of node states throughout the diffusion process.\n\nFirst, the reconstruction of the diffusion path offers valuable support in managing information spread. When provided with the observed infection status graph $X_T$, early source identification is crucial when T is small as the information diffusion scale is limited, allowing effective control through timely identification. However, as T grows larger, information has already propagated to various parts of the network, making it challenging to effectively control its dissemination even with source identification. In such cases, focus shifts to previously infected nodes or those with a high infection probability.  For instance, in the context of the global Covid-19 pandemic, a common control strategy following a major outbreak involves identifying infected individuals and their close contacts at $T-1$, $T-2$, and $T-s$ based on observed infections in $X_T$, and subsequently isolating them. This strategy has proven to be highly effective in controlling epidemic outbreaks, despite being time-consuming and labor-intensive. Here, our reconstruction of node states at $T-1$, $T-2$, ... and $T-s$ can provide direct assistance in this context.\n\nAdditionally, reconstructing the diffusion path can deepen our understanding of information diffusion processes in complex networks and offer insights applicable to other fields. DDMSL generates possible diffusion paths from observed graphs, guiding network control for specific information. It has applications in disease and rumor control, as well as influence maximization. The research on reconstructing diffusion pathways holds immense practical value, and we are optimistic about its future prospects.\n\n(2)\"Does DDMSL output a range of possible source nodes with [S,I,R] labels?\"\n\nThe answer to this question is affirmative.DDMSL can generate a series of possible source nodes, each accompanied by probability labels representing three states:S,I, and R.\n\nIn practice, we can only observe $X_t$, which corresponds to the infection data.Therefore,we must infer $X_{t-1}$ from $X_t$ iteratively until we obtain the inferred result of $X_0$.\n\nVarious methods exist for inferring $X_{t-1}$ from $X_t$. For example, DDIM predicts the noise between $X_t$ and $X_{t-1}$ and subtracts it from $X_t$ to obtain $X_{t-1}$. In our case, we use a discrete denoising diffusion model (D3PM), which utilizes a neural network model to predict an intermediate $X_0^{'}$ (different from $X_0$) based on $X_t$. Then, $X_0^{'}$ is substituted into equation 8 to derive $X_{t-1}$.Appendix D details the DDMSL model's process.\n\nIt is important to note that during the training and testing of DDMSL, Gumbel Softmax is utilized to sample $X_{t-1}$ after obtaining its distribution (shown in lines 10 and 7 of Algorithm1 and Algorithm2 in Appendix D, respectively). Therefore, the inferred results of DDMSL at each time step are sampled from a probability distribution, and the state distribution of the next time step depends on the inferred results from the previous time step. As a result, DDMSL can generate a series of diverse source nodes and diffusion paths that conform to different diffusion patterns.\n\n(3)\"the efficiency issue of the proposed framework.\"\n\nDDMSL uses a residual graph convolutional network as its backbone. For a graph with N nodes and E edges, the complexity of the network during inference from time t=T to t=0 is roughly $O(KT|E|)$(K is the number of GCN layers).\n\nDuring experiments, we found that the generation of $Q_t$ has the most significant impact on time complexity. $Q_t$ is derived by performing m SIR Monte Carlo simulations to calculate $Q_t$, resulting in a time complexity of approximately $O(mTN^2)$(m is the number of Monte Carlo simulations). This calculation can be time-consuming, especially for large-scale graphs.\n\nTo reduce inference time in DDMSL, we employ the following strategies:\n\n1)To optimize the computation of $Q_t$ using Monte Carlo simulation, we support code optimization for parallel computation. Our code enables accelerated computation through multi-threading. By utilizing CUDA operators, we can reduce inference time by 20-30 times (GPU version still undergoing testing). We also provide a comparison of the training and testing times of DDMSL using 12 threads alongside other benchmark algorithms (Due to time constraints, we used default settings for baseline algorithms.) in Table 3, where DDMSL demonstrates acceptable inference speed. Please refer to the attached PDF.\n\n2)An alternative approach involves leveraging existing deep learning models(see Appendix D) to fit $Q_t$, requiring only a portion of the generated data for accurate predictions. This substitution can roughly reduce testing and training times by one-third.\n\n(4)\"Limitations\"\n\nWe believe that the negative impact of our research on society is limited. One potential concern regarding DDMSL is its impact on privacy when reasoning about source nodes or diffusion paths. This issue may lead to the unintended exposure of sensitive information associated with certain nodes. For instance, in the context of an AIDS transmission network, patients who wish to keep their AIDS diagnosis confidential could potentially be identified. We acknowledge this concern and will address it in the upcoming revised version of our paper.\n\nOnce again, we express  gratitude for your thorough and meticulous review, as well as for the valuable suggestions you have provided for our paper. We hope that our explanation adequately addresses your inquiries."}]

Author rebuttal:
I would like to extend my sincere appreciation to the esteemed reviewers for their invaluable suggestions on our paper. It has come to our attention that several reviewers have expressed interest in understanding the rationale behind our adoption of the denoising diffusion model. We will address this and share additional experiments conducted based on reviewer feedback, including testing DDMSL on real-world diffusion data, generalization experiments, time complexity analysis, and supplementary ablation experiments. Please refer to the attached PDF file for the experimental results.

We will introduce our motivation from the following aspects:

a. "Differences between discrete diffusion denoising models and classical diffusion models (such as DDIM)"

Firstly, it is important to clarify that we are employing the D3PM [2] diffusion denoising model in a discrete space, which distinguishes it from classical diffusion models like DDIM [1]. D3PM leverages Markov chains to model discrete data, where the data between two consecutive time steps are interconnected through the state transition matrix $Q_t$. In contrast to DDIM, which introduces Gaussian noise directly to the data to establish a forward process, D3PM incorporates noise into $Q_t$ to account for the stochastic nature of the diffusion process.

b. "Comparison of SIR diffusion and noise diffusion on images"

The application process of diffusion models in image analysis can be summarized as follows: starting with an original image, noise is continuously introduced to approximate a Gaussian distribution. DDIM, in particular, performs continuous denoising on the Gaussian distribution noise to generate a new image. In information dissemination models such as SIR , the process of information diffusion bears similarities to the noise introduction process in image analysis. Given a set of initial seed nodes $x_0$, their states gradually become perturbed under the influence of SIR rules. Subsequently, the noise imposed by SIR rules is eliminated from the final noisy infection graph. We hypothesize that the state of $x_0$ evolves over time based on the impact of SIR rules, consequently causing changes in the states of $x_0$'s neighbors and even higher-order neighbors. This progressive alteration obscures the original information encoded in $x_0$. From this perspective, the node state modification at each time step induced by SIR can be interpreted as noise introduced during the evolution of information diffusion. 

Models like SIR,SIS and SI epitomize the most natural diffusion processes.	For instance, in the SIS model, which can also be processed b DDMSL, when specific network topology conditions are met, and given a set of seed nodes, as the propagation time increases, each node eventually converges to a stable distribution where the probabilities of being in the susceptible(S) and infected(I) states approach 0.5. In fact, Theorem 3.1 demonstrates that the node state distribution of both the SI  and SIR  models at each time step during the diffusion process exhibits convergence.

Hence, it is our contention that the dissemination of information on a graph and the propagation of noise in an image are two distinct occurrences transpiring in discrete and continuous spaces, respectively. This observation serves as one of the motivations behind employing diffusion models for source localization. As previously discussed, we leverage the D3PM to address this task. The SIR, SI, and SIS models align seamlessly with the utilization requirements of D3PM, and the formulation of the state transition matrix (as specified in Equation 2) for each node is straightforward.

c. "Why use generative models for source localization?"

In the context of an observed infection graph ($Y_T$), numerous sets of source nodes ($x_0$) can exist, as depicted in Fig1 of the paper. Inferring the distribution of source nodes using posterior diffusion data constitutes a critical task. However, existing source localization algorithms such as LPSI, OJC, and other methods employed in Baseline generate deterministic and imprecise outcomes. This limitation motivates the utilization of VAE in SLVAE. In fact, this forms the foundation of DDMSL and SLVAE's approach to source localization.

d. "Why use diffusion models for source localization? What are their advantages?"

Firstly, the diffusion model demonstrates superior performance compared to traditional generative models like VAE in terms of task generation accuracy. In our experimental evaluation, DDMSL outperformed generative models such as SLVAE and DDMIX for source localization purposes.

Secondly, the unique advantage of the diffusion model lies in its ability to leverage the inference results from previous time steps to inform the inference process at each subsequent time step. As a result, it can construct comprehensive node information at each time step, aligning perfectly with our objective of reconstructing the communication path. This aspect serves as an additional motivation for adopting the discrete diffusion model for our task.

In summary, the reasons for using the discrete diffusion model are as follows:

1) The phenomenon of information diffusion is a pervasive occurrence in various domains, making the adoption of D3PM suitable for modeling forward processes and learning reverse processes.
2) DDMSL exhibits remarkable performance in reconstructing the states of individual nodes at each time step throughout the propagation process, enabling successful accomplishment of the two primary tasks outlined in the paper: source localization and path reconstruction.
3) When compared to alternative algorithms like SLVAE and DDMIX, which also employ generative approaches for source localization, DDMSL demonstrates superior performance.

[1] Song, Jiaming ,  C. Meng , and  S. Ermon . "Denoising Diffusion Implicit Models." (2020).

[2] Austin, Jacob , et al. "Structured Denoising Diffusion Models in Discrete State-Spaces." (2021).

Meta-review: The paper introduces a novel reversible residual network construction for source identification. Extensive experiments involving multiple networks and baselines demonstrate DDMSL's superior performance. This is a borderline paper. The original reviewer scores were quite mixed. After the rebuttal, some reviewers seemed to be satisfied with the authors' rebuttal and slightly raised the score. I hope in the final version, the authors can further justify the motivation to use generative model for sources localization and improve the overall presentation of this paper.
Decision: Accept (poster)