SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}, \boldsymbol {x} _ {0} ^ {i}\right) = q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}\right) = \frac {\sum_ {\boldsymbol {x} _ {0} ^ {i}} q \left(\boldsymbol {x} _ {t - 1} ^ {i} , \boldsymbol {x} _ {t} ^ {i} , \boldsymbol {x} _ {0} ^ {i}\right)}{q \left(\boldsymbol {x} _ {t} ^ {i}\right)} \\ = \frac {\sum_ {\boldsymbol {x} _ {0} ^ {i}} q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i} , \boldsymbol {x} _ {0} ^ {i}\right) q \left(\boldsymbol {x} _ {0} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}\right) q \left(\boldsymbol {x} _ {t} ^ {i}\right)}{q \left(\boldsymbol {x} _ {t} ^ {i}\right)} \tag {20} \\ = \sum_ {\boldsymbol {x} _ {0} ^ {i}} q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}, \boldsymbol {x} _ {0} ^ {i}\right) q \left(\boldsymbol {x} _ {0} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}\right) \\ = \mathbb {E} _ {q \left(\boldsymbol {x} _ {0} ^ {i} | \boldsymbol {x} _ {t} ^ {i}\right)} q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}, \boldsymbol {x} _ {0} ^ {i}\right) \\ \end{array}
$$
Fitting this distribution using a neural network:
$$
\begin{array}{l} q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}\right) \approx p _ {\theta} \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}\right) = \mathbb {E} _ {\boldsymbol {x} _ {0} ^ {i} \sim p _ {\theta} \left(\boldsymbol {x} _ {0} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}\right)} q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}, \boldsymbol {x} _ {0} ^ {i}\right) \\ = \mathbb {E} _ {p _ {\vartheta} \left(\boldsymbol {x} _ {0} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}\right)} \frac {q \left(\boldsymbol {x} _ {t} ^ {i} \mid \boldsymbol {x} _ {t - 1} ^ {i} , \boldsymbol {x} _ {0} ^ {i}\right) q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {0} ^ {i}\right)}{q \left(\boldsymbol {x} _ {t} ^ {i} \mid \boldsymbol {x} _ {0} ^ {i}\right)} \\ = \mathbb {E} _ {p _ {0} \left(\boldsymbol {x} _ {0} ^ {i} \mid \boldsymbol {x} _ {t} ^ {i}\right)} \frac {q \left(\boldsymbol {x} _ {t} ^ {i} \mid \boldsymbol {x} _ {t - 1} ^ {i}\right) q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {0} ^ {i}\right)}{q \left(\boldsymbol {x} _ {t} ^ {i} \mid \boldsymbol {x} _ {0} ^ {i}\right)} \tag {21} \\ = \frac {\sum_ {j} \left[ q \left(\boldsymbol {x} _ {t} ^ {i} \mid \boldsymbol {x} _ {t - 1} ^ {i}\right) q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {\mathbf {0}} ^ {i ^ {(j)}}\right) p _ {\theta} \left(\boldsymbol {x} _ {\mathbf {0}} ^ {i ^ {(j)}} \mid \boldsymbol {x} _ {t} ^ {i}\right) \right]}{q \left(\boldsymbol {x} _ {t} ^ {i} \mid \boldsymbol {x} _ {\mathbf {0}} ^ {i}\right)} \\ = \frac {q \left(\boldsymbol {x} _ {t} ^ {i} \mid \boldsymbol {x} _ {t - 1} ^ {i}\right) \left[ \sum_ {j} q \left(\boldsymbol {x} _ {t - 1} ^ {i} \mid \boldsymbol {x} _ {\mathbf {0}} ^ {i (j)}\right) p _ {\theta} \left(\boldsymbol {x} _ {\mathbf {0}} ^ {i (j)} \mid \boldsymbol {x} _ {t} ^ {i}\right) \right]}{q \left(\boldsymbol {x} _ {t} ^ {i} \mid \boldsymbol {x} _ {0} ^ {i}\right)} \\ \end{array}
$$
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 -  $\left(\boldsymbol{x}_t^i\boldsymbol{Q}_t^{iT}\right)$  and  $\left(\boldsymbol{x}_0^i\bar{\boldsymbol{Q}}_{t-1}^i\right)$  - that are crucial in preserving the propagation rule.
The three rows of  $Q_{t}^{i^{T}}$  correspond to the distribution of  $q(x_{t}^{i}|x_{t - 1}^{i},x_{0}^{i})$  when  $x_{t}^{i}$  is in one of three states:  $S, I$ , and  $R$ . Similarly, the three rows of  $\bar{Q}_t^i$  represent the distribution of  $q(x_{t - 1}^{i}|x_{0}^{i})$  when  $x_0^i$  is in one of three states:  $S, I$ , and  $R$ . Let  $\left[\bar{Q}_t^i\right]_{12}$  and  $\left[\bar{Q}_t^i\right]_{13}$  be denoted by  $q_{t}^{a}$  and  $q_{t}^{b}$  respectively, then  $q_{1}^{a} = \gamma_{1}^{i},q_{1}^{b} = 0$ .
$$
\bar {Q} _ {t} ^ {i} = \left[ \begin{array}{c c c} \prod_ {k = 1} ^ {t} (1 - \beta_ {k} ^ {i}) & \prod_ {k = 1} ^ {t - 1} (1 - \beta_ {k} ^ {i}) \beta_ {t} ^ {i} + q _ {t - 1} ^ {a} (1 - \gamma_ {t} ^ {i}) & q _ {t - 1} ^ {a} (\gamma_ {t} ^ {i}) + q _ {t - 1} ^ {b} \\ 0 & \prod_ {k = 1} ^ {t} (1 - \gamma_ {k} ^ {i}) & \sum_ {j = 1} ^ {t} \prod_ {k = 1} ^ {t - j} (1 - \gamma_ {k} ^ {i}) \gamma_ {k + 1} ^ {i} \\ 0 & 0 & 1 \end{array} \right] \tag {22}
$$
When  $x_{t}^{i}$  and  $x_{0}^{i}$  are in different states, the results of  $q(x_{t - 1}^{i}|x_{t}^{i},x_{0}^{i}) = \left(\pmb{x}_{t}^{i}\pmb{Q}_{t}^{i^{T}}\right)\odot \left(\pmb{x}_{0}^{i}\bar{\pmb{Q}}_{t - 1}^{i}\right)$  are shown in Table 4. The table reveals that  $q(x_{t - 1}^{i}|x_{t}^{i},x_{0}^{i}) = [0,0,0]$  when the propagation rule is violated. Since we are predicting  $x_{0}^{i}$ , there are only two possible states for  $x_{0}^{i}$ :  $S$  (Susceptible) and  $I$  (Infected), with  $x_{0}^{i} = R$  (Recovered) being excluded. In such instances,  $q(x_{t - 1}^{i}|x_{t}^{i},x_{0}^{i})$  can be set to  $[1,0,0]$  to resolve the issue. Furthermore, if  $x_{t}^{i} = R$  and  $x_{0}^{i} = S$ , then  $x_{t - 1}^{i} = S$  would violate the propagation rules. However, as shown in the 9th row of Table 4, the probability of  $x_{t - 1}^{i} = S$  is much smaller than that of  $x_{t - 1}^{i} = I$  and  $x_{t - 1}^{i} = R$ , 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 supervision of the propagation rule. Specifically, when using this supervision function, nodes that have a state of  $R$  are set to  $I$  to enforce the propagation rule.
$$
L _ {\text {c o n s t r a i n 1}} = \operatorname {R e l u} \left(\mathbf {X} _ {\mathbf {t} - \mathbf {1}} - (\mathbf {A} + \mathbf {I}) \mathbf {X} _ {\mathbf {0}}\right) \tag {23}
$$
Table 4: The distribution of the unnormalized  $q(x_{t - 1}^{i}|x_{t}^{i},x_{0}^{i})$  in different cases.
xti	x0i	q(xt-1|i x0i)
S	I	R
S	S	Πt k=1 (1 - βkt) Πt-1 k=1 (1 - βkt)	0	0
I	0	0	0
R	0	0	0
I	S	qtia Πt-1 k=1 (1 - βkt)	Πt k=1 (1 - γkt) qta t-1	0
I	0	Πt k=1 (1 - γkt) Πt-1 k=1 (1 - γkt)	0
R	0	0	0
R	S	qtib Πt-1 k=1 (1 - βkt)	qtia Σt j=1 Πt-1 k=1 (1 - γkt) γt k+1	qtb t-1
I	0	Σt j=1 Πt-1 k=1 (1 - γkt) γt k+1 ·Πt-1 k=1 (1 - γkt)	Σt j=1 Πt-1 k=1 (1 - γkt) γt k+1
R	0	0	1
where  $(\mathbf{A} + \mathbf{I})\mathbf{X}_0$  represents the total number of infected nodes in a given node's first-order neighborhood. Equation 23 penalizes  $\mathbf{X}_0$  if the node is infected and there are no other infected nodes within its first-order neighborhood. To maintain the stability of inferred  $\mathbf{X}_0$  generating  $\mathbf{X}_{t-1}$ , we apply the same constraint to the process. Specifically, we utilize the monotonicity regularization of information diffusion from [22]. If the source set  $\mathbf{X}_0^{(i)}$  is a superset of  $\mathbf{X}_0^{(j)}$ , then the generated  $\mathbf{X}_{t-1}$  resulting from their diffusion needs to satisfy the following equation.
$$
L _ {\text {c o n s t r a i n 2}} = \left\| \max  \left(\mathbf {0}, \mathbf {X} _ {\mathbf {t} - \mathbf {1}} ^ {(\mathbf {j})} - \mathbf {X} _ {\mathbf {t} - \mathbf {1}} ^ {(\mathbf {i})}\right) \right\| ^ {2}, \forall \mathbf {X} _ {\mathbf {0}} ^ {(\mathbf {i})} \supseteq \mathbf {X} _ {\mathbf {0}} ^ {(\mathbf {j})} \tag {24}
$$
D Proofs of lemmas and theorems
D.1 The proof of theorem 3.1
Proof Set the initial infection seed set as:  $\mathbb{S} = \{x_0^{s_1}, x_0^{s_2}, \ldots, x_0^{s_m}\}$ . At the initial moment, the infection status distribution of node  $i$  is:
$$
\left\{ \begin{array}{l} P _ {S} ^ {i = S _ {m}} (0) = 0, P _ {I} ^ {i = S _ {m}} (0) = 1 \\ P _ {S} ^ {i \neq S _ {m}} (0) = 1, P _ {I} ^ {i \neq S _ {m}} (0) = 0 \\ P _ {R} ^ {i} (0) = 0 \end{array} \right. \tag {25}
$$
At time  $t$ , the infection status distribution of node  $i$  is:
$$
\left\{ \begin{array}{l} P _ {I} ^ {i} (t) = P _ {I} ^ {i} (t - 1) \left(1 - \gamma_ {R} ^ {i} (t - 1)\right) + P _ {S} ^ {i} (t - 1) \left[ 1 - \prod_ {j} \left(1 - \beta_ {I} ^ {j} (t)\right) A _ {i j} P _ {I} ^ {j} (t - 1) \right] \\ P _ {S} ^ {i} (t) = P _ {S} ^ {i} (t - 1) \left[ \prod_ {j} \left(1 - \beta_ {I} ^ {j} (t)\right) A _ {i j} P _ {I} ^ {j} (t - 1) \right] \\ P _ {R} ^ {i} (t) = P _ {I} ^ {i} (t - 1) \gamma_ {R} ^ {i} (t) \end{array} \right. \tag {26}
$$
where  $A$  is the adjacency matrix and  $j$  is the neighbor of node  $i$ . When dealing with a static graph  $\mathcal{G}$ ,  $A$  is fixed, allowing for determination of the state distribution at any time based on the initial node state. Specifically, if both graph  $G$  and the seed node set  $\mathbb{S}$  are known, it becomes possible to calculate the state distribution of each node at any given time using Equation 26. Utilizing Equations 2 and 3,  $Q_{t}^{i}$  can also be determined.
D.2 Proof of lemma 3.1
Proof The graph convolution layer with batch normalization  $\mathbf{BN}(g(\xi))$  can be abbreviated as GConv. In our approach, we apply spectral normalization to both the linear transformation  $\mathbf{U}$  and convolutional layers GConv. As a result, the weight parameters of both the networks  $f_w$  and GConv possess 1-Lipschitz continuity after spectral normalization, as described in [28].
Let  $GConv: \mathbb{R}^n \to \mathbb{R}^m$ ,  $x_1, x_2 \in \mathbb{R}^n$ . Note that the nonlinear activation function  $\sigma(\cdot)$  is set to  $Mish(\cdot)$ , which obviously possesses 1-Lipschitz continuity.
$$
\begin{array}{l} \left\| f _ {\theta} \left(x _ {1}\right) - f _ {\theta} \left(x _ {2}\right) \right\| _ {p} = \left\| \sigma (G C o n v \left(x _ {1}\right)) - \sigma (G C o n v \left(x _ {2}\right)) \right\| _ {p} \\ \leq \left\| G C o n v \left(x _ {1}\right) - G C o n v \left(x _ {2}\right) \right\| _ {p} \tag {27} \\ <   \left\| x _ {1} - x _ {2} \right\| _ {p} \\ \end{array}
$$
where  $\| \cdot \| _p$  represents the  $p$ -norm ( $p = 2$  or  $p = \infty$ ). Therefore, the Lipschitz constant of  $f_{\theta}$  is less than 1.
D.3 Proof of theorem 3.2
Proof To prove that  $\mathbf{Y}_{\mathbf{T}} = \mathbf{F}(f_w(\mathbf{X}_0))$  is reversible, it is necessary to ensure that  $F$  and  $f_{w}$  are reversible [43].
$$
\left\{ \begin{array}{l} f _ {w} \left(\mathbf {X} _ {\mathbf {0}}\right) = \mathbf {X} _ {\mathbf {0}} + f _ {w} \left(\mathbf {X} _ {\mathbf {0}}\right) - \mathbf {X} _ {\mathbf {0}} = \xi \\ \xi + f _ {\theta} (\xi) = \mathbf {Y} _ {\mathbf {T}} \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \mathbf {X} _ {\mathbf {0}} = \xi + \mathbf {X} _ {\mathbf {0}} - f _ {w} \left(\mathbf {X} _ {\mathbf {0}}\right) \\ \xi = \mathbf {Y} _ {\mathbf {T}} - f _ {\theta} (\xi) \end{array} \right. \tag {28}
$$
We construct the following iterative formula:
$$
\left\{\begin{array}{l}\mathbf {X} _ {\mathbf {0}} ^ {\mathbf {k} + 1} = \xi + \mathbf {X} _ {\mathbf {0}} ^ {\mathbf {k}} - f _ {w} \left(\mathbf {X} _ {\mathbf {0}} ^ {\mathbf {k}}\right), \mathbf {X} _ {\mathbf {0}} ^ {\mathbf {0}} = \xi\\\xi^ {k + 1} = \mathbf {Y} _ {\mathbf {T}} - f _ {\theta} \left(\xi^ {k}\right), \xi^ {0} = \mathbf {Y} _ {\mathbf {T}}\end{array}\Rightarrow \left\{\begin{array}{l}\lim  _ {k \rightarrow \infty} \mathbf {X} _ {\mathbf {0}} ^ {\mathbf {k}} = \mathbf {X} _ {\mathbf {0}}\\\lim  _ {k \rightarrow \infty} \xi^ {k} = \xi\end{array}\right. \right. \tag {29}
$$
By Lemma 2, we can ensure that  $L_{w} < 1$  and  $L_{\theta} < 1$ . Moreover, the Lipschitz constant of  $(\mathbf{X}_0^{\mathbf{k}} - f_w(\mathbf{X}_0^{\mathbf{k}}))$  is  $1 - L_{w}$ , which is less than 1. Thus, when the number of iterations  $k$  is sufficiently large, it follows that the transformation  $\mathbf{Y}_{\mathbf{T}} = \mathbf{F}(f_w(\mathbf{X}_0))$  is reversible according to the Banach fixed point theorem [4].
D.4 Proof of theorem 3.3
Proof Specifically, Theorem 3.2 proves that  $(\mathbf{F}_1 \circ \mathbf{F}_2 \circ \ldots \circ \mathbf{F}_n)(\xi)$  is invertible when  $n = 1$ . Therefore, we are currently examining whether  $(\mathbf{F}_1 \circ \mathbf{F}_2 \circ \ldots \circ \mathbf{F}_n)(\xi)$  retains its reversibility for  $n > 1$ . To make the notation simpler, we denote  $(\mathbf{F}_1 \circ \mathbf{F}_2 \circ \ldots \circ \mathbf{F}_i)(\xi)$  as  $\hat{\mathbf{F}}_i$ .
$$
\hat {\mathbf {F _ {n}}} = \left(\mathbf {F _ {1}} \circ \mathbf {F _ {2}} \circ \dots \circ \mathbf {F _ {n}}\right) (\xi) = \underbrace {\mathbf {D P} \circ \cdots \circ \mathbf {D P}} _ {n} \left[ \xi + f _ {w} (\xi) + \sum_ {i = 1} ^ {n - 1} f _ {w} \left(\hat {\mathbf {F _ {i}}} (\xi)\right) \right] \tag {30}
$$
The application of the dropout function  $\mathbf{DP}$  will limit the Lipschitz constant of any function  $f$  to  $1 - r$  times its original value:  $L_{\mathbf{DP}(f)} = (1 - r)L_f$ . Additionally, we have  $L_{\hat{\mathbf{F}}_i} \leq \prod_{j=1}^i \left(L_{F_j}\right) \leq (1 - r)^i (1 + L_{f_w})^i$  [13]. Therefore, the Lipschitz constant of  $\hat{\mathbf{F}}_n$  is expressed as:
$$
\begin{array}{l} L _ {\mathbf {\hat {F} _ {n}}} \leq (1 - r) ^ {n} \left[ 1 + L _ {f _ {w}} + \sum_ {i = 1} ^ {n - 1} L _ {f _ {w}} \cdot L _ {\hat {\mathbf {F} _ {i}}} \right] \\ \leq (1 - r) ^ {n} \left[ 1 + L _ {f _ {w}} \sum_ {i = 0} ^ {n - 1} [ (1 - r) (1 + L _ {f _ {w}}) ] ^ {i} \right] \tag {31} \\ = (1 - r) ^ {n} \left[ 1 + L _ {f _ {w}} \cdot \frac {1 - [ (1 - r) (1 + L _ {f _ {w}}) ] ^ {n}}{1 - (1 - r) (1 + L _ {f _ {w}})} \right] \\ \end{array}
$$
Note that when  $L_{f_w} < 1$  and  $n > 1$ , we only need to set  $r = 1/2$  to ensure that  $L_{\hat{\mathbf{F}}_n} < 1$ , thereby guaranteeing that  $\hat{\mathbf{F}}_n$  is reversible.
E DDMSL implementation details
The DDMSL approach has been previously explained, and we will now provide further details on the implementation of DDMSL. The linear transform  $\mathbf{U}$  refers to a fully connected layer, and in Figure 2, the dense layer is composed of two fully connected layers that undergo spectral normalization. The
final output of the  $nn_{\theta}$  is represented by an  $N \times 1$  matrix, indicating the probability that each node is in an infected state at  $t = 0$ . We designate nodes with an infection probability higher than a certain threshold as infected nodes.
Given a complete information diffusion instance  $\mathbf{X} = \{\mathbf{X}_0, \dots, \mathbf{X}_{\mathbf{T}}\}$  where  $\mathbf{X}_{\mathbf{t}} = \left\{x_{t}^{1}, \dots, x_{t}^{N}\right\}$ , we sample  $t \in \{1, \dots, T\}$  to be included in the neural network  $nn_{\theta}$  based on the probability distribution  $p(t) \sim \frac{t}{\sum_{t=1}^{T} t}$ , and use the sine-cosine position encoding [41] to embed  $t$ . The training and inference processes are shown in Algorithm 1 and Algorithm 2, respectively. Additionally, the variable  $T$  remains consistent with the information diffusion step size.
Algorithm 1: Training
Input:  $\mathbf{X}_0, \mathcal{G}, \mathbf{X}_t$ , threshold  $\alpha$   
1 repeat  
2  $Q_t \gets$  Equation 2 and Equation 3  
3  $q(x_{t-1}|x_t) \gets$  Calculate Equation 6 using  $\mathbf{X}_t$  and  $\mathbf{Q}_t$   
4  $t \sim P(\{1, \dots, T\}), P(t) = \frac{t}{\sum_{t=1}^{T} t}$   
5  $\mathbf{X}_0^{t-1} = n n_\theta(\mathcal{G}, \mathbf{X}_t, t)$  // Using data from  $t$ ,  $n n_\theta$  infers the source node  $\mathbf{X}_0^{t-1}$ , which is then used to reconstruct the diffusion graph at  $t-1$ .  
6  $\mathbf{X}_0^{t-1}[\mathbf{X}_0^{t-1} > \alpha] = 1$   
7  $\mathbf{X}_0^{t-1}[\mathbf{X}_0^{t-1} != 1] = 0$   
8  $Q_t' \gets$  Equation 2 and Equation 3  
//  $Q_t'$  is generated by  $\mathbf{X}_0^{t-1}$ .  
9  $P_\theta(x_{t-1}|x_t) \gets$  Calculate Equation 8 using  $\mathbf{X}_t$ ,  $\mathbf{X}_0^{t-1}$  and  $\mathbf{Q}_t'$   
10  $X_{t-1} \gets Gumbel - Softmax(P_\theta(x_{t-1}|x_t))$   
11 Take gradient descent step on:  
12  $\nabla_\theta(L_{simple} + L_{strain})$
Algorithm 2: Infering
9 end
Input:  $\mathbf{X}_{\mathrm{t}},\mathcal{G}$  Empty set  $\mathbb{X}$  , threshold  $\alpha$  Output: X   
1 for t starts from T to 1 do   
2  $\mathbf{X}_0^{t - 1} = nn_\theta (\mathcal{G},\mathbf{X}_t,t)$    
3  $\mathbf{X}_0^{t - 1}[\mathbf{X}_0^{t - 1} > \alpha ] = 1$    
4  $\mathbf{X}_0^{t - 1}[\mathbf{X}_0^{t - 1}! = 1] = 0$    
5  $\mathbf{Q}_{\mathrm{t}}^{\prime}\gets$  Equation 2 and Equation 3   
6  $P_{\theta}(x_{t - 1}|x_t)\gets$  Calculate Equation 8 using  $\mathbf{X}_{\mathrm{t}},\mathbf{X}_{0}^{t - 1}$  and  $\mathbf{Q}_{\mathrm{t}}^{\prime}$    
7  $\mathbf{X}_{\mathrm{t - 1}}\gets Gumbel - Softmax(P_{\theta}(x_{t - 1}|x_t))$    
8  $\mathbb{X}[t]\gets \mathbf{X}_{t - 1}$
In Algorithm 1, we have  $\mathbf{Q}_{\mathbf{t}} = \{Q_{t}^{1},\dots ,Q_{t}^{N}\}$ , where  $Q_{t}^{i}$  can be calculated using Equations 2 and 3. Additionally, the  $P_{S}^{i}(t)$ ,  $P_{I}^{i}(t)$ , and  $P_{R}^{i}(t)$  in Equation 3 can be computed using various methods. Monte Carlo simulations provide the most accurate results, but require at least  $10^{5}$  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  $[P_S^i (t),P_I^i (t),P_R^i (t)]$ , which significantly reduces the training time of the model. When applied to the SI model, DDMSL utilizes the state transfer matrix by:
$$
Q _ {t} ^ {i} = \left[ \begin{array}{c c} 1 - \beta_ {I} ^ {i} (t) & \beta_ {I} ^ {i} (t) \\ 0 & 1 \end{array} \right] \tag {32}
$$
where  $\mathbf{X}_0^{t-1}$  represents the predicted source node using  $\mathbf{X}_t$ , while  $Q_t'$  is generated using the same method as above. Ultimately,  $P_\theta(x_{t-1}|x_t)$  is calculated using  $\mathbf{X}_0^{t-1}$ ,  $\mathbf{X}_t$ , and  $\mathbf{X}_0^{t-1}$ . We obtain  $\mathbf{X}_{t-1}$  by sampling from  $P_\theta(x_{t-1}|x_t)$ , and label nodes as  $0(S)$ ,  $1(I)$ , or  $2(R)$  based on their state. Algorithm 2 proceeds in a similar manner to the process described above.
F Additional algorithms and dataset parameters
F.1 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 as default is common to 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.
Algorithms	Hyper-parameter	karate	jazz	cora_ml	power grid	PGP	Search space	Description
DDMSL	Initial learning rate	2 × 10-3	2 × 10-3	2 × 10-3	2 × 10-3	3 × 10-3	[2 × 10-3,4 × 10-3,5 × 10-3]	Learning rate decreases by 0.97 times the set epoch.
Learning Rate	[1200,1500]	[200,1000]	[500, 1200]	[500, 1200]	[200,500,800,1200]	Determined by the LOSS curves of the training and validation sets.
Decline Interval
n	6	6	6	6	8	[min = 3,max = 9,step = 1]	Number of residual blocks
Dropout rate	0.5	0.5	0.5	0.5	0.5	Determined by Theorem \3.3
α in SIR model	0.4	0.6	0.4	0.4	0.45	[min=0.3,max=0.7,step=0.05]	Threshold
α in SI model	0.4	0.45	0.4	0.4	0.45	[min=0.3,max=0.7,step=0.05]
Epoch	2000	1600	1600	1600	1600	Determined by the LOSS curves of the training and validation sets.
SLVAE	α	0.55	0.5	0.55	0.55	0.45	[min=0.3,max=0.7,step=0.05]	Threshold
Learning rate	2 × 10-3	2 × 10-3	2 × 10-3	2 × 10-3	3 × 10-3	[2 × 10-3,4 × 10-3,5 × 10-3]
GCN-based encoder parameters	[64,128,256]	[64,128,256]	[64,128,256]	[64,128,256]	[64,128,256]	[64,128,256],[128,256,512]	The hidden dim of the encoder
MLP-based decoder parameters	[256,128,1]	[256,128,1]	[256,128,1]	[256,128,1]	[256,128,1]	[256,128,1],[512,256,1]	The hidden dim of the decoder
Epoch	200	200	200	200	200	\
DDMIX	α	0.5	0.5	0.5	0.5	0.5	[min = 3,max = 9,step = 1]	Threshold
Learning rate	2 × 10-3	2 × 10-3	2 × 10-3	2 × 10-3	2 × 10-3	[2 × 10-3,4 × 10-3,5 × 10-3]
Epoch	100	100	100	100	100	\
algorithms are both running on Windows 10 systems and trained using a 4090 graphics card. The code for DDMSL is already open source, please refer to: https://github.com/Ashenone2/DDMSL.
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 [48]: It includes a network of interrelationships between 34 members of the Karate club, comprising 34 nodes and 78 edges. The Karate dataset is a real dataset, widely employed in complex network community discovery research.
- Jazz [12]: The Jazz dataset is 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 [27]: Cora-ML is a citation network dataset containing papers from the field of machine learning. Nodes represent papers and edges represent citation relationships between papers.
- Power Grid [45]: The Power Grid dataset is a network dataset describing the topology of the Northeastern US power grid, containing 4,941 nodes and 6,594 edges.
- PGP [5]: It is a User network of the Pretty-Good-Privacy algorithm for secure information exchange, consisting of 10,680 nodes and 24,316 edges.
Table 6: Statistics of 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
G Additional experiments
Experiments on Real Diffusion Datasets. In order to gauge the efficacy of DDMSL on real-world propagation datasets, we opted for the Twitter [6] and Douban [6] 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 7.
Table 7: Additional experiments on real diffusion datasets.
Twitter	Douban
Methods	PR	RE	F1	AUC	PR	RE	F1	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 on datasets of varying scales to assess the generalization performance of both DDMSL and SLVAE algorithms. The comparative results are presented in Table 8, revealing that DDMSL demonstrates commendable generalization performance across a majority of scenarios.