Model development
In this study, the mini network is established based on the property of small-worldness. Then the robustness of the mini network is evaluated by Monte Carlo Simulation and disruption parameters. In addition, our model can identify the potential vital biomarkers in the disease progression. The framework of the model is shown in Figure 2.
Figure 2  The framework of model.

Modeling mini network
The structure of mini network is shown in Figure 3. The mini network is established based on small-worldness property which means that the markers in the network can affect each other. The interaction among markers involves multiple middle links. For example, Aβ increases the activity of AChE. Then the AChE activates the GSK-3β inducing the hyperphosphorylation of tau. In contrast, P-tau can also affect Aβ by elevating AChE. In our model, these middle links between the biomarkers are represented by transit compartments. The mathematic model of mini network is given in Supplementary Material.
Figure 3  Structure of mini network. Transit compartments with a mean transit time constant τ are used to descript the indirect interactions between these markers.

Estimation of mini network integral disruption parameters
In this study, three mini network integral disruption parameters U, K, and φ are used to evaluate the integral variation of the mini network.
(1)  K = | V a   |   | V b   |          (2)  φ = cos  − 1   V a  · V b   | V a   |  | V b   |          (3)  U = ( V  a   - V  b    )  ( V  a   - V  b    )   T          Va is a vector including levels of all the markers in the mini network in pathogenic state. Vb is a vector including levels of all the markers in the mini network in normal state. |Va| and |Vb| are their modular. The symbol “T” in Equation (3) represents vector transposition. Moreover, a simulation experiment is performed for investigating the physiological significance of the disruption parameters (shown in Investigation of Disruption Parameters Physiological Significance).

Estimation of mini network disruption probability
In this study Monte Carlo Simulation is used to estimate probability of mini network disruption. At the beginning, generate random perturbations for all the biomarkers in the mini network and assess the mini network disruption by U, K, and φ after random perturbation. Finally, calculate probability of network disruption (Equation 4) and its relative error (Equation 5), and repeat the above steps until the relative error is less than 5%.
(4)  p  f   = d  D        (5)  ε  p   = t  α ∕ 2   p  f   ( 1 - p  f    )   p  f   n          tα∕2 is unilateral threshold of t distribution. pf is the probability of mini network disruption. n is the predefined iterative number. d is number of network disruption. D is the current iterative number.

Recognition of potential vital biomarkers
We define the marker with the greatest contribution to the mini network disruption as the potential vital biomarker during the disease progression, and its contribution can be measured by the probability of mini network disruption calculated when only a single marker or a group of markers with interaction is disturbed. For the recognition of potential vital biomarkers, the first step of Monte Carlo simulation needs a minor modification. When evaluating the contribution of the ith marker, it needs to be disturbed and the other markers remain invariant. The joint contribution of multi-marker can be estimated in the same way: disturb the group of the markers to be evaluated and keep the other markers constant.

Model performance evaluation
In this study, we evaluate model performance in two ways. First, check if the mini network disorder probability can be used as a proxy for the disease progression by performing regression analysis of mini network and MMSE.
Second, check if the model can improve the accuracy of AD diagnosis by comparing classification accuracy of AD vs. normal and MCI vs. normal. Two Support vector machines (SVM) are trained for measure the classification performance: SVM based on mini network disruption parameters (U, K, φ) and mini network disorder probability and SVM based on CSF markers (tau, P-tau, and Aβ). The classification performance is evaluated by 10-fold cross-validation.

Investigation of disruption parameters physiological significance
To illustrate significance of U, K, and φ, we perform a simulation experiment. The change of biomarkers is simulated in three cases. The three situations are shown as follows: Single marker changing simulation: Only one marker changes as the gradient.
Multi-marker changing simulation 1: All the markers increase as the gradient. For instance, the gradient is 10%. At this situation, all three markers increase 10%.
Multi-marker changing simulation 2: The first marker increases as the gradient and the other two markers decrease as the gradient. For instance, marker A increases 10% and markers B and C decrease 10%.
Finally, observe the change of the three dynamic parameters U, K, and φ.
For a further insight into the significance of mini network integral disruption parameters, an addition test is performed. Three SVMs are trained for evaluating the contribution of mini network integral disruption parameters in the classification: SVM based on K and φ for assessing the contribution of parameter U, SVM based on K and U for assessing the contribution of parameter φ, SVM based on φ and U for assessing the contribution of parameter K. Then observe the performance of these three SVMs.