# Brain Network Modeling Based on Mutual Information and Graph Theory for Predicting the Connection Mechanism in the Progression of Alzheimer’s Disease

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Acquisition and Participants Selection

#### 2.2. Data Preprocessing

**Slice timing correction**was performed to ensure all remaining time points in the correct time domain. (3)

**Realignment**was then executed to eliminate the movement artifact in the BOLD time series. Participants whose head translation exceeded 3.0 mm and participants whose head rotated more than ${3.0}^{\circ}$ were discarded. (4) The functional volumes would subsequently be

**Spatial normalized**to the standard EPI template and re-sliced to $3\times 3\times 3$ mm${}^{3}$ resolution in Montreal Neurological Institute (MNI) space. (5)

**Spatially smoothing**was further performed on the normalized images using a Gaussian kernel of 4 mm full width at half-maximum (FWHM). (6) To reduce the influence of low-frequency drifts and high-frequency noise, temporal band-pass

**filtering**in the frequency range 0.06–0.11 Hz was achieved over each smoothed images. (7) Both linear and quadratic trends were removed. (8) Nuisance covariates such as six head motion parameters, whole-brain signal, cerebrospinal fluid, and white matter were regressed out from the preprocessed data.

#### 2.3. Construction of Real Brain Network

#### 2.4. Synthetic Brain Network Modeling

#### 2.4.1. Network Modeling Steps

**Initialization**step, followed by the

**Connection probabilities calculation**step and the

**Evolution**step. The detailed network modeling steps are listed in the following:

- Initialization: At the beginning of modeling, all participants in groups of NC, MCI and AD were preprocessed and the corresponding inter-regional correlation matrices ${G}_{n}=({g}_{n1},{g}_{n2},\dots ,{g}_{n{k}_{1}})$, ${G}_{m}=({g}_{m1},{g}_{m2},\dots ,{g}_{m{k}_{2}})$, ${G}_{a}({g}_{a1},{g}_{a2},\dots ,{g}_{a{k}_{3}})$ were obtained, where ${k}_{1}$, ${k}_{2}$ and ${k}_{3}$ represent the number of participants in NC, MCI, and AD, respectively. We calculated the average correlation matrix of each group, and each average correlation matrix was threshold with the same $\theta =0.15$ to construct the corresponding real brain networks, i.e., $\overline{{G}_{n}}$ for NC, $\overline{{G}_{m}}$ for MCI and $\overline{{G}_{a}}$ for AD. The real brain network in each group consisted of a constant number of nodes, $\left|V\right|=90$. The connection number of $\overline{{G}_{n}}$, $\overline{{G}_{m}}$ and $\overline{{G}_{a}}$ were represented by $|{E}_{n}|$, $|{E}_{m}|$ and $|{E}_{a}|$, respectively. In current work, we studied the evolution process of AD networks from two stages, i.e., the stage from NC to MCI and the stage from NC to AD. By comparing the elements in the binary graphs $\overline{{G}_{n}}$, $\overline{{G}_{m}}$ and $\overline{{G}_{a}}$, we obtained a constant number ${\alpha}_{1}$ of connections that need to be added; another constant number ${\beta}_{1}$ of connections that need to be deleted in the stage from NC to MCI; and ${\alpha}_{2}$ connections to be added and ${\beta}_{2}$ connections to be deleted from NC to AD. Here, ${\alpha}_{1}\le {\beta}_{1}$ and ${\alpha}_{2}\le {\beta}_{2}$ because a declining number of connections was found when comparing $|{E}_{m}|$ and $|{E}_{a}|$ with $|{E}_{n}|$ ($|{E}_{a}|\le |{E}_{n}|\le |{E}_{m}|$). It is worth mentioning that we call the real brain network $\overline{{G}_{n}}$ of NC the initial network in our modeling, and call $\overline{{G}_{m}}$ and $\overline{{G}_{m}}$ the real target brain networks (TN).
- Connection probabilities calculation: After initialization, we calculated the connection probabilities of any node pairs in $\overline{{G}_{n}}$ according to the proposed connection probabilities models, i.e., ECM and MINM introduced in the following subsection. Then, we sorted each node pair in line with its connection probability. The node pair with the largest connection probability and the node pair with the smallest connection probability were recorded, respectively.
- Evolution: Our model started to evolve from the initial graph $\overline{{G}_{n}}$. In each iteration, a random number was generated to decide whether to add or delete one connection in $\overline{{G}_{n}}$. The node pair with the largest connection probability will establish a link if its two nodes disconnected with each other. Meanwhile, the node pair with the smallest connection probability would cut off its link if there were a connection between its two nodes. It should be noted that each node must have a connection to ensure the connectivity of the synthetic network. Therefore, a new pair of nodes must be chosen according to the sorted connection probabilities, if either node’s connection number in the node pair is equal to 1 when deleting the link between them. We upgraded connection set in $\overline{{G}_{n}}$ at the end of this step.
- End of the modeling: Our model ran the above steps of connection probabilities calculation and evolution round by round. The simulation did not proceed to the end, if ${\alpha}_{1}$ connections were established and ${\beta}_{1}$ connections were deleted successfully for $\overline{{G}_{n}}$ in the stage from NC to MCI; or ${\alpha}_{2}$ connections were established and ${\beta}_{2}$ connections were deleted in the stage from NC to AD. Finally, we obtained two synthetic networks with the same connection size as $\overline{{G}_{m}}$ and $\overline{{G}_{a}}$, respectively.

#### 2.4.2. Connection Probabilities Models

**topological similarity**of common neighbors (CN) and the

**Euclidean distance similarity**between two brain regions are treated as essential impactors in brain networks modeling. The connection probability of ECM is defined in the following function:

- Connection probability of ECM: The more common neighbors (CN) that node u and node v have, the higher topological similarity to one another [48]. The number of common neighbors between node u and node v can be described by$${S}_{(u,v)}^{CN}=|\Gamma \left(u\right)\bigcap \Gamma \left(v\right)|$$$$P(u,v)={{S}_{(u,v)}^{CN}}^{\gamma}\xb7E{(u,v)}^{-\eta}.$$

**Definition**

**1.**

**Definition**

**2.**

- Connection probability of MINM: Given a pair of node $(u,v)$, whose common neighbors can be represented by ${\omega}_{u,v}=\Gamma \left(u\right)\bigcap \Gamma \left(v\right)$, the connection probability of MINM between them can be given by$$P(u,v)={{S}_{(u,v)}^{MI}}^{\gamma}\xb7E{(u,v)}^{-\eta}.$$$${S}_{u,v}^{MI}=-I\left({L}_{u,v}^{1}\right|{\omega}_{u,v})$$

#### 2.5. Evaluation of Synthetic Networks

## 3. Results

#### 3.1. Topological Differences in Brain Networks of NC, MCI and AD

#### 3.2. Network Modeling of the Stage from NC to MCI

#### 3.3. Network Modeling of the Stage from NC to AD

#### 3.4. Degree Distribution

#### 3.5. Topological Properties of the Synthetic Networks Generated by MINM with Different $\lambda $ and $\eta $

#### 3.6. Connections Deleted in the Early Stage from NC to AD

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Topological differences among the real brain networks of NC, MCI and AD. NC represents the real brain network of normal control (NC) group; MCI and AD are the real brain networks of Mild cognitive impairment (MCI) and Alzheimer’s disease (AD) group, respectively.

**Figure 2.**Topological properties of the synthetic brain networks generated by various models and the real target brain network (TN) of MCI.

**Figure 3.**Topological properties of the synthetic brain networks generated by various models and the real target brain network (TN) of AD.

**Figure 4.**Degree distributions of the real target brain networks (TN) and the synthetic brain networks.

**Figure 5.**The changes of topological properties of the synthetic networks generated by MINM with different $\lambda $ and $\eta $ in the stage from NC to AD.

**Table 1.**Demographic and clinical characteristics of the participants in Normal controls (NC), Mild cognitive impairment (MCI) and Alzheimer’s disease (AD) groups.

NC | MCI | AD | |
---|---|---|---|

Number | 62 | 45 | 40 |

Gender (Male/Female) | 27/35 | 20/25 | 21/19 |

Age | 73.95 ± 4.83 | 74.38 ± 4.92 | 74.86 ± 5.52 |

MMSE score | 28.72 ± 1.06 | 27.68 ± 1.86 | 22.36 ± 2.77 |

CDR score | 0.00 ± 0.00 | 0.51 ± 0.17 | 0.93 ± 0.16 |

Property Name | Symbol | Description |
---|---|---|

Clustering coefficient | C | It is a measure of the number of triangles in a graph. |

Local efficiency | Eloc | It is a measure to quantify the efficiency of local information transmission. |

Global efficiency | Eglob | It is a measure to quantify the efficiency of global information transmission. |

Characteristic path length | L | L is the average shortest path length between all node pairs in the network. |

Modularity | M | It is used to detect the strength of the division of a network into communities. |

Transitivity | T | It measures the probability that the adjacent nodes of a node are connected. |

Degree | k | It indicates the number of links connecting with a node. |

Models | Abbreviation | Mathematical |
---|---|---|

Preferential Attachment [49] | PA | ${S}_{u,v}^{PA}=|\Gamma \left(u\right)|\times |\Gamma \left(v\right)|$ |

Jaccard [50] | JC | ${S}_{u,v}^{JC}=\frac{|\Gamma (u)\bigcap \Gamma (v\left)\right|}{|\Gamma (u)\bigcup \Gamma (v\left)\right|}$ |

Adamic–Adar [51] | AA | ${S}_{u,v}^{AA}={\displaystyle \sum _{\xi \in |\Gamma (u)\bigcap \Gamma (v\left)\right|}}\frac{1}{log|\Gamma (\xi \left)\right|}$ |

Resource Allocation [52] | RA | ${S}_{u,v}^{RA}={\displaystyle \sum _{\xi \in |\Gamma (u)\bigcap \Gamma (v\left)\right|}}\frac{1}{|\Gamma (\xi \left)\right|}$ |

**Table 4.**The optimal $SI$-value of different brain network models for networks modeling in the stage from NC to MCI. ${\xi}_{C}$ is the relative error of clustering coefficient between synthetic networks and the real target brain network (TN); ${\xi}_{Eloc}$ and ${\xi}_{Eglob}$ represent the relative errors of local efficiency and global efficiency; and ${\xi}_{M}$, ${\xi}_{L}$, and ${\xi}_{T}$ are the relative errors of modularity, the characteristic path length, and transitivity, respectively. A larger value of $SI$ indicates that the model could generate synthetic networks with properties more similar to the real target brain network of MCI.

Models | $\mathit{\lambda}$ | $\mathit{\eta}$ | ${\mathit{\xi}}_{\mathit{C}}$ | ${\mathit{\xi}}_{\mathit{Eloc}}$ | ${\mathit{\xi}}_{\mathit{M}}$ | ${\mathit{\xi}}_{\mathit{L}}$ | ${\mathit{\xi}}_{\mathit{Eglob}}$ | ${\mathit{\xi}}_{\mathit{T}}$ | SI |
---|---|---|---|---|---|---|---|---|---|

ECM | 0.2 | 1.6 | 0.0687 | 0.0302 | 0.0713 | 0.1851 | 0.1413 | 0.0205 | 1.9339 |

PA | 0.2 | 1.8 | 0.0213 | 0.0187 | 0.0639 | 0.1464 | 0.1356 | 0.0480 | 2.4010 |

AA | 0.2 | 1.4 | 0.0811 | 0.0578 | 0.1210 | 0.1324 | 0.0514 | 0.0173 | 2.1692 |

RA | 0.4 | 2.0 | 0.0975 | 0.0638 | 0.0465 | 0.1345 | 0.0502 | 0.0164 | 2.4358 |

JC | 0.2 | 0.2 | 0.0844 | 0.0515 | 0.1082 | 0.0861 | 0.0789 | 0.0126 | 2.3714 |

MINM | 0.4 | 2.0 | 0.0816 | 0.0077 | 0.0727 | 0.0598 | 0.0292 | 0.0010 | 3.9683 |

Random | – | – | 0.1406 | 0.1132 | 0.1796 | 0.1558 | 0.1204 | 0.2068 | 1.0912 |

**Table 5.**The optimal $SI$-value of different brain network models for networks modeling in the stage from NC to AD. ${\xi}_{C}$ represents the relative error in clustering coefficient between synthetic networks and the real target brain network (TN); ${\xi}_{Eloc}$ represents the relative error in local efficiency; ${\xi}_{M}$ is the relative error in modularity; ${\xi}_{L}$ is the relative error in the characteristic path length; ${\xi}_{Eglob}$ is the relative error in global efficiency; and ${\xi}_{T}$ is the relative error in transitivity.

Models | $\mathit{\lambda}$ | $\mathit{\eta}$ | ${\mathit{\xi}}_{\mathit{C}}$ | ${\mathit{\xi}}_{\mathit{Eloc}}$ | ${\mathit{\xi}}_{\mathit{M}}$ | ${\mathit{\xi}}_{\mathit{L}}$ | ${\mathit{\xi}}_{\mathit{Eglob}}$ | ${\mathit{\xi}}_{\mathit{T}}$ | SI |
---|---|---|---|---|---|---|---|---|---|

CN | 0.4 | 1.2 | 0.0426 | 0.0491 | 0.1263 | 0.0665 | 0.1460 | 0.0419 | 2.1169 |

PA | 0.2 | 1.6 | 0.0194 | 0.0289 | 0.0395 | 0.0551 | 0.1377 | 0.0515 | 3.0111 |

AA | 0.2 | 1.6 | 0.0386 | 0.0339 | 0.1291 | 0.0576 | 0.0551 | 0.0881 | 2.4851 |

RA | 0.2 | 2.0 | 0.0448 | 0.0245 | 0.1370 | 0.0732 | 0.0496 | 0.0758 | 2.4697 |

JC | 0.4 | 0.4 | 0.0653 | 0.0661 | 0.1249 | 0.0364 | 0.0786 | 0.0709 | 2.2614 |

MINM | 0.2 | 1.8 | 0.0432 | 0.0358 | 0.0835 | 0.0253 | 0.0303 | 0.0368 | 3.9231 |

Random | – | – | 0.0888 | 0.1355 | 0.2304 | 0.0534 | 0.0611 | 0.1476 | 1.3951 |

**Table 6.**Detailed connections deleted in the early stage of transition. It should be noted that (85,2) in this table means there is a connection between Region 85 and Region 2. The first column “Deleted connections number = 10” records the first ten connections deleted from the NC brain network; the second and the third columns record the additional connections deleted when our model evolved to the further steps.

Deleted Connections Number = 10 | Deleted Connections Number = 20 | Deleted Connections Number = 30 |
---|---|---|

(85,2) (53,4) | (54,7) (53,2) | (51,10) (54,23) |

(77,12) (85,4) | (51,14) (49,26) | (51,23) (54,24) |

(51,4) (86,85) | (52,7) (54,3) | (50,7) (49,14) |

(51,8) (51,2) | (53,14) (51,24) | (53,24) (54,13) |

(53,8) (85,8) | (49,8) (49,10) | (49,24) (52,9) |

**Table 7.**Description of topological properties changes in the progress of deleting a different number of connections.

Deleted Connections Number | C | $\mathit{Eloc}$ | M | L | $\mathit{Eglob}$ | T |
---|---|---|---|---|---|---|

0 | 0.5906 | 0.7667 | 0.2691 | 2.0082 | 0.5651 | 0.5672 |

10 | 0.5829 | 0.7626 | 0.2723 | 2.0109 | 0.5639 | 0.5492 |

20 | 0.5784 | 0.7626 | 0.2760 | 2.0254 | 0.5616 | 0.5459 |

30 | 0.5764 | 0.7626 | 0.2772 | 2.0397 | 0.5575 | 0.5438 |

AD | 0.5190 | 0.7258 | 0.3045 | 2.2821 | 0.5277 | 0.5027 |

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**MDPI and ACS Style**

Si, S.; Wang, B.; Liu, X.; Yu, C.; Ding, C.; Zhao, H. Brain Network Modeling Based on Mutual Information and Graph Theory for Predicting the Connection Mechanism in the Progression of Alzheimer’s Disease. *Entropy* **2019**, *21*, 300.
https://doi.org/10.3390/e21030300

**AMA Style**

Si S, Wang B, Liu X, Yu C, Ding C, Zhao H. Brain Network Modeling Based on Mutual Information and Graph Theory for Predicting the Connection Mechanism in the Progression of Alzheimer’s Disease. *Entropy*. 2019; 21(3):300.
https://doi.org/10.3390/e21030300

**Chicago/Turabian Style**

Si, Shuaizong, Bin Wang, Xiao Liu, Chong Yu, Chao Ding, and Hai Zhao. 2019. "Brain Network Modeling Based on Mutual Information and Graph Theory for Predicting the Connection Mechanism in the Progression of Alzheimer’s Disease" *Entropy* 21, no. 3: 300.
https://doi.org/10.3390/e21030300