# A Generative Network Model of the Human Brain Normal Aging Process

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## Abstract

**:**

## 1. Introduction

#### 1.1. Brain Ageing

#### 1.2. Brain Network Modelling

#### 1.3. Contributions of the Research

## 2. Materials and Methods

#### 2.1. Data Acquisition and Processing

#### 2.2. Construction of fMRI Brain Networks

_{ij}, an element of the matrix A, represents the functional connection between node i and node j. Several studies proved different thresholds θ affect functional brain network [32,33], according to previous findings, θ was set to 0.5, as it is a relatively practical and optimized threshold.

#### 2.3. Topological Measures in Functional MRI Brain Networks

## 3. Generative Network Model of Functional Brain Network

#### 3.1. Generative Scheme of Artificial Brain Network

_{i,j}denotes the link between node i and j, e

_{i,j}∈ E. The artificial brain network formation starts from the brain network of young adults (group Young), and it evolves either towards mid-age or old. Each initial brain network of group Young contains a fixed number of nodes, │V│ = 90. Each network contains different numbers of edges. In each iteration (growth and test), one node out of 90 is selected and one connection is established between two neighbours of the chosen node based on connection likelihood model.

- Initialization: Brain network extraction is started by image pre-processing. Experimental brain fMRI datasets of participants across all age groups (group Young, group Mid-age and group Old) are processed, and the relevant initial fMRI brain networks G
_{y}= (g_{y}_{1}, g_{y}_{2}, …, g_{yn}), G_{m}= (g_{m}_{1}, g_{m}_{2}, …, g_{mn}) and G_{o}= (g_{o}_{1}, g_{o}_{2}, …, g_{on}) are obtained. Each initial network consists of a fixed number of nodes │V│ = 90. Edge sets of participants in groups young, mid-age and old are represented as E_{y}= (e_{y}_{1}, e_{y}_{2}, …, e_{yn}), E_{m}= (e_{m}_{1}, e_{m}_{2}, …, e_{mn}), E_{o}= (e_{o}_{1}, e_{o}_{2}, …, e_{on}), with the threshold θ value is being set to 0.5. - Growth: After initialization, artificial brain networks start to evolve from the status of young age. For each brain network, one interested node v out of 90 ROIs is selected according to probability p
_{v}-see Equation (1) below—and the degree of node v is requested no less than 2. A connection is expected to be established between node v′s two neighbour nodes i and j. Node i is selected firstly from v′s neighbour nodes set |Γ(v)|. Moreover, node i is not connected with at least one node in |Γ(v)|. Otherwise, a new interested node v must be selected again based on p_{v}. The network evolution process will not progress until the new qualified node v is found. Connection probability p_{i,j}of node i and the remaining neighbours (which are necessarily unconnected to node i) is calculated by the connection likelihood model. A link will then be established according to the highest connection likelihood, in between node i and j. This step continues to the next step when each brain network within the group succeeds in adding a link. - Two-sample t-test: After a link is established in each network, the t-test is executed to test the difference between the artificial brain networks and the fMRI brain networks (group Mid-age and group Old, respectively), in comparing the number of edges, which is used to evaluate how significant the difference is.
- Process of iterations: Artificial brain networks formation is a process of iterations. Step 2 and step 3 constitute one iteration, which adds one link to the network and performs a two-sample t-test. As new links are being added to artificial brain networks, the p-value of t-test increases. Therefore, the formation process ends when p-value stops increasing. The artificial brain network with the greatest p-value at the current iteration is considered as the final artificial brain network.The selection probability p
_{v}of selecting an interested node v is expressed by Equation (1):$${p}_{v}=\frac{{k}_{v}}{{{\displaystyle \sum}}_{i}^{RO{I}_{s}}{k}_{i}}$$_{v}represents the degree of node v, the p_{v}is positive correlated to the degree of node v. This strategy allows nodes with larger degrees to get a larger probability of being selected.

#### 3.2. Connection Likelihood Model-LNBE

_{i}

_{,j}, taking the CN model as a basis, we investigate whether classifying common neighbours of node i and node j can help with a more practical generative network model for human fMRI brain networks in the process of ageing.

_{i}

_{,j}is the probability that nodes i and j are connected. S

_{i}

_{,j}denotes the contribution of topological similarity to connection probability, and is computed by model LNB. In this study, 1/d

_{i,j}refers to the similarity of anatomical distance similarity of nodes i and j. A comprehensive illustration of how S

_{i}

_{,j}of the LNB works is shown in the following part.

_{1}, …, x

_{n}), in which C represents a dependent class, X = (x

_{1}, …, x

_{n}) denotes the features. By using Bayes’ theorem, the posterior probability P (C|X), can be calculated by P (C), P (X) and P (X|C), given by Equation (3):

_{i}of a given class C is conditionally independent from any other features x

_{j}, for i ≠ j. Thus, posterior probability can be rewritten as Equation (4):

_{i,j}denotes a common neighbour node set of any pair of node i and j. $p\left({O}_{i,j}\right)$ is the probability of node i and j having any common neighbours. $p\left({O}_{i,j}\right)$ and $p\left({\widehat{C}}_{i,j}\right)$ denote prior probabilities of node i and node j being connected and unconnected, respectively. Both prior probabilities can be calculated according to a given graph G (V, E) as follows I Equations (7) and (8):

_{i,j}of node i and node j is defined as the ratio of $p\left({C}_{i,j}|{O}_{i,j}\right)$ to $p\left({\widehat{C}}_{i,j}|{O}_{i,j}\right)$ in Equation (13):

_{i,j}│ξ) can be calculated by the clustering coefficient of ξ, which is given by Equation (14):

_{ξ}denotes the neighbour’s set of node ξ, |E

_{ξ}| represents the number of existing edges among N

_{ξ}, |N

_{ξ}|(|N

_{ξ}| − 1)/2 is the number of edges could possibly exist among N

_{ξ}. As $p\left({C}_{i,j}|\xi \right)+p\left({\widehat{C}}_{i,j}|\xi \right)=1$, we have:

^{−1}can both be neglected in the calculation. It is obvious, if R

_{ξ}= 1, the contribution of topological similarity to connection probability, which is computed by the LNBE model, is equal to the CN model.

_{i,j}obtain different values of ${R}_{\xi}$, which means they make a different contribution to the topological similarity while the connection likelihood of nodes i and j are being calculated.

## 4. Statistical Evaluation of Generative Network Models

_{edges}is the p-value associated with two-samples t-test for the difference in the number of network edges of a set of 90 model-generated artificial brain networks vs. a set of 90 fMRI brain networks. Similarly, P

_{C}, P

_{Q}, P

_{L}, P

_{Eglbl}and P

_{Elcl}are the p-values of two-sample t-tests for the difference in average clustering coefficient, modularity, characteristic path length, global efficiency and local efficiency between artificial brain networks vs. fMRI brain networks, respectively. A high Likeness value indicates a high similarity between the generated artificial brain networks and the observed fMRI brain networks. The Likeness function applied to those artificial networks when the p-values stop increasing. In other words, the Likeness function is applied to mid-age artificial brain networks and to old artificial brain networks.

## 5. Results

#### 5.1. Ageing of the Functional Brain Network

#### 5.2. Evaluation Results—Comparison of the LNBE Model Versus Mechanistic Generative Network Models

_{CAR}, p-value = 0.7095 > 0.05, while C

_{CRA}, p-value = 0.5151 > 0.05). For model CAR, p-values were all above 0.05 in group mid-age. In contrast, among group old, it performed poorly in the index of modularity (Q, p-value = 0.0442 < 0.05) as well as local efficiency (Elcl, p-value = 0.0162 < 0.05), but had a considerably good performance in characteristic path length (L, p-value = 0.9599 > 0.05). For the CRA model, all the observed topological properties showed acceptable performance, as their p-values are above 0.05, apart from the index of local efficiency in both group mid-age (Elcl, p-value = 0.0041 < 0.05) and group old (Elcl, p-value = 0.0298 < 0.05). The Likeness value of the artificial brain network generated by the LNBE model was 4.4794 in group mid-age, and 3.4021 among group old, with both higher than the other models.

#### 5.3. Model LNBE—In Comparison with Functional Brain Networks

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Illustration of the real fMRI brain network topology. (

**a**) Brain network topology of group young, (

**b**) brain network topology of group mid-age, (

**c**) brain network topology of group old. The colours of nodes vary with the node′s modularity classes. Node sizes vary with the node′s degree. Node no. 1 of each brain network was marked.

**Figure 2.**Illustration of six observed topological properties of fMRI brain network. Subfigure (

**a**) is the number of edges of functional brain network, while (

**b**) is for average clustering coefficient, (

**c**) is modularity, (

**d**) characteristic path length, (

**e**) global efficiency and (

**f**) local efficiency. Each subfigure is demonstrated by boxplot, where the comparison is made among different groups, from young through mid-age to old. Small windows within each illustration is a linear regression plot of property value, which represents a general evolution trend of each topological property. Orange and blue represent the increase and decrease of the property value, respectively.

**Figure 3.**Comparison of artificial networks generated by LNBE and fMRI brain networks on six studied topological properties. Subfigure (

**a**) is the comparison of the number of edges between artificial network and functional brain network, while subfigure (

**b**) is for average clustering coefficient, (

**c**) is modularity, (

**d**) characteristic path length, (

**e**) global efficiency and (

**f**) local efficiency. The shaded part is for the fMRI brain network, while the non-shaded part is for the LNBE generated network.

Group Classification | Number of Participants | Sex Ratio (M/F) | Age Range | Mean Age |
---|---|---|---|---|

Young | 30 | 13/17 | 19–30 | 22.7 |

Mid-age | 30 | 16/14 | 31–53 | 42.6 |

Old | 30 | 12/18 | 54–79 | 61.3 |

**Table 2.**A comparison between artificial brain networks and fMRI brain networks among group mid-age and group old.

Comparision | Model | E | C | Q | L | Eglbl | Elcl | Energy |
---|---|---|---|---|---|---|---|---|

(Group Mid-age) artificial brain network vs. fMRI brain network | LNBE | 0.9863 | 0.3766 | 0.8346 | 0.7501 | 0.9953 | 0.5365 | 4.4794 |

CN | 0.9863 | 0.1694 | 0.7246 | 0.7338 | 0.9866 | 0.4997 | 4.1004 | |

CAR | 0.9863 | 0.2761 | 0.4875 | 0.0590 | 0.9158 | 0.5279 | 3.2525 | |

CRA | 0.9863 | 0.2761 | 0.3229 | 0.9799 | 0.8923 | 0.0041 | 3.4617 | |

BA | 0.9863 | 0.4812 | 0.4770 | 0.0092 | 0.8800 | 0.1510 | 2.9847 | |

JC | 0.9863 | 0.3902 | 0.4371 | 0.0119 | 0.2643 | 0.1430 | 2.2328 | |

Random | 0.9863 | 0.1274 | 0.6284 | 0.0152 | 0.0382 | 0.0965 | 1.8920 | |

(Group Old) artificial brain network vs. fMRI brain network | LNBE | 0.9927 | 0.0215 | 0.5624 | 0.6437 | 0.8775 | 0.3043 | 3.4021 |

CN | 0.9927 | 0.0104 | 0.4046 | 0.5923 | 0.8316 | 0.1836 | 3.0132 | |

CAR | 0.9863 | 0.7095 | 0.0442 | 0.9599 | 0.2632 | 0.0162 | 2.9794 | |

CRA | 0.9863 | 0.5151 | 0.1492 | 0.5650 | 0.9346 | 0.0298 | 3.1800 | |

BA | 0.9863 | 0.0496 | 0.0017 | 0.2778 | 0.0756 | 0.1008 | 1.4919 | |

JC | 0.9863 | 0.4574 | 0.0321 | 0.2294 | 0.0223 | 0.5938 | 2.3212 | |

Random | 0.9927 | 0.0022 | 0.1502 | 0.0051 | 0.0407 | 0.4003 | 1.5912 |

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Liu, X.; Si, S.; Hu, B.; Zhao, H.; Zhu, J.
A Generative Network Model of the Human Brain Normal Aging Process. *Symmetry* **2020**, *12*, 91.
https://doi.org/10.3390/sym12010091

**AMA Style**

Liu X, Si S, Hu B, Zhao H, Zhu J.
A Generative Network Model of the Human Brain Normal Aging Process. *Symmetry*. 2020; 12(1):91.
https://doi.org/10.3390/sym12010091

**Chicago/Turabian Style**

Liu, Xiao, Shuaizong Si, Bo Hu, Hai Zhao, and Jian Zhu.
2020. "A Generative Network Model of the Human Brain Normal Aging Process" *Symmetry* 12, no. 1: 91.
https://doi.org/10.3390/sym12010091