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SCIENTIA SINICA Informationis, Volume 48, Issue 9: 1214-1226(2018) https://doi.org/10.1360/N112018-00038

Reconstruction of probabilistic Boolean networks

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  • ReceivedApr 21, 2018
  • AcceptedMay 30, 2018
  • PublishedAug 23, 2018

Abstract

Probabilistic Boolean control networks (PBCNs) have received a great amount of attention in the field of opinion dynamics in social networks and gene (or genetic) regulatory networks. PBCNs have been transferred to state transition probability matrices. Using a Markov chain theory, the PBCN is investigated under a state space framework. In this paper, we address the problem of constructing a probabilistic Boolean control network from a prescribed transition probability matrix. First, an algorithm is given to obtain the realization of a PBCN. Second, because of the non-uniqueness of the logical realization of a PBCN, a modified algorithm is introduced to obtain other realizations of PBCNs. Finally, an illustrative example is given to demonstrate both the efficiency and effectiveness of the proposed algorithms. In addition, the future direction of the research is discussed.


Funded by

国家自然科学基金(61640315,61603125)

河南省高等学校青年骨干教师资助计划(2017GGJS-243)

河南省高等学校重点科研项目(18A110003,17A120001)

河南财经政法大学学术创新骨干支持计划和 河南财经政法大学青年拔尖人才资助计划(hncjzfdxqnbjrc201607)


Acknowledgment

感谢编委和审稿人对本文的初稿提出宝贵的修改意见和建议.


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  •   

    Algorithm 1 概率布尔网络的重构

    Step 1 基于概率逻辑矩阵$L$, 计算第$i$个结点的概率逻辑矩阵 $M_i$, $i=1,2,\ldots,n$;Step 2 从 $M_i$构造$M_i^j$, $j=1,2,\ldots,\ell_i$和相应的概率$p_i^{j}$, $j=1,2,\ldots,\ell_i$, 使得 $$M_i=\mathop{\sum}\limits_{j=1}^{\ell_i}{\rm Pr}(f_i=f_i^j)M_i^j;$$ Step 3 从 $M_i^j$构造 $f_i^j$, $j=1,2,\ldots,\ell_i$.

  •   

    Algorithm 2 概率逻辑矩阵$L$的分解算法

    设$R_0=L$, $k=0$;

    Do while $R_k\neq~0$;

    $k=k+1$;

    Step 1 记 $R_k$ 的第1行中最小的非零元为$p_k$;

    Step 2 构造矩阵 $L_k$,

    \begin{align} & L_k(1,j)=\left\{ \begin{aligned} & 1,\quad j\in\{j|m_j\neq 0\},\\ & 0,\quad j\in\{j|m_j= 0\}, \end{aligned} \right.\\ & L_k(2,j)=1-L_k(1,j); \end{align}

    Step 3 记 $R_{k+1}=R_{k}-p_kL_k$;

    End while

  •   

    Algorithm 3 概率逻辑矩阵$L$的分解算法(改进)

    设$R_0=L$, $k=0$;

    Do while $R_k\neq~0$;

    $k=k+1$;

    Step 1 记 $R_k$ 中最小的非零元为$p_k$;

    Step 2 构造矩阵$L_k$, 考虑$j=1,2,\ldots,2^n$. 记$L_k$的第$j$ 列为$\tiny{\big[\begin{matrix} R_k(1,j)~\\ R_k(2,j)~~\\ \end{matrix}\big]} $,

    当$R_k(1,j)R_k(2,j)\neq~0$, 则$L_k$的第$j$列可取为$\delta_2^1$ 或$\delta_2^2$;

    当$R_k(1,j)=~0$, 则$L_k$的第$j$列可取为$\delta_2^2$;

    当$R_k(2,j)=~0$, 则$L_k$的第$j$列可取为$\delta_2^1$;

    Step 3 记 $R_{k+1}=R_{k}-p_kL_k$;

    End while

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