Optimal stabilization of Boolean networks through collective influence

Boolean networks have attracted much attention due to their wide applications in describing dynamics of biological systems. During past decades, much effort has been invested in unveiling how network structure and update rules affect the stability of Boolean networks. In this paper, we aim to identify and control a minimal set of influential nodes that is capable of stabilizing an unstable Boolean network. For locally treelike Boolean networks with biased truth tables, we propose a greedy algorithm to identify influential nodes in Boolean networks by minimizing the largest eigenvalue of a modified nonbacktracking matrix. We test the performance of the proposed collective influence algorithm on four different networks. Results show that the collective influence algorithm can stabilize each network with a smaller set of nodes compared with other heuristic algorithms. Our work provides a new insight into the mechanism that determines the stability of Boolean networks, which may find applications in identifying virulence genes that lead to serious diseases.

Figure 1

Normalized average Hamming distance $\langle H\rangle$ plotted against the fraction of controllers $q$ in $N-K$ network. The network consists of 20 000 nodes and its average degree is 3. The performances of collective influence, high degree, eigenvector centrality, PageRank, voter rank, and degree product are represented in different colors. The small panel shows the results of $q_c$ with the increase of average degree.

Figure 2

Normalized average Hamming distance $\langle H\rangle$ plotted against the fraction of controllers $q$ in a heterogeneous network. The networks is constructed using the configuration model. The in-degrees of nodes follow a Poisson distribution and the out-degrees are scale-free. The network consists of 1000 nodes and its average degree is 3. The performances of collective influence, high degree, eigenvector centrality, PageRank, voter rank, and degree product are represented in different colors.

Figure 3

Normalized average Hamming distance $\langle H\rangle$ plotted against the fraction of controllers $q$ in the physician network. The network consists of 241 nodes and 1098 edges. The performances of collective influence, high degree, eigenvector centrality, PageRank, voter rank, and degree product are represented in different colors. Simulations are performed for an initial Hamming distance $H\left(t_0\right)=0.01$ and the results of $\langle H\rangle$ are averaged over 100 random initial values.

Figure 4

Normalized average Hamming distance $\langle H\rangle$ plotted against the fraction of controllers $q$ in student social network. The network consists of 1000 nodes and 4175 edges, which is part of the adolescent social network constructed from the survey. The performances of collective influence, high degree, eigenvector centrality, PageRank, voter rank, and degree product are represented in different colors. The small panel shows the performance of the collective influence algorithm when larger radii are applied.

Figure 5

Normalized average Hamming distance $\langle H\rangle$ plotted against the fraction of controllers $q$ in (a) Kauffman's model with $N=5000$ and $K=3$; (b) configuration model with Poisson distributed in-degrees and scale-free out-degrees; (c) physician network consisting of 241 nodes and 1098 edges; (d) student social network with 1000 nodes and 4175 edges.