Abstract
Computational analysis of the structure of intra-cellular molecular interaction networks can suggest novel therapeutic approaches for systemic diseases like cancer. Recent research in the area of network science has shown that network control theory can be a powerful tool in the understanding and manipulation of such bio-medical networks. In 2011, Liu et al. developed a polynomial time optimization algorithm for computing the size of the minimal set of nodes controlling a given linear network. In 2014, Gao et al. generalized the problem for target structural control, where the objective is to optimize the size of the minimal set of nodes controlling a given target within a linear network. The working hypothesis in this case is that partial control might be “cheaper” (in the size of the controlling set) than the full control of a network. The authors developed a Greedy algorithm searching for the minimal solution of the structural target control problem, however, no suggestions were given over the actual complexity of the optimization problem. In here we prove that the structural target controllability problem is NP-hard when looking to minimize the number of driven nodes within the network, i.e., the first set of nodes which need to be directly controlled in order to structurally control the target. We also show that the Greedy algorithm provided by Gao et al. in 2014 might in some special cases fail to provide a valid solution, and a subsequent validation step is required. Also, we improve their search algorithm using several heuristics, obtaining in the end up to a 10-fold decrease in running time and also a significant decrease of the size of the minimal solution found by the algorithms.
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Notes
- 1.
An intuitive description of those systems for which a linking is not translated to a valid controlling path is when there exist two targets \(t_1\) and \(t_2\) such that for every path from a driver note d to \(t_1\) there exists another path from d to \(t_2\) using the exact same collection of edges (as a multiset).
- 2.
Note that there is a one-to-one correspondence between genes and proteins; thus, having as target a set of essential genes means the equivalent set of essential proteins.
- 3.
Similar analyses were performed for networks of average degree from 2 to 6, but due to space limitations we concentrate here over average degree 4 networks; similar results were obtained in all cases, with more pronounced differences (for the normalized values) in the case of higher degree networks.
- 4.
Note that computing the driven target control is different than computing the driver control, as for the latter one there is a known polynomial time algorithm computing the size of the minimal (total) driver control set, see [10]. In practice however, we observed that the driver and driven control values are very close to one-another in the case of randomly generated networks and real-life bio-medical networks.
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Czeizler, E., Gratie, C., Chiu, W.K., Kanhaiya, K., Petre, I. (2016). Target Controllability of Linear Networks. In: Bartocci, E., Lio, P., Paoletti, N. (eds) Computational Methods in Systems Biology. CMSB 2016. Lecture Notes in Computer Science(), vol 9859. Springer, Cham. https://doi.org/10.1007/978-3-319-45177-0_5
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