# 网络可控性

Controlling a simple network.

Controlling a simple network.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al.[1] on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

Network Controllability is concerned about the structural controllability of a network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene- gene, miRNA- gene, Protein-protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.

## 背景 Background

Consider the canonical linear time-invariant dynamics on a complex network

Consider the canonical linear time-invariant dynamics on a complex network

$\displaystyle{ \dot{\mathbf{X}}(t) = \mathbf{A} \cdot \mathbf{X}(t) + \mathbf{B}\cdot \mathbf{u}(t) }$

where the vector $\displaystyle{ \mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T} }$ captures the state of a system of $\displaystyle{ N }$ nodes at time $\displaystyle{ t }$.

where the vector $\displaystyle{ \mathbf{X}(t)=(x_1(t),\cdots,x_N(t))^\mathrm{T} }$ captures the state of a system of $\displaystyle{ N }$ nodes at time $\displaystyle{ t }$.

The $\displaystyle{ N \times N }$matrix $\displaystyle{ \mathbf{A} }$ describes the system's wiring diagram and the interaction strength between the components. The $\displaystyle{ N \times M }$ matrix $\displaystyle{ \mathbf{B} }$ identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector $\displaystyle{ \mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T} }$ that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by $\displaystyle{ N_\mathrm{D} }$, whose control is sufficient to fully control the system's dynamics, Liu et al.[3] attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed[3] that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict $\displaystyle{ n_\mathrm{D} =N_\mathrm{D}/N }$ for scale-free and Erdős–Rényi Graphs.[3] However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, $\displaystyle{ \mathbf{A} }$, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.[4]

The $\displaystyle{ N \times N }$matrix $\displaystyle{ \mathbf{A} }$ describes the system's wiring diagram and the interaction strength between the components. The $\displaystyle{ N \times M }$ matrix $\displaystyle{ \mathbf{B} }$ identifies the nodes controlled by an outside controller. The system is controlled through the time dependent input vector $\displaystyle{ \mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T} }$ that the controller imposes on the system. To identify the minimum number of driver nodes, denoted by $\displaystyle{ N_\mathrm{D} }$, whose control is sufficient to fully control the system's dynamics, Liu et al. attempted to combine the tools from structural control theory, graph theory and statistical physics. They showed that the minimum number of inputs or driver nodes needed to maintain full control of the network is determined by the 'maximum matching’ in the network, that is, the maximum set of links that do not share start or end nodes. From this result, an analytical framework, based on the in-out degree distribution, was developed to predict $\displaystyle{ n_\mathrm{D} =N_\mathrm{D}/N }$ for scale-free and Erdős–Rényi Graphs. However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, $\displaystyle{ \mathbf{A} }$, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.

$\displaystyle{ N \times N }$ 的矩阵 $\displaystyle{ \mathbf{A} }$ 描述了系统的连接图以及各组分之间的交互强度。$\displaystyle{ N \times M }$ 矩阵 $\displaystyle{ \mathbf{B} }$ 列出了由外部控制器控制的节点。控制器通过施加给系统的时间相关向量 $\displaystyle{ \mathbf{u}(t) = (u_1(t),\cdots,u_M(t))^\mathrm{T} }$ 来实现对系统的控制。为了确定驱动节点的最小数目，$\displaystyle{ N_\mathrm{D} }$，亦即确定对最少几个节点施加控制便足以完全控制系统的动力学进程，在这方面，Liu等人尝试了将结构控制理论 Structural Control Theory 图论 Graph Theory 统计物理 Statistical Physics 的工具的结合。[3] 他们发现，完全控制一个网络所需要的最少输入或驱动节点，取决于网络的最大匹配 maximum matching，也就是不共享起始顶点和终止顶点的最大边集。从这个结论出发，一个基于入-出度分布的分析框架被开发出来，用于预测无标度网络 scale-free networkER随即图 Erdős–Rényi Graph$\displaystyle{ n_\mathrm{D} =N_\mathrm{D}/N }$值。

--Ricky：“ However, more recently it has been demonstrated that network controllability (and other structure-only methods which use exclusively the connectivity of a graph, $\displaystyle{ \mathbf{A} }$, to simplify the underlying dynamics), both undershoot and overshoot the number and which sets of driver nodes best control network dynamics, highlighting the importance of redundancy (e.g. canalization) and non-linear dynamics in determining control.”这句读不懂，怀疑有语法错误。


It is also notable, that Liu's et al. formulation[3] would predict same values of $\displaystyle{ {n_\mathrm{D}} }$ for a chain graph and for a weak densely connected graph. Obviously, both these graphs have very different in and out degree distributions. A recent unpublished work,[5] questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of $\displaystyle{ {n_\mathrm{D}}^{real} }$ and $\displaystyle{ {n_\mathrm{D}}^\mathrm{rand\_degree} }$ calculated by Liu et al.[3] is notable. If controllability is decided mainly by degree, why are $\displaystyle{ {n_\mathrm{D}}^{real} }$ and $\displaystyle{ {n_\mathrm{D}}^\mathrm{rand\_degree} }$ so different for many real world networks? They argued [3] (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown[5] that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

It is also notable, that Liu's et al. formulation questions whether degree, which is a purely local measure in networks, would completely describe controllability and whether even slightly distant nodes would have no role in deciding network controllability. Indeed, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of $\displaystyle{ {n_\mathrm{D}}^{real} }$ and $\displaystyle{ {n_\mathrm{D}}^\mathrm{rand\_degree} }$ calculated by Liu et al. is notable. If controllability is decided mainly by degree, why are $\displaystyle{ {n_\mathrm{D}}^{real} }$ and $\displaystyle{ {n_\mathrm{D}}^\mathrm{rand\_degree} }$ so different for many real world networks? They argued (arXiv:1203.5161v1), that this might be due to the effect of degree correlations. However, it has been shown that network controllability can be altered only by using betweenness centrality and closeness centrality, without using degree (graph theory) or degree correlations at all.

A schematic diagram shows the control of a directed network. For a given directed network (Fig. a), one calculates its maximum matching: a largest set of edges without common heads or tails. The maximum matching will compose of a set of vertex-disjoint directed paths and directed cycles (see red edges in Fig.b). If a node is a head of a matching edge, then this node is matched (green nodes in Fig.b). Otherwise, it is unmatched (white nodes in Fig.b). Those unmatched nodes are the nodes one needs to control, i.e. the driver nodes. By injecting signals to those driver nodes, one gets a set of directed path with starting points being the inputs (see Fig.c). Those paths are called "stems". The resulting digraph is called U-rooted factorial connection. By "grafting" the directed cycles to those "stems", one gets "buds". The resulting digraph is called the cacti (see Fig.d). According to the structural controllability theorem,[6] since there is a cacti structure spanning the controlled network (see Fig.e), the system is controllable. The cacti structure (Fig.d) underlying the controlled network (Fig.e) is the "skeleton" for maintaining controllability. 示意图显示了一个有向向网络的控制。对于给定的有向网络（图a），可以计算其最大匹配：没有共同头部或尾部的最大边集。最大匹配将由一组顶点不相交的有向路径和有向环组成（请参见图b中的红色边缘）。如果一个节点是匹配边的头部，则该节点是匹配的（图b中的绿色节点）。否则，它是不匹配的（图b中的白色节点）。那些不匹配的节点是需要施加控制的节点，即驱动节点。通过向这些驱动节点注入信号，可以得到一组以起点为输入的有向路径（见图c）。这些路径被称为“茎”。得到的有向图称为U根因子连接。通过将定向周期“嫁接”到那些“茎”，人们就会得到“芽”。得到的有向图称为仙人掌 cacti（见图d）。根据结构可控性定理，由于存在跨越受控网络的cacti结构（参见图e），因此该系统是可控的。受控网络（图e）下的cacti结构（图d）是维持可控性的“骨架”。

### 结构可控性 Structural Controllability

The concept of the structural properties was first introduced by Lin (1974)[6] and then extended by Shields and Pearson (1976)[7] and alternatively derived by Glover and Silverman (1976).[8] The main question is whether the lack of controllability or observability are generic with respect to the variable system parameters. In the framework of structural control the system parameters are either independent free variables or fixed zeros. This is consistent for models of physical systems since parameter values are never known exactly, with the exception of zero values which express the absence of interactions or connections.

### 最大匹配 Maximum Matching

In graph theory, a matching is a set of edges without common vertices. Liu et al.[3] extended this definition to directed graph, where a matching is a set of directed edges that do not share start or end vertices. It is easy to check that a matching of a directed graph composes of a set of vertex-disjoint simple paths and cycles. The maximum matching of a directed network can be efficiently calculated by working in the bipartite representation using the classical Hopcroft–Karp algorithm, which runs in O(E模板:Radic) time in the worst case. For undirected graph, analytical solutions of the size and number of maximum matchings have been studied using the cavity method developed in statistical physics.[9] Liu et al.[3] extended the calculations for directed graph.

By calculating the maximum matchings of a wide range of real networks, Liu et al.[3] asserted that the number of driver nodes is determined mainly by the networks degree distribution $\displaystyle{ P(k_\mathrm{in}, k_\mathrm{out}) }$. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al.[3] would predict same values of $\displaystyle{ {n_\mathrm{D}} }$. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of $\displaystyle{ {n_\mathrm{D}}^{real} }$ and $\displaystyle{ {n_\mathrm{D}}^\mathrm{rand\_degree} }$ calculated by Liu et al.[3] is notable. If controllability is decided purely by degree, why are $\displaystyle{ {n_\mathrm{D}}^{real} }$ and $\displaystyle{ {n_\mathrm{D}}^\mathrm{rand\_degree} }$ so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality[5] over degree (graph theory) based metrics.

By calculating the maximum matchings of a wide range of real networks, Liu et al. asserted that the number of driver nodes is determined mainly by the networks degree distribution $\displaystyle{ P(k_\mathrm{in}, k_\mathrm{out}) }$. They also calculated the average number of driver nodes for a network ensemble with arbitrary degree distribution using the cavity method. It is interesting that for a chain graph and a weak densely connected graph, both of which have very different in and out degree distributions; the formulation of Liu et al. would predict same values of $\displaystyle{ {n_\mathrm{D}} }$. Also, for many real-word networks, namely, food webs, neuronal and metabolic networks, the mismatch in values of $\displaystyle{ {n_\mathrm{D}}^{real} }$ and $\displaystyle{ {n_\mathrm{D}}^\mathrm{rand\_degree} }$ calculated by Liu et al. is notable. If controllability is decided purely by degree, why are $\displaystyle{ {n_\mathrm{D}}^{real} }$ and $\displaystyle{ {n_\mathrm{D}}^\mathrm{rand\_degree} }$ so different for many real world networks? It remains open to scrutiny whether control robustness" in networks is influenced more by using betweenness centrality and closeness centrality over degree (graph theory) based metrics.

While sparser graphs are more difficult to control,[3][5] it would obviously be interesting to find whether betweenness centrality and closeness centrality[5] or degree heterogeneity[3] plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

While sparser graphs are more difficult to control, it would obviously be interesting to find whether betweenness centrality and closeness centrality or degree heterogeneity plays a more important role in deciding controllability of sparse graphs with almost similar degree distributions.

## 复合量子系统的控制与代数图论 Control of composite quantum systems and algebraic graph theory

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively.[10] This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.[11][12]

A control theory of networks has also been developed in the context of universal control for composite quantum systems, where subsystems and their interactions are associated to nodes and links, respectively. This framework permits to formulate Kalman's criterion with tools from algebraic graph theory via the minimum rank of a graph and related notions.

## References

1. Sharma, Ankush; Cinti, Caterina; Capobianco, Enrico (2017). "Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment". Frontiers in Immunology (in English). 8: 918. doi:10.3389/fimmu.2017.00918. ISSN 1664-3224. PMC 5536125. PMID 28824643.
2. Sharma, Ankush; Cinti, Caterina; Capobianco, Enrico (2017). "Multitype Network-Guided Target Controllability in Phenotypically Characterized Osteosarcoma: Role of Tumor Microenvironment". Frontiers in Immunology (in English). 8: 918. doi:10.3389/fimmu.2017.00918. ISSN 1664-3224. PMC 5536125. PMID 28824643.
3. Liu, Yang-Yu; Slotine, Jean-Jacques; Barabási, Albert-László (2011). "Controllability of complex networks". Nature. Springer Science and Business Media LLC. 473 (7346): 167–173. doi:10.1038/nature10011. ISSN 0028-0836.
4. Gates, Alexander J.; Rocha, Luis M. (2016-04-18). "Control of complex networks requires both structure and dynamics". Scientific Reports. Springer Science and Business Media LLC. 6 (1): 24456. doi:10.1038/srep24456. ISSN 2045-2322.
5. Banerjee, SJ; Roy, S. "Key to Network Controllability". arXiv:1209.3737.
6. C.-T. Lin, IEEE Trans. Auto. Contr. 19 (1974). 引用错误：无效<ref>标签；name属性“Lin-74”使用不同内容定义了多次
7. R. W. Shields and J. B. Pearson, IEEE Trans. Auto. Contr. 21 (1976). 引用错误：无效<ref>标签；name属性“Shields-76”使用不同内容定义了多次
8. K. Glover and L. M. Silverman, IEEE Trans. Auto. Contr. 21(1976).
9. L. Zdeborová and M. Mezard, J. Stat. Mech. 05 (2006).
10. Burgarth, Daniel; Giovannetti, Vittorio (2007-09-05). "Full Control by Locally Induced Relaxation". Physical Review Letters. American Physical Society (APS). 99 (10): 100501. arXiv:0704.3027. doi:10.1103/physrevlett.99.100501. ISSN 0031-9007.
11. Burgarth, Daniel; D'Alessandro, Domenico; Hogben, Leslie; Severini, Simone; Young, Michael (2013). "Zero forcing, linear and quantum controllability for systems evolving on networks". IEEE Transactions on Automatic Control. 58 (9): 2349–2354. doi:10.1109/TAC.2013.2250075. MR 3101617.
12. S. O'Rourke, B. Touri, https://arxiv.org/abs/1511.05080.