Network controllability concerns 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, and protein–protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment.
Background Consider the canonical linear time-invariant dynamics on a complex network
X ˙ ( t ) = A ⋅ X ( t ) + B ⋅ u ( t ) {\displaystyle {\dot {\mathbf {X} }}(t)=\mathbf {A} \cdot \mathbf {X} (t)+\mathbf {B} \cdot \mathbf {u} (t)}
where the vector X ( t ) = ( x 1 ( t ) , ⋯ , x N ( t ) ) T {\displaystyle \mathbf {X} (t)=(x_{1}(t),\cdots ,x_{N}(t))^{\mathrm {T} }} captures the state of a system of N {\displaystyle N} nodes at time t {\displaystyle t} . The N × N {\displaystyle N\times N}
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![Network controllability: 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,[5] 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.](https://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/YYL2.pdf/page1-1280px-YYL2.pdf.jpg?utm_source=en.wikipedia.org&utm_campaign=parser&utm_content=thumbnail)
