Controlling edge dynamics in complex networks

The interaction of distinct units in physical, social, biological and technological systems naturally gives rise to complex network structures. Networks have constantly been in the focus of research for the last decade, with considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Here we introduce and evaluate a dynamical process defined on the edges of a network, and demonstrate that the controllability properties of this process significantly differ from simple nodal dynamics. Evaluation of real-world networks indicates that most of them are more controllable than their randomized counterparts. We also find that transcriptional regulatory networks are particularly easy to control. Analytic calculations show that networks with scale-free degree distributions have better controllability properties than uncorrelated networks, and positively correlated inand out-degrees enhance the controllability of the proposed dynamics. The last decade has witnessed an explosive growth of interest in the descriptive analysis of complex natural and technological systems that permeate many aspects of everyday life [3, 11, 42]. Research in network science has mostly been focused on measuring [6, 8, 62], modeling [28, 44] and decomposing [22, 43, 47] network representations of existing natural phenomena in order to deepen our understanding of the underlying systems. Considerably less attention has been dedicated to the various types of network dynamics [19,34,46,52] and even less to the problem of controllability [33, 53, 63], i.e. determining the conditions under which the dynamics of a network can be driven from any initial state to any desired final state within finite time [26,32,59,60]. Structural controllability [31] has been proposed recently as a framework for studying the controllability properties of directed complex networks [32]. In this framework, a linear time-invariant nodal dynamics is assumed on the network, governed by the following equation: ẋ(t) = Ax(t) + Bu(t) (1) where A is the transpose of the (weighted) adjacency matrix of the network, x(t) is a timedependent vector of the state variables of the nodes, u(t) is the vector of input signals, and B is the so-called input matrix which defines how the input signals are connected to the Department of Biological Physics, Eötvös Loránd University, Pázmány Péter sétány 1/a, 1117 Budapest, Hungary. Statistical and Biological Physics Research Group of the Hungarian Academy of Sciences, Pázmány Péter sétány 1/a, 1117 Budapest, Hungary. ∗Corresponding author: [email protected] 1 ar X iv :1 11 2. 59 45 v1 [ ph ys ic s. so cph ] 2 7 D ec 2 01 1 nodes of the network. The dynamics is said to be structurally controllable if there exists a matrix A∗ with the same structure as A such that the network can be driven from any initial state to any final state by appropriately choosing the input signals u(t) [31]. Here, structural equivalence of A and A∗ means that A∗ is not allowed to contain a non-zero entry when the corresponding entry in A is zero. Structural controllability is a general property in the sense that almost all weight combinations of a given network are controllable if the network is structurally controllable for a given B [31, 57]. The minimum number of input signals is then determined by finding a maximum matching in the network, i.e. a maximum subset of edges such that each node has at most one inbound and at most one outbound edge from the matching. The number of nodes without inbound edges from the matching is then equal to the number of input signals required for structural controllability [32]. Perhaps the most striking feature of the structural controllability approach to linear nodal dynamics is that input signals tend to control the hubs of the network only indirectly. In addition, real-world networks that seem to have evolved to control an underlying process (such as transcriptional regulatory networks) need many input signals [32]. This is due to the fact that driven nodes (i.e. those which receive an input signal directly) are not able to control their subordinates independently from each other. However, these results apply only for linear nodal dynamics. In this paper, we examine and describe a dynamics that takes place on the edges of the network, and show that this dynamics leads to significantly different controllability properties for the same real-world networks. 1 Switchboard dynamics in complex networks We study a dynamical process on the edges of a directed complex network G(V,E) as follows. Let x = [xj ] denote the state vector of the process, where one state variable corresponds to each edge of the network. Let y− i and y + i be vectors consisting of those xj values that correspond to the inbound and outbound edges of vertex i, respectively, and let Mi denote a matrix with the number of rows being equal to the out-degree and the number of columns being equal to the in-degree of vertex i. Furthermore, we assume that the dynamics can be influenced from the environment by adding an offset vector ui to the state vector of the outbound edges of any node i. The equations governing the dynamics of the network are then as follows: ẏ i (t) = Miy − i (t)− τ i ⊗ y + i (t) + σiui(t) (2) where τ i is a vector of damping terms corresponding to the edges in y + i (t), σi is 1 if vertex i is a so-called driver node and zero otherwise, and ⊗ denotes the entry-wise product of two vectors of the same size. We call the above the switchboard dynamics (SBD) since each vertex i acts as a small switchboard-like device mapping the signals of the inbound edges to the outbound edges using a linear operator Mi, which is called the mixing or switching matrix from now on. To simplify the equations, state variables and signals like y i , y − i and ui are implicitly considered as time-dependent, even if the time variable t is omitted. Furthermore, note that for an edge v → w, exactly one of the coordinates of uv affects the state of this edge, therefore we can simply introduce a unified input vector u where the jth element uj is simply the component of the offset vectors that affects edge j directly. In some sense, the SBD provides a simplified representation of the underlying dynamic processes of many real-world networks. For instance, in social communication networks, a