Emergence of complexity in controlling simple regular networks

Quantifying the capacity of a given node or a bunch of nodes in maintaining a system’s controllability is a crucial problem in complex networks and control theory. We give a systematic analysis of the ability of a single node or a pairs of nodes to control an undirected unweighted chain and ring. By combining algebraic theory and graph spectrum analysis, we derive analytic expressions for the control range of some given control inputs and find that complex phenomena emerge even from these simplest graph structures. Specifically, the control range is sensitive to the location of driver nodes and shows complex periodic behaviors. Our findings have implications for evaluating the control range and practically controlling complex networks. Copyright c © EPLA, 2016 Introduction. – Controlling collective dynamics is the ultimate goal of investigating complex natural or technological systems, in which the interaction patterns among dynamical elements modeled as complex networks play crucial roles [1,2]. The past few years have witnessed the rapid development of control theory for complex networked systems [3], among which two representative theoretical frameworks are the structural control theory (SCT) [2] and the exact control theory (ECT) [4]. A great deal of efforts have been motivated by the development of the two control frameworks [5–17]. The central issue in network control is identifying a minimum set of driver nodes, by directly controlling which the network can be fully controlled to reach any target state under certain constraints. Practically, however, some driver node discerned by the control theories may be not externally accessible, rendering full control of a network from a minimum set of driver nodes impossible. On the other hand, in some real situations, it is only necessary to control a fraction of nodes rather than all nodes. Thus, an alternative question is how to determine which nodes are indirectly controllable by directly controlling a given set of externally accessible nodes. Despite some pioneer efforts dedicated to addressing this question [18,19], we still lack a general and theoretical approach available for any types of networks. Even for a very simple regular network, a full understanding of (a)E-mail: xindonggao@mail.bnu.edu.cn (b)E-mail: shenzhesi@mail.bnu.edu.cn (c)E-mail: wenxuwang@bnu.edu.cn the control range in terms of directly controlling a small fraction of nodes is yet lacking. In this letter, we explore the control range of two simple regular graphs —chain and ring— for a given set of external inputs. We offer theoretical predictions for the control range, i.e., the number of controllable nodes for a given set of inputs at any locations. Among the nodes two types of interactions are considered which are characterized by the Laplacian matrix and the adjacency matrix, respectively. We find that complex phenomena emerge in these simple graphs. In particular, the control range is very sensitive to the location of imposed input signals and several periodic behaviors are observed. The analytical prediction based on eigenvector decomposition indicates that the control range is closely related to integer factoring and prime numbers, which accounts for the complex phenomena. Numerical results show an exact agreement with the analytical results for both regular graphs. Our findings gain insight into the complexity in controlling complex networks and are valuable for developing a general theoretical framework for evaluating control range and practically controlling complex networks. Model and method. – Consider a controlled network that has N nodes, as described by the following linear ordinary differential equations [20]: