## 1 Introduction
For example, the neighbors graph (Neigh) constructs the topology with probability [20], the low interference forest establisher (LIFE) uses the value of nodes covered by link to reestablish the minimum spanning tree (MST) [21], and the fault-tolerant local s panning subgraph (FLSS) [22] guarantees the capacity of fault tolerance. [PERSON] et al. [23] proposed a distributed battery recovery effect aware connected dominating set constructing algorithm (BRE-CDS) to prolong the network lifetime of wireless sensor networks. However, most topology control algorithms in these fields are established in two-dimensional (2D) planes while SCNs is a kind of dynamic networks working in three-dimensional (3D) space. In [24], [PERSON] et al. extended several 2D geometric topologies to 3D case and proposed a new 3D flexible Yao-based graph (FIYG) algorithms for sensor networks. Nevertheless, FYG only considered the static nodes, not involving dynamic situations, and is not appropriate for SCNs with dynamic relative movement [25, 26]. Considering the deficiency of present research, this article focuses on the problem of topology control strategy in SCNs when working in some emergency EO missions. First, we describe the SCNs with time snapshot by using the Graph Theory, and give the definition of SCNs by considering its bounded relative motion. Then, we suppose an emergency situation with cooperative-working SCNs in EO missions and give its mathematical model. Because of the network properties of relative motion, unstable and periodic changes in topology, SCNs requires reliable topology and ISLs to ensure the stability of data transmission. Thus, based on the equation of relative motion of satellites [27], we propose a novel link cost metric using the properties of prediction and periodicity of satellites movement. Also, a general reliability model of SCNs is analyzed to give a selection criterion of choosing suitable ISLs in the proposed topology control strategy. Briefly, the proposed topology control strategy, based on the novel link metric with reliability and the selection criterion, consists of three phases: (i) Satellites in SCNs analyze the network links' connections and obtains the reliability-optimized topology matrix; (ii) The topology matrix is distributed to every satellite in SCNs; (iii) In each topology updating period, SCNs changes the link connections according to the topology matrix. The validity of the proposed strategy will be demonstrated by using comprehensive experiments. In conclusion, the main contributions of this paper, given the state-of-the-art research work, are mainly summarized in the following three aspects:  
* A cooperative work model in SCNs with emergency EO missions is proposed to give an application case when facing emergency situations. The large scale, complex and dynamic SCNs is decoupled into three kinds of satellites through different working modes, which make the topology control of SCNs easier. * A novel link cost metric of reliability is proposed to give the selection criterion for choosing the most reliable data routing paths. Through the laws of satellites relative motion, the periodicity and predictability of satellites' relative position in SCNs are involved in the link cost metric to enhance each chosen ISLs' reliability. * A topology control strategy with reliability assurance is proposed to solve the problem of topology management in the dynamic and unstable SCNs. Through some numeric simulations, the proposed reliable strategy has been tested in term of effectiveness of improving data transmission performance and extending average topology lifetime. The rest of this paper is organized as follows: Section 2 gives a network description about the satellite cluster with EO missions. In Section 3, the link cost metric with reliability assurance is discussed. Section 4 presents a topology control strategy for a satellite cluster network with an EO mission. An example and simulation to furtherly illustrate the performance of the proposed topology is shown in Section 5.