Dynamic topology control in optical satellite networks based on algebraic connectivity

Journal Pre-proof Dynamic Topology Control in Optical Satellite Networks Based on Algebraic Connectivity Xianfeng Liu, Xiaoqian Chen, Lei Yang, Quan Chen, Jianming Guo, Shuai Wu PII:

S0094-5765(19)31269-X

DOI:

https://doi.org/10.1016/j.actaastro.2019.09.011

Reference:

AA 7657

To appear in:

Acta Astronautica

Received Date: 10 May 2019 Revised Date:

28 July 2019

Accepted Date: 8 September 2019

Please cite this article as: X. Liu, X. Chen, L. Yang, Q. Chen, J. Guo, S. Wu, Dynamic Topology Control in Optical Satellite Networks Based on Algebraic Connectivity, Acta Astronautica, https:// doi.org/10.1016/j.actaastro.2019.09.011. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 IAA. Published by Elsevier Ltd. All rights reserved.

Dynamic Topology Control in Optical Satellite Networks Based on Algebraic Connectivity Xianfeng Liu1, Xiaoqian Chen2, Lei Yang1, Quan Chen1, Jianming Guo1, Shuai Wu1 1. College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, China 2. Chinese Academy of Military Science, Beijing, China

Abstract Optical communication satellite network (OCSN) has been considered as a superior solution to broadband applications of space-based information, since it has many advantages such as less power and higher data rate. Topology framework is the foundation and premise of OCSN. However, frail links, limited number of optical transceivers and time-varying network topology deteriorate topology control of OCSN. Usually, topology control is mainly conducted in processes of network bootstrapping and reconfiguring. In this paper, we investigate the problem of topology control about constructing robust spanning trees in above two processes. The model of OCSN based on space-time graph is built and link availability is proposed to represent the capability of choosing strong links. In addition, the algebraic connectivity is introduced to measure robustness of spanning trees and then the problem is formulated as a 0-1 multi-objective mixed-integer nonlinear programming (MOMINP) which has been proven to be NP-hard. For fast constructing a spanning tree with large algebraic connectivity and average edge weight in OCSN, we develop the subtree merging and maximizing algebraic connectivity (SMMAC) algorithm that is effectively appropriate for distributed network bootstrapping and reconfiguring, and also discuss the properties of SMMAC containing outcome existence, computational complexity and approximation analysis in this paper. The algorithm is based on a primal decomposition method to solve MOMINP, which can efficiently reduce the complexity. Some simulations are conducted in two typical distributed scenarios. Results demonstrate that, compared with other alternative algorithms, our algorithm can significantly improve algebraic connectivity of OCSN and meanwhile guarantee relatively high average edge weight for the topology of OCSNs.

Key words: optical communication satellite network; topology control; degree constrained spanning trees; algebraic connectivity

1 Introduction Satellite networks have been utilized for providing communication services to fixed and mobile users in various scenarios due to the wide geographical coverage. Recently, increasing applications that require higher data transmission rate and communication security pose a challenge on design of communication satellite network [1]. Fortunately, laser communication can efficiently support these applications since it has many merits such as less power and mass requirements, free licensed spectrum, high bandwidth, among others [2]. With the fast development of satellite communication technologies, increasing attention has been drawn by developing optical communication satellite networks [3] to satisfy various service demands, such as enhanced navigation, emergency communications for disaster management, environmental monitoring, global mobile communication and deep space communication. Actually, OCSN is an optical wireless communication system that uses laser to transmit optical data

signals at high bit rates. Wireless optical communication with high rate and reliability is crucial for networking. Recent studies about investigating channel modeling and modulation schemes for wireless optical communications are helpful for improving performance of optical communication. Escribano et al. [4] propose an index modulation system suitable for optical communications which performs well in Bit Error Ratio (BER) and/or power efficiency. The literature [5] investigates the optimum beam size for free space optical uplink, which is useful for forming optical link and improving uplink data rate. However, it is well known that topology framework is the foundation and premise of network. OCSN consists of various satellites which are distributed in their respective orbits. Compared with terrestrial optical networks, some factors such as the movement of satellites and the inaccuracy of pointing, acquisition and tracking (PAT) systems [6] may cause the OCSNs different from terrestrial optical networks:


Based on orbit dynamics, the trajectory of each satellite node can be predicted;




Inter-satellite communication distance is long and network topology is time-varying;




The optical inter-satellite links are intermittent and fragile;




Each satellite node has limited optical transceivers, which seriously affects the network connectivity. All of these characteristics bring about the challenges on designing suitable topology control

mechanisms for OCSNs. In the past, researches about topology control mainly concentrate on terrestrial ad hoc and sensor networks. However, with the large scale of optical satellite network appearing, topology control in OCSNs gives rise to more attention. Topology control mainly focuses on how nodes form a high-connectivity topology and make the network auto-reconfigurable to adapt various scenarios. During the operation of topology control, there are two important processes which are bootstrapping network and reconfiguring network respectively. The process of bootstrapping is that when each singular node in OCSN is powered on, an effective strategy is applied to schedule all member nodes to form a connected network. The reconfiguration process is that when nodes or links fail in the formed network after the bootstrapping process, an efficient strategy is used to maintain the connectivity of the network. For OCSNs, there exists a variety of topology architectures such as the cluster satellite networks, large-scale satellite networks, mixed satellite networks, etc. Although the real-time movement of each node can be predicted, channel conditions and node states change rapidly, which leads to each node’s having no knowledge of the whole network information at every moment. Hence, the distributed topology control mechanism is more suitable for OCSNs. Nowadays, some researchers adopt graph theory to model optical communication networks and solve topology control problem. Due to the character of no loop, the spanning tree derived from graph theory has been widely studied in many communication networks. For instance, the spanning tree is introduced into OCSNs so as to avoid bridge loops or routing loops [7]. In the graph theory, assuming an undirected graph G with N nodes, a spanning tree T is a connected sub-graph of G with N nodes and N-1 edges [8]. In topology control of communication networks, researchers focus more on the Minimum Weight Spanning Tree (MWST) which is a spanning tree with minimum average edge weight [9]. Usually the edge weight is the cost of establish link between two nodes. An example of spanning tree is presented in Fig.1 (a) where the number inserted edge represents edge weight. The graph contains 6 nodes which can be connected by black edge. Nodes and related red lines form the MWST. Fig.1 (b) indicates that the same graph may produce different MWSTs. Actually, minimum weight spanning tree is equivalent to maximum weight spanning tree in searching strategy. The only difference depends on the mean of edge weight. Thus, minimum weight spanning tree and maximum

weight spanning are uniformly called MWST in following discussions.

(a) a minimum weight spanning tree

(b) a minimum weight spanning tree

Fig.1 An example of minimum weight spanning tree During the process of constructing a spanning tree, each satellite has limited number of optical transceivers, which makes MWST more challenging and the problem is converted into the Degree Constrained Spanning Tree (DCST) problem. Actually, DCST problem has been proven to be NP-hard [10]. Some researchers have proposed heuristic algorithms to obtain an approximate solution of DCST problem [8]. However, these algorithms are mostly centralized algorithms that may be not suitable for bootstrapping a distributed OCSN with initially singular nodes. In [11], a distributed bottom-up bootstrapping algorithm (BUBA) is presented for bootstrapping Free Space Optics (FSO) networks. BUBA can construct a spanning tree with maximal node degree while BUBA does not consider the link availability and node synchronization. Actually, these links with high link availability should be chosen in priority when constructing the spanning tree in order to obtain a network of high connectivity and robustness. In [7], the Distributed Minimum Weight Spanning Tree (DMWST) algorithm is proposed for obtaining a spanning tree with minimal average edge weight in OCSNs. The edge weight represents the cost of link establishment. Nevertheless, DMWST neglects the connectivity and robustness of the dynamic network. Recent researches adopt the algebraic connectivity [12] as a measure for network connectivity and robustness. It is well-known that the same set of nodes and potential links may produce a network with quite different algebraic connectivity since the structure of forming graph is various. In other words, diverse connection among nodes in network might make big difference in connectivity and robustness of the whole network. In a distributed wireless networks, the heuristic Fragment Selection and Merging (FSM) algorithm is presented for topology control whose aim is to construct a robust spanning tree in the network [13]. During bootstrapping, FSM can form a spanning tree with relatively good performance on connectivity and average link weight. In [14], a model of mobile double-tier FSO network is introduced and the Localized Delaunay Triangulation (LDT) algorithm is proposed for the topology control of the upper tier. However, these algorithms are not suitable for OCSN scenarios since OCSNs have salient features compared with terrestrial optical networks such as large scale, topology variation with time and vulnerable links. According to above, we exploit algebraic connectivity as robustness measure and propose a primal decomposition method to construct a Maximum Weight Spanning Tree with Large Algebraic Connectivity (MWSTLAC) in OCSNs. The rest of this paper is organized as follows. Section 2 describes the OCSN system model. The topology control is constructing a spanning tree with high robustness and mainly contains two aspects: bootstrapping network and reconfiguring network. We formulate the problems of topology control as

MOMINP problem in two distributed scenarios respectively. The distributed algorithm based on decomposition for topology control is presented in Section 3. Simulation and results are presented in Section 4. Section 5 concludes this paper.

2 System Model and Problem Formulation 2.1 System Model We consider an OCSN including N satellites in their respective orbit. The location of satellites is varying with space and time while predictable. Thus, we introduce the space-time graph [15] which can capture both the space and time dimensions of the network topology to describe the variation of the OCSN’ topology. An example is shown in Fig. 2.

Fig.2 Space-Time Graph The operating time of OCSN is divided into a series of time slots t = {τ1 ,τ 2 ...τ n } . The motion of satellite is periodical and each satellite has knowledge of which satellites keep in the field-of-view of its terminal in advance. Thus each node might have some potential links with neighboring nodes and nodes except these without potential links can form a potentially connected graph G’ in each time slot. n is equal to the number of finite states of the OCSN. The length of a slot is determined by adjustment time of transceivers and dynamic characteristic of satellites. Due to the PAT system, we can consider relative location of satellites unchangeable within a certain time slot. Global Navigation Satellite System (GNSS)[16] is employed to achieve time synchronization of all nodes and correct orbit perturbation. Thus, topology control algorithm can uniquely operate in each time slot. Moreover, according to features of OCSN, there are some reasonable assumptions and definitions:


Each satellite has a unique identifiers number;




Optical channel between two satellites is full duplex;




Each satellite has limited number of optical transceivers denoted as the degree limitation constraint;




We define that two satellites are ‘connectable’ with each other if and only if they meet both two conditions: (1) link attenuation, the azimuth angle rate and the elevation angle rate are all less than certain threshold values; (2) the link between two satellites is not blocked by anything. According to graph theory, an OCSN can be modeled as a simple and weighted graph G(V t , E t )

at a given time slot t, where V t = {1, 2...N} is the set of vertices representing all satellite nodes. E t = {eijt | i, j ∈ V t , i ≠ j} is the edge set representing potential links among all pair of satellites at slot t.

According to above assumptions and characteristics of optical inter-satellite links [17], we quantify link availability as edge weight shown in formula (1), and the greater the value is, the higher the link availability is.

 0 ε t ≥ ε thr OR φ&t ≥ φ& thr wt =  t t t & & % α ( t ) ε ε ε AND φ φ < <  thr thr

OR ϕ&t ≥ ϕ&thr AND ϕ&t < ϕ&thr

(1)

t where α (t ) is scale factors corresponding to the normalized link attenuation ε% , azimuth angle rate

&t and elevation angle rate ϕ% &t respectively; ε thr is the threshold of link attenuation; φ& & φ% thr and ϕ thr denote the threshold of azimuth angle rate and elevation angle rate respectively. The azimuth angle rate and elevation angle rate are determined by orbital elements, which are discussed in [18]. The normalized processing for link attenuation is shown in equation (2)

ε%=

ε thr − ε t ε thr

(2)

Inter-satellite optical links are not subject to weather conditions or cloud outrages as the satellite orbit is far above the atmosphere [19]. Hence, ε% mainly depends on optical transceiver and link distance. The link attenuation in OCSN [20], [21] is formulated as following

ε=

L2θT2 1 2 DR TT (1 − L p )TR

L≥

DT 2

λ

(3)

where DT and DR are transmitter and receiver telescope diameter respectively; L is link distance between a pair of two satellites; wavelength is denoted as λ ; TT and TR denote the transmission factors ( ≤ 1) of the transmitting and receiving telescopes respectively, LP is the PAT loss caused by misalignment of transmitter and receiver; And the divergence angle resulting from the transmit telescope can be approximated by

θT =

λ DT

(4)

2.2 Problem Formulation In OCSNs, each satellite node shares its ID and channel information with neighbors by antennas based on PAT function. According to predictability of orbit, the location of each node in OCSN can be estimated in ahead of time by other nodes. Hence, in a certain slot, we have a set of potential edges denoted as E pot . The MWSTLAC problem is to choose N-1 edges from E pot to form a spanning tree with relatively large average edge weight meanwhile making the algebraic connectivity maximal. For a weighted graph G, the algebraic connectivity is the second smallest eigenvalue of the weighted Laplacian matrix which is a good indication of graph connectivity and robustness, and the greater the

value is, the higher the network robustness is [22]. The weighted Laplacian matrix is given by − wij , aij = 1, i ≠ j N  L = ∑ win , aij = 1, i ≠ j  n =1 0, i= j 

(5)

Where wij is edge weight between node i and j. aij is the element of the adjacent matrix A which represents potential connection among nodes [12]. In this section, we formulate two separated problems about bootstrapping and reconfiguring in OCSNs. High link availability is corresponding to large edge weight. With respect to the process of bootstrapping, the optimization problem is constructing a spanning tree with maximal the average edge weight and algebraic connectivity denoted as λ2 ( X ) ,which is expressed as:

max

1 E

∑

( i , j )|eij ∈E

wij

(6)

λ2 ( X )

(7)

1T X 1 = 2*( N − 1)

(8)

1 ≤ ∑ xij ≤ d i

(9)

s.t

∀i

i

X ≤A

(10)

xij ∈{0,1}

(11)

The constraint (8) guarantees the network being structure of a tree. There are N satellites in OCSN. Satellite i has at most di optical transceivers shown in formula (9). And X is a matrix which consists of optimal variables xij . In real space environment, the network structure might be broken because of nodes or links failures. For reconstructing new network, satellites affected by link or node failures will adjust their optical antennas to point at other satellites and form new connection. If these satellites independently connect with other satellites, conflicting may happen, which reduces the efficiency of network reconfiguration. Fortunately, the main structure of spanning tree has been formed after the process of bootstrapping. Network information is shared among the whole nodes in OCSN. Reconfiguring strategies based on the information of original spanning tree can be applied to quickly form a new spanning tree. Thus, for reducing adjustment time of optical antennas as much as possible, the objective of the network reconfiguration is that minimizing the transceiver movement while maximizing the algebraic connectivity of new spanning tree. The problem is formulated as

max

λ2 (T )

(12)

min

1T *( X1 ⊕ X 2 )* 1

(13)

1 ≤ ∑ e ∈κ ( v ) xij ≤ d v

(14)

1T X 2 1 = 2*( N − 1)

(15)

X 1 ≤ A1

(16)

X 2 ≤ A2

(17)

xij ∈{0,1}

(18)

s.t ij

where ⊕ is a loop-sum operator in graph theory [23]. For instance, X = {xij }m*n , Y = { yij }m*n and

Z = {zij }m*n , then zij = ( xij + yij ) mod 2 . A1 and A2 are adjacent matrices of original tree and new tree after reconfiguration respectively. On the account of the integer variable xij and the nonlinearity of λ2 ( X ) , this optimization problem is MOMINP which is proven to be NP-hard [24]. For solving MOMINP, some methods based on various evolutionary algorithms are used to compute solutions and choose the optimized solution upon different criterions [25]. However, substantial computation resources are utilized to find Pareto Front, which significantly increases time of topology control in OCSN. In addition, although exhaustive algorithms can find the optimal solution in a small network, these exhaustive algorithms similarly result in high computation complexity. Some centralized algorithms [26] might perform well but not for the large-scale and distributed OCSN. However, our target is to quickly find an optimized solution for topology control in distributed OCSN. Hence, in bootstrapping of OCSN, we decompose the problem into two sub-problems: (1) constructing degree constraint spanning tree; (2) improving the algebraic connectivity respectively. A heuristic algorithm is proposed to construct a spanning tree where an iterative mechanism applied to improve the algebraic connectivity. And for reconfiguring network, a centralized algorithm is proposed to solve it since global network information has been collected in process of bootstrapping.

3 Topology Control Algorithms The spanning tree is structure of no loop containing all vertices of graph G. At beginning, there exist many fragments such as singular node and subtree in graph G. Our algorithm is to asynchronously merge singular nodes or subtrees into a whole spanning tree including all nodes. Note that a subtree is a fragment with a tree structure and a singular node can be also regarded as a special subtree. During the operation of this topology control, each node uses local information to asynchronously connect with its ‘best’ neighbor node or subtree and merge with it, until a whole spanning tree is formed. For better describing the topology control algorithm, some assumptions and definitions are presented below.

3.1 Assumptions and Definitions


ID: Each node is randomly allocated to unique node ID and subtree ID in beginning. The subtree ID of a node is consistent with its root node; Specially, for a singular node, its node ID is equal to subtree ID;




Root: Usually the node with smallest ID plays the role of root. The root node controls all nodes in the subtree;




Candidate Set: Each singular node or subtree has its own potential connecting nodes which form candidate node set (CNS); Accordingly, each edge on the tree has its own candidate edge set (CES)

Eca which is similar to the literature [11];


Short Message: Two nodes use the short message to exchange controlling information about whether establishing link with each other or not; During networking of optical satellites, all optical satellites will produce a large number of short

messages which require a low data rate. However, it is wasting for optical link to send these short messages. Thus, our model employs a way of collaborative RF and optical communication [27] where the optical transmitters are responsible for delivering data information, while the RF acts as a complementary for controlling information.

3.2 Bootstrapping Algorithm This bootstrapping algorithm consists of two phases: (1) construct degree constrained spanning tree based on subtree merging; (2) maximize the algebraic connectivity.

3.2.1 Phase 1: Construct Degree Constrained Spanning Tree based on Subtree Merging The first phase is asynchronously constructing degree constrained spanning tree. Given a tagged node or subtree choosing one of its neighboring to merge, three cases might occur: (1) node-to-node merging; (2) node-to-subtree merging; (3) subtree-to-subtree merging. Since the objective of topology control is to build a tree with large average edge weight and algebraic connectivity, we describe the merging selection mechanisms based on some characteristics of the algebraic connectivity. We consider the case (1) that the node i connects the node j and the resulting sub-graph is denoted as T. The edge weight is wij . λk (T ) is the k-th eigenvalue of the Laplacian matrix about T based on ascending ranking. On account of λ1 (T )=0 , the algebraic connectivity follows as

λ2 (T ) = ∑ i =1 λi (T ) = 2 wij 2

(19)

The second case is that a subtree T1 connects with a singular node i and resulting new subtree is T ' . According to [28], λ2 (T ' ) is bounded as

λ2 (T ' ) ≤ λ2 (T1 )

(20)

In the third case, subtree T1 merges subtree T2 and forms a new subtree T " with λ2 (T " ) bounded as [29]

λ2 (T " ) ≤ min{λ2 (T1 ), λ2 (T2 )}

(21)

From the formula (19) to (21), we can draw a conclusion that a singular node is preferred to connect with another singular node from its candidate set Eca . The selection and merging strategy is that: node i firstly searches ‘best’ neighbor to connect. If there is more than one singular node in i’s candidate set, i will choose the largest edge weight. The incident node is regarded as its ‘best’ neighbor. Assume that

‘best’ neighbor is node j at the moment. Subsequently, node i sends ‘request’ message to j. After receiving ‘request’ message from i, j makes a decision on whether connecting i or not, according to j’s degree constraint and ‘request’ messages from other nodes. And then j would send ‘response’ message to i. Only when the ‘best’ neighbor of j is i, i and j would be connected and a new subtree will be formed. Moreover, we mark the node with the smallest ID as root node of the subtree. After merging, all nodes kept in the new subtree will update the neighbor information and new subtree ID which is equal to that of root node. Note that node ID is unchangeable in a slot. The process iteratively executes until all the cases of node-to-node merging are done. Notably, for avoiding conflict among nodes, the algorithm only executes in the root node. Hereafter, we consider the cases of node-to-subtree and subtree-to-subtree merging. The process of exchanging information among subtrees is almost the same as the case of node-to-node. The only difference is that root node controls all nodes in subtree sending and receiving short messages. According to formula (20) and (21), the algebraic connectivity of new subtree is less than that of original subtree and the algebraic connectivity would decrease with more merging. Thus, the tagged subtree T1 prefers merging singular neighbor nodes. After the process of node-to-subtree, T1 will choose a neighboring subtree with the largest algebraic connectivity to merge with. Finally, only if all nodes are connected or no potential links can be chosen, the algorithm will stop.

(a) Connecting N1 with N 3

(b) Connecting N1 ∪ N 3 with N 5

(a) Merging N1 ∪ N 3 ∪ N 5 with T2

(b) The whole spanning tree

Fig .3 An example of subtree merging A simple example with 7 nodes is illustrated in Fig.3. Each node has a pair of numbers where the first one represents subtree ID and the other is node ID. The edges painted by dotted line represent potential links and solid edges mean selected links. Each link has its own weight denoted as the number next to each edge. According to definitions in section 2.1, links with larger edge weight are always chosen in priority. The red node represents root node. Fig.3 shows that this example contains 4 steps and hypothetically begins at node 1. Each node reads its local neighbor information. According to edge weight, node 1 sends ‘request’ message to node 3. After receiving ‘request’, node 3 checks its local neighbor information and whether its degree exceeds constraint. If node 1 is the ‘best’ neighbor of node 3 and vice versa, node 3 will send ‘response’ message to node 1 and agrees to establish a link with node 1. Notably, if node 5 simultaneously sends ‘request’ message to node 3, node 3 will sent ‘reject’ message to node 5 since the link between node 3 and node 1 has larger link availability than that between node 3 and node 5. Hence, in first step, node 1 connects with node 3 and updates the subtree ID of node 3. Then there still exists a singular node 5. According to the criterion that singular node is preferred than subtree neighbors, node 5 connects with node 3. In the third step, Fig.3 (c) shows the process of subtree merging. The subtree T2 is chosen since it has larger algebraic connectivity than that of subtree T6 . In the example, each subtree has two nodes, the root node selects the link with the largest edge weight. Thus, node 1 connects with node 2. Similarly, the edge between node 3 and node 6 is inserted. Note that in each step, root node controls member nodes updating subtree ID and corresponding neighbor information. The process of connection among nodes is asynchronous. After the fourth step, the algorithm of subtree merging stops and a degree constrained spanning tree is constructed. The phase 1 SMMAC algorithm is presented in Table 1.

Table 1 Phase 1 of SMMAC algorithm

Phase 1: Construct degree constrained spanning tree based on subtree merging 1: Input current member and neighbor information 2: While the whole spanning tree is not constructed do 3:

If there exist singular node neighbors unconnected then

4:

Get rid of nodes not satisfying themselves degree constraint and generate candidate set;

5:

Choose the edge with the largest weight;

6:

If (receive message of connecting confirmation) then

7:

Insert the selected edge;

8:

Update ID and neighbor information;

Else

9: 10:

Go to line 2;

End

11: 12: End

13: If there exist subtree neighbors to be connected then 14: 15:

Get rid of nodes not satisfying degree constrained and form candidate set; Choose the edge connecting to the subtree neighbor with the largest λ2

16:

If (receiving message of connecting confirmation) then

17:

Insert the selected edge;

18:

Update ID and neighbor information;

Else

19:

Go to line 2;

20:

End

21: 22: End 23: End

3.2.2 Phase 2: maximize the algebraic connectivity After a degree constrained spanning tree is built, root node will dominate all nodes on the spanning tree. In other words, for each node, its state information and neighbor information are shared with each other. From the perspective of overall network, each edge belong to the spanning tree might have its own CES. Choosing and exchanging appropriate edges from CES leads to changing the structure of the spanning tree, thereby improving the algebraic connectivity in some way. Thus, we can iteratively choose and exchange edge to maximize algebraic connectivity. For the sake of quickly finding an ideal candidate edge, we introduce the bottleneck matric of branch which is defined in [30]. For a branch B at vertex v in weighted tree T, the bottleneck matric M v ( B ) is a k*k matrix, where k is the number of vertices in the branch. The value of the entry (i, j) is equal to

∑

e∈Pi , j

w−1 ( B ) , where Pi , j represents an edge set of the paths from ui to v and u j to v.

Suppose that T is a weighted tree and T% is a new tree after replacing edges of T. According to [31, 32], if M v ( B ) is entry-wise dominated by M v ( B%) , λ2 (T ) ≥ λ2 (T%) . This theorem means that larger bottleneck matrix for the branch B will give us smaller algebraic connectivity. Then we always can find a branch including a candidate edge which may give us a smaller bottleneck matrix. For reducing the

computation complexity, we only need to compare the of

∑

e∈Pi , j

∑

e∈Pi , i

w−1 ( B ) for each candidate edge instead

w−1 ( B ) , where Pi ,i is the edge set of path ui to v [33]. Moreover, increasing edge weight

contributes to improving the algebraic connectivity on the condition that the structure of graph is unchangeable [12]. The maximum the algebraic connectivity algorithm (MACA) searches candidate edges with larger weight to form ‘real candidate edges’ set and chooses a real candidate edge which will give us the best bottleneck matrix until the algebraic connectivity is not increasing. MACA can be considered as a centralized algorithm since it is based on the outcome generated by phase 1. The root node has controlled all nodes in the tree and executes MACA which is presented in Table 2. Table 2 Phase 2 of SMMAC algorithm

Phase 2: Maximum the algebraic connectivity algorithm 1: Input information of degree constrained spanning tree; 2: While for each edge in the tree do 3:

Establish the set of ‘real candidate edges’ set;

4:

If real candidate edges exist then

5:

Find a real candidate edge which will give us a branch where

6:

∑

e∈Pi ,i

w−1 ( B ) is the smallest;

Else

7:

go to step 2;

8:

End

9:

Insert the selected edge and remove the original edge;

10: End

3.3 Reconfiguration Algorithm The complex space environment might cause might cause troubles in OCSNs: (1) some links are broken; (2) some OCSN nodes fail. In essence, these cases both lead to one or more links failures. An efficient scheduling method is necessary for OCSN reconfiguring. Fortunately, the knowledge of the set of candidate edges for every edge from the bootstrapping process helps reconfiguration algorithm rapidly construct a new reliable spanning tree. Each node saves the state information of the spanning tree before broken. In the case of one link failure, the reconfiguration algorithm establishes a set of remaining candidate edge denoted as Ere and finds a remaining candidate edges which give us the smallest bottleneck matrix. This method is similarly suitable for the case where multiple links are all incident on one failed node. In the case of node failure and the case of multiple links failures in different nodes, more than one connected components are formed and we can make use of the bootstrapping algorithm to reconstruct a new spanning tree due to the fact that SMMAC algorithm can separately operate at any singular node or subtree. The reconfiguration algorithm is presented in Table 3

Table 3 Reconfiguration Algorithm Reconfiguration Algorithm 1:

IF (a link fails and multiple failure links are all incident on one failed node)

2:

Establish the set of ‘remaining candidate edges’;

3:

IF remaining candidate edges exist then

4:

Find a remaining candidate edge which will give us a branch

5:

where

6:

∑

e∈Pi ,i

w−1 ( B ) is the smallest;

END

7:

Insert the selected edge;

8:

END

9:

IF (nodes failure and multiple failure links are not incident on one failed node)

10: 11:

Bootstrapping the OCSN;

END

4 Algorithm Analysis The SMMAC algorithm contains not only a computing process but also a physical process where transceivers change their directions. For further exploring characteristics of our algorithm, existence analysis, computational complexity and approximation guarantee will be presented below.

4.1 Existence Analysis During the dynamically networking about OCSN, it is vital for SMMAC algorithm to quickly find a solution. We will prove existence of solution generated by SMMAC algorithm in this section. Proposition 1: In each slot, given a connected graph G, the proposed SMMAC algorithm guarantees a spanning tree solution. Proof for proposition 1: During the merging, singular node will be preferentially chosen to form subtree and then the merging of subtrees begins. That guarantees that all singular nodes will be connected in early iteration. Moreover, under the premise of satisfying degree constraint, each subtree or node chooses at most one potential edge to connect with other at each iteration, which guarantees that no redundant edge exists in the newly forming tree. From the point of a given subtree, all nodes and subtree will be added to the subtree and then the algorithm of phase 1 terminates. Thus, algorithm of phase 1 eventually generates a spanning tree under the degree constraint. In phase 2, algorithm is based on centralized way to improve the performance of spanning tree. The algorithm of phase 2 will terminate after all potential edges for each edge are traversed. More importantly, for the spanning tree from a connected graph G, the number of potential edges is finite. Thus, our algorithm converges at an optimized solution lastly. Proposition 1 proves the existence of tree structure and the fact that SMMAC will terminate

eventually. The SMMAC of convergence time will be discussed in Section 5.

4.2 Computational Complexity As discussed above, the SMMAC algorithm operates in distributed scenarios. In phase 1, considering the worst case where all nodes are only connected one at a time, the SMMAC takes (N-1) iterations to build a tree. At each iteration, the SMMAC compares at most Eca to find ‘best’ neighbor meanwhile computing algebraic connectivity. The time complexity of computing eigenvalues of N x N matrix is O ( N 2 ) [13]. Hence, the computational complexity of phase 1 is O( N 3 * Eca ) . In phase 2, the tree has (N-1) edges to be replaced by their candidate edges in worst case. For (N-1) edges, the SMMAC takes at most Eca

times searching candidate edge and finding the best bottleneck matrix

takes O ( N ) in worst case. Thus, the computational complexity of phase 2 is O( N 2 * Eca ) . Consequently, the SMMAC algorithm yields an O( N 3 * Eca ) computational complexity bound. For the case of one link failure and the case where multiple failure links are all incident on one failed node, this algorithm takes Ere

to find the best remaining candidate edge. The best bottleneck

matrix takes O ( N ) in worst case. Thus, in above two cases, the complexity is O( N 2 * Eca ) . Moreover, the complexity of bootstrapping algorithm is O( N 2 * Ere ) and Ere ≤ Eca . In sum up, the whole complexity of reconfiguration algorithm is O( N 2 * Eca ) .

4.3 Approximation Analysis The SMMAC algorithm is designed for distributed scenarios and operates on satellite. For building a degree constraint spanning tree, some exact algorithms such as Dual method, Lagrange relaxation and branch and bound method have been proposed. However, they are not suitable for bootstrapping distributed optical satellite network, since a satellite will only have knowledge of its neighboring’s information [13, 33]. In order to reduce computational complexity thereby guaranteeing online computation, SMMAC decomposes the problem and applies some heuristic rules. Node or subtree greedily chooses the best object to merge in each iteration. Moreover, orbit dynamics constraints lead to variation of the number of satellites joining constructing a spanning tree over the time. Thus, the optimality is hard to guarantee. According to existence analysis in Section 4.1, the solution generated by SMMAC algorithm is a structure of tree. We can make use of theorems from spectral graph theory to prove that there exists an upper bound about algebraic connectivity of spanning tree. In previous work, Field [34] demonstrates the algebraic connectivity of a path Pn in given connected graph G. λ2 ( Pn ) = 2*(1 − cos(π / n)) (22)

Where n is the number of vertices traversed by the path. Obviously, for an unweighted path or tree, the diameter d and the number of vertices n satisfy the equation [23] n = d +1

And the diameter is equal to the number of edges traversed by the path. Thus the equation (22) is converted to following formula λ2 ( Pn ) = 2*(1 − cos(π / (d + 1))) (23) More generally, the conclusion can be extended to weighted graph [35]. In other word, Pn is considered as a path with edge weight equal to 1 in some way. Consider a weighted tree T ’ with edge weight less than 1, the inequality d w ≤ d holds. The dimeter of T ’ denoted by d w is numerically equal to sum of edge weight traversed by the longest path in the tree. The lemma from the literature [12] indicates that under the same structure of tree, the operation of decreasing edge weight will reduce the algebraic connectivity. Thus, λ2 (T ) ≥ λ2 (T ' ) holds under the condition that T has the same structure with T ' . Moreover, a tree can be constructed from Pn by attaching pendant vertices and this process does not increase the algebraic connectivity [36]. Base on the above, consider a weight tree T‘ bound to as following

with edge weight, the algebraic connectivity of weighted tree is λ (T ' ) ≤ 2 * (1 − cos(π / ( d + 1)))

(24)

Thus, our proof is completed and the upper bound will be presented in Section 5. Actually, under the unchangeable edge weighted and the same number of nodes, the structure of star is a special tree structure which has the largest algebraic connectivity [35]. There exists a node with degree of N-1 in the structure of star. which greatly exceeds node’s degree constraint in real OCSN. Thus, the optimal algebraic connectivity of spanning tree is far less than the upper bound due to the degree constraint of each satellite and distributed scenario limitation. This challenge is similar for finding a spanning tree with optimal average edge weight. However, some exact methods are proven to efficiently find a spanning tree with optimal average edge weight under certain conditions. Thus, regardless of degree constraint and distributed scenario limitation, we utilize Kruskal method [37] to present the upper bound of average edge weight in Section 5.

5 Simulation Studies For the sake of exploring the performance of proposed algorithm, we conduct some simulations in two typical scenarios. The first scenario is a cluster of satellites based on the project called Edsion Demonstration of Satellite Networks (EDSN) [38]. This scenario 1 consists of 8 satellites and these satellites operate in 500km orbit at 40.5°orbital inclination. In addition, the second scenario is a multi-layer optical satellite network (MLOSN) [39] including 2 GEO satellites, 4 MEO satellites and 12 LEO satellites. The main network topology parameters of two scenarios are presented in Table 3. We omit the argument of perigee, true anomaly and right ascension of ascending node since satellites is even-distributed in each orbit. And Inter-satellite link analyses about azimuth angle rate, elevation angle rate and single-hop distance are illustrated in Fig.3.

Table 3 Network topology parameters Constellation

EDSON

MLOSN

Type

LEO

LEO

MEO

GEO

Number of orbits

8

3

3

1

Number of satellites

1

4

2

2

Orbit height(km)

500

1000

13892

35786

Orbit inclination(°)

26°-33°

89.7

40.78

0

in each orbit

Fig.4 (a) The average azimuth angle rate in EDSN

Fig.4 (d) Part of azimuth angle rates in MLOSN

Fig.4 (b) The average elevation angle rate in EDSN

Fig.4 (e) Part of elevation angle rates in MLOSN

Fig.4 (c) Average inter-satellite distance in EDSN

Fig.4 (f) Part of average inter-satellite distance in MLOSN

Fig.4 Single-hop inter-satellite link analyses For better investigating the topology features of the cluster satellite, we analyze the average azimuth

angle rate and elevation angle rate about EDSN illustrated in Fig.4 (a) and Fig.4 (b) respectively. The fluctuation of the average azimuth angle rate is relatively larger than that of the elevation angle rate. Fig.4 (c) shows that the average inter-satellite distance is from 185km to 425km. For MLOSN, we present part of inter-satellite link analyses in Fig.4 (d), Fig.4 (e) and Fig.4 (f) respectively. We set thresholds for three parameters on the basis of different scenarios in order to guarantee above three parameters being within certain range so that the simulation conforms more to reality. The laser with 800nm wavelength is adopted. We set transceiver telescope diameter as 0.1m and transmission factors of transceiver is 0.8 [21]. The simulating time of first scenario is set as 96 min since EDSON’s period is about 48min shown in Fig.4(c) and similarly according to results in Fig.4 (f), simulating time of the second scenario lasts 24 hours. All simulations are conducted on STK and Matlab. Then in order to evaluate the performance of the SMMAC algorithm, we provide a comparison between alternative schemes: (1) DMWST algorithm [7]; (2) BUBA [11]. For comparison purpose, we employ the algebraic connectivity of resulting spanning tree as a measure of robustness. Each satellite equips 4 optical transceivers. We set the length of a time slot as 1min and sample outcomes duration the whole simulating time. With regard to bootstrapping of network, the algebraic connectivities and the average edge weights achieved by the three algorithms for two scenarios are plotted in Fig.5 and Fig. 6 respectively. In Fig.5 (a), it seems that SMMAC can achieve a larger algebraic connectivity than DMWST and BUBA. For the EDSN, on average, SMMAC achieves 44.31% normalized improvement of algebraic connectivity over DMWST and 34.46% normalized improvement of algebraic connectivity over BUBA. However, the topology with large algebraic connectivity does not always guarantee large average edge weight. As we can see from Fig.5 (b), SMMAC provides about 8.62% reduction of average edge weight than DMWST. But the average edge weight presented by SMMAC still improves 3.61% over that achieved by BUBA. For the MLOSN, compared with DMWST, SMMAC provides a remarkable normalized improvement of algebraic connectivity (78.35%) at slight cost of 8.29% reduction of average edge weight shown in Fig.6 (a) and Fig.6 (b). In addition, SMMAC improves not only 32.28% normalized algebraic connectivity but also 5.34% improvement of average edge weight over BUBA. Although DMWST performs well in average edge weight, it does not guarantee robustness and connectivity of the whole network. According to analyses in Section 4, we also present corresponding upper bounds about the algebraic connectivity and average edge weight in Fig.5 and Fig.6. In the aspect of algebraic connectivity, the gap between SMMAC with the upper bound in EDSN is less than that in MLOSN, which denotes that the algebraic connectivity of the spanning tree built with fewer nodes is more likely to approach the upper bound. For the average edge weight of spanning tree, DMWST derives from Kruskal method in essence. In other word, the average edge weight generated by DMWST is equivalent to that achieved by Kruskal method regardless of degree constraint. Thus the average edge weight generated by DMWST more approaches the upper bound. However, the upper bound about the algebraic connectivity and average edge weight of are inaccessible due to the degree constraint of each satellite and distributed scenario limitation in real OCSN. From Fig.5 (a) and Fig.6 (a), the algebraic connectivities achieved by three algorithms in EDSN are larger than that in MLOSN. That is accord with the fact that more nodes joining network will decrease the algebraic connectivity, which is supported by the lemmas (22) and (23). Moreover, in two scenarios, we also find that BUBA performs better than DMWST in algebraic connectivity. That is because that smaller graph diameter causes better connectivity [29]. During the process of searching,

DMWST only chooses the edge with the largest weight. BUBA may choose an edge with low weight to insert but possibly generating larger algebraic connectivity after two subtrees merging. However, in EDSN, BUBA compared with DMWST, the improvement of algebraic connectivity is not obvious. It may attribute to the fact that fewer nodes reduce the number of feasible solutions and degree constraints similarly impedes algorithms searching the optimal solution discussed in Section 2.2. Nevertheless, SMMA algorithm can achieve the largest algebraic connectivity in two scenarios and meanwhile constructing a spanning with relatively large average edge weight.

(a) Algebraic connectivity of spanning trees in EDSN

(b) Average edge weight of spanning trees in EDSN

Fig.5 Simulation results of spanning trees in EDSN

(a) Algebraic connectivity of spanning trees in MLOSN

(b) Average edge weight of spanning trees in MLOSN

Fig.6 Simulation results of spanning trees in MLOSN For the case of edges interrupted or nodes disabled, we pay more attention to the robustness of new spanning tree. Thus, for two scenarios, the algebraic connectivities of reconstructed tree are illustrated in Fig.7. Similarly, to investigate the performance of proposed algorithm, a comparison with several algorithms is provided. We randomly make edges or nodes disabled in two scenarios and the algebraic connectivities achieved by three algorithms are presented in Fig.7 (a) and Fig.7 (b). The value of each point is the average of 100 randomly operation. In EDSN, SMMAC provide 14.23% normalized improvement of algebraic connectivity over DMWST and 10.61% normalized improvement of algebraic connectivity over BUBA. In MLOSN, the improvement is more obvious. Compared with DMWST and BUBA, SMMAC achieves 47.29% and 29.24% improvement of algebraic connectivity respectively.

(a) Reconfiguration in EDSN

(b) Reconfiguration in MLOSN

Fig. 7 Algebraic connectivities of reconstructed spanning tree SMMAC is an online algorithm and designed for distributed satellite context. The time for convergence is important for SMMAC’s operation in satellite. Our simulations are conducted on the joint software platform. For better describing convergence of SMMAC, we only consider the convergence time of SMMAC apart from time consumed by software interaction and delay caused by information exchanging within nodes. These corresponding results in two scenarios are illustrated in Fig.8 (a) and Fig.8 (b) respectively. For EDSN, results of convergence time about bootstrapping are less than 0.6s seconds and that about reconfiguration not exceeds 0.45 seconds in different time slot. SMMAC in MLOSN takes longer convergence time due to more nodes where bootstrapping and reconfiguration need on more than 1.15 seconds and 0.92 seconds respectively. Moreover, in two scenarios, each convergence time of bootstrapping is less than that of reconfiguration. That is because that reconfiguration is based on overall network information after bootstrapping, which can decrease iterations of algorithm.

(a) Convergence time in EDSN

(b) Convergence time in MLOSN

Fig. 8 Convergence time of SMMAC

6 Conclusions In this paper, we investigate the problem of dynamic topology control in optical communication satellite network. The study of OCSN topology control faces many challenges such as the node degree constraints, frail link and time-varying topology. Firstly, for describing the time-varying topology of

network, OCSN model based on space-time graph is introduced in this paper. The topology over time is divided into a series of independent states. Then, in topology control, we investigate the problem of constructing robust spanning trees in OCSN. This problem mainly lies on how to form a spanning with high robustness and connectivity in distributed scenarios. Link availability is defined and quantized as edge weight in graph. Meanwhile, we use algebraic connectivity as a measure of robustness and formulate a 0-1 MOMINP. Due to its NP-hardness, we propose the SMMAC algorithm for topology control in OCSN. Some properties of SMMAC containing existence of spanning tree, computational complexity and approximation analysis have been discussed. This algorithm is designed to be suitable not only for bootstrapping but also for auto-reconfiguration in distributed OCSN. Finally, in the simulations, we choose two typical OCSNs. The first scenario is that a cluster of satellite network consists of satellites which are distributed in respective orbits and the second scenario is that satellites nodes are initialized in multi-layer satellites system. Simulation results indicate that our SMMAC algorithm outperforms other prior algorithms. In bootstrapping of EDSN, SMMAC provide 44.31% normalized improvement of algebraic connectivity at 8.62% reduction of average edge weight over DMWST and 34.46% normalized improvement of algebraic connectivity at 3.61% improvement of average edge weight over BUBA. For MLOSN, 78.35% and 32.28% normalized improvements of algebraic connectivity are achieved by SMMAC compared with these provided by DMWST and BUBA respectively. For reconfiguration of network, SMMAC finishes approximately 14.23% and 10.61% improvement in EDSN, 47.29% and 29.24% improvement in MLOSN over the two alternative algorithms. According to the relevant simulations and analyses, it is reasonable to believe that our algorithm can effectively solve the problem of topology control in OCSN. ACKNOWLEDGMENT This paper is supported by program ‘National Key R&D Program of China’ (2016YFB0502402)

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School of

Highlights





We investigate topology control in optical communication satellite network (OCSN). A mathematical model of topology control in distributed OCSN is first developed. The problem is formulated as multi-objective mixed-integer nonlinear programming. An effective algorithm based on a primal decomposition is designed for the problem.

Highlights    

We investigate topology control in optical communication satellite network (OCSN). A mathematical model of topology control in distributed OCSN is first developed. The problem is formulated as multi-objective mixed-integer nonlinear programming. An effective algorithm based on a primal decomposition is designed for the problem.