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Abstract

Space–ground integrated network, a strategic, driving, and irreplaceable infrastructure, guarantees the development of economic and national security. However, its natures of limited resources, frequent handovers, and intermittently connected links significantly reduce the quality of service. To address this issue, a quality-of-service-aware dynamic evolution model is proposed based on complex network theory. On one hand, a quality-of-service-aware strategy is adopted in the model. During evolution phases of growth and handovers, links are established or deleted according to the quality-of-service-aware preferential attachment following the rule of better quality of service getting richer and worse quality of service getting poor or to die. On the other hand, dynamic handover of nodes and intermittent connection of links are taken into account and introduced into the model. Meanwhile, node heterogeneity is analyzed and heterogeneous nodes are endowed with discriminate interactions. Theoretical analysis and simulations are utilized to explore the degree distribution and its characteristics. Results reveal that this model is a scale-free model with drift power-law distribution, fat-tail and small-world effect, and drift character of degree distribution results from dynamic handover. Furthermore, this model exerts well fault tolerance and attack resistance compared to signal-strength-based strategy. In addition, node heterogeneity and quality-of-service-aware strategy improve the attack resistance and overall quality of service of space–ground integrated network.

Keywords

Space–ground integrated network scale-free network model dynamic evolution model quality-of-service-aware power-law distribution dynamic handover node heterogeneity complex network Poisson birth and death mechanisms

Introduction

Space–ground integrated network (SGIN)^1,2 is a highly heterogeneous complex network which adopts satellite network as backbone network and consists of deep-space network, near-space network, and terrestrial network. It has received increasing attentions^1,2 for its potential in achieving all-round development of strategic information services in fields such as national security, aerospace engineering, and military. However, its natures of limited resources, frequent handovers, and intermittently connected links greatly reduce quality of service (QoS) of SGIN. What’s more, limitation of costs and technologies aggravates the challenges in investigating practical SGIN. Thus, it is of great significance to build a SGIN evolution model in order to optimize its topology and address the QoS issue.

Complex network theory, a useful tool for exploring the macroscopic statistical characteristics, microcosmic evolution mechanism, and dynamic features of complex systems, provides a new idea for optimizing QoS of SGIN. Barabási and Albert³ successfully constructed a scale-free network model with power-law distribution considering the growth and preferential attachment of practical network in 1999. Later, considerable studies focus on the generation mechanism of vertexes, preferential attachment, and robustness of complex network. In terms of node generation mechanism, a linear growth evolution model (LGEM) and an accelerated growth evolution model (AGEM) are proposed for wireless sensor networks (WSNs) by Zheng et al.⁴ Considering that the time of nodes entering the network are not uniform, a Poisson dynamics of WSNs is explored by Jiang et al.⁵ Luo et al.⁶ take the potential extinction of nodes into account in its evolution model. Wen et al.⁷ investigated the dynamical behaviors of a class of weighted local-world evolving networks with aging nodes. Feng et al.⁸ proposed three novel models based on the homogeneous Poisson, nonhomogeneous Poisson, and birth death processes, revealing the influence of generation mechanisms. Master equations for evolution model with birth and death are analyzed by Alvarez-Martínez et al.⁹ As for preferential attachment, popular strategies include fitness model,^10,11 attractiveness model,^12,13 energy-aware model,^6,14–16 and threshold preferential attachment model.¹⁷ Moreover, Jin et al.¹⁸ proposed a survival topology evolution model in consideration of node survivability. A local-world model based on random walk and policy attachment is presented by Jiang.¹⁹ Newly added nodes are connected to existing nodes through strength-age preferential attachment by Wen et al.⁷ Spatial positions of nodes are taken into account by Jacob and Peter,²⁰ and a spatial preferential attachment model is proposed. From the perspective of network architecture, evolution model of hierarchical network,²¹ peer-to-peer (P2P) network,²² and hypernetworks^23,24 are investigated respectively.

Although existing evolution models provide great references for topology research, there are several drawbacks. First, most evolution models focus on energy and survivability, QoS requirements are ignored, especially for SGIN with frequent handovers. Second, node heterogeneity, dynamic handover, and intermittent connection which pose significant effects on topology are overlooked in previous researches. Moreover, most models consider variable such as generation rates of vertices and number of connections as constants.

To address these issues, a QoS-aware dynamic evolution (QoSa) model is proposed. QoSa introduces evolution phases of dynamic handover and intermittently connecting, and endows heterogeneous nodes with different interactions considering nodes’ heterogeneity. During the evolution, a local world is constructed based on coverage signal intensity (CSI) and a QoS-aware strategy is adopted while selecting nodes to establish or delete links in evolution phases. Meanwhile, QoSa employs Poisson process to mimic the generation and extinction of vertices, and adopts general distribution to produce the number of nodes for connection, disconnection, and handover. Mean field theory and simulations are utilized to analyze the degree distribution and topology features. Results suggest that QoSa is a scale-free model with drift power-law distribution and heavy tail, and drift phenomenon is positively related to handover frequency. Meanwhile, QoSa possesses a small-world character and excellent robustness under attacks. Furthermore, compared to signal-strength preferential attachment, QoS-aware preferential attachment and node heterogeneity promote the attack resistance and network performance.

QoSa evolution model for SGIN

Node heterogeneity and generation mechanism

Topology of SGIN can be formularized as graph $G (V, E)$ based on graph theory, where $V = {v_{i} | i = 1, 2, \dots, n}$ is vertexes set and $E = {e (v_{i}, v_{j}) | v_{i}, v_{j} \in V, i \neq j}$ is links set. For any node $v_{i}$ , attribute set is defined as follows

AS = {dm, tp, loc, str, ra, ql, ca}

(1)

where $dm, tp, loc, str, ra, ql, and ca$ , respectively, denote network domains, types, spatial position, signal strength, coverage radius, quality, and capacity of nodes. In detail, $dm (v_{i})$ , network domain of node $v_{i}$ could be space network domain $d m_{s}$ , near-space network domain $d m_{h}$ , and terrestrial network domain $d m_{g}$ , and $tp (v_{i})$ node type of node $v_{i}$ can be $no d_{B}, no d_{R}, and no d_{N}$ , which, respectively, represent backbone node, relay node, and normal node. For any node, the probabilities that $dm (v_{i})$ equals $d m_{s}, d m_{h}, and d m_{g}$ are $p^{*}, q^{*}, and 1 - p^{*} - q^{*}$ . Meanwhile, the probabilities that $tp (v_{i})$ equals $no d_{B}, no d_{R}, and no d_{N}$ are $p', q', and 1 - p' - q'$ . Note that the definitions of all symbols mentioned in this article are explained in Appendix 1.

Thus, heterogeneity is taken into account to mimic different interactions of nodes. To simplify the problem, only differences in node type and network domain are considered in heterogeneity, because they are the key factors leading to heterogeneity. Then, heterogeneous nodes are endowed with different interactions. For instance, if node $v_{i}$ locates in $d m_{s}$ , it could only link to backbone nodes and relay nodes in $d m_{s}$ ; however, node $v_{j}$ in $d m_{h}$ may link to backbone nodes and relay nodes in both $d m_{s}$ and $d m_{h}$ . What’s more, if node $v_{i}$ is the backbone node, it can establish more links with existing nodes than that of normal nodes.

So far, most network models overlook the degeneration of vertices, but these phenomena do exist in SGIN and significantly influence network topology. Therefore, QoSa takes both the generation of vertices and their extinctions into consideration. What’s more, given the randomness of nodes’ entering or leaving SGIN, Poisson process⁸ is adopted to mimic the generation and extinctions of vertices.

QoS-aware strategy for preferential attachment

In our model QoSa, the QoS-aware-strategy-based preferential attachment is proposed for establishing links and disconnecting links. The QoS-aware strategy can be described as the rule of “better QoS getting richer and worse QoS getting poorer.” In other words, QoS-aware strategy includes two parts: nodes with better QoS are more likely to be linked with, and nodes with poorer QoS are more likely to delete a link or perform handover or die.

When a newly added node connects to existing nodes, it can only observe a subset of the whole nodes in SGIN due to the limitation of situation perception. A local world where newly added nodes decide to connect with should be constructed. QoSa introduces CSI to be the criteria to generate such a local world.

Definition 1

CSI denotes the signal strength of nodes in a given position. Thus, if signal strength of node $v_{i}$ at $loc (v_{j})$ is denoted as $str (v_{i}, v_{j})$ , then CSI of node $v_{i}$ at $loc (v_{j})$ is

str (v_{i}, v_{j}) = \frac{str (v_{i})}{\sqrt{| \vec{lo c_{j}} - \vec{lo c_{i}} |}} \cdot \frac{Δ h}{| Δ h |}

(2)

where $str (v_{i})$ denotes the signal strength of node $v_{i}$ . $\vec{lo c_{i}} and \vec{lo c_{j}}$ indicate the vector coordinates of $v_{i}$ and $v_{j}$ , respectively, and $Δ h = r a_{i} - \sqrt{| \vec{lo c_{j}} - \vec{lo c_{i}} |}$ . If $str (v_{i}, v_{j}) \leq 0$ , position of $v_{j}$ is out-of-signal coverage range of $v_{i}$ . We can see that the lager the distance $\sqrt{| \vec{lo c_{j}} - \vec{lo c_{i}} |}$ , the worse the CSI.

Local world is generated based on CSI using algorithm GenLocalWorld as follows:

Get alternative nodes set. Let $no d_{B} = 3, no d_{R} = 2, no d_{N} = 1$ and $d m_{s} = 3, d m_{h} = 2,$ $d m_{g} = 1$ . Considering that $tp (v_{i}) \neq no d_{B}$ , if there is a node $v_{j}$ satisfying $tp (v_{i}) \leq tp (v_{j})$ , $dm (v_{i}) \leq dm (v_{j})$ and $tp (v_{j}) \neq no d_{N}$ , then $v_{j}$ belongs to alternative node set. When $tp (v_{i}) = no d_{B}$ , if $v_{j}$ satisfies $tp (v_{j}) \neq no d_{N}$ and $dm (v_{i}) \leq dm (v_{j})$ , then $v_{j}$ belongs to alternative node set.

Remove nodes out of coverage. Calculate CSI of nodes in alternative node set at $loc (v_{i})$ and discard nodes that have negative CSI.

Generate local world. Select M nodes with highest CSI from alternative node set as local world $L (v_{i})$ , where M is the size of local world.

Assuming $N (t)$ to be the number of nodes at time t, according to the algorithm, the probability that vj belongs to $L (vi)$ at time t is decided by $Π (v_{j} \in L (v_{i}))$

\begin{matrix} Π (v_{j} \in L (v_{i})) \\ = \sum_{\binom{t p_{0} = no d_{B}, no d_{R}, no d_{N}}{d m_{0} = d m_{s}, d m_{h}, d m_{g}}} p (t p_{0}) p (d m_{0}) \underset{\binom{tp (v_{j}) = t p_{0}}{dm (v_{j}) = d m_{0}}}{Π} (v_{j} \in L (v_{i})) \\ \overset{t \to \infty}{=} \frac{M}{N (t)} \end{matrix}

(3)

where values of $t p_{0}$ can be $no d_{B}, no d_{R}, and no d_{N}$ , and $d m_{0}$ can be $d m_{s}, d m_{h}, and d m_{g}$ . Meanwhile, $p (t p_{0})$ denotes the probability that $t p_{0}$ equals a certain node type, and $p (d m_{0})$ refers to as the probability that $d m_{0}$ equals a certain domain. And $\underset{\binom{tp (v_{j}) = t p_{0},}{dm (v_{j}) = d m_{0}}}{Π} (v_{j} \in L (v_{i}))$ represents the probability that vj belongs to $L (vi)$ under the conditions of $tp (v_{j}) = t p_{0}, dm (v_{j}) = d m_{0}$ .

Given the features of highly exposed links and limited computing and communication resources, the QoS can be significantly influenced by the environment of links and limited resources of nodes. If the number of links is more, it would cost many node resources such as channels, CPUs, memories, and storage. As the number of links increases, all nodes linked will experience poor QoS. Meanwhile, when the distance is far from each other, then nodes will receive poor CSI, which leads to bad quality of links and data transmission services. Therefore, in our model QoSa, only CSI and limited resources of nodes are considered as shown in equation (4), and factors such as network traffic and routing strategies are ignored to simplify the problem.

Definition 2

QoS $η$ is defined as the function of CSI, node quality, and node capacity. Then, $η_{j}$ , the QoS of $v_{j}$ , at $loc (v_{i})$ is defined as follows

\begin{matrix} η_{j} = f (str (v_{j}, v_{i}), ql (v_{j}), ca (v_{j}), k_{j}) \\ = \frac{str (v_{j}, v_{i}) \cdot ql (v_{j}) \cdot ca (v_{j})}{k_{j}^{α}} \end{matrix}

(4)

where $ql (v_{j})$ is $v_{j}$ ’s node quality and $ca (v_{j})$ denotes $v_{j}$ ’s total capacity of links. Coefficient $α$ indicates the degree of influence that number of links $k_{j}$ have on QoS of the node $v_{j}$ .

When establishing a link to node $v_{k}$ , the probability that node $v_{i}$ is selected from node set $S (S \subseteq V)$ to be one end of the link that depends on QoS $η_{i}$ and degree $k_{i}$ , namely

\underset{v_{i} \in S}{Π} i = \frac{η_{i} k_{i}}{\sum_{v_{j} \in S} η_{j} k_{j}}

(5)

While deleting links, the probability that $v_{i}$ is selected to delete a link depends on the derivative of the product of QoS $η_{i}$ and degree $k_{i}$ , namely

Π_{i}^{*} = \frac{{(η_{i} k_{i})}^{- 1}}{\sum_{v_{j} \in V} {(η_{j} k_{j})}^{- 1}}

(6)

Evolution process of QoSa

In QoSa, evolution phases of dynamic handover and intermittent connection are introduced on the basis of Barabási–Albert (BA) model, which have a significant influence on network topology and are often overlooked in most evolution models. Handover of nodes is mainly caused by QoS of existing links getting worse due to their high-speed movement. A handover process includes two steps: first, a node is selected to move to a new location obeying Lévy flight theory²⁵ and its links are deleted; then, this node reconstructs its links according to preferential attachment, which can be regarded as a newly added node.

Meanwhile, a BA scale-free network obtains permanent m edges for each vertex added in, but in real situation, connections of different types of vertices are not the same and m is correspondently set to be $m_{b}, m_{r}, or m_{n}$ , when node type equals $no d_{B}, no d_{R}$ , or $no d_{N}$ . Furthermore, the number of nodes for handover $m_{h}$ , links for connection $m_{c}$ , and links for disconnection $m_{d}$ are mutable in real situation. And $m_{h}, m_{c}, and m_{d}$ are variables which are different from previous works regarding them as constant. QoSa assumes that $m_{h}, m_{c}, and m_{d}$ are subjected to the general distribution with expectation $μ'$ . Record the probabilities of growth, handover, extinction, connecting, and disconnecting events as $p_{0}, q_{0}, r_{0}, s_{0}, and u_{0}$ , respectively. Note that $p_{0}, q_{0}, r_{0}, s_{0}, u_{0} \geq 0$ and $p_{0} + q_{0} + r_{0} + s_{0} + u_{0} = 1$ . The values of $p_{0}, q_{0}, r_{0}, s_{0}, and u_{0}$ can be determined by statistical analysis of history services data in SGIN such as nodes accessing SGIN, exiting SGIN, and motion trails. Especially, the values of $s_{0}$ and $u_{0}$ can be inferred according to the physical link models of SGIN, and the values of $q_{0}$ can be calculated according to the motion model of nodes. In addition, at time t, growth, handover, extinction, connecting, and disconnecting events are independent events.

The QoSa evolves from an initial configuration consisting of $m_{0}$ backbone nodes of space network domain as a nearest-neighbor coupled networks.

Growth and extinction. QoSa performs growth and extinction with probability at $p_{0}$ and $r_{0}$ , respectively. During growth, the newly added vertices obey Poisson process with $λ$ . For each new vertex, m edges are connected to the existing vertices. Meanwhile, the existing vertices are gone by Poisson process with $μ$ and the selection of one vertex to die following the rule of the worse QoS getting to die, where the probability $Π_{i}^{*}$ is decided by equation (6).

Local preferential attachment. For each new vertex $v_{j}$ , QoSa first obtains the local world of $v_{j}$ and then connects $v_{j}$ to the existing nodes in its local world. Local world $L (v_{j})$ is generated with algorithm GenLocalWorld. Then, node $v_{j}$ is linked to m existing vertexes in $L (v_{j})$ using QoS-aware strategy with probability $Π_{v_{i} \in L (v_{j})} i$ decided by equation (5). Then, the probability $Π_{i}$ that $v_{i}$ is selected to be linked is

Π_{i} = Π (v_{i} \in L (v_{j})) E (m) \underset{v_{i} \in L (v_{j})}{Π} i {(1 - \underset{v_{i} \in L (v_{j})}{Π} i)}^{m - 1}

(7)

where $E (m)$ refers to the expectation of number of a node’s initial links, which is calculated in equation (8).

Dynamic handover. At time t, handover occurs with probability at $q_{0}$ and $m_{h}$ existing vertices experience handover. According to QoS-aware strategy of “Worse QoS getting poor,” an existing vertex is selected to handover with probability $Π_{i}^{*}$ decided by equation (6). And then, the selected nodes reconstruct its links with probability $Π_{i}$ decided by equation (7).

Connecting and disconnecting. Connecting and disconnecting happens with probability at $s_{0}$ and $u_{0}$ , respectively. In detail, $m_{c}$ links are newly added while connecting and an existing vertex is chosen as one side of a new link with probability $Π_{v_{i} \in V} i$ decided by equation (5). $m_{d}$ existing links are deleted when disconnecting, and an existing vertex is chosen as one side of a new link with probability $Π_{i}^{*}$ decided by equation (6).

Termination. If time reaches T, algorithm terminates and the network is output.

Theoretical analysis and proof

Degree distribution

Degree distribution is an important statistical characteristic of complex network. The mean field theory²⁶ is utilized to prove the degree distribution of QoSa.

Lemma 1

Given fixed growth rate $λ$ , extinction rate $μ$ , and probabilities $p_{0} and r_{0}$ , the expectation of total number of vertices at time t is $(p_{0} λ - r_{0} μ) t$ .

Lemma 2

Given fixed growth rate $λ$ , extinction rate $μ$ , interaction parameters $m_{b}, m_{r}$ , and $m_{n}$ , and proportion of node types $p' and q'$ , expectation of total number of vertices’ degrees at time t is $2 t (p_{0} λ - r_{0} μ) (p' \cdot m_{b} + q' \cdot m_{r} + (1 - p' - q') \cdot m_{n})$ .

Proof

Let $N (t)$ be the number of nodes at time t, and $E [N (t) m]$ denotes the total links. According to node heterogeneity, $\forall v_{i}$ , expectation of number of its initial links is

E (m) = p' \cdot m_{b} + q' \cdot m_{r} + (1 - p' - q') \cdot m_{n}

(8)

Considering that the total number of degree is twice that of links, and the number of links is independent of that of nodes. Thus, expectation of total number of vertices’ degrees is

\begin{matrix} E_{sgin} & = E [2 N (t) m] = 2 E (m) E [N (t)] \\ = 2 t (p_{0} λ - r_{0} μ) (p' \cdot m_{b} + q' \cdot m_{r} + (1 - p' - q') \cdot m_{n}) \end{matrix}

(9)

The result follows.

Lemma 3

Given fixed interaction parameters $m_{b}, m_{r}, and m_{n}$ and proportions of node types $p' and q'$ , the average degree of vertices at time t is $2 (p' \cdot mb + q' \cdot mr + (1 - p' - q') \cdot mn)$ .

Theorem 1

Degree distribution of QoSa follows a power-law distribution and is independent of time t.

Proof

Suppose degree of node $v_{i}$ at time t is a continuous real-value variable $k_{i} (t)$ . Meanwhile, according to nodes’ heterogeneity, it is reasonable to assume that M satisfies the inequality $m < M < N (t)$ at time t.

We can find that changes of $k_{i} (t)$ occur in the three situations and can be calculated based on the mean field theory as follows.

Situation 1

Let $λ$ nodes enter SGIN with rate $p_{0}$ and $μ$ nodes go with rate $r_{0}$ . Then

{(\frac{\partial k_{i}}{\partial t})}_{(i)} \approx p_{0} λ Π_{i} - r_{0} μ (k_{i} Π_{i}^{*} + \sum_{j \in linked (i)} Π_{j}^{*})

(10)

where $Π_{i}$ denotes the probability that $v_{i}$ is selected to be connected with newly added nodes $v_{j}$ . And we can obtain the following

\begin{matrix} Π_{i} & = Π (v_{i} \in L (v_{j})) E (m) \underset{v_{i} \in L (v_{j})}{Π} i {(1 - \underset{v_{i} \in L (v_{j})}{Π} i)}^{m - 1} \\ \approx E (m) \underset{v_{i} \in L (v_{j})}{Π} i Π (v_{i} \in L (v_{j})) \\ = \frac{E (m) \cdot M}{N (t)} \cdot \frac{η_{i} k_{i}}{\sum_{v_{h} \in L (v_{j})} η_{h} k_{h}} \end{matrix}

(11)

According to continuous field theory, $\sum_{v_{h} \in L (v_{j})} η_{h} k_{h}$ could be calculated as follows

\begin{matrix} \sum_{v_{h} \in L (v_{j})} η_{h} k_{h} & = \sum_{v_{h} \in L (v_{j})} \frac{str (v_{h}, v_{j}) \cdot ql (v_{h}) \cdot ca (v_{h})}{k_{h}^{α}} k_{h} \\ \approx M \cdot \bar{k} \cdot \bar{η} = M \cdot {\bar{k}}^{1 - α} \cdot \bar{str} \cdot \bar{ql} \cdot \bar{ca} \end{matrix}

(12)

where $\bar{η} and \bar{k}$ , respectively, denote the average QoS and average degree of nodes. And $\bar{str}, \bar{ql}, and \bar{ca}$ refer to the average value of CSI, node quality, and node capacity of all nodes in node’s local world. At time t, we can get $Π_{i}^{*} \approx 1 / N (t)$ when t is large enough. And then, there exists $\sum_{j \in linked (i)} Π_{j}^{*} \approx k_{i} / N (t)$ , where $\sum_{j \in linked (i)} Π_{j}^{*}$ is the probability that neighbor nodes connected with $v_{i}$ are selected to die and we can get. Thus, equation (10) can be converted into

\begin{matrix} {(\frac{\partial k_{i}}{\partial t})}_{(i)} & = \frac{p_{0} λ E {(m)}^{α}}{2^{1 - α} N (t)} \cdot \frac{str (v_{i}, v_{j}) ql (v_{i}) ca (v_{i}) k_{i}^{1 - α}}{\bar{str} \cdot \bar{ql} \cdot \bar{ca}} \\ - 2 r_{0} μ \cdot \frac{k_{i}}{N (t)} \end{matrix}

(13)

Situation 2

Select $m_{h}$ nodes to perform handover with rate $r_{0}$

{(\frac{\partial k_{i}}{\partial t})}_{(i i)} \approx q_{0} m_{h} (\prod_{i}^{*} (E (m) - k_{i}) - \sum_{j \in l i n k e d (i)} \prod_{j}^{*} + \prod_{i} \cdot \sum_{j \neq i, j \notin l i n k e d (i)} \prod_{j}^{*})

(14)

Handover phase includes three cases: (1) $v_{i}$ is selected to handover with probability at $Π_{i}^{*}$ ; (2) a neighbor node connected with $v_{i}$ is selected with probability $\sum_{j \in linked (i)} Π_{j}^{*}$ ; and (3) other nodes are selected with probability $\sum_{j \neq i, j \notin linked (i)} Π_{j}^{*}$ . Therefore, we can obtain equation (14).

Take $Π_{i}^{*} \approx 1 / N (t)$ and $\sum_{j \in linked (i)} Π_{j}^{*} \approx k_{i} / N (t)$ into equation (14), we can get the following

\begin{matrix} {(\frac{\partial ki}{\partial t})}_{(ii)} & \approx q_{0} m_{h} \frac{E (m)}{N (t)} - q_{0} m_{h} \frac{2 k_{i}}{N (t)} \\ + \frac{q_{0} m_{h} E {(m)}^{α}}{2^{1 - α} N (t)} \cdot \frac{str (v_{i}, v_{j}) ql (v_{i}) ca (v_{i}) k_{i}^{1 - α}}{\bar{str} \cdot \bar{ql} \cdot \bar{ca}} \end{matrix}

(15)

Situation 3

Establishing $m_{c}$ links with rate $s_{0}$ and disconnecting $m_{d}$ links with rate $u_{0}$

\begin{matrix} {(\frac{\partial ki}{\partial t})}_{(iii)} & = 2 s_{0} m_{c} \underset{v_{i} \in V}{Π} i \\ - u_{0} md [Π_{i}^{*} + \sum_{j \in linked (i)} Π_{j}^{*} K_{j}^{- 1} Π_{i}^{*}] \end{matrix}

(16)

where $\sum_{j \in linked (i)} Π_{j}^{*} K_{j}^{- 1} Π_{i}^{*}$ is the probability that neighbor nodes connected with $v_{i}$ are selected as one end of a deleted link and $K_{i}^{- 1} = \sum_{j \in Linked (i)} Π_{j}^{*}$ . When t is large enough, we can get $\sum_{j \in linked (i)} Π_{j}^{*} K_{j}^{- 1} Π_{i}^{*} \approx \sum_{j \in linked (i)} Π_{j}^{*} \cdot \frac{1}{\bar{k}} \approx \frac{k_{i}}{\bar{k} N (t)}$

\begin{matrix} {(\frac{\partial ki}{\partial t})}_{(iii)} & \approx \frac{2 s_{0} mE {(m)}^{α}}{2^{1 - α} N (t)} \cdot \frac{str (v_{i}, v_{j}) ql (v_{i}) ca (v_{i}) k_{i}^{1 - α}}{\bar{str} \cdot \bar{ql} \cdot \bar{ca}} \\ - \frac{u_{0} m_{d}}{2 E (m)} \cdot \frac{k_{i}}{N (t)} - \frac{u_{0} m_{d}}{N (t)} \end{matrix}

(17)

Combining the above three situations, $k_{i} (t)$ satisfies the following dynamic equation

\begin{matrix} \frac{\partial ki}{\partial t} & \approx \frac{E {(m)}^{α}}{2^{1 - α} N (t)} \cdot \frac{str (v_{i}, v_{j}) ql (v_{i}) ca (v_{i}) k_{i}^{1 - α}}{\bar{str} \cdot \bar{ql} \cdot \bar{ca}} \\ (\frac{2 s_{0} m}{E (m)} + q_{0} m_{h} + p_{0} λ) - (2 r_{0} μ + 2 q_{0} m_{h} + \frac{u_{0} m_{d}}{2 E (m)}) \cdot \\ \frac{k_{i}}{N (t)} + (q_{0} m_{h} E (m) - u_{0} m_{d}) \cdot \frac{1}{N (t)} \end{matrix}

(18)

Case 1. $α = 0$

As $\bar{η} = \frac{\bar{str} \cdot \bar{ql} \cdot \bar{ca}}{k^{α}}$ , then $\bar{η} = \bar{str} \cdot \bar{ql} \cdot \bar{ca}$ . Equation (18) could be simplified as follows

\frac{\partial k_{i}}{\partial t} = ω_{1} \frac{k_{i}}{N (t)} + ω_{2} \frac{1}{N (t)}

(19)

where $ω_{1} = \frac{p_{0} λ η_{i}}{2 \bar{η}} - 2 r_{0} μ + \frac{q_{0} m_{h} η_{i}}{2 \bar{η}} - 2 q_{0} m_{h} + \frac{s_{0} m_{c} η_{i}}{\bar{η} m} - \frac{u_{0} m_{d}}{2 m}$ and $ω_{2} = q_{0} m_{h} m - u_{0} m_{d}$ . According to QoSa model, initial degree $k_{i} (t_{i}) = m$ , thus the solution

\begin{matrix} k_{i} (t) & = (m + \frac{ω_{2}}{ω_{1} (p_{0} λ - r_{0} μ)}) {(\frac{t}{t_{i}})}^{\frac{ω_{1}}{p_{0} λ - r_{0} μ}} \\ - \frac{ω_{2}}{ω_{1} (p_{0} λ - r_{0} μ)} \end{matrix}

(20)

As the growth and extinction process is a Poisson process with $λ$ and $μ$ , $t_{i}$ satisfies the Gamma distribution $Γ (i, p_{0} λ - r_{0} μ)$ . Thus, we can get

\begin{matrix} P (k_{i} (t) < k) = P (t_{i} > t \cdot {A_{0}}^{B_{0}}) = 1 - P (t_{i} \leq t \cdot {A_{0}}^{B_{0}}) \\ = e^{- (λ - μ) \cdot t \cdot {A_{0}}^{B_{0}}} \sum_{0}^{i - 1} \frac{{((λ - μ) \cdot t \cdot {A_{0}}^{B_{0}})}^{l}}{l!} \end{matrix}

(21)

where $A_{0} = \frac{ω_{2}}{ω_{1} (p_{0} λ - r_{0} μ)}$ and $B_{0} = \frac{p_{0} λ - r_{0} μ}{ω_{1}}$ .

Then, existence of stable average degree distribution is proved below

\begin{matrix} P (k_{i} (t) = k) & \approx \frac{\partial P (k_{i} (t) < k)}{\partial k} = \frac{{(p_{0} λ - r_{0} μ)}^{2} \cdot t}{ω_{1}} \cdot \\ \frac{{(m + A_{0})}^{B_{0}}}{{(k + A_{0})}^{B_{0} + 1}} e^{- λ \cdot t \cdot {A_{0}}^{B_{0}}} \frac{{(λ \cdot t \cdot {C_{0}}^{- B_{0}})}^{i - 1}}{(i - 1)!} \end{matrix}

(22)

where $C_{0} = \frac{m + A_{0}}{k + A_{0}}$ . Then, degree distribution of QoSa is as follows

\begin{matrix} P (k) & \approx lim_{t \to \infty} \frac{1}{E [N (t)]} \cdot \sum_{i = 1}^{\infty} P (k_{i} (t) = k) \\ = B_{0} \cdot {(m + A_{0})}^{B_{0}} \cdot {(k + A_{0})}^{- B_{0} - 1} \end{matrix}

(23)

Thus, QoSa is a scale-free network with drift power-law distribution. The exponent

\begin{matrix} γ = \frac{p_{0} λ - r_{0} μ}{ω_{1}} + 1 \\ = \frac{p_{0} λ - r_{0} μ}{\frac{p_{0} λ η_{i}}{2 \bar{η}} - 2 r_{0} μ + \frac{q_{0} m_{h} η_{i}}{2 \bar{η}} - 2 q_{0} m_{h} + \frac{s_{0} m_{c} η_{i}}{\bar{η} m} - \frac{u_{0} m_{d}}{2 m}} + 1 \end{matrix}

(24)

With assumptions that $p_{0}, q_{0}, r_{0}, s_{0}, u_{0}, λ, μ \geq 0$ , $E (m_{h}) = E (m_{c}) = E (m_{d}) = μ'$ , and let $η_{i} = \bar{η}$ , the exponent is simplified as follows

γ = \frac{p_{0} λ - r_{0} μ}{\frac{p_{0} λ}{2} - 2 r_{0} μ - \frac{3 q_{0} μ'}{2} + \frac{s_{0} μ'}{m} - \frac{u_{0} μ'}{2 m}} + 1

(25)

With the growth characteristic, $ω_{1} > 0, p_{0} λ \geq r_{0} μ$ . Due to $s_{0} = 1 - p_{0} - q_{0} - r_{0} - u_{0}$ , it can be inferred that $ω_{1}$ increases as $p_{0}$ increases and $0 < ω_{1} < λ / 2$ . Thus, we get $1 \leq γ \leq 3$ .

Case 2. $α = 1$

Let $str (v_{i}, v_{j}) ql (v_{i}) ca (v_{i}) = \bar{str} \cdot \bar{ql} \cdot \bar{ca}$ , and equation (18) could be simplified as follows

\frac{\partial k_{i}}{\partial t} = ω_{1} \frac{k_{i}}{N (t)} + ω_{2} \frac{1}{N (t)}

(26)

where $ω_{1} = - 2 r_{0} μ - 2 q_{0} m_{h} - \frac{u_{0} m_{d}}{2 E (m)}$ and $ω_{2} = 2 s_{0} m + E (m) (2 q_{0} m_{h} + p_{0} λ) - u_{0} m_{d}$ . Similarly, with case 1, degree distribution of QoSa is as follows

\begin{matrix} P (k) & \approx lim_{t \to \infty} \frac{1}{E [N (t)]} \cdot \sum_{i = 1}^{\infty} P (k_{i} (t) = k) \\ = B_{0} \cdot {(m + A_{0})}^{B_{0}} \cdot {(k + A_{0})}^{- B_{0} - 1} \end{matrix}

(27)

Thus, QoSa is a scale-free network with drift power-law distribution. The exponent

γ = \frac{p_{0} λ - r_{0} μ}{ω_{1}} + 1 = \frac{p_{0} λ - r_{0} μ}{- 2 r_{0} μ - 2 q_{0} m_{h} - \frac{u_{0} m_{d}}{2 m}} + 1

(28)

Due to $p_{0}, q_{0}, r_{0}, u_{0}, s_{0} \geq 0$ and $λ, μ, m_{h}, m_{d} > 0$ , we get $γ \leq 1$ .

In summary, the QoSa model is a scale-free network with power-law character, and power exponent $γ$ is lower than 3 depending on node type, generation mechanism, and evolution parameter such as $m_{h}, m_{c}, and m_{d}$ , but independent of time.

The result follows.

Inference 1

The QoSa model equals BA model in some degree.

According to equation (23), set $p_{0} = 1, q_{0} = r_{0} = s_{0} = u_{0} = 0$ , then there exist $ω_{1} = λ η_{i} / 2 \bar{η}$ , $ω_{2} = 0$ , and $B_{0} = 2 \bar{η} / η_{i}$ , thus the stable average degree distribution $P (k)$ is

P (k) \approx \frac{2 \bar{η}}{η_{i}} \cdot m^{\frac{2 \bar{η}}{η_{i}}} \cdot k^{- \frac{2 \bar{η}}{η_{i}} - 1}

(29)

Assume that nodes are homogeneous and set $η_{i} = \bar{η}$ , then we get $P (k) = 2 \cdot m^{2} \cdot k^{- 3}$ . Thus, the QoSa model equals BA model under certain conditions.

The result follows.

Average path length

Definition 3

Average path length is defined as the average number of steps along the shortest paths for all possible pairs of network nodes, namely

L = \frac{1}{C_{N}^{2}} \cdot \sum_{1 \leq i < j \leq N} d_{ij}

(30)

where $C_{N}^{2}$ denotes the total links of complete graph with size N, and $d_{ij}$ indicates the steps of shortest path between $v_{i}$ and $v_{j}$ . Average path length L is a measure of the efficiency of data transmission on a network. The shorter the L, the faster the transmission speed.

Theorem 2

The QoSa possesses the small-world characteristic.

Proof

Assuming that the average path lengths of the giant component of space network, near-space network, and terrestrial network are, respectively, $L_{s}, L_{h}$ , and $L_{g}$ .

Without losing generality, suppose that $v_{i}$ and $v_{j}$ are nodes in terrestrial network $d m_{g}$ and assume $L_{1}, L_{2}, and L_{3}$ are distances of three shortest paths via each layer between $v_{i}$ and $v_{j}$ . If such a path $L_{i} (i = 1, 2, 3)$ does not exist, set $L_{i} = \infty$ . Thus, we can get the following

min (L_{g}, L_{h}, L_{s}) \leq d_{ij} = min (L_{1}, L_{2}, L_{3}) \leq 2 L_{g} + 2 L_{h} + L_{s}

(31)

Therefore, the average path length L satisfies the following equation

min (L_{g}, L_{h}, L_{s}) \leq L = \frac{1}{C_{N}^{2}} \cdot \sum_{1 \leq i < j \leq N} d_{ij} \leq 2 L_{g} + 2 L_{h} + L_{s}

(32)

According to the evolution model, any connected subgraphs of SGIN can be approximated as BA model with average path length $LBA \propto In (N) / In (In (N))$ ,²⁷ where $In (N)$ is the natural logarithm of N with base e and $In (In (N))$ is the natural logarithm of $In N$ with base e. Thus, we can figure out that $L = O (In (N) / In (In (N)))$ . Thus, the small-world character of QoSa is verified.

This is the end of proof.

Simulations and discussion

Simulations are conducted to verify the degree distribution and vulnerabilities of QoSa under conditions of variable node heterogeneity, growth and extinction mechanism, and preferential attachment. Global parameters in simulations are set as follows. Let end time $T = 2000$ , initial number of nodes $m_{0} = 4$ , and number of links $e_{0} = 3$ . Set $m_{b} = m_{r} = 4, m_{n} = 1$ and $p_{0} = 0.7, q_{0} = 0.15, r_{0} = 0.05, s_{0} = 0.05, and u_{0} = 0.05$ . Unless specifically stated, set $μ' = 1$ , $α = 0$ , and $p' = 0.25, q' = 0.35, and 1 - p' - q' = 0.4$ .

Degree distribution

Figure 1(a) plots the degree distribution of QoSa under the conditions $λ = 6, μ = 4$ , and a power-law fitting with γ = 2.1. This figure reveals that $P (k)$ displays a trend of increasing first and then decreasing with the increase in degree k and that its degree distribution is an unimodal power-law distribution. Scale-free fitted curve proves that QoSa is subjected to a power-law distribution with γ = 2.1 which is consistent with theoretical calculation. Meanwhile, deviation from the fitted curve reveals the phenomenon of drift power-law distribution and heavy tail of QoSa. The former is caused by dynamic handover which results in rapid change in degree and the latter by the statistical nature of the empirical distribution.

Figure 1.

(a) Degree distribution of SGIN and a power-law fitting with γ = 2.1 (λ = 6, μ = 4) and (b) degree distribution with different preferential attachment strategies (λ = 8, μ = 6).

Figure 1(b) displays the influence of different preferential attachment on degree distribution with $λ = 8 and μ = 6$ . Compared to signal-strength-based preferential attachment, the degree distribution curve of QoSa adopting QoS-aware strategy drops slower with the increase in degree k, displaying less significant power-law characteristics. Signal-strength-aware preferential attachment tries to connect to nodes with higher signal strength and overlook their QoS, leading to the phenomenon of “The rich getting richer.” However, QoS-aware strategy takes QoS of nodes into account and selects nodes with better QoS to connect which balances the degree distribution of nodes.

Figure 2(a) explores the degree distribution with different growth rate $λ$ and extinction rate $μ$ . From this figure, it could be found that under the condition of same $λ$ and k, smaller $μ$ makes the points wider. Meanwhile, under the condition of same $μ$ and k, larger $λ$ makes the points wider. The reason is that $p_{0} λ - r_{0} μ$ determines the number of newly added nodes, the high value of $λ$ or low value of $μ$ will increase the degree of vertex, and enlarge the distance of different degrees of k, which makes the plot fatter.

Figure 2.

(a) Degree distribution of SGIN with different birth and death mechanisms and (b) degree distribution of SGIN with different nodes type (λ = 8, μ = 6).

Figure 2(b) draws the degree distribution with different proportion of node type which reflects nodes’ heterogeneity and different network model. Actually, the closer the proportion, the more heterogeneous the model. Meanwhile, if $p', q', and 1 - p' - q' \neq 0$ , normal nodes could connect to relay nodes or backbone node if necessary. But, if $q' = 0$ , normal nodes can only connect to backbone node, which indicates a different network model of SGIN. From Figure 2(b), it could be observed that the less heterogeneous the model, the fatter the plot and the slower the degree distribution curve drops which indicate less significant power-law characteristic. Meanwhile, when $q'$ drops to 0 gradually, and the degree distribution curve drops faster and faster.

Figure 3 plots the degree distribution with different coefficient $α$ . It can be spotted that the degree distribution curve of $α = 0$ drops faster than that of $α = 1$ with the increase in degree k, displaying less significant power-law characteristics. When $α = 1$ , degree distribution is more balanced than that of $α = 0$ , because it avoids selecting nodes with higher degree to connect with.

Figure 3.

Degree distribution of SGIN with different coefficient α.

Connectivity and vulnerability

Average path length and the giant component are the key metrics to measure connectivity and vulnerability of complex network. The influences of nodes’ generation mechanism, preferential attachment strategy, and node heterogeneity on the average path length and giant component are investigated in this section.

Figure 4(a) plots the average path length with different growth rate $λ$ and extinction rate $μ$ . From Figure 4(a), we can find that the average path length is between 2 and 5, which indicates the small-world characteristic of QoSa. And average path length decreases with the increasing number of nodes. According to QoSa model, the number of nodes is small at the beginning of the network evolution and nodes are randomly distributed. With the increase in the number of nodes, adjacent nodes can be connected directly leading to the decrease in the average path length. In addition, birth and death mechanisms have little effect on average path length.

Figure 4.

(a) Average path length of SGIN with different birth and death mechanisms and (b) average path length of SGIN with different nodes type (λ = 8, μ = 6).

Figure 4(b) plots the trend of average path length as of network size increases with different node type. From Figure 4(b), it can be found that the more heterogeneous the nodes, the shorter the average path length and the faster the data are transmitted. Furthermore, as $q'$ drops to 0 gradually, the average path length changes little.

Definition 4

Connection robustness is defined as the ability to keep links connected while attacks or faults occur. Criterion of connection robustness R is defined as follows

R = \frac{C}{(N - Nr (n))}

(33)

where N is the size of initial network, $Nr (n)$ denotes the number of nodes removed from the network, and C is the size of the giant component after removal.

The trend of scale of giant component under random faults and deliberate attack with different growth rate $λ$ and extinction rate $μ$ are plotted in Figure 5. It could be seen that while faults occur, scale of giant component decreases slowly in Figure 5(a) and that scale of giant component decreases rapidly under degree-based deliberate attack in Figure 5(b), indicating that QoSa enjoys good fault tolerance and poor attack robustness. Meanwhile, it can be inferred that birth and death mechanisms have little influence on the connection robustness under random attack and deliberate attack.

Figure 5.

Connection robustness with different birth and death mechanisms: (a) size of giant component under random attack with different ratios to networks and (b) size of giant component under deliberate attack with different ratios to networks.

Figure 6 uncovers the trend of size of giant component with different node types under random attack and deliberate attack. From Figure 6(a), it can be found that the more heterogeneous the nodes, the more rapidly the size of giant component decreases. Thus, the node’s heterogeneity reduces the robustness of the network under random attacks. However, in Figure 6(b), the more heterogeneous the nodes, the less rapidly the size of giant component decreases, which means that node heterogeneity improves the robustness of network under deliberate attacks. The reason is that nodes’ heterogeneity balances the degree distribution of nodes and weakens the importance of the hub node.

Figure 6.

Connection robustness with different node types (λ = 8, μ = 6): (a) size of giant component under random attack with different ratios to networks and (b) size of giant component under deliberate attack with different ratios to networks.

Figure 7 reveals the trend of size of giant component with different preferential attachment under random attack and deliberate attack. From Figure 7(a), it can be spotted that as the attack ratio of random attack increases, the size of giant component of QoS-aware strategy is smaller and decreases faster than that of signal-strength-aware preferential attachment. From Figure 7(b), QoS-aware strategy is lager and decreases slower than that of signal-strength-aware preferential attachment. Therefore, QoS-aware strategy performs better under deliberate attack.

Figure 7.

Connection robustness with different preferential attachment (λ = 8, μ = 6): (a) size of giant component under random attack with different ratios to networks and (b) size of giant component under deliberate attack with different ratios to networks.

Figure 8 plots the trend of size of giant component with different coefficient $α$ under random attack and deliberate attack. These figures suggest that compared to situation under $α = 0$ , the size of giant component under random attack when $α = 1$ is smaller and decreases faster, but the size of giant component under random attack when $α = 1$ is larger and decreases slower. Thus, it possesses better attack resistance when $α = 1$ , because it avoids selecting nodes with higher degree to connect with and balances degree distribution, which results in weaker fault toleration.

Figure 8.

Connection robustness with different coefficient α (λ = 8, μ = 6): (a) size of giant component under random attack with different ratios to networks and (b) size of giant component under deliberate attack with different ratios to networks.

Conclusion

In order to improve QoS of SGIN, a QoS-aware evolution model is proposed. A QoS-aware strategy of nodes with better QoS getting richer and nodes with worse QoS getting poorer is adopted when establishing and deleting links in order to optimize network QoS. To mimic the natures of SGIN, QoSa introduces dynamic handover and intermittent connection into the evolution process. Heterogeneity in types and layers of nodes is taken into account, and heterogeneous nodes are endowed with discriminating interaction behaviors. Theoretical analysis and simulations are conducted to demonstrate the degree distribution and verify the influences of heterogeneity, dynamic handover, and generation mechanism of nodes on the topology characteristics.

This article reveals that SGIN generated by QoSa is a scale-free network, the degree distribution of which possesses drift power-law distribution and heavy tail, and its power exponent is lower than 3 which varies with the heterogeneity and generation mechanism of nodes and parameters such as number of links and handover nodes, but is independent of time. And the drift power-law distribution results from frequent handovers. Meanwhile, QoSa enjoys a small-world character and efficient data transmission. Its average path length is positively related to nodes’ heterogeneity and has little to do with node generation mechanism. In addition, QoSa has high fault tolerance and attack resistance. Specifically, its fault tolerance is negatively related to nodes’ heterogeneity, but its attack resistance is positively related to nodes’ heterogeneity. What’s more, QoS-aware strategy balances the degree distribution and promotes the attack resistance of QoSa at the same time of improving QoS.

Compared to existing scale-free evolution models, QoSa focuses mainly on improving network QoS. To mimic SGIN more accurately, it not only takes nodes’ heterogeneity, dynamic handover, and intermittent connection into account during evolution but also adopts Poisson process and Lévy flight to simulate generation mechanism and handover. This article provides strong reference for access and networking, topology optimization, and security protection of SGIN. Results of this article can also be expanded to WSNs and IoVs.

Footnotes

Appendix 1 Acknowledgements

The authors would like to thank the Associate Editor and all the anonymous reviewers for their insightful comments and constructive suggestions.

Academic Editor: Haibo Zhou

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant nos 61502531, 61403400 and 61773399) and Industry Academia Research Project of Henan Province (grant no. 152107000026).

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A quality-of-service-aware dynamic evolution model for space–ground integrated network

Abstract

Keywords

Introduction

QoSa evolution model for SGIN

Node heterogeneity and generation mechanism

QoS-aware strategy for preferential attachment

Definition 1

Definition 2

Evolution process of QoSa

Theoretical analysis and proof

Degree distribution

Lemma 1

Lemma 2

Proof

Lemma 3

Theorem 1

Proof

Situation 1

Situation 2

Situation 3

Inference 1

Average path length

Definition 3

Theorem 2

Proof

Simulations and discussion

Degree distribution

Connectivity and vulnerability

Definition 4

Conclusion

Footnotes

Appendix 1

Acknowledgements

Declaration of conflicting interests

Funding

References