esa-sceva/satcom-chunk-collection huggingface dataset:
# FANET Routing Protocol Analysis for Multi-UAV-Based Reconnaissance Mobility Models
## 4 Experimental Setup
### Performance Metrics
#### 4.3.2 Network Connectivity Ratio  
The possibility that packets are successfully delivered to a destination node is limited by the percentage of the nodes that are connected in the network. Additionally, the ratio of connecting nodes in a network at each time point varies according to the mobility model. Therefore, we measured how many nodes are connected in a mobility model at each time point.  
The network connectivity ratio counts the number of packets that have a possibility of being delivered while considering the nodes that are connected in FANETs. Different from PDR, NCR does not consider a specific routing protocol but considers the locations of nodes and their communication distances when a source node sends a packet. We express the network connectivity ratio metric as the ratio of the nodes in connection over the total number of packets as follows:  
\(P_{k}\) = 1 when the \(K\)th packet could be delivered in the network connection; 0 otherwise  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Network simulator & NS-3 (Version 3.30) \\ Mobility model simulator & Direct Development \\ Routing protocols & AODV, DSDV, OLSR \\ Mobility models & RDPZ, SRWP, EGM, DPR, MP \\ Propagation model & Friis (Free Space Path Loss) \\ Simulation area & 4000 \(\times\) 4000 m\({}^{2}\) \\ Simulation time & 3600 s \\ Simulation runs & 10 \\ Transmission range & 1000 m \\ Number of nodes & 5, 10, 15, 20, 25, 30 \\ Node speed & 10, 20, 30, 40, 50 m/s \\ Number of packet transmissions per node & 100 \\ GCS location & Left Top, Top, Middle, Bottom \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the network simulator.  
\(P_{s}\) = Total number of packets sent  
\[NCR=\frac{\sum_{k=1}^{P_{s}}P_{k}}{P_{s}} \tag{4}\]
# Analysing Urban Traffic Patterns with Neural Networks and COVID-19 Response Data
## 4 Results
### Comparison with Other Methods  
In this section, we compare the results of predicting the number of vehicles using ensemble neural network models with those generated by models created using different techniques. As mentioned in the methodology section, we employed four methods suitable for solving regression data mining tasks: decision trees of the CART and CHAID types, nearest neighbours, and multivariate linear regression. To ensure comparability with the ensemble neural networks, all these models were created also as ensembles, containing up to ten component models. The predictions of these models for the testing data were compared with the actual values, and their performance was evaluated using correlation coefficients. Table 8 presents a comparison of these models with the neural network models. The nearest neighbour technique did not provide a solution for some types of vehicles due to the data dimensionality and the memory demand on computations (denoted as \"NA\" in the table). The correlations of actual and predicted values show the ensemble neural network models' high performance compared to the other used methods.
# Robust Cross-Validation of Predictive Models Used in Credit Default Risk
## 2 Credit Default Modeling Use Case
### Area Under ROC Curve  
Predictive model performance metrics, such as accuracy or misclassification error, have been commonly used in many research papers. However, these are not the most convenient metrics to assess the performance of credit default models, and the area under the receiver operating characteristic curve (area under ROC curve or AUR) should be used instead [18; 19; 20].  
The AUR is suitable for imbalanced data [9] because it considers the trade-off between the benefit of right predictions and the cost of errors. Normally, a binary classifier that increases the number of true positives (captured events) will also increase the number of false positives, and this relationship should be taken into consideration in use cases where the cost of a false negative is very different from the cost of a false positive. This is the reason why AUR has been adopted as the best practice in the field when modeling rare events with high severity, as happens in credit defaults.  
To calculate the AUR for a particular PD model over a sample dataset, all the observations in the sample must be ranked first according to the predicted PD, from highest to lowest probability of default. Then, a two-way table of the observed versus predicted defaulting events must be created for each different PD value that is considered as a _cut-off_ value to differentiate defaults from non-defaults. For any specific _cut-off PD_ value, correctly predicted defaults are those defaulted observations with a predicted PD greater than or equal to the cut-off PD, while incorrectly predicted defaults are the non-defaulted observations with a predicted PD also above or equal to the cut-off value.  
For each PD value, the corresponding ROC curve point (\(X_{ROC}\), \(Y_{ROC}\)) is obtained from its two-way table as the pair \"sensitivity\" vs. \"false alarm rate\" in the Y-axis and X-axis, respectively. Sensitivity at a specific cut-off PD is calculated as the correctly predicted defaults divided by all the observed defaults of the sample. The false alarm rate is calculated at a particular cut-off PD value as the number of observations incorrectly predicted as defaults divided by all the observed non-defaulters of the sample.  
Then, if there are \(M\) different PD values predicted by the model \(PD_{1}>PD_{2}>\ldots>PD_{M}\), the AUR can be simply calculated as the area under the discrete ROC curve by:  
\[AUR=\sum_{i=1}^{M-1}\frac{Y_{ROC}(PD_{i+1})-Y_{ROC}(PD_{i})}{2}\left(X_{ROC}( PD_{i+1})-X_{ROC}(PD_{i})\right)\]  
There are several ways to approximate the area under the curve; it is not of great importance which is chosen as long as the formula used is always the same when comparing model performances. In any case, the greater the AUR, the better the model performance. AUR equals 1 in a perfect model with no predicted errors, while the AUR of a pure random model is 0.5, where the response variable is independent from the covariates.
# Revisiting the Meandering Instability During Step-Flow Epitaxy
## 2 Stability Analysis  
In this paper, we employ a linear stability analysis to study step-meandering. For our convenience, let \(u=\mu-\frac{F}{\sigma}\). The steady-state reaction-diffusion Equation (5) becomes the Helmholtz equation  
\[\Delta u-\frac{u}{x_{s}^{2}}=0, \tag{8}\]  
where \(x_{s}=\sqrt{\frac{M}{\sigma}}\) is the surface diffusion length. We introduce a constant parameter \(\Gamma=\frac{\Psi^{s}}{a\rho^{b}}\) and a dimensionless parameter \(\Theta:=\frac{\rho_{eq}}{a\rho^{b}}\), the equilibrium adatom coverage, which measures the equilibrium adatom density \(\rho_{eq}\) relative to the bulk atomic density \(a\rho^{b}\). For the adatom grand canonical potential \(\omega\) in (4), we adopt the linear approximation \(\omega\approx\omega_{0}-\mu\rho_{eq}\) with a positive constant \(\omega_{0}\) in [17]. Therefore, the step-edge boundary conditions (6) can be written as  
\[M(\partial_{n}u)^{+}=C_{+}[u^{+}+\Phi-\Theta(u^{+}-u^{-})+ \Gamma\kappa],\] \[-M(\partial_{n}u)^{-}=C_{-}[u^{-}+\Phi-\Theta(u^{+}-u^{-})+ \Gamma\kappa], \tag{9}\]  
where \(\Phi=\frac{F}{\sigma}-\mu^{b}\). And the evolution equation for \(u\) is  
\[V=MN[(\partial_{n}\mu)^{+}-(\partial_{n}\mu)^{-}] \tag{10}\]  
with \(N=1/(a\rho^{b})\).  
The general solution to the Helmholtz Equation (8) is given by the form of  
\[u(x,z)=\sum_{k}[A_{k}\sinh(\Lambda_{k}z)+B_{k}\cosh(\Lambda_{k}z)][\sin(kx)+C _{k}\cos(kx)], \tag{11}\]  
where \(\Lambda_{k}=\sqrt{(1/x_{s}^{2})+k^{2}}\). Assume there is a small amplitude sinusoid perturbation with a wave vector \(k\) for the step. Then the position of the step is given by  
\[z(x)=z+\varepsilon\sin(kx). \tag{12}\]  
Therefore the curvature \(\kappa=-\epsilon k^{2}\sin(kx)\) and the normal vector \(n=(-\epsilon k\cos(kx),1)^{T}\). If we allow the accuracy of the solution \(u\  
To compute the velocity of the step, we take the time derivative of (12) and get  
\[V(x)=V_{0}+\frac{d\epsilon}{dt}\sin(kx). \tag{14}\]  
On the other hand, we can also use (10) to compute the velocity. Later we can see it has the form of  
\[V(x)=V_{0}+\omega(k)\epsilon\sin(kx), \tag{15}\]  
where the stability function \(\omega(k)=g(k)-k^{2}f(k)\) with certain functions \(f\) and \(g\). Equations (14) and (15) imply \(\frac{d\epsilon}{dt}=\omega(k)\epsilon\). It shows that the sign of \(\omega(k)\) determines the growth or decay of perturbation.  
It is necessary to define the kinetic coefficients \(d_{\pm}=M/C_{\pm}\) for the attachment-detachment of adatoms from the lower and upper terrace onto the step edge and a surface capillary length \(\xi=\frac{\Gamma}{\Phi}\). These notations will be used in the following section.
## 3 System Model  
### Communication Model  
As shown in Figure 1, the communication scenario consists of a pair of transceivers and an active jammer, where the transmitter and receiver continuously communicate. The intelligent agent at the receiver end perceives the electromagnetic environment in real time to acquire the features of the jamming signals, and the transmitter dynamically selects anti-jamming strategies. When the transmitter sends signals to the receiver, the jammer performs electromagnetic jamming on the receiver's channel access, blocking legitimate communications between the two parties.  
The system uniformly divides the communication frequency band into \(N\) continuous non-overlapping channels with a bandwidth of \(b\), and the channel set is \(f\in\{0,1,2,\cdots,N-1\}\). In the discrete time model, the communication time is divided into multiple time slots with a length of \(T\). A time slot is the minimum time unit for continuous data transmission, and the system can communicate within the given wireless communication frequency band by changing the communication channel in the time slot. The receiver determines the jamming degree of this communication according to the Signal-to-Jamming-plus-Noise Ratio (SINR) of the received signal and evaluates the communication effect, which can be represented by  
\[\theta_{t}=10\lg\frac{P_{S}}{P_{S}+P_{J}+P_{N}}. \tag{3}\]  
where \(P_{S}\) is the transmission power of the signal transmitter, \(P_{J}\) is the transmission power of the jammer, and \(P_{N}\) is the radio frequency environmental noise. The SJNR in the communication system needs to satisfy \(\theta_{t}\geq\theta_{th}\) in any time slot \(T\) of the time period to ensure successful signal transmission. Here, \(\theta_{t}h\) represents the SJNR threshold.
# Inverse Kinematics of a Serial Manipulator with a Free Joint for Aerial Manipulation
## 2 Background Theory
### Differential Kinematics of Free-Flying Robots  
For a serial fixed-base manipulator, the velocity of the end-effector \(\mathbf{v}_{ce}\) can be computed from the joint velocities \(\mathbf{\dot{q}}\) by using the Jacobian matrix \(\mathbf{J}\)[28]:  
\[\mathbf{v}_{ce}=\mathbf{J}(\mathbf{q})\mathbf{\dot{q}}. \tag{6}\]  
The end-effector acceleration can be obtained by deriving Equation (6), yielding  
\[\dot{\mathbf{v}}_{ce}=\mathbf{J}(\mathbf{q})\mathbf{\dot{q}}+\mathbf{\dot{J}}(\mathbf{q})\mathbf{\dot{q}} \tag{7}\]  
If the base is not fixed, linear (\(\mathbf{v}_{b}\)) and angular (\(\mathbf{\omega}_{b}\)) velocities of the base (collected in the vector \(\dot{\mathbf{x}}_{b}=\{\mathbf{v}_{b},\mathbf{\omega}_{b}\}^{T}\)) contribute to end-effector velocity \(\mathbf{v}_{ce}\). In this case, Equation (6) becomes:  
\[\mathbf{v}_{ce}=\mathbf{J}\mathbf{\dot{q}}+\mathbf{J}_{b}\mathbf{\dot{x}}_{b}. \tag{8}\]  
From the momentum balance equations (Equation (5)) it is possible to express base velocities as a function of \(\mathbf{\dot{q}}\):  
\[\dot{\mathbf{x}}_{b}=-\mathbf{H}_{b}^{-1}\mathbf{H}_{bm}\mathbf{\dot{q}}+\mathbf{H}_{b}^{-1} \begin{bmatrix}\mathbf{p}\\ \mathbf{K}\end{bmatrix} \tag{9}\]  
Combining Equation (9) with the forward kinematics formula (Equation (8)) yields:  
\[\mathbf{v}_{ce}=\mathbf{J}\mathbf{\dot{q}}+\mathbf{J}_{b}\begin{pmatrix}-\mathbf{H}_{b}^{-1}\mathbf{H }_{bm}\mathbf{\dot{q}}+\mathbf{H}_{b}^{-1}\begin{bmatrix}\mathbf{p}\\ \mathbf{K}\end{bmatrix}\end{pmatrix}=\begin{pmatrix}\mathbf{J}-\mathbf{J}_{b}\mathbf{H}_{b}^{- 1}\mathbf{H}_{bm}\mathbf{\dot{b}}\end{pmatrix}\mathbf{\dot{q}}+\mathbf{J}_{b}\mathbf{H}_{b}^{-1} \begin{bmatrix}\mathbf{p}\\ \mathbf{K}\end{bmatrix}. \tag{10}\]  
The matrix \(\begin{pmatrix}\mathbf{J}-\mathbf{J}_{b}\mathbf{H}_{b}^{-1}\mathbf{H}_{bm}\mathbf{\dot{b}}\end{pmatrix}\) is called the Generalized Jacobian matrix (\(\mathbf{J}_{gen}\)) [8]. The end-effector velocity can thus be written as:  
\[\mathbf{v}_{ce}=\mathbf{J}_{gen}\mathbf{\dot{q}}+\mathbf{J}_{b}\mathbf{H}_{b}^{-1} \begin{bmatrix}\mathbf{p}\\ \mathbf{K}\end{bmatrix} \tag{11}\]  
Differentiating this equation gives the end-effector acceleration \(\dot{\mathbf{v}}_{ce}\):  
\[\dot{\mathbf{v}}_{ce}=\mathbf{J}_{gen}\mathbf{\dot{q}}+\mathbf{\dot{J}}_{gen}\mathbf{\dot{q}}+\mathbf{ J}_{b}\mathbf{H}_{b}^{-1}\begin{bmatrix}\dot{\mathbf{p}}\\ \mathbf{\dot{K}}\end{bmatrix}+\mathbf{\dot{J}}_{b}\mathbf{H}_{b}^{-1}\begin{bmatrix}\mathbf{p} \\ \mathbf{K}\end{bmatrix} \tag{12}\]  
where the derivatives of the momenta can be calculated from the sum of the external forces and torques acting on the system.
# Real-Time Tracking and Position Control of an Elastic Cable-Driven Winch System
## 2 Mathematical Model of an Elastic Cable Winch System
If the initial height of the cable is denoted by \(h\), its height at any given moment can be expressed as \(h-y\). The stiffness of the cable is given by \(k=\frac{EA}{h-y}\). Assuming both pulleys rotate equally, the relationship between the drum's and pulley's angular velocities is \(\dot{\theta}_{m}=\dot{\theta}\frac{R}{R_{m}}\). Within the framework of simplified system dynamics modeling, the original kinetic energy expression Equation (2) has been reduced to Equation (8), while the potential energy expression Equation (4) has been simplified to Equation (9), resulting in a more streamlined formulation. \[T=\frac{1}{2}(m+m_{c})\dot{y}^{2}+\frac{1}{2}\left(I+2I_{m}\frac{R^{2}}{{R_{m }}^{2}}\right)\dot{\theta}^{2} \tag{8}\]  
\[U=\frac{1}{2}k\Delta^{2}+mgy+\frac{m_{c}}{2}g(y+h+h_{m}) \tag{9}\]  
In this system, there are two generalized coordinates: \(y(t)\), which defines the vertical position of the load, and \(\theta(t)\), which defines the angular position of the drum. Thus, the generalized coordinates in Equation (7) are \(q_{1}=\theta\) and \(q_{2}=y\), respectively. On the other hand, the generalized force is composed of the frictional force at the direction pulleys and the driving torque. \[Q_{1}=-2b\dot{\theta}_{m}+T_{s} \tag{10}\]  
As a result, the equations of motion of the two-degree-of-freedom winch system with elastic cable are as follows, where nonlinear effects due to the cable elasticity are noticeable. \[ \begin{array}{c}\left(I+2I_{m}\frac{R^{2}}{R_{m}^{2}}\right)\ddot{\theta}+2b \dot{\theta}\frac{R}{R_{m}}+\frac{EAR}{h-y}(\theta R-y)=T_{s}\\ (m+m_{c})\ddot{y}+\frac{EA}{2(h-y)^{2}}(\theta R-y)^{2}-\frac{EA}{h-y}(\theta R -y)+\left(m+\frac{m_{c}}{2}\right)g=0\end{array}  \tag{11}\]
# GPS + BDS Network Real-Time Differential Positioning Using a Position Domain Estimation Method
## 3 Results
### Static Experiment
#### 3.1.2 Data Set II
In GPS-only solution, the maximum improvement of RMS errors can be found in the E component for station ZQGT, with a value of 33.7%, when replacing RD with PD. The corresponding value in GPS + BDS solution is 14.2% in the E component for station ZQGT. The improvements of PD over RD in GPS-only solution are larger than those in GPS + BDS solution. Further, the improvements of PD over RD in Figure 7 are smaller than those in Figure 4, whether in GPS-only or GPS + BDS solution. This is because the varying observed satellites have a greater impact on RD. The STD values of the network real-time differential positioning for each user station in Data set II are shown in Figure 8. The STD values of PD are smaller than those of RD, and the STD values of GPS + BDS solution are smaller than those of GPS-only solution. Remarkable improvements of PD over RD can be found when using GPS-only solution. Moreover, the improvements of PD with respect to RD in Figure 8 are smaller than those in Figure 5.  
Figure 6: Position errors of the network real-time differential positioning and number of satellites for ZQGT. Left panels: GPS-only; right panels: GPS + BDS. Table 4 gives the RMS errors and STD values for all user stations in Data set II. The positioning accuracy and precision of PD are better than those of RD. The overall improvements when replacing RD with PD are about 9.5%, 18.3% and 14.8% in the N, E and U components, respectively, which are smaller than those in Tables 1 and 2. Further, for both PD and RD, the values in GPS + BDS solution are smaller than those in GPS-only solution. The RMS errors of GPS + BDS solution for PD are 0.217 m, 0.159 m and 0.330 m in the N, E and U components, respectively, and the corresponding STD values are 0.173 m, 0.155 m and 0.319 m.  
Figure 8: STD values of the network real-time differential positioning for each user station in the N (top), E (middle) and U (bottom) components, respectively. Left panels: GPS-only; right panels: GPS + BDS. The percentages on each bar are the improvements of PD over RD. Figure 7: RMS errors of the network real-time differential positioning for each user station in the N (top), E (middle) and U (bottom) components, respectively. Left panels: GPS-only; right panels: GPS + BDS. The percentages on each bar are the improvements of PD over RD. The improvements of GPS + BDS solution over GPS-only solution are about 20%-30% for PD, and about 35%-43% for RD. Moreover, the improvements of PD over RD in GPS + BDS solution are smaller than those in GPS-only solution. It is also indicated that PD is less affected by the satellite constellation. The percentages of the position errors falling within predefined thresholds are given in Table 5 for all user stations in Data set II. It can be seen that the percentages of the position errors of PD are larger than those of RD in almost all thresholds. For both PD and RD, the percentage of the position errors of GPS + BDS solution is larger than that of GPS-only solution in each threshold.
# Image Recovery with Data Missing in the Presence of Salt-and-Pepper Noise
## 3 Transform Domains  
### Orthogonal Basis-Wavelet  
The wavelet transform (WT) is a useful tool to provide a time-frequency representation of a signal. For a mother wavelet \(\psi(t)\), the WT of the signal of interest (SOI) of \(x(t)\) is given by [25]  
\[\text{CWT}_{x}(a,b)=\frac{1}{a^{-1/2}}\int_{-\infty}^{\infty}x(t)\psi^{*}(\frac {t-b}{a})dt, \tag{5}\]  
where \(*\) is the complex conjugate, and \(a\) and \(b\) respectively represent the scaling and translation factors. From Equation (5), the scale parameter \(a\) compresses the wavelet when it is small. By doing so, the rapidly changing details of the signal are revealed. When it is large, it stretches the wavelet and, therefore, the coarse features of the signal are revealed.  
The inverse WT is expressed  
\[x(t)=\frac{C_{\bar{\psi}}^{-1}}{a^{-1/2}}\int_{-\infty}^{\infty}\int_{-\infty} ^{\infty}\text{CWT}_{x}(a,b)\widetilde{\psi}(\frac{t-b}{a})db\frac{da}{a^{2}}, \tag{6}\]  
where \(\widetilde{\psi}(t)\) denotes the dual function of \(\psi(t)\) and \(C_{\bar{\psi}}^{-1}\) represents the admissible constant.  
In a vector representation, matrix \(\mathcal{W}_{WT}\) is utilized to represent the wavelet decomposition and \(\mathcal{W}_{WT}^{\text{T}}\) denotes the wavelet reconstruction. Using these notations, the image in wavelet domain is  
\[\boldsymbol{\alpha}_{WT}=\mathcal{W}_{WT}\mathbf{x}, \tag{7}\]  
and correspondingly the image is reconstructed by  
\[\mathbf{x}=\mathcal{W}_{WT}^{\text{T}}\boldsymbol{\alpha}_{WT}. \tag{8}\]  
It is well known that the wavelet coefficient \(\boldsymbol{\alpha}_{WT}\) of the image contains many negligible components, which means \(\boldsymbol{\alpha}_{WT}\) is sparse--for more information, see [25; 26].
## 4 Conclusions  
The conventional measurement methods for obtaining the effective focal length, such as the nodal slide and magnification methods, are not useful for center-obstructed omni-directional optical systems. Therefore, the focal length of a center-obstructed omni-directional reflective optical system with an FOV of 80-135\({}^{\circ}\) was measured by applying the novel measurement method described in this paper. The EFL calculated using the measurements from our proposed measurement system was 0.3739 mm, which was only \(\sim\)11 \(\upmu\)m from the EFL obtained using the design software. In addition, the tolerance was calculated and compared with the theoretical value obtained for the omni-directional optical system to verify the validity of the measurement method. By applying the tolerance level commonly used in practice to the optical system, we concluded that the measured EFL falls within the tolerance range of \(\pm\)15 \(\upmu\)m. The measurement reliability could be increased if the sources of measurement error mentioned in this paper were reduced.  
Supplementary materials are available online [[http://www.mdpi.com/2076-3417/9/11/2350/s1](http://www.mdpi.com/2076-3417/9/11/2350/s1)]([http://www.mdpi.com/2076-3417/9/11/2350/s1](http://www.mdpi.com/2076-3417/9/11/2350/s1)). Table S1: Specifications of our designed omni-directional optical system; Table S2: Number of pixels for each zone and real ray heights on the sensor; Table S3: Measured and theoretical values; Table S4: The calculated focal lengths; Table S5: Focal length sensitivity for tolerance; Table S6: Permissible errors for focal length (in mm).  
**Author Contributions:** Conceptualization, H.C., J.-M.R.; methodology, J.-Y.J., J.-M.R.; software, J.-Y.J., J.-M.R.; validation, J.-M.R.; data curation, J.-Y.J., J.-M.R.; writing--original draft, H.C., J.-Y.J., J.-M.R.; supervision, J.-M.R.; funding acquisition, J.-M.R.  
**Funding:** This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2017R1D1A3B03029119).  
**Acknowledgments:** The authors thank the Editing Service for editing the English of this scientific paper.  
**Conflicts of Interest:** The authors declare no conflict of interest.
## 4 Simulation Environment and Methodology
We discarded the results of DODAGs not formed in less than 10,000 s. We applied this measure since we observed that, in certain scenarios, the time until all nodes receive at least one DIO message may be significantly longer than 10,000 s, whereas network convergence requirements for most deployments are 3-5 orders of magnitude below that threshold [6, 7, 8]. In order to characterize this phenomenon, we evaluated the percentage of networks formed in less than 10,000 s in the scenarios considered in subsection 5.1: small, medium and large network scenarios, for different node density and \(k\) values, when \(I_{min}\) is set to the default value, _i.e._, 8 ms (see Figure 3). In the described conditions, the minimum and maximum percentage of times that networks formed in less than 10,000 s are 50.14% and 99.97%, which correspond to the large network scenario when the node degree is 5 and \(k\) is 1, and the small network scenario when the node degree and \(k\) are 15, respectively. As shown in Figure 3, the percentage of networks not formed in less than 10,000 s grows as the network density decreases (since nodes have a lower amount of neighbors from which DIO messages can be received), as the redundancy constant decreases (because a greater number of DIO message transmissions are suppressed), and as the network size increases (since the average number of end-to-end hops between the root and the rest of nodes increases as well).