esa-sceva/satcom-chunk-collection huggingface dataset:
## 2 Theoretical Modeling
### Four equations of \(x\)-\(z\) planar motion.  
\[\frac{\partial M_{y}}{\partial t}+EI_{p}\frac{\partial\dot{g}_{y}}{\partial l }-EI_{p}\frac{\dot{g}_{z}}{R_{p}}=0 \tag{5}\]  
\[\frac{\partial\dot{w}_{x}}{\partial t}+\frac{1}{m}\frac{\partial f_{x}}{ \partial l}-\frac{2m_{f}V}{m}\frac{\partial\dot{w}_{x}}{\partial l}-\frac{m_{ f}V^{2}+PA_{f}}{m}\left(\frac{1}{EI_{p}}M_{y}-\frac{\theta_{z}}{R_{p}}\right)+g \cos\beta-\frac{1}{m}f_{ex}=0 \tag{6}\]  
Figure 1: Coordinate system of the fluid-filled pipe section.  
\[\frac{\partial f_{x}}{\partial t}+\kappa GA_{p}\frac{\partial\dot{w}_{x}}{ \partial l}-\kappa GA_{p}\dot{\varrho}_{y}=0 \tag{7}\]  
\[\frac{\partial\dot{\varrho}_{y}}{\partial t}+\frac{1}{T_{y}}\frac{\partial M_{ y}}{\partial l}+\frac{1}{T_{y}}f_{x}-\frac{1}{T_{y}R_{p}}M_{z}-\frac{1}{T_{y}}M_{ey}=0 \tag{8}\]
3. Four equations of \(y\)-\(z\) planar motion. \[\frac{\partial M_{x}}{\partial t}+EI_{p}\frac{\partial\dot{\varrho}_{x}}{ \partial l}=0\] (9)  
\[\frac{\partial\dot{w}_{y}}{\partial t}+\frac{1}{m}\frac{\partial f_{y}}{ \partial l}+\frac{2m_{f}V}{m}\frac{\partial\dot{w}_{y}}{\partial l}+\frac{m_{ f}V^{2}+PA_{f}}{m}\bigg{(}\frac{M_{x}}{EI_{p}}-\frac{1}{R_{p}}\bigg{)}-\frac{1}{mR _{p}}f_{z}+g\cos\beta-\frac{1}{m}f_{ey}=0 \tag{10}\]  
\[\frac{\partial f_{y}}{\partial t}+\kappa GA_{p}\frac{\partial\dot{w}_{y}}{ \partial l}+\kappa GA_{p}\dot{\varrho}_{x}-\frac{\kappa GA_{p}}{R_{p}}\dot{w}_ {z}=0 \tag{11}\]  
\[\frac{\partial\dot{\varrho}_{x}}{\partial t}+\frac{1}{T_{x}}\frac{\partial M _{x}}{\partial l}-\frac{1}{T_{x}}f_{y}-\frac{1}{T_{x}}M_{ey}=0 \tag{12}\]
4. Two equations of torsion motion. \[\frac{\partial M_{z}}{\partial t}+G_{Ip}\frac{\partial\dot{\varrho}_{z}}{ \partial l}+\frac{G_{Ip}}{R_{p}}\dot{\varrho}_{y}=0\] (13)  
\[\frac{\partial\dot{\varrho}_{z}}{\partial t}+\frac{1}{\rho_{p}l_{p}}\frac{ \partial M_{z}}{\partial l}+\frac{1}{\rho_{p}l_{p}R_{p}}M_{y}-\frac{1}{\rho_{ p}l_{p}}M_{xz}=0 \tag{14}\]  
where  
\[\tau_{w}=\tau_{ws}+\tau_{wuz}+\tau_{wub}=\frac{\rho fV|V|}{8}+k\frac{\rho V}{ D}\int_{0}^{t}\frac{\partial V(t)}{\partial t}\frac{C_{1}e^{-C_{2}t}}{\sqrt{ \pi t}}dt+(1-k)\frac{\rho D}{4}\cdot k_{3}\bigg{(}\frac{\partial V}{\partial t }-\alpha\frac{\partial V}{\partial z}\bigg{)} \tag{15}\]  
\[C_{1}=\frac{R}{2\sqrt{v_{w}}},C_{2}=\frac{0.135v_{w}}{R^{2}}Re^{k_{4}} \tag{16}\]  
\[k_{3}=\sqrt{\frac{7.41/Re^{k_{4}}}{2}},k_{4}=\lg(14.3/Re^{0.05}) \tag{17}\]  
\[\alpha=\frac{\sqrt{K/\rho_{f}}}{\sqrt{1+DK/eE}} \tag{18}\]  
Equation (15), \(\tau_{ws}\) is the wall shear friction in steady flow, \(\tau_{wuz}\) is the [PERSON] additional shear friction term in unsteady flow and \(\tau_{wub}\) is the [PERSON] additional shear friction term in unsteady flow. For the FSI analysis of pipe conveying fluid within a wide range of the Reynolds number, the selection coefficient \(k\) needs to be defined. When the fluid inside the pipe is in the transient laminar flow and the low-Reynolds-number turbulence (Reynolds number is less than 2320), \(k=1\), instead, when the fluid under the unsteady turbulence state and high-Reynolds-number turbulence (Reynolds number is greater than 2320), \(k=0\).
## 2 Materials and Methods  
### Biodegradability Experiment  
For this experiment, seven 500 mL flasks were used: one as a control that only contained inoculum and water (BLANK), two for conventional anaerobic digestion (AD1 and AD2), two for anaerobic digestion using PLA/Carbon Black Protopasta pellets (NB1 and NB2) as fillers and two for anaerobic digestion using PLA/Graphene pellets (OB1 and OB2) as fillers. The PLA/Carbon Black Protopasta (polylactic acid) material has a reported resistance of 15 \(\Omega\) cm and PLA/Graphene BlackMagic3D (BlackMagic3D, Ronkonkoma, NY, USA) of 0.6 \(\Omega\) cm. Furthermore, once the pellets were printed (Figure S1), the former had a density of 0.97 g/mL and the latter had a density of 1.05 g/mL. PLA/Graphene and PLA/Carbon Black were selected since previous abiotic characterisation tests were carried out by this group, resulting in promising results [30; 31]. One hundred and twenty-five 6 mm diameter pellets were used as fillers into each of the corresponding reactors. The experiment consisted of two cycles, which lasted from the introduction of the inoculum and substrate until the biogas production in the digesters started to decrease. In the first cycle, inoculum and substrate were introduced in a 1:1 ratio (\(w\)/\(w\)) of volatile solids (VS) for all digester types, leaving about 200 mL of headspace. The final volume was 303 mL, with a volatile solids content of 5.20 g and a water content of about 96%. For the second cycle, 78 mL of the digestate resulting from the first cycle was replaced with fresh sludge (substrate).  
Digested sludge from the anaerobic digester of the Leon WWTP was used as inoculum. The substrate used was a mixture of undigested primary and secondary sludge with an average VS content of 2.5%. During both cycles, a bath was used to maintain the digesters at a temperature of 36 \({}^{\circ}\)C and constant agitation. During the experiment, gas measurements were performed every 2-3 days and volatile solids were analysed at the end of each cycle, as specified in Section 2.3.
## 8 Conclusions  
This study addressed the challenges of understanding the results of multi-class classification of network intrusions in highly imbalanced data, including the CIC-IDS2017 and CSE-CIC-IDS-2018 datasets. In the research, machine learning models of different complexity and explainability were included in order to evaluate and understand whether more complex ML models were indeed capable of providing higher classification results for the classification of 28 classes of intrusions. The study compared the classification performance of six machine learning models using various measures of classification accuracy. The results revealed that the DT model utilizing the CART algorithm achieved the highest performance in the multi-class classification task, achieving an _F1-score_ of 0.998 and an average macro _F1-score_ of 0.969. The lowest results were obtained with the dense-layer network, with an _F1-score_ of 0.833 and an average macro _F1-score_ of 0.063. Another decision tree algorithm, namely ID3, along with the RF model, resulted in slightly lower but still significant results, achieving an _F1-score_ of 0.985.  
In addition, experiments were carried out with the XAI methods, LIME and SHAP, to assess the potential and reliability of identifying the most important features of the dataset. Although these methods provide a list of the most influential features, it has been observed that the local explanation is often unstable, and each regeneration may lead to a completely different result. Even a specially added column that has no relevance to the problem may have the greatest influence on the decision. Another observation is that a global interpretation of all classes is not very explicit, and it is quite difficult to understand the visualization. Therefore, it is likely that when we have more than multi-class and multi-feature datasets, it would be more useful to use numerical or aggregated results of explanations.  
Conceptualization, A.P.-T.; data curation: M.B. and A.P.-T.; investigation, M.B. and A.P.-T.; methodology, M.B. and A.P.-T.; software, M.B. and A.P.-T.; resources M.B. and A.P.-T.; validation M.B.; writing--original draft, M.B. and A.P.-T.; writing--review and editing, M.B. and A.P.-T.; supervision, A.P.-T. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Data Availability StatementData is available upon reasonable request to the corresponding author. The authors declare no conflict of interest. References
# DNS Firewall Based on Machine Learning
## 2 State of the Art
Regarding the used dataset, there are some related works about the creation process. In [21], the authors collected more than 100 numerical features of a domain. The collected data were based on DGA and legitimate domain names to provide a large dataset to the scientific community. The lack of features from OSINT sources are a potential limitation thatwe took into account in our dataset creation process. Contrarily, in [22] the authors collected around 1.5 million domain names and extracted 11 features mainly based on host-based and web-based characteristics of a domain, such as the IP address, TLD and information from WHOIS, but not using numerical features. Therefore, we create a dataset with 34 features using numerical, host-based, and web-based characteristics of a domain name. Although there are numerous studies on DNS firewall systems, the need to analyze and predict newly malicious domains over DNS communications in a timely manner is a valuable cybersecurity countermeasure to mitigate, prevent and alert both organizations and end-users against malicious DNS traffic.
# Markov Chain Model-Based Optimal Cluster Heads Selection for Wireless Sensor Networks
## 2 Related Work and Background
### Clustering
#### 2.1.2 Single-Hop vs. Multiple-Hop Clustering  
In single-hop clustering, each and every node in a cluster can exchange data among themselves, and each node can communicate with its concerned CH directly [7; 29]. On the other hand, in the multiple-hop strategy, a node needs to go through many hops to communicate with its concerned CH [23; 30]. In this categorization, hybrid clustering is one in which the size of a cluster is changeable as follows: few clusters can be adjusted at the two-hop distance and the remaining could be adjusted to less or more hops. An example of hybrid clustering is discussed in [8], where the authors recommend larger hopping distance between far away clusters. However, there is a tradeoff between cluster organization, intra-communication and inter-communication ranges, and allocation of resources.
## 3 Method
### BiGRU Layer  
Gated Recurrent Unit (GRU) is a special recurrent neural network developed on the basis of Long Short Term Memory (LSTM), which has a simpler structure and solves the problem of gradient disappearance and gradient explosion in LSTM and can better capture semantic dependencies and achieve long-term memory. Structurally, GRU combines the forgetting gate and the input gate into an update gate, so there are only update and reset gates and a mixture of cell states and hidden states, so GRU has fewer training parameters, which can speed up the training and improve the model performance [26].  
The structure of the GRU is shown in Figure 3, where \(r_{t}\) is a reset gate to control whether the current content is remembered, and \(z_{t}\) is an update gate to control the proportion of preorder memory and candidate memory in the current state [27].  
The GRU network forward propagation weight parameter update formula [28] is:  
\[z_{t}=\sigma(w_{xz}x_{t}+w_{hz}h_{t-1}+b_{z}) \tag{10}\]  
\[r_{t}=\sigma(w_{xx}x_{t}+w_{hr}h_{t-1}+b_{r}) \tag{11}\]  
\[\widetilde{h}_{t}=\tanh(w_{xht}x_{t}+r_{t}\otimes w_{hh}h_{t-1}) \tag{12}\]  
\[h_{t}=z_{t}\otimes h_{t-1}+(1-z_{t})\otimes\widetilde{h}_{t} \tag{13}\]  
where \(x_{t}\) denotes the input at moment \(t\); \(h_{t-1}\) denotes the input of the hidden state at moment \(t-1\); \(\widetilde{h}_{t}\) denotes the candidate hidden state; \(h_{t}\) denotes the hidden state, which is the output of the GRU unit; \(\oplus\) denotes vector splicing; \(\otimes\) denotes Hadamard product, i.e., multiplying by the corresponding elements; \(\sigma\) is the sigmoid function; \(w\) is the weighting factor; \(b\) is the offset term.  
The hidden state \(h_{t}\) of GRU can only obtain information from the past, but in the process of NER, the context of the text is important for predicting the labels, so this paper adopts BiGRU as part of the NER model. The forward GRU captures the above information of the text, the reverse GRU extracts the below information of the text, and the output of the network is obtained by stitching the output of the forward and reverse GRUs, which fully extracts the contextual information of the text and can effectively improve the effect of NER. The output of the decoding layer BiGRU can be expressed in Equation (14).  
\[h_{i}=\left[\overset{\rightarrow}{h_{i}}\oplus\overset{\leftarrow}{h_{i}}\right] \tag{14}\]  
Figure 3: Structure of GRU.
# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 4 Harnessing the Domain Trade-Off
### Capacity of a Tensor Channel
The constant \(\mu\) is chosen to satisfy the powerconstraints, and \(()^{+}\) indicates that each element in the enclosed entity is non-negative, i.e., \((a)^{+}=\max\{0,a\}\). The solution in (24) can be seen as a higher order generalization of the classical water-filling solution for the MIMO channel to a tensor setting. Assigning the optimum covariance tensor to the input performs power allocation across all the domains and also prescribes a multi-domain input precoding strategy. Notice that the eigenvalues of \((\mathcal{H}^{H}*_{M}\mathcal{H})\) depend on the multitude of domains in the tensor channel. This paves the way for certain flexibility to exploit the trade-off between different domains, depending on the specific tensor channel eigenvalues. The capacity is obtained by summation of a log function over all such multi-domain eigenvalues. The lack of resources in a specific domain can be compensated with resources in alternate domains, which would allow a more judicious and efficient resource allocation mechanism. In the next section, we elaborate on such a domain trade-off through numerical examples.
## 3 Description of the Environment
### The Reward Function  
A reward function is critical for guiding the AUV agent's behavior in the environment, incentivizing actions that lead to successful goal-reaching. In this work, a _dense reward_ was employed that included three principal components:  
1. **A penalty for distance from the goal**: The farther the vehicle is from the target point \(\big{(}x_{\text{goal}},y_{\text{goal}}\big{)}\), the larger the negative portion of the reward. This encourages the agent to continually decrease its distance to the goal.
2. **A penalty for excessive control changes**: To limit abrupt maneuvers, a weighted cost \(\sum_{j=1}^{M}\lvert a_{j}\rvert\) is added. The parameter \(w_{a}\) controls the influence of this term on the reward.
3. **A penalty** for collisions with port infrastructure elements, denoted by \(c\). Commonly, a large negative reward is assigned upon collisions, strongly dissuading risky trajectories. The total reward at time \(t\) is given by  
\[r\ =\ -\sqrt{\big{(}x_{\text{goal}}-x_{t}\big{)}^{2}+\big{(}y_{\text{goal}}-y_ {t}\big{)}^{2}}\ -\ w_{a}\,\sum_{j=1}^{M}\lvert a_{j}\rvert\ +\ c, \tag{35}\]  
where  
* \(\big{(}x_{t},\,y_{t}\big{)}\) is the AUV's position at time \(t\).
* \(\big{(}x_{\text{goal}},\,y_{\text{goal}}\big{)}\) is the target position.
* \(w_{a}\) is the weight of the penalty on control actions, regulating abrupt or frequent changes \(\sum_{j=1}^{M}\lvert a_{j}\rvert\).
* \(M\) denotes the number of decision actions (e.g., heading changes, speed changes).
* \(c\) is the penalty (zero/negative reward) for collisions; it takes a large negative value when a collision occurs or 0 if no collisions occur.  
This reward design motivates the agent to approach its goal promptly and smoothly (minimizing the distance), constrains sudden control deviations (through the penalty on large changes), and severely penalizes collisions.
# Value of Information Based Data Retrieval in UWSNs
## 4 Path Planning Algorithms
### Greedy Path Planner with Intermediate-Node-Visitation-**G**ipp  
It is a similar algorithm to GPP but with the added notion that while the AUV is on its way to the next node with the highest VoI offer, it can visit nodes that (by some definition e.g., distance) lie on the prescribed path.  
It might be a reasonable idea to visit intermediate nodes, i.e., nodes that lie on the path to the destination node. This will help in minimizing tour time and, hence, improving VoI accumulation.  
Algorithm 3 lays out the steps for GIPP. It is similar to GPP except for the intermediate node visitation which is a slight detour. For the intermediate node visitation, the algorithm selects the next node to be visited from the neighbors of the current node in a manner that the distance between the neighbor node and the node with the maximum VoI (destination node) is minimum among all the neighboring nodes. This algorithm is inspired by concepts behind both the GPP and LPP, i.e., visit sequence is greedy but tour time is minimized by visiting nodes that lie across the path. However, when it comes to reporting catastrophes at the earliest, it is a tad slower than GPP because of the intermediate node visitations. A potential path planned using the GIPP algorithm is depicted by the green colored trail in Figure 1.  
The complexity of GIPP is \(\mathcal{O}(n^{2}+n\times d\times m)\) where \(n\) is the number of sensor nodes, \(d\) is the count of nodes in the longest intermediate tour, and \(m\) is the maximum number of nodes in a neighborhood definition. Because \(d<n\) and also \(m<n\), therefore, it follows that the complexity of GIPP is \(\mathcal{O}(n^{3})\).  
```
1:procedureGIPP(\(S,s_{o}\))
2:\(S\leftarrow\{s_{1},s_{2},...,s_{n}\}\)\(\triangleright\) Set of sensor nodes
3:\(V\leftarrow\emptyset\)\(\triangleright\) Visitation sequence
4:\(i\gets s_{o}\)\(\triangleright\) Tour starting node
5:while\(S\
eq\emptyset\)do
6:\(j\leftarrow\)getNodeThatHasMaxVOI(S)
7:\(T\leftarrow\emptyset\)
8:\(T\leftarrow\)tourIntermediate(\(i,j,S\))
9:\(V\gets V+T\)
10:\(S\gets S-T\)
11:\(i\gets j\)
12:endwhile
13:return\(V\)
14:endprocedure
15:proceduretourIntermediate(source,destination,S_{r}\)
16:\(p\gets\)source
17:\(q\gets\)destination
18:\(T_{I}\leftarrow\emptyset\)\(\triangleright\) Intermediate Visitation Sequence
19:while\(p\
eq q\)do
20:\(N\leftarrow\)getNeighborhood(\(p,S_{r}\))
21:\(i\gets s_{x}\in N\) such that eculideanDistance(\(s_{x},q\)) is minimized
22:tourEculidean(\(p,i\))
23:\(p\gets i\)
24:\(T_{I}\gets p\)
25:endwhile
26:return\(T_{I}\)
27:endprocedure
```  
**Algorithm 3** Greedy Path Planner with Intermediate-Node-Visitation-**GIPP**.
# The Impact of Gravity on Different Height Systems: A Case Study on Mt. Medvednica
## 3 Leveling Survey
### Benchmarks Coordinates  
The points were located at distances 200 to 500 m apart depending on the openness of the horizon and tree canopies, all with the aim of better determining positions and minimizing interference during GNSS surveying. The points were measured three times, each for 30 s. One repetition consists of three consecutive measurements, with each measurement lasting 30 s, representing 30 measurement epochs. A total of 40 benchmarks were established along the trajectory (Figure 8).  
The coordinates of all the points in the leveling line were stabilized with a steel bolt and determined by GNSS using the real-time high-precise positioning service of the Croatian Positioning System (CROPOS VPPS) [18; 19; 20]. By using the CROPOS VPPS service, plane coordinates (E, N) in the HTRS96/TM (EPSG:3765) map projection and normal-orthometric height in HVRS71 were obtained.  
Along with HVRS71, based on the Decree on establishing new official geodetic data and map projections of the Republic of Croatia on 4 August 2004 [12], the Republic of Croatia established new official geodetic data and map projections of the Republic of Croatia.  
Figure 8: Trajectory of the measurements (the background is an orthophoto map obtained from the Croatian Geoportal [17]).  
According to the Decree, a new horizontal reference system called the Croatian Terrestrial Reference System 1996 (HTRS96) was introduced based on the European Terrestrial Reference System 1989 (ETRS89) [21] with GRS80 as the reference ellipsoid.  
In addition to the coordinates of the 40 newly stabilized benchmarks measured using the GNSS method, the coordinates of one gravimetric eccentric point, AGT03-E3, and three existing height benchmarks--two at the beginning and one at the end of the route--were required for this study. The coordinates of these points were obtained from the positional records compiled at the time of their stabilization.
## 4 Implementation
### Management Server: MindsCare PC Application  
The aggregated patient data were sent to the server for monitoring. The received data were integrated and mined through the MindsCare PC application and shown as a visual UI, as illustrated in Figures 4 and 5.  
On the Dashboard page (Figure 4a), the SUD, SAPS, BAS/BIS, DFDII, and BSCS information is shown in each visual circular chart. The users can view the number of samples when they place the mouse cursor over the circular chart (Figure 4b). The group distribution by age and sex is shown in the bar graphs, and the managers can view the number of samples when they place the mouse cursor over the bar graphs (Figure 4c).  
The user smartphone application monitoring information that is obtained via the background-running app Internet Detox is observed on the Smartphone Usage (SMU) page (Figure 5a). The top five smartphone application lists are shown in a circular chart (e.g., Kakao Talk, Chrome, and Google  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Characteristics**} & \multicolumn{2}{c}{**SUD (\(n=29\))**} & \multicolumn{2}{c}{**SUC (\(n=67\))**} & \multicolumn{2}{c}{**Total (\(n=96\))**} \\ \cline{3-8} \multicolumn{2}{c}{} & \(n\) & \% & \(n\) & \% & \(n\) & \% \\ \hline \multirow{2}{*}{Gender} & Male & 22 & 75.9 & 65 & 97.0 & 87 & 90.6 \\  & Female & 7 & 31.8 & 2 & 3.4 & 9 & 9.4 \\ \hline \multirow{3}{*}{Age} & 10–19 & 15 & 51.7 & 57 & 85.1 & 72 & 75.0 \\  & 20–29 & 7 & 24.1 & 4 & 6.0 & 11 & 11.5 \\  & 30–39 & 7 & 24.1 & 6 & 9.0 & 13 & 13.5 \\ \hline \multirow{3}{*}{Education} & Undergraduate & 20 & 69.0 & 59 & 88.1 & 79 & 82.3 \\  & Graduate & 8 & 27.6 & 7 & 10.4 & 15 & 15.6 \\  & Postgraduate & 1 & 3.4 & 1 & 1.5 & 2 & 2.1 \\ \hline \multirow{2}{*}{Job} & Employed & 9 & 31.0 & 6 & 9.0 & 15 & 15.6 \\  & Unemployed & 20 & 69.0 & 61 & 91.0 & 81 & 84.4 \\ \hline \multirow{3}{*}{Marital Status} & Married & 5 & 17.2 & 5 & 7.5 & 10 & 10.4 \\  & Unmarried & 24 & 82.8 & 62 & 92.5 & 86 & 89.6 \\ \hline \multirow{3}{*}{SES} & High & 3 & 10.3 & 15 & 22.4 & 18 & 18.8 \\  & Middle & 16 & 55.2 & 32 & 47.8 & 48 & 50.0 \\ \cline{1-1}  & Low & 8 & 27.6 & 18 & 26.9 & 26 & 27.1 \\ \cline{1-1}  & Unknown & 2 & 6.9 & 2 & 3.0 & 4 & 4.2 \\ \hline \hline \end{tabular} Unemployed: Student, Housewide; Abbreviation: SES, Socio-economic Status.  
\end{table}
Table 2: Socio-demographic status of participants in the smartphone usage disorder (SUD) and smartphone usage control (SUC) groups.  
apps). A manager can view the aggregate usage time of each application when they place the mouse cursor over the circular chart (Figure 5b).
# Influence Maximization with Priority in Online Social Networks
## 2 Model and Problem Definition
### Problem Definition  
We investigate Influence Maximization with Priority (IMP) defined as follows:  
**Definition 1** (Imp problem).: _Given a graph \(G=(V,E)\) under IC model, a positive integer \(k\) (budget), the priority set \(U\subset V\), and the threshold \(T\) with \(T\leq k,T\leq|U|\). IMP problem asks to find the seed set \(S\subset V\), with \(|S|\leq k\) and \(\sigma_{U}(S)\geq T\) so that influence spread, \(\sigma(S)\), is maximized, i.e, find \(S\) that is the solution to the following optimization problem:_  
**maximize:**: \[\sigma(S)\] (6)
**subject to:**: \[|S|\leq k\] (7) \[\sigma_{U}(S)\geq T\] (8)IMP becomes IM problem when \(U=\emptyset\). Therefore, IM is a special case of IMP and IMP is also NP-hard. In addition, the calculation of the influence function from the seed set is proven to be #P-hard [12]. Thus finding the solution to the problem within the time allowed is very challenging.