Robustness of Real-World Networks after Weight Thresholding with Strong Link Removal

Abstract

Weight thresholding (WT) is a method intended to decrease the number of links within weighted networks that may otherwise be excessively dense for network science applications. WT aims to remove links to simplify the network by holding most of the features of the original network. Here, we test the robustness and the efficacy of the node attack strategies on real-world networks subjected to WT that remove links of higher weight (strong links). We measure the network robustness along node removal with the largest connected component (LCC). We find that the real-world networks under study are generally robust when subjected to WT. Nonetheless, WT with strong link removal changes the efficacy of the attack strategies and the rank of node centralities. Also, WT with strong link removal may trigger a more significant change in the node centrality rank than WT by removing weak links. Network science research with the aim to find important/influential nodes in the network has to consider that simplifying the network with WT methodologies may change the node centrality.

1. Introduction

Weight thresholding is a simple technique that aims to reduce the number of edges in weighted networks that are otherwise too dense for applying standard graph-theoretical methods [1]. WT is a methodology in sparsification approaches to reduce link density in different real-world networks [2]. WT has many real-world applications, such as sparsifying ecological, financial, brain, and biological networks [3,4,5]. The principal aim of WT is to remove links to simplify the networks and make them easier to analyze. Therefore, the WT policy should guarantee that the significant traits of the original network are retained intact. In short, the objective of the WT procedure is to prune the highest number of links, avoiding drastic alterations in the critical structure of the original real-world network. Unfortunately, many conventional network properties quickly change under the WT procedure [1,6].

WT finds applications in research focusing on neural networks (NNs) or other machine learning models. In essence, WT involves applying a threshold to the weights of links in a NN. Links with weights below the threshold are considered less significant and can be eliminated or considered inactive. This process reduces the overall number of connections in the model, making it simpler and often more computationally efficient [7,8].

Network robustness is an essential field of research in network science [9]. Robustness is the property of a system to maintain functioning when perturbed or attacked [10]. The seminal research of Albert et al. [10] explores the error and attack resilience of complex networks of both real-world and model networks. The author investigates how networks react against random node removal (error and failures) and the targeted removal of the most connected nodes (attack). The findings provide valuable insights into understanding the robustness of real-world networks by opening a vast and important area of research.

A recent study investigated how weight thresholding procedures, which remove weak links (links of lower weight), affect the robustness of real-world networks to node attacks, and the rank of node centrality [2]. The study found that real-world networks subjected to the WT procedure have a robust connectivity structure to node attacks.

Here, we test whether WT with strong link removal changes the efficacy of the node attack strategies and how it affects the robustness of a set of real-world networks. To do this, we perform a sparsification procedure by removing a fixed fraction of higher-weight links. After sparsification, we execute a network attack by removing nodes using different node centrality indicators from the literature.

Performing the removal of strong links followed by a node attack can clarify the role that links of higher weights play in maintaining network connectivity. Previous studies have analyzed how the removal of strong links affects network connectivity. These studies removed links in decreasing (or increasing) order of weight and measured the resulting network connectivity using network structural indicators [11,12,13,14,15,16]. Here, we adopt a different and novel approach by removing strong links and then testing the resulting network structure with further node removals (attacks).

Generally, the real-world networks under study show robust connectivity against the WT procedure. Differently, the WT procedure removing strong links induces a more significant change in the ranking of nodes than the weak WT procedure.

2. Methods

2.1. Real-World Networks

We implemented five different node attack strategies on nine real-world weighted networks from different domains.

Table 1. Statistics of real-world networks. N number of nodes; L number of links; <k> average node degree; <w> average link weight; <CC> global clustering coefficient; LCC size of the largest connected component.

Networks	Key	Ref.	Type	Node	Link	Weight	N	L	<k>	<w>	<CC>	LCC
C. Elegans	Eleg	[17,18]	Biological	Neurons	Neurons connection	Number of Connections	297	2344	15.8	3.761	0.181	297
Cargoship	Cargo	[19]	Transport	Ports	Route	Shipping journeys	834	4348	10.4	97.709	0.222	821
US airport	Air	[20]	Transport	Airports	Route	Passengers	500	2979	11.9	152320.2	0.351	500
E. Coli	Coli	[19,21]	Biological	Metabolites	Common reaction	Number of Common reactions	1100	3636	6.61	1.364	0.139	1100
Netscience	Net	[22]	Social	Authors	Coauthorship	Number of Common papers	1461	2741	3.75	0.434	0.693	379
Human 12a	Hum	[23,24]	Biological	Brain regions	Connection between regions	Connection density	501	6038	24.1	0.01	0.457	501
Caribbean	Carib	[25,26]	Ecological Food web	Species	Trophic relation	Amount of biomass	249	3503	28.13	0.067	0.172	249
CypDry	Cyp	[16,27]	Ecological Food web	Species	Trophic relation	Amount of biomass	66	503	15.24	0.358	0.421	65
Budapest	Buda	[28]	Biological	Brain regions	Neural connection	Amount of track flow	480	1000	4.167	5.024	0.120	467

summarizes the statistics of these real-world networks, with node, link, and link weight meaning.

2.2. Attack Strategies

We simulated the following centrality-based node attacks in the networks: nodes with the highest centrality were removed first.

• Random (Ran): Nodes are removed randomly. Selecting nodes at random is analogous to simulating errors or failures in the network [9,10].
• Degree (Deg): The degree of a node is the number of links connected to it [10,29,30,31,32]. The degree $k_i$ of node i is given by the following:

  $$k_i={\sum }_{j=1}^N{a}_{ij},$$

  (1)

  where $a_{ij}=1$ indicates the presence of a link between nodes i and j and is 0 otherwise. $N$ is the number of nodes in the network.

• Strength (Str): A node’s strength is the total weight of the links connected to that node [33], also called a weighted degree.

Mathematically, the strength$s_i$ of node i is as follows:

$$s_i={\sum }_{j=1}^N{a}_{ij}.w_{ij},$$

(2)

where $a_{ij}=1$ indicates the presence of a link between nodes i and j and is 0 otherwise. $w_{ij}$ is the weight of the link between i and j.

• Betweenness (Bet): Betweenness of a node is the number of shortest paths passing through it [29,30,31]. This binary metric defines the shortest path between two nodes as the minimum number of links needed to travel between them.

Mathematically, the betweenness $b_i$of node i is as follows:

$$b_i={\sum }_{s,t=1}^N\frac{{\sigma }_{st}\left(i\right)}{{\sigma }_{st}},$$

(3)

where ${\sigma }_{st}\left(i\right)$ is the number of shortest paths between nodes s and t passing through the node i.${\sigma }_{st}$ is the total number of shortest paths between nodes s and t.

• Weighted Betweenness (WBet): Weighted betweenness of a node is defined as the number of weighted shortest paths passing through that node [34].

Weighted betweenness $b_i^w$ of node i is as follows:

$$b_i^w={\sum }_{s,t=1}^N\frac{{\sigma }_{st}^w\left(i\right)}{{\sigma }_{st}^w},$$

(4)

where ${\sigma }_{st}^w\left(i\right)$is the number of weighted shortest paths between nodes s and t passing through the node i.${\sigma }_{st}^w$ is the total number of weighted shortest paths between nodes s and t.

While computing shortest paths, it is fundamental to consider whether the link weight corresponds to “flows” or “costs” [35]. If link weight means flow, then the shortest path is computed by summing the inverse of link weights. If link weights are costs, shortest paths are computed directly by summing the link weights.

These attacks are performed by removing nodes and the links incident on them by targeting the nodes according to the decreasing order of their centrality values (Deg, Str, Bet, WBet). First, the node with the highest centrality is targeted, and the attack is continued on lesser centrality nodes until the network collapses. Attacking the nodes based on their pre-calculated rank is known as an initial (not recalculated) or simultaneous attack strategy [29]. However, the network structure may change after each attack, and the nodes’ importance may also change. In such a scenario, the pre-calculated ranking of nodes may no longer be valid. Here, we recalculated the node centrality values and updated the node’s rank after each attack [29]. This attack strategy is known as a recalculated (also named adaptive) attack strategy. In the case of ties (i.e., nodes with equal centrality value), we randomly selected the node to remove. These node ties were randomized by averaging the outcomes over 100 simulations.

2.3. Weight Thresholding

We investigated the effect of strong link removal on the robustness of real-world networks under various node attack strategies. This analysis was performed using the weight thresholding (WT) technique. The WT is performed by removing a fraction of the strong links. Given a weighted network G, the first step is to apply the weight thresholding on G. In our study, we took nineteen discrete threshold values WT = {0.0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9} (i.e., from 0% to 90% removal of strong links). In the case of ties (links having the same weight), the links were selected randomly. These ties were randomized by averaging the outcomes over 100 simulations. The thresholded network G’ was the subgraph of G with the same number of nodes. Then, node attack strategies on G’ were applied by identifying the nodes in the decreasing order of their centrality measures (Deg, Bet, Str, and WBet) computed from G’. This procedure was repeated for each WT.

2.4. Network Robustness Indicator

The largest connected component (LCC) is the simplest binary measure of the network’s functioning along node removal. It is defined as the highest number of connected nodes in the network [9,10,32]. Here, normalized LCC against the fraction (q) of nodes removed was used to measure network damage. The normalization was performed in two ways.

• One way was to normalize the LCC after node removal using the initial LCC value (before node attack) of the network after WT. In this case, we considered each thresholded network an independent network, and we did not account for the LCC decrease directly caused by the WT procedure.
• A second way was to normalize the LCC after node removal using the initial LCC at WT = 0, i.e., we normalized using the LCC of the original network. In this second case, we also considered the LCC decrease triggered by the link removal of the WT procedure. This normalization was intended to analyze the joint effect of the weight thresholding and node attack to decrease the LCC (total LCC decrease).

For ease of comparison, the response of networks to each attack strategy was represented by a single number called robustness (R). It was defined as the area under the curve of network functioning measure (here, LCC) against the fraction (q) of nodes removed. From now on, we refer to ‘robustness’ as the R measure computed with the first LCC normalization (R) and ‘total robustness’ (R_tot) as the measure computed with the second LCC normalization.

Table 2. List of the abbreviations used in this manuscript.

Abbreviation	Full Name
WT	Weight thresholding
LCC	Size of largest connected component
N	Number of nodes
L	Number of links
<w>	Average link weight
<k>	Average node degree
<CC>	Global clustering coefficient
Ran	Random node attack
Deg	Degree node attack
Str	Strength node attack
Bet	Betweenness node attack
WBet	Weighted Betweenness node attack
G	Weighted network
G’	Thresholded network
L’	Number of links in G’
q	Fraction of nodes removed
R	Robustness
Rtot	Total Robustness
<s>	Average node strength
Initial_Weak WT	WT by weak link removal with initial node attack strategy
Initial_Strong WT	WT by strong link removal with initial node attack strategy
Recalculated_Weak WT	WT by weak link removal with recalculated node attack strategy
Recalculated_Strong WT	WT by strong link removal with recalculated node attack strategy

lists the abbreviations used in this manuscript.

3. Results and Discussion

3.1. Robustness against WT

We investigate the role of strong links on the robustness of networks to node attack strategies. The WT removes a fixed fraction of strong links, and then, we perform the node attack strategies on each thresholded network. These strategies are performed using initial and recalculated node attack methods.

Figure 1 and Figure 2 show the LCC and the robustness R as a function of WT for different real-world networks. First, we analyze the LCC decrease induced by the WT procedure. The bar plots in the first column of Figure 1 and Figure 2 depict this LCC decrease. The networks C. Elegans, Caribbean, Human12a, and US airports show the slowest LCC decrease when subjected to the WT procedure. The WT procedure corresponds to the classic strong link removal [36]. Specifically, C. Elegans and the Caribbean keeps 80% of the LCC even up to 75% removal of strong links (WT = 0.75), and Human12a keeps 85% of the LCC for 80% removal of strong links (WT ≤ 0.85). The smallest network in our study, Cypdry (N = 66), and the air transportation network, US airports, also maintain a comparable LCC up to WT = 0.7.

The other networks, such as E. Coli, Budapest, Cargoship, and Netscience, present a lower robustness against the WT procedure, showing a faster LCC decrease than other networks. Budapest and Netscience networks show a faster LCC disruption under the WT procedure. Removing strong links accelerates the fragmentation of the science co-authorship networks (Netscience). In this network, dense local neighborhoods of scientists are primarily composed of weak links. In contrast, the strong links represent more significant and enduring connections among leading scholars, bridging distant research communities and thus playing a crucial role in overall network connectivity [14].

In summary, the real-world networks under study are robust to the strong WT procedure regarding the LCC. For this reason, the real-world networks under study unveil general robustness to strong link removal [36].

3.2. Robustness to WT and Node Attack

We investigate the network robustness against the coupled effect of the WT and node attack strategies in two ways.

First, we normalize the LCC along node removal with the initial LCC of the network after the WT procedure. This normalization does not consider the LCC decrease triggered by the WT link removal. This normalization evaluates the network after WT as an independent system and accounts only for the LCC decrease caused by the node attack. The trends of the robustness R with this normalization procedure for the node attack strategies, Ran, Deg, Str, Bet, and WBet are represented in Figure 1 and Figure 2.

Another way is to compute the relative robustness normalizing the LCC over the original LCC size, i.e., before WT and a node attack. In this manner, we can understand the decrease in network functioning by the joint effect of WT and a node attack (i.e., total robustness R_tot). The R_tot for the node attack strategies, Ran, Deg, Str, Bet, and WBet, is represented in Figure 1 and Figure 2.

We find a gradual change in R along the WT in both the initial and recalculated strategies for most of the networks. The C. Elegans network almost maintains steady robustness in all the attack strategies up to WT = 0.75. After removing 75% of the strong links, we can see a drop in the robustness of the network. The C. Elegans network, with the remaining 25% weak links, is highly vulnerable to all the attack strategies. The networks Caribbean, Human12a, E. Coli, Cargoship, and US airports show gradual changes in robustness after each thresholding even up to WT = 0.90. Instead of a smooth change in R, the network Cypdry shows some spikes in R, especially towards the Bet (red) and Str (purple) attack strategies.

The total robustness R_tot (solid lines) follows a similar pattern of robustness decrease for all the attack strategies except Ran (see green dotted and solid lines). In networks such as C. Elegans, Human12a, and E. Coli, the joint effect of thresholding and node attacks (R_tot) returns roughly the same robustness computed with the first normalization procedure (R). In all other networks, we can observe only a small difference in the values of these two types of robustness when focusing on targeted attacks. Differently, the robustness of the networks against random removal is always lower when considering the joint effect of WT and random node attacks. The solid green lines describing the R_tot decrease with increasing WT in Figure 1 and Figure 2 are significantly lower than the dotted green lines (R).

The principal aim of WT is to remove links to simplify the networks, making them easier to analyze and reducing the simulation time. Previous analyses showed that many standard network features quickly change under the WT procedure [1,6]. Here, we test whether WT with strong link removal changes the robustness of real-world networks when subjected to a node attack. Combining these results leads to the point that the real-world networks analyzed here hold comparable robust connectivity using both the two normalization procedures of the LCC.

There are some exceptions in Budapest, Netscience, and CypDry networks when we consider the normalization with the initial LCC of the network after the WT procedure. The Budapest network shows a higher robustness structure towards the end of thresholding (WT>0.7) (see Figure 2). Figure 3 shows the LCC as a function of the fraction of nodes removed q for Ran, Deg, Str, Bet, and WBet attacks in the Budapest network for WT values 0.75, 0.8, 0.85, and 0.9. It clearly shows that a higher WT value returns a slower LCC decrease.

In Figure 2, the Netscience also shows a higher robustness structure for some thresholding (WT > 0.2). The effect is also visible in Figure 4. The Cyp network also exhibits an increase in robustness when nodes are removed randomly (Ran). This interesting and counterintuitive result reveals that the network structures after WT may show a more robust LCC connectivity structure to node removal. In other words, the strong link removal performed by applying WT can strengthen networks against node attacks.

Scientific collaboration networks present links of higher weight connecting different communities of nodes [14]. Removing the strong links could fragment the scientific social network (Net) into smaller communities. Figure 5 shows that the Net network’s node clustering coefficient (<CC>) increases as a function of WT; that is, <CC> decreases when strong links are removed. Figure 5 shows an analogous <CC> increase at the end of the WT procedure for the Buda (WT > 0.8) and Cyp networks (WT>0.75). The <CC> rise can explain why an increase in robustness is also observed for different node attacks (Ran, Str, WBet) in the Buda network (Figure 2) and for the Cyp network for random node removal (Ran) (Figure 2) at the end of the WT procedure. The Buda network is a complex brain network where nodes are brain regions and links indicate electrical activity between them [28]. The Cyp network is a food web ecological network in which nodes are species and links depict trophic interactions among them [16]. Global node clustering <CC> is a simple measure evaluating the presence of communities of nodes in networks [28], and it is a measure that counts node triplets in the network. A triplet is three nodes connected by either two (open triplet) or three (closed triplet) links. <CC> is the ratio between the number of closed triplets and the total number of triplets (both open and closed) in the network [28]. The higher the <CC>, the higher the node’s tendency to cluster in communities.

Taking together the results would suggest that the removal of strong links can lead, in some cases, to the fragmentation of the network into communities (clusters of nodes) that are more resistant to node removal than the original network. This last pattern may explain the counterintuitive finding of increased network robustness in these real-world networks after applying strong WT. At the same time, this result would indicate that in highly clustered networks, removing bridge links (here, the strong links) connecting different communities of nodes may lead to a sparser network that is more resistant to node removal than the original one. Nonetheless, for the Air network, that is, the network of US airports [20], we observe a <CC> increase with WT but not a corresponding increase in robustness to node removal. For this reason, further mechanisms must be elucidated to understand why, in some real-world networks, the removal of strong links is associated with an increased robustness of the remaining network.

3.3. The Efficacy of the Node Attack Strategies

Figure 6 and Figure 7 list the best attack strategy, returning the lowest R value for each real-world network and each WT value. We find that with increasing WT, the efficacy of the attack strategy changes as well, and this is for both the normalization procedures of the LCC. For example, for initial node attack strategies, the best attack strategy for the C. Elegans network is the degree-based strategy (Deg) for WT≤0.1, whereas for WT > 0.1, the betweenness attack strategy (Bet) becomes the best method to dismantle the LCC (Figure 7). For the Cargo network, the best attack strategy is Str for 0.25 ≤ WT ≤ 0.4; in the remaining WT parameter space, the best attack strategy is Deg.

These findings show that the strong link removal performed using the WT procedure changes the efficacy of the node attack strategies. This last result has two important consequences. (I) Finding the best node attack strategies in real-world networks is a fundamental problem in network science with many real applications [31,35,37,38]. The WT procedure aiming to simplify the network by reducing the number of links induces structural changes that affect the efficacy of the node attack strategies. For this, network science research focusing on node attack strategies must consider that applying WT may significantly change the node attack efficacy. (II) Finding the best attack strategies is a heuristic way to select important nodes in the network [35]. Here, we show that WT performed with strong link removal changes the efficacy of the attack strategies. Therefore, strong WT is likely affecting the node rank in the network [2]. To test how WT affects the rank of the different node centralities, we use Kendall’s tau coefficient (τ) to evaluate the change in node rank after weight thresholding [39]. The τ coefficient is a measure of the magnitude of correspondence between two ranked pieces of data, i.e., the higher the Kendall’s τ coefficient, the more similar the two ranking sequences. The range of Kendall’s τ coefficient is from −1 to 1. We depict the results of this analysis in Figure 8. The τ coefficient decreases by increasing WT, indicating changes in the node rank after the WT procedure. Comparing the τ coefficient for strong WT (Figure 8, solid lines) with the τ coefficient discovered in a previous work by applying weak WT [2] (Figure 8, dashed lines), we find that strong WT produces a faster decrease in the τ coefficient. John et al. [2] found that applying the WT weak link removal decreases the τ coefficient to around 0.3 for most networks. By applying strong WT, we can lower the τ coefficient to 0 or even negative values (Figure 8, solid lines). This indicates that sparsification procedures based on strong link removal may trigger a greater change in the node centrality rank concerning the sparsification procedures removing weak links. Network science research focusing on developing algorithms to find important influential nodes [40] has to consider that simplifying the network with WT methodologies may also change the node importance evaluated by different node centrality indicators in the network.

3.4. Comparing Strong and Weak WT Procedures

John et al. [2] investigated the effect of weight thresholding (WT) on the robustness of real-world complex networks against various node attack strategies by removing a fixed fraction of weak links. In this study, we investigate the opposite perspective and perform WT by removing strong links. Figure 9 compares the R_tot against the initial node attack when weak and strong WT procedures simplify networks. We do not find a clear trend; in some cases, weak WT triggers a faster robustness decrease, and in others, it is to the contrary. For example, the weak WT induces a higher decrease in robustness concerning the strong WT for the Eleg, Cyp (except under WBet initial attack), Air, and Cargo networks for both the initial (Figure 7, red lines) and recalculated attacks (Figure 9, green lines). These results agree with the study by Onnela et al. [13] on mobile communication networks [13]. Onnela et al. [13] show the counterintuitive consequence that real-world social networks are robust to removing the strong links but fall apart quickly if the weak links are removed. Onnela et al. [13] analyzed the network’s robustness to removing links only. Our study, on the other hand, analyzes the combined effect of removing links and then attacking the network by removing nodes. Despite the differences between Onnela et al. [13] and our approaches, similar systems’ responses are observed: for certain types of real-world networks, removing weak links can induce greater fragility. Our results show that this may happen not only in social networks [13] but also in transportation and biological networks.

However, the strong WT returns lower robustness in the Carib and Hum networks, especially for initial node attacks. The Car network is a food web ecological network depicting who eats whom in the ecosystem. Hum is the human brain network modeling the electrical communication activities sharing information among brain regions [23]. Hence, from very different domains of science, these real-world networks show a higher fragility when combining strong WT and node attacks (Figure 9). From these results, it is possible to infer possible dynamics of these real-world networks. In food webs, removing (or disrupting) the higher magnitude trophic connections between species may trigger the remaining ecological network to be more sensitive to species removal. The removal of species in food webs models the case of species extinction [16]. Our results would suggest that removing strong trophic interactions would make the ecosystem more prone to biodiversity loss. The higher vulnerability to strong WT in the brain network would indicate that once the connections with the highest electrical activity between different brain regions are removed, the remaining brain network is more prone to brain region malfunctioning (node removal) and becoming more easily disconnected. This may help in understanding the mechanisms by which brain networks and which brain regions play the main routing information.

Our latest results show the difficulty in predicting how different sparsification procedures may affect the robustness of node attacks on real-world networks. Different real-world networks may exhibit opposite behaviors regarding sparsification through removing the heaviest-weight links (strong WT) compared to removing the links of lower weights (weak WT).

4. Conclusions

We analyzed the impact of weight thresholding on the robustness of real-world networks to different node attack strategies. Here, weight thresholding is performed by removing a fixed fraction of strong links. Generally, the real-world networks under study show robust connectivity against the WT procedure. In other words, real-world networks maintain a robust structure regarding the LCC to strong link removal. These results suggest that strong link removal can be used as a method for the sparsification of networks for applications in which the robustness to node attacks is important.

Then, we find that applying WT may significantly change the node attack efficacy and the rank of different node centrality measurements. The strong WT procedure induces a greater change in the ranking of nodes than the weak WT procedure. For this reason, network research focusing on finding the efficacy of node attack strategies or finding important nodes in the network has to consider the network structural changes caused by the weight thresholding (sparsification) procedures.

Studying the robustness against node attacks after strong WT may have different real-world applications. Removing links with higher weights and then performing node attacks could help identify the parts of the network that are more robust (or less affected) when removing key connections. In the real world, this can be useful for designing network protection or reinforcement strategies in critical infrastructure networks, such as those for energy, transportation, or communications. These vital systems can benefit from identifying the robustness of network components resulting from attacks on strong links, planning their protection, and developing risk mitigation strategies.

Our research also has significant implications for understanding ecological networks. By identifying keystone species in food web ecological networks, we can gain insights into the mechanisms of biodiversity loss in ecosystems. Food webs are networks of species and their trophic interactions [16,41]. Strong WT can simulate the deletion of strong trophic interactions that occur with the extinction or decreasing abundance of the most general species/resources in ecosystems. The subsequent node removal can then model the occurrence of species extinction in the remaining parts of the food web ecological network, providing a deeper understanding of biodiversity loss mechanisms.

Moreover, the emergence of the role of strong and weak links is associated with the local structure of the social networks [42], and understanding the specific embedding of strong links is important to comprehend complex social systems.

For example, scientific collaboration networks present links of higher weight connecting different communities of nodes [14]. Removing the strong links could fragment the scientific social network into smaller communities. Subsequently, removing nodes from these communities can help us better understand the robustness and relationships within specific groups of scientists.

Last, the results presented in this study can be useful in network science research that needs to simplify complex networked systems and in machine learning and neural network research that needs to reduce model complexity or eliminate less important network connections.