Novel Node Centrality-Based Efficient Empirical Robustness Assessment for Directed Network

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Introduction
With rapid increasing online social network, the network structure of online social networks has become more complicated than before. Analysis and explaining the dynamics and properties of social networks has become an interesting researching task with plenty of applications in social sciences and many other web application scripts. In some social networks, it is very common for some users who decide to leave the network or begin to stop being active in the activities of their community [1]. is phenomenon is also called as quitting or churn and has absorbed much research attention in social networks. And how to analyze and evaluate network robustness and resilience [2][3][4] after and the latest found rules of complex network which can be found in Figure 1 for Toy example. And it would be helpful to find some new discipline and cast new light on robustness and vulnerability assessment of directed social network.
In detail, our contributions are as follows: (1) Basing on classic probability graph theory and clustering coefficient definition, we have proposed two new node centralities named ITG (information transfer gain) and IM p (v) (directed node importance). It can be used to measure the robustness and vulnerability in directed networks especially directed social networks which have been seldom tested before. (2) We have proposed enough experiment results on undirected artificial networks and directed online social networks to make robustness assessment comparison of ITG and IM p (v), which was mentioned scarcely in former-related works. (3) Our ITG-and IM p (v)-based centrality has been proved to be more accurate, efficient, and faster than classical centrality methods such as degree and betweenness with sufficient experiments results for node robustness assessment in directed social networks. (4) We have firstly proposed a new rigorous proving process of directed node importance centrality IM p (v) and its implementation method. We have found that, in the more densely connected directed social network, the IM p (v) node removal strategy is the most harmful to the network connecting structure. And to the best knowledge of us, we attain the varying trend of iteration times k to the marginal difference ε on directed social networks for the first time based on strict mathematical proof. e outline of the paper is as follows: Section 2 introduces the related work on robustness and resilience of scale-free network which includes some traditional complex network and social network. Section 3 presents the definitions of network structure quantities which we used to evaluate the robustness and vulnerability of network datasets. And our novel ITG-and IM p (v)-based directed node centrality will be introduced. Section 4 proposes the experiments results in synthetic signed network and real-life large networks derived from directed online social media and directed human mobile phone calling network with former undirected node centrality measures and our directed ITG centrality measure and IM p (v). Section 5 gives the final conclusion of this article and draws some new light on the future work.

Related Work on Network Robustness
ere have been many important research studies on vulnerability assessment and robustness assessment in network structures [11,12] after node departure or network attack and other significant research areas. e robustness of ER random network and BA scale-free network was analyzed firstly by Albert [13] in 2000. He used the multiple correlation relationship data of l −1 (inverse geodesic length), S (size of the largest connected subgraph), and removed node ration under node attack to make vulnerability and robustness assessment. He discovered that collapse of scale-free network may reach a high price because of the selection and removal of a few nodes which play an important role in maintaining the connectivity of network.
Also, in 2000, Cohen and Callaway [10,14] observed that real networks demonstrating power-law degree distribution are robust against random node removal but easy to crash in case of attacks to high-degree nodes.
Holme et al. [8] used four different removal strategies which will be introduced in Section 4 in 2002, and he found details of the response of networks according to these attacks on vertices and edges. Holme and his research team observed that the removals by the recalculated degrees and betweenness centralities are always more harmful than the attack strategies based on degrees and betweenness centralities of the initial network. But they only use one real communication network and did not propose some application points.
en, some researchers have discovered a few valuable results on directed networks. Xu and Wang [6] have done some experiments on the cascading crash on weighted complex networks in 2008. Newman and Ghoshal [2] found that removal of some special single node in the network may cause the bicomponents in the graph to be disconnected.
Malliaros and Vazirgiannis [4] proposed a model to capture this cascading effect node departure of social network, based on engagement dynamics of social networks in 2015. Fragkiskos introduced a new concept of robustness assessment method under cascades triggered by the quitting of nodes based on their engagement level. His results indicated that social networks are very robust and strong under cascades triggered by randomly selected nodes but highly vulnerable in cascades caused by targeted departures of nodes with high engagement level.
In 2016, Gao et al. [10] developed a set of analytical tools with which to identify the natural control and state parameters of a multidimensional complex system for stimulation, and it can attain effective one-dimensional dynamics that can accurately predict the system's resilience. eir proposed analytical framework tool can systematically separate the roles of the system's dynamics and topology, collapsing the behavior of different networks onto a single universal resilience function.
But in the related research mentioned above, these researchers did not model on the information transfer process of directed social networks and cannot fulfill the network robustness and vulnerability assessment requirement of directed social networks nowadays, and it needs to construct some new model and methods to bridge the requirement gap in some efficient ways.

Definitions of Network Structure Quantities and Node Centrality Measures
In a scale-free network which includes undirected and directed ones, after some very important nodes are intentionally selected and removed, the network would suffer a serious collapse. It is necessary to research on the network 2 Complexity resilience in directed networks, so we adopted three key performance quantitative indicators and some famous centralities indicators to make robust assessment of simulated and real networks.

Network Structure Quantitative Indicator Definition.
In this paper, we used two types of network which are undirected and directed networks. On the one hand, for the undirected networks, the network model is unweighted and undirected, which can be demonstrated as G � (V, E). V is the set of nodes with number N � |V|, and E is the set of edges with number L � |E|. On the other hand, the directed network model is always defined as G → � (V, E) with each edge having its own weight and direction to supply the directed vivid information and material spreading.

Average Inverse Geodesic Length.
In an undirected network, average inverse geodesic length l is the important network structure quantities after node failure. And in the directed network especially in social networks, the average inverse geodesic length l can be calculated by removal of the direction of edges: In formula (1), d(v, w) stands for the geodesic path length between different nodes v and w. In traditional social networks of human relationship which is a small-world network, l is always around 6. In the new social networks such as Twitter and Facebook, it would decline to 4 which has been proved by Robert and Sebastiano in Laboratory for web algorithm [15] in 2015. After some nodes are removed, if there is no path between nodes v and w, d(v, w) would reach +∞. And there is another length quantity l −1 instead of l: e value of 1/d(v, w) is zero where there does not exist path from node v to node w.

Network Average Cluster Coefficient.
In most network models, the node cluster coefficient C i reflects the density of connection around some focus nodes. In the whole network, the network average cluster coefficient C G�(V,E) demonstrates the macroscopically cluster characteristics of the whole network to reveal the density of links among neighbor nodes and the cluster coefficient of node can be found in the following formula: where k i stands for stands for the neighbor node number of node i and E i stands for the actual number of existing links among neighbor nodes of node i. Network average cluster coefficient can be found in the following formula:  Complexity 3.1.3. Relative Size of Largest Connected Subgraph. In some disconnected large networks, there would exist some separated subgraphs and cannot connect to each other. Size of the largest connected subgraph is also called the node number of the largest connected subgraph in the whole graph which is important to reveal inner connectivity characteristics of graph. In the real network, it can assume that if the connected subgraphs set are g 1 , g 2 , g 3 , g 4 , . . . . . . , g n in G � (V, E), then definition of the relative size of the largest connected subgraph can be found in the following formula:

Node Centrality Measures.
In network, there are many node centrality measures to calculate the importance of node and each node has its own status and influence to its neighbors. Particularly, in the social relations network, it can demonstrate the relations among nodes. Based on node centrality measures, they can be used to analyze the closeness in the each other dependent relations between different nodes. Furthermore, node centrality measures can be used to calculate the role and graph position of every node for information dissemination influence analysis and other advancing applications [16][17][18][19][20][21]. Among these researches, node centrality is always the focus of analysis and research results of social network researchers [22] to find the role and status of node. Among these well-known node centralities, we have chosen three most common node centrality statistics and proposed two new node centrality measures which can be found as follows.

Betweenness Centrality.
Betweenness is a very famous and vital node centrality measure in network statistics computing. Its detailed definition is as follows: in graph G, if there exists a path between every node v i and v j while ∀v i , v j ∈ V, G is a connected graph and there must be a shortest graph to connect v i and v j which is always called the geodesic path. us, we can use the assuming shortest paths number between two vertices to quantify the importance of a vertex or an edge in terms of its betweenness centrality [23] which can be found in the following formula: In formula (6), σ(i, u, j) is the number of shortest paths between vertices v i and v j that pass through vertex u and σ(i, j) is the total number of shortest paths between nodes v i and v j .

PageRank Centrality.
PageRank is a typical link analysis centrality measure invented by Larry Page in 1997 with the purpose of "calculating" its relative importance within the linking data set. And PageRank also can be applied to calculate node centrality. e simple definition of PageRank can be found in formula (7). In our calculation, damping factor d equals 0.85 as a constant value. e calculation formula of PageRank is as follows.
When page number increases and the damping factor is used to recalculate the PageRank mark of every page, damping factor stands for the real mark when one page is linked to the other page with the range from zero to 1. In our calculation, the damping factor value equals 0.85 as a constant value. e calculation formula of PageRank is as follows: In formula (7), PR(t 1 ) stands for PR (PageRank) marks in which t 1 brings to page A (t 1 is linked to A). In the same way, C(t 1 ) stands for pages which are linked to page t 1 .

Degree Centrality.
In social network, the degree centrality is that central users have the most ties to other actors in the network. If the central user is more powerful and influential in the connecting network, it will have much more links to other users including in and out links. e number of adjacent edges a node has is defined as a degree. In graph G, if the degree of node v i is deg(v i ) and the total degree of nodes in the graph is v i ∈V deg(v i ), the degree centrality of node v i is D(v i ) which defined in as follows [24]: From the definition above in formula (8), it can be found that node degree centrality expressed the ratio of degree of the node u to the total degree in the whole graph and some nodes with a higher degree centrality reflect that this node may have some powerful role linking to other node in the network and it may be the most vital person living in the focus of attention.

Directed Path-Based Node Importance Centrality.
With the rapid development of information technology, our life is becoming more and more networked. e graph information is closely related to our daily life, such as the directed WeChat relation network and directed Facebook relation network with such complex structures. And most important of all is that the scale of these graphs is enormous and it is hard to handle these so complex and massive graphs. In a directed graph network, how to quantitatively analyze and calculate the importance of each vertex has become an important problem to be solved urgently for those directed graph networks, while they all have some common features [25,26]. Among these statistical indicators for directed graph networks, directed path-based directed 4 Complexity node importance centrality has become more prominent for its easy and fast calculation.
In the directed graph network G, if there exists a path e i � (v i , v j ) between nodes v i and v j , directed path-based directed node importance centrality of node v j can be expressed by the paths across node v j . e more the across paths are, the more important the node v j is. Directed pathbased node importance centrality IM p (v) of node v has been posed in the famous book with the name "Computer Science eory for the Information Age" by John Hopcroft and Kannan [27], but this book did not demonstrate some fast and efficient computing methods of IM p (v) which we have found in formula (9). And definition of directed path-based node importance centrality IM p (v) of node v can be found in formula as follows: In formula (9), I p (v) stands for the directed paths number which finally ends in node v where Basing on formula (9), we can calculate the IM p (v) of nodes in directed networks when we solve two critical problems first. One problem is that, in complex structured graphs with huge number nodes, the time complexity of the iterative calculation will be very high and the other problem is that the convergence of the calculation process of formula (9) cannot be reached if we do not use some approximate calculation methods.
For (9), it can be converted to iterative computation in the following formula: In formula (10), I p (v) k stands for the k th iteration calculation result, and for any node v, the initial value of I p (v) k is I p (v) 0 � 1. For the reason that computation of formula (10) cannot converge by its diverging computation process, the iterative results of formula (10) need to be standardized in formula (11a) in which α is the damping factor with the value range of 0 < α < 1 just like the damping factor in the PageRank computation model: For the similar computation principle, the standardized form of O p (v) in formula (9) can be written as follows: In computation of formulas (11a) and (11b), we can use M to stand for the adjacency matrix of directed graph network G and I p to stand for the vector composed by all the can be changed as follows: In formula (12a), I � [1, 1, . . . , 1] T and I p can be initialized to I p � I. For the similar computation principle, formula (11b) can be changed as follows: By using formulas (12a) and (12b), the computation of directed path-based node importance centrality IM p (v) of node v can be transferred to the computation of adjacency matrix M of directed graph network G with great promotion. Furthermore, we will demonstrate the whole proof process which using the Power Iteration eory [28] to prove that formulas (12a) and (12b) can converge to a definite vector by finite times of computations. For the adjacency matrix M � (m ij ) ∈ R n * n of directed graph network G, it has the eigenvalue set of λ 1 , . . . , λ n (λ i ∈ R), while λ 1 > |λ 2 | ≥ · · · ≥ |λ n | and λ 1 � 1. At the same time, We can assure that y (0) � n i�0 c i x i and y (k) can be expressed as y (k) � n i�0 c i λ k i x i . Here, [y (0) , y (1) , . . . , y (k) ] can be used to replace For where a > 0, a � (n − 1)|c| max |λ 2 − 1|‖x i ‖ max .
We can also prove that the iteration times k (k � [1, N] of computation for ‖y k+1 − y k ‖) are around a constant level value in the following formula: And the constant level value of k is 3.2.5. ITG-Based Directed Node Centrality. In most actual social networks such as Twitter, Facebook, and WeChat, the two friends of someone can be friends with each other. And this phenomenon can be found in both undirected networks and directed networks. So this attribute of social networks sometimes is called the clustering characteristic of network [7]. Network average clustering coefficient reflects the microscopically clustering characteristic of network and has become the very important measure of adjacent nodes which are connected closely. e definition of network node clustering coefficient can be found in Section 3.1.2. Based on the classic probabilistic graphical model (PGM) theory from Turing Award Owner Pearl [29], we have made a detailed research on directed node influence clustering coefficient [30] to propose a new vector influence clustering coefficient model with both information propagation direction and information propagation probability.
e new vector influence clustering coefficient model starts from the two basic directed triple forms of vertex i in the directed graph which can be found in Figure 2.
Basing on deducting from Figures 2(a) and 2(b), we can get the related edge information propagation probability and direction of different two types including 36 subgraphs of all different triples [30]. And then we can calculate all numerical values of directed ITG node centrality in directed social networks in all kinds of situations by the following formula: Because there are three possible directional statuses for each edge in Figures 2(a) and 2(b), the adjacent edge of node i has three different definitions, which are friends relationship (i ⟷ j), following relationship (i ⟶ j), and fan relationship (i ⟵ j). At the same time, the opposite edge i ⟷ k of node i also has three definitions. Furthermore, the edge j ⟷ k has three types of relationships, in which node j and node k are friends, node j follows node k, and node k follows node j. We can use 0, 1, and 2 to stand for the relationships and substitute the three different definitions, and we obtain the following 27 arrangements in Table 1 [30].
In the above 27 arrangement cases, because node i is the source node, we can find some symmetry results, and finally, we can get 15 independent results. And the ITG-based directed node centrality value from information transfer gain clustering coefficient (ITGC) of node i in a directed network can be finally summed by the 15 different independent results in formula (16) as follows: ITGC i is the ITG value of node i in a directed network.
15 t�1 ITG i triangle (t) × Number(t) is the weighted number of triangles which use node i as the top vertex (i.e., the information transfer source node), and its weight is the ITG (information transfer gain) contribution ITG i triangle (t) from the 15 different types of weighted triangles multiplied by its counted number Number(t). 6 t′�1 ITG i triple (t ′ ) × Number(t ′ ) is the weighted number of the triples using node i as the top vertex; its weight is the weighted sum of the six ITG i triple (t ′ ) values of different types of triples multiplied by its counted number Number(t ′ ). Similar to undirected clustering coefficients, ITGC has the same characteristic, measuring the tightness of the graph to form tight communities.

Simulation of Node Failure Generation Method.
Basing on the importance and connectivity of different nodes in the network, there are eight node attack strategies [8] always chosen by researchers to evaluate network robustness such as follows: (1) e ID removal strategy: the attack starting from the node with the highest degree and node attack strategy uses the initial node degree distribution. (2) e IB removal strategy: the attack starting from the node with the highest betweenness and node attack strategy uses the initial node betweenness distribution. (3) e ITG removal strategy: the attack starting from the node with the highest ITG centrality and node attack strategy uses the initial node ITG centrality distribution. (4) e IMP removal strategy: the attack starting from the node with the highest IM p (v) centrality and 6 Complexity node attack strategy uses the initial node IM p (v) centrality distribution. (5) e RD removal strategy: using the recalculated node degree distribution at every removal step. (6) e RB removal strategy: using the recalculated node betweenness at every step. (7) e RTG removal strategy: using the recalculated node ITG centrality distribution at every removal step. (8) e RIMP removal strategy: using the recalculated node IM p (v) centrality distribution at every removal step.

Experiment Result.
Selection of typical and persuasive dataset is very important to experiment results, and we selected some classical undirected network and directed network dataset including synthetic signed network and real-life large networks used by Albert et al. [13] and Holme et al. [8]. And these typical datasets can be used to validate the network robustness and node vulnerability assessment in undirected and directed networks. From Figures 3 to 21, all the prominence in experiment result which was created by ITG, RTG, IMP, and RIMP strategies have be pointed out by red ellipses and text tag. Based on these marks, we can easily find the more strong effects by our proposed ITG and IM p (v) centralities. Table 2, the network dataset includes the classical BA scale-free network (a) proposed by Albert et al. [13] (generation parameter is m 0 � 5, m � 4, p t � 0.8, and n � 490) and the undirected call community [31] graph (b) from cellphone calling records in one month in China of a southern city, the LFR (Lancichinetti Fortunato Radicchi) [32] benchmark network with generation parameter of N � 1000, k degree � 2, C min � 20, C max � 100, u � 0.3, and C degree � 1. For the reason that the LFR benchmark has presented a much solid testing dataset for algorithms and having good performance from other dataset, we used LFR benchmark to generate testing dataset having typical attributes compared with real networks, such as real node degree distribution and heterogeneous distribution of community size. Because the above networks are important undirected networks, the three network datasets are chosen for our research. Now we will give the brief introduction of vertical and horizontal coordinates of the graph, in which the x-axis in Figures 3 to 8 stands for the removed node ration (Nrm: the number of removed nodes) to initial graph. e y-axis stands for the relative value of S and l −1 (the ration of S and l −1 after every step of node removal to initial S and l −1 in the network). Figures 7 and 11 consist of 400 calculated results each in 200 networks generated by the four simulated node generation mechanisms above to avoid random deviation (every mechanism generated 50 fault nodes).

Undirected Dataset and Experiment Results. In
In calculation of ITG-based node coefficient in undirected networks, all the information transfer gain probability a between nodes i and j for undirected edges will be equal to 0.5 and we would calculate ITG node centrality coefficient of each node and the whole graph.

Experiment Results Analysis for Undirected Dataset.
After processing and analyzing the experimental data above, it can be clearly found in Figures 3, 5, and 7 that the influence to l −1 is as follows: (1) When the number of removed nodes is very small that is to say when 0.0 < N rm /N < 0.005, it is the very  early stage of collapse time by node removal of the six different removal strategies and the l −1 curve falls quickly to make the whole graph shrink with a high speed. (2) During the fall process of l −1 , the RTG curve, RB curve, and RD curve look like falling faster than other three removal strategies.
(3) Among the six different curves of S, the curves of ITG, IB, and ID are more likely a linear curve than other ones, and the experimental result has shown the correctness of ITG in its almost linear curve with IB and ID.
In the common circumstances, the network will be more denser while having the larger value. But in generally, the value of network average cluster coefficient having a sudden increase after the continuous decreasing trend would tell that the node removal can make the network suddenly into disconnected subnetworks. Besides that, Albert et al. [13] and other researchers have also found that when the node having a very high betweenness value leaves the network, it may trigger the huge collapse in the whole network on the system level with a sudden unpredicted speed. Figures 4, 6, and 8 demonstrate the corresponding variation trend of C G�(V,E) : (1) It can be also clearly found in Figures 4, 6, and 8 that, among the six curves of S, the RTG, ITG, RB, and IB removal strategies are more harmful than RD and ID where in some datasets, the RTG removal strategy is much more harmful than others such as in S and C G�(V,E) curve of LFR in Figures 7 and 8. (2) ere have been much more rises in ITG and IB curves than other curves for the reason that it can break the whole network into disconnected subnetworks with much higher numbers. It has also proved that our ITG node centrality is a very good node centrality by the performance of curve S, C G�(V,E) , and l −1 .    Complexity was provided by the INFOMAP algorithm [34] which was a subject reference network from the research of Physics, Chemistry, Biology, and Ecology in 6,128 journals connected by 6,434,916 citations. e SinaBlog Tweet (c) dataset came from the tweeting message reposting chain of one famous Chinese scholar in SinaBlog. e Calling Record (d) dataset was provided by cellphone calling records from another city in one month in China. All the above datasets are typical directed social networks. e x-axis in Figures 9 to 16 stands for N rm . e y-axis stands for the relative value of S and l −1 . Specially, all the network datasets in Table 3 would be processed as undirected graph in calculation of node betweenness and degree for IB, RB, ID, and RD strategy besides ITG and IM p (v) strategy. But, all the network datasets in Table 3 would be processed as directed graph in calculation of ITG node centrality coefficient. For the reason that there are only 40 nodes in subject reference (b) dataset, we only calculated the top five nodes.

Directed Dataset and Experiment
(1) Experiment Results of ITG with IM p (v) to IB, RB, ID, and RD.
(2) Experiment Results Analysis for Directed Dataset of ITG with IM p (v) to IB, RB, ID, and RD. Figures 9 to 16 display the experiment results on directed social network, and we can find that the ITG, IMP, RTG, and RIMP strategies have amplified their harm in directed networks: (1) l −1 and S: ITG, IMP, RTG, and RIMP are most harmful strategies, and the RIMP strategy is more powerful than RTG especially in the late stage of Figures 11, 13, and 15.  (3) Directed Experiment Dataset of ITG to IM p (v). In this part, we added two typical directed networks with much nodes inside which can be found in Table 4. e first one is the DBLP directed network dataset which is a famous directed heterogeneous information network that contains a dataset of author-centric English literature in the field of computer science with 14736 papers and 14475 authors [28]. We selected the data related to the field of computer, including database, data mining, artificial intelligence, and information retrieval including titles of papers published in various fields, authors who published more than five papers, abstracts of papers, and conferences titles. e another directed network dataset is the well-known ENRON e-mail network among employees of ENRON company from May 11, 1999    structure clearly shows some important moments in the company's development, such as the company's collapse and the suicide of its former CEO. Dataset C is a Facebook dataset which was collected from survey participants using Facebook app. e dataset includes node features (profiles), circles, and ego networks [37].
e Facebook dataset has been anonymously processed by replacing the Facebook-internal ids for each user with a new value. It is obviously that, in these directed networks, the influence of IB, ITG, and IM p (v) has demonstrated a different varying trend in the undirected networks from Table 1 Figure 20 used the logarithmic coordinates. Figure 21 demonstrates the comparison result of iteration times in different datasets of Table 4 when computing the directed path-based node importance centrality IM p (v). By calculating formula ⟹ k > ((log ε − log a)/log|λ 2 |) (13b), we attain the varying curve of iteration times k to the marginal difference ε, while ε can be calculated by ε ≥ ‖y (k+1) − y (k) ‖. Furthermore, each largest eigenvalue value λ 1 and second largest eigenvalue value λ 2 of dataset DBLP, ENRON, and Facebook also can be computed.
(5) Experiment Results Analysis for Directed Dataset of ITG to IM p (v). Figures 16 to 21 display the compared experiment results of IB, ITG, and IM p (v) node removal strategies on directed social networks. In addition, we explore the varying trend of iteration times k to the marginal difference ε.
(1) Generally after analyzing the demonstrated data in Figures 17 and 18, we can find that the influence of node with high ITG values have the most harmful effect to the l −1 in directed networks with low edge density such as DBLP. But in directed networks with high edge density such as ENRON, IB strategy and ITG strategy are just like doing the same effect. (2) But in the directed network Facebook with a more higher edge density, we can find that the ITG node removal strategy suddenly lost its magic and the IM p (v) removal strategy does the best harmful effect to the Facebook network, while the IB removal strategy followed. And it maybe needs to be varied in more densely connected directed social networks such as Twitter and WeChat. (3) In Figure 21, we can clearly find that the relationship of k and ε. e more densely connected directed network Facebook has the larger iteration times number k. And in dataset Facebook, when ε is more close to zero, iteration times rises to around twenty which has shown high efficiency of our node IM p (v) computing algorithm.

Conclusion
In this paper, we have proved new information transfer gain-(ITG-) based probability clustering coefficient and directed node importance centrality IM p (v) for measuring directed graph. Our comparisons in the variation trend of some key performance quantities of network robustness and node vulnerability assessment are useful and helpful. Comparison could help us to capture this cascading effect in directed online social networks. Experiments results showed that node RIMP and RTG strategies are more harmful than node betweenness-based strategies such as RB and IB in directed social networks including real Sina Blogging and Calling Record network. With sufficient experiments in synthetic signed networks and real networks derived from directed online social media and directed human mobile phone calling network, it has been proved that our ITG-and IM p (v)-based directed social network robustness and node vulnerability assessment method is more accurate, efficient, and faster than several classical traditional centrality methods such as degree and betweenness. Furthermore, we will carry out our ITG centrality on more types and more large-scale directed social networks.
In addition, we propose a new proving process of directed node importance centrality IM p (v) in Section 3.2.4. By rigorous mathematical derivation and approximate calculation, to the best knowledge of us, we attain the varying trend of iteration times k to the marginal difference ε on directed social networks for the first time to our best knowledge.