Inspired by the well-known percolation theory in statistical physics, the robustness and resilience of a network is usually defined as the network structural degradation caused by the removal of some critical nodes [19]. Tolerance to random removals and intentional attacks is understood as the ability of the network to maintain operations and connectivity under the loss of some nodes or links [8]. In order to ensure the efficiency of attacks, it is necessary to identify the most vulnerable nodes in a network. Indeed, the identification of critical nodes is an influence maximization problem [18]. The quantification algorithms can be roughly classified into three categories: centrality-based algorithms, random-walk algorithms, and greedy-based algorithms.

Centrality-based algorithms perform a fundamental quantification by considering geodesics between nodes to evaluate nodal importance. Up to now, many centrality-based algorithms have been proposed, such as degree centrality [22, 23], betweenness centrality [22, 24], closeness centrality [22, 25], eigenvector centrality [26], and other improved centrality-based algorithms [27–29]. The random-walk algorithms include the well-known PageRank [30] and other improved algorithms [31]. Random-walk algorithms work only well for directed networks. Greedy-based algorithms formulate the influence measurement as a discrete optimization problem, and their elementary strategies are to select the spreaders that contribute the largest incremental influence one by one, according to a specified influence cascade model [32]. In terms of algorithm construction, although greedy-based algorithms can achieve excellent results, they also have very high computational complexity and are not suitable for large-scale social networks.

Previous studies have shown that the adaptability of networks behaves differently from various attack strategies [20, 33]. Thus, in this article, we will study tolerance to various attacks in real online social networks and further find a minimized set of nodes triggering the collapse of network. We will consider four straightforward and efficient centrality indices as attack strategies to identify the importance of nodes.

1) Degree centrality: The algorithm measures a node’s influence according to the number of edges attached to it, which reflects the ability of a node to connect directly with other nodes.

Because the removal of nodes under intentional attacks changes the balance of structure and leads to a global redistribution over all the networks, we recalculate the degree centrality, the betweenness centrality, the closeness centrality, and the eigenvector centrality, every time a small fraction of nodes is removed.

Numerous empirical results on real networks have revealed that the heterogeneous topology structure may be fit for most real networks [35–37], where degree distribution significantly deviates from a Poisson and low degree nodes are far more abundant than the nodes with high degrees. Due to the inhomogeneity of general networks, removing some critical nodes will decrease the network connectivity and lead to the loss of the global information-carrying ability of the network [6, 20]. Generally, when assessing the vulnerability of a network under intentional attacks, three performance criteria should be concerned in the framework of graph theory [38]:

1) The number of components that are being removed.

Most of the online social networks can be abstracted as a non-complete connected graph G = ( V , E ) , where individual members and personal relationships can be defined by a set of nodes V and a set of edges E , respectively. In general, a good social network should have short distance between nodes, average distance, and high connectivity. In fact, there has been much effort directed at developing methods for evaluating network adaptability to attacks. In most instances, the characteristic path length and the network efficiency as evaluation metrics can only provide a useful topological snapshot for connected networks [39]. But for disconnected networks, the geodesic distance between any two nodes belonging to two disconnected subgroups is identically zero or infinity, which will directly affect the accuracy of evaluation results. In graph theory, some helpful indicators of evaluating network vulnerability have been proposed. These metrics relates to network topology and attributes, such as toughness [40], integrity [41], tenacity [42], and scattering number [43]. The detailed description of each indicator is shown in Table 1.

As one of the basic concepts of graph theory, connectivity plays a vital role in network performance and is fundamental to vulnerability measures. The concept of connectivity K ( G ) of G is defined as K ( G ) = min { | S | : S ⊂ V } , (1) where | S | is a cutset of V ( G ) . In Eq. 1, the connectivity K ( G ) asks for the minimum number of nodes whose removal renders the graph G disconnected. As one of the graph theoretical concepts, connectivity deals with the criterion (1).

Toughness and integrity are two other graph concepts used in the vulnerability assessment. The notion of the toughness T ( G ) of G , originally introduced by Chvátal [40], is defined as follows: T ( G ) = min { | S | w ( G − S ) : S ⊂ V , w ( G − S ) ≥ 2 } , (2) where w ( G − S ) stands for the number of components of G − S . Unlike the connectivity K ( G ) , the toughness T ( G ) incorporates the relationship between the size of the cutset and the number of components after destruction and takes into account of the criteria (1) and (2). But the toughness T ( G ) is still insufficient to measure the network vulnerability. Considering the graphs G 1 and G₂ (see Figure 1), two graphs have the same connectivity and toughness, where K ( G 1 ) = K ( G 2 ) = 1 and T ( G 1 ) = T ( G 2 ) = 1 3 , but they are really different in the vulnerability of graphs. For instance, after the minimum cutset { u 1 } is removed, we find that the G 1 has been divided into three small components, while the vast majority of nodes of G 2 have been retained in the largest connected component { u 2 , u 6 , u 5 } , within which mutual communication among the remaining nodes can still occur. It implies that the connectivity K ( G ) and toughness T ( G ) cannot accurately reveal the global damage done to the network.

The notion of integrity introduced as another vulnerability parameter of graphs [41] focuses on the criteria (1) and (3). For a non-complete connected graph G , its integrity I ( G ) is defined as follows: I ( G ) = min { | S | + m ( G − S ) : S ⊂ V } , (3) where m ( G − S ) denotes the giant component size after destruction.

Obviously, the disruption is more successful if the disconnected network contains more components and is much more successful if, in addition, the components are small. Unfortunately, connectivity and toughness give the minimum cost to disrupt a network but fail to indicate accurately what remains after the disruption. Although Barefoot’s integrity has taken the size of the largest remaining component after destruction into account, it cannot indicate the extent of the damage.

The notion of tenacity was originally proposed in Ref. [42], where they introduced the mix-tenacity to measure the vulnerability of Harary graphs. The precise definition of tenacity is defined as follows: R ( G ) = min { | S | + m ( G − S ) w ( G − S ) : S ⊂ V , w ( G − S ) ≥ 2 } (4) The tenacity R ( G ) of graphs directly integrates all three criteria, such as the cost of network breakage, the number of components, and the giant component size, and is considered to be a reasonable measure for the vulnerability of graphs. As shown in Figures 2A,B, it is easy to know that R ( G 1 ) = min { 1 + 2 3 } = 1 and R ( G 2 ) = min { 2 + 1 4 } = 3 4 , R ( G 1 ) > R ( G 2 ) , which indicates the adaptability to attacks for G 1 is better than G 2 .

In general, if the network remains more disconnected subgroups and smaller connected component size after destruction, the disruption is more successful. As shown in Figures 3A,B, G 3 and G 4 all have the same number of nodes and edges. After { u 1 , u 2 } and { u 2 , u 1 , u 4 } are removed respectively, the minimum toughness and the minimum tenacity can be obtained, which are T ( G 3 ) = T ( G 4 ) = 1 2 and R ( G 3 ) = R ( G 4 ) = 4 3 . The cutsets { u 1 , u 2 } and { u 2 , u 1 , u 4 } are the minimum removal costs of the graph G 3 and the graph G 4 , respectively. But we find that there are differences both in the attack efficacy in the graph G 3 and graph G 4 , where for the graph G 3 , the minimum removal cost is 2 and the giant component size also is 2; while for the graph G 4 , the minimum removal cost is 3 and the giant component size is 1. As discussed earlier, the tenacity R ( G ) is still an imperfect criterion to assess network vulnerability.

In Ref. [43], Hendry used the concept of scattering number to measure the vulnerability of extremal non-Hamiltonian graphs and found that it was more efficient for measuring the degree of global destruction. The scattering number S ( G ) of G is defined as follows: S ( G ) = max { w ( G − S ) − | S | : S ⊂ V , w ( G − S ) ≥ 2 } . (5) So we think that it is necessary to add the criterion to reveal the global damage done to networks under attacks, and keep its priority in assessing the vulnerability of networks. Therefore, we propose an improved tenacity based on the concepts of scattering number and tenacity, which is named R s c a ( G ) and defined as follows: R s c a ( G ) = min { m ( G − S ) | w ( G − S ) − | S | | : S ⊂ V , w ( G − S ) ≥ 2 } . (6) As shown in Figures 3A,B, R s c a ( G 3 ) = min { 1 + 2 3 } = 1 and R s c a ( G 4 ) = min { 2 + 1 4 } = 3 4 , R s c a ( G 3 ) > R s c a ( G 4 ) , which indicates the anti-interference capability of G 3 is better than G 4 .

Compared with existing evaluation metrics [6, 20, 21, 39–42], our notion of improved tenacity is not to measure a single property performance such as giant component size or the number of components after destruction, which is insufficient to evaluate the network vulnerability by only considering whether or not it is a disconnected network, and how fragmental the network becomes, but to pay special attention to the potential equilibrium between the giant component size and the number of components under intentional attacks. As shown in definition (6), the number of components w ( G − S ) and the size of the largest connected component m ( G − S ) are re-calculated after each iteration, so the whole computational complexity of the proposed method is O ( n 2 ) . The result demonstrates that the proposed algorithm has relatively less computational burden in evaluating the vulnerability of networks and can be applicable to large-scale networks.

Because online social networks serve as much social function as other kinds of social interaction, including e-mail exchanging, text messaging, instant messaging, digital video sharing, and so on, each edge meaning a connection in online social networks is complex and changeable, whereas each node meaning an individual of online social networks is constant. To measure the vulnerability of networks more properly, our method will pay more attention to the importance of nodes, rather than edges. Our data set is composed of six undirected and unweighted networks, that is, networks that have a binary nature, where the edges between nodes are either present or not, and each edge has no directional character. Table 2 indicates that the six real social network include the following: OClinks is a representative online community network, where users are from the University of California, Irvine, which is from Panzarasa et al.’s [44]; Twitter, Wiki-Vote, and Facebook can be downloaded from the Stanford network dataset (http://snap.stanford.edu/data/index.html); and Lilac and RenRen are collected from the online social networks by us.

In Table 1, we show the main topological properties of a series of online social networks, belonging to two different types: the online community service based on group-centered service and the social network service based on individual-centered service. The first three networks, Lilac, OClinks, and Wiki-vote, are three examples of online community services, where the users have a common interest and purpose and can exchange information or seek help. In these networks, the nodes are the registered users, and the links represent a relationship between two users existing message exchange. In the latter part of Table 1, we show three examples of social network services. Twitter is a social news website, where users may mention other people or follow other people to make his/her posts, so the links imply communication between users existing mention or comment. RenRen is a real-name social networking internet platform in China, where users can connect and communicate with each other or enjoy a wide range of other features and services, so the links reflect different kinds of social relationships between the users. Facebook is an ego network consisting of friends lists from Facebook.

The most basic topological characterization of networks can be obtained in terms of the degree distribution P ( k ) , defined as the probability that a node is chosen uniformly at random has degree k or, equivalently, as the fraction of nodes in the graph having degree k [19]. In recent years, scientists approached the study of real networks from the available databases and found most of the real networks having inhomogeneous structure, where the connections within nodes of the highest degree are rather sparse, and a large number of nodes just have a few connections. Moreover, most of real networks exhibit power law–shaped degree distribution P ( k ) ∼ k − γ , with exponents varying in the range 2 < γ < 3 [35, 36], and a little of them follow shifted power law distribution, stretched exponential distribution, or more complicated distributions [37].

The empirical results demonstrate that the degree distributions P ( k ) of aforementioned networks are subjected to two types: the segmented power law distribution (Lilac, OClinks, and Wiki-Vote) and the stretched exponential distribution (Twitter, RenRen, and Facebook). The insets in Figures 4A–C show the degree distributions P ( k ) of Lilac, OClinks, and Wiki-Vote are heavy-tailed, and approximately follow the power law distribution, where γ = 1.72 , γ = 0.99 , and γ = 1.31 , respectively. However, when the size of data set is small, it may happen that the data have a rather strong intrinsic noise due to the finiteness of the sampling. In order to avoid the statistical fluctuations, one better possibility is to measure the cumulative degree distributions P ( x ≥ k ) . The cumulative degree distributions P ( x ≥ k ) of Lilac, OClinks, and Wiki-Vote show two different scaling regions: a slow region and a rapid decaying region, and are well-approximated by the segmented power law distribution. The crossover takes place between k = 10 and k = 100, and the cumulative degree distributions P ( x ≥ k ) of Lilac network can be defined by the following equation: P ( x ≥ k ) ∼ k − ( r − 1 ) { r − 1 = 0.89 , if   k ≤ 16 r − 1 = 2.11 , if   k > 16 . The similar trend is shown in Figure 4B, where the probability P ( x ≥ k ) of the OClinks network can be fitted by the following equation: P ( x ≥ k ) ∼ k − ( r − 1 ) { r − 1 = 0.51 , if   k ≤ 20 r − 1 = 2.02 , if   k > 20 . As shown in Figure 4C, the probability P ( x ≥ k ) of the Wiki-Vote network can be fitted by the following equation: P ( x ≥ k ) ∼ k − ( r − 1 ) { r − 1 = 0.45 , if   k ≤ 63 r − 1 = 2.26 , if   k > 63 . Otherwise, the insets in Figures 5A–C show the degree distributions of Twitter, RenRen, and Facebook, where R 2 = 0.815 , R 2 = 0.811 , and R 2 = 0.809 , respectively, and the curves of the degree distributions are not well-approximated by a power law distribution. Figures 5A–C indicate that the degree distributions of Twitter, RenRen, and Facebook obey the stretched exponential distribution P ( x ≥ k ) = e − ( k k 0 ) c , where the graph of log   P ( x ≥ k ) versus k k 0 is characteristically stretched, and a stretching exponent c takes a value between 0 and 1. The stretching exponent c can be obtained from P ( x ≥ k ) , as we add the slope of log   P ( x ≥ k ) in a log–log plot. As shown in Figure 5, the distribution functions of Twitter, RenRen, and Facebook can be defined by P ( x ≥ k ) = e − ( k k 0 ) 0.79 , P ( x ≥ k ) = e − ( k k 0 ) 0.89 , and P ( x ≥ k ) = e − ( k k 0 ) 0.94 , respectively.

The stretched exponential distribution is obtained by inserting a fractional power law distribution into the exponential distribution: as c = 1 , the usual exponential function is recovered; as c → 0 , the distribution follows the power law distribution. In general, the power law distribution is characterized by a slower than exponentially decaying probability tail. In contrast with Twitter, RenRen, and Facebook, where c = 0.79 , c = 0.89 , and c = 0.94 approaching c = 1 , some extremum nodes in Lilac, OClinks, or Wiki-Vote can occur with a more non-negligible probability.

In this section, we have examined the static properties of a variety of online social networks empirically and found that two types of networks, the online community service and the social network service, have completely different structural properties. As shown in Figures 4A–C, Lilac network, OClinks network, and Wiki-Vote network exhibit a highly inhomogeneous degree distribution, where the simultaneous presence of a few nodes tending to link many other nodes, and a large number of poorly connected nodes. But in Figures 5A–C, the curves of degree distributions have witnessed that the nodes in Twitter, RenRen, and Facebook networks are more evenly distributed than those in Lilac, OClinks, and Wiki-Vote networks, where extreme value still runs at a relatively low level. As it can be noticed, Twitter, RenRen, and Facebook as social network services are mainly based on an individual-centered online platform for organizing feature and technical feature themselves and have lower centralization than the Lilac, OClinks, and Wiki-Vote network based on group-centered service.