Percolation of localized attack on complex networks

The robustness of complex networks against node failure and malicious attack has been of interest for decades, while most of the research has focused on random attack or hub-targeted attack. In many real-world scenarios, however, attacks are neither random nor hub-targeted, but localized, where a group of neighboring nodes in a network are attacked and fail. In this paper we develop a percolation framework to analytically and numerically study the robustness of complex networks against such localized attack. In particular, we investigate this robustness in Erd\H{o}s-R\'{e}nyi networks, random-regular networks, and scale-free networks. Our results provide insight into how to better protect networks, enhance cybersecurity, and facilitate the design of more robust infrastructures.

The functioning of complex networks such as the Internet, airline routes, and social networks is crucially dependent upon the interconnections between network nodes. These interconnections are such that when some nodes in the network fail, others connected through them to the network will also be disabled and the entire network may collapse. In order to understand network robustness and design resilient complex systems, one needs to know whether a complex network can continue to function after a fraction of its nodes have been removed either through node failure or malicious attack [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. This question is dealt with in percolation theory [20,21,22,23] in which the percolation phase transition occurs at some critical occupation probability p c . Above p c , a giant component, defined as a cluster whose size is proportional to that of the entire network, exists; below p c the giant component is absent and the entire network collapses. Only nodes in the giant component continue to function after the node-removal process.
The robustness of complex networks under attack is dependent upon the structure of the underlying network and the nature of the attack. Previous research has focused on two types of initial attack: random attack and hub-targeted attack. In a random attack each node in the network is attacked with the same probability [1,2,3,8,20,10]. In a hub-targeted attack the probability that high-degree nodes will be attacked is higher than that for low-degree nodes [1,3,4,7,12]. An important feature of the network structure is its degree distribution, P (k), which describes the probability that each node has a specific degree k. Networks with different degree distributions behave very differently under different types of attack. For instance, the Internet, which shows a power law degree distribution, is extremely robust against random attack but vulnerable to hub-targeted attack [1,2].
However these two types of attack-random attack and hub-targeted attack-do not adequately describe many real-world scenarios in which complex networks suffer from damage that is localized, i.e., a node is affected, then its neighbors, and then their neighbors, and so on (see Fig. 1). Examples include the effects of earthquakes, floods, or military attacks on infrastructure networks and the effects of a computer virus or malware on computer networks. Recent occurrences of the latter include attacks carried out by cybercriminals who create a "botnet", a cluster of neighboring "zombie computers" in a computer network and, by using them, are able to damage the entire network. An understanding of the effect of this kind of attack on the functioning of a network is still lacking.
Here we will analyze the robustness of complex networks sustaining this kind of localized attack in order to determine how much damage a network can sustain before it collapses, i.e., to find the percolation threshold p c . We also want to predict the fraction of nodes that keep functioning after an initial attack of a fraction of 1 − p nodes, i.e., the relative size of the giant component (the order parameter), P ∞ . Note that localized attack has been studied only on specific network structures [24] or on interdependent spatially embedded networks [25], but a general theoretical formalism for studying localized attacks on complex networks is currently missing.
Here we develop a mathematical framework for studying localized attacks on complex networks with arbitrary degree distribution and we find exact solutions for percolation properties such as the critical threshold p c and the relative size of the giant component P ∞ . In particular, we apply our framework to study and compare the robustness of three types of random networks, (i) Erdős-Rényi (ER) networks with a Poissonian degree distribution (P (k) = e − k k k /k!) [26], (ii) random-regular (RR) networks with a Kronecker delta degree distribution (P (k) = δ k,k 0 ), and (iii) scale-free (SF) networks with a power law degree distribution (P (k) ∼ k −λ ) [5]. We find that the effect of a localized attack on an ER network is identical to that of a random attack. For an RR network, we find that the p c of a localized attack is always smaller (i.e., more robust) than that of a random attack. However, the robustness of a SF network against localized attack is found to be critically dependent upon the power law exponent λ. Surprisingly, a critical exponent λ c exists such that when λ < λ c , for localized attack the network is significantly more vulnerable compared to random attack, with p c being larger. While for λ > λ c , the opposite is true.
Consider a random network with a degree distribution P (k), which indicates the probability that a node in the network has k neighbors. The generating function of the degree distribution is defined as G 0 (x) = ∞ k=0 P (k)x k [27,28]. We start from a randomly chosen "root" node. All nodes in the random network are listed in ascending order of their distances from this root node (see Fig. 1(a)). The shell l is defined as the set of nodes that are at distance l from the root node [29,30]. Within the same shell, all nodes are at the same distance from the root node and are positioned randomly.
We initiate the localized attack process by removing the root node, then the nodes in the first shell, and so on. We remove nodes in the ascending order of their distances from the root node. Within the same shell we remove nodes randomly and, after nodes in shell l are fully removed, we begin removing nodes in shell l + 1. We continue the localized attack process until a fraction 1 − p of nodes in the entire network are removed. Thus a "hole" of attacked nodes forms around the root node. The remaining p fraction of nodes in the network are those at greater distances from the root node (see Fig. 1(b)). After the initial removal of 1 − p fraction of the network nodes and all links connected to them, the remaining network fragments into connected clusters. As in percolation theory [21,22], only nodes in the giant component (the largest cluster) are still functional. Nodes belonging to other small clusters are considered non-functional and are also removed (see Fig. 1(c)). Note that for localized attack on a regular lattice, as the number of network nodes N → ∞, p c → 0, i.e., one has to attack all nodes in the regular lattice in order to collapse the lattice (see Fig. 1

(d)).
We find that the generating function of the degree distribution of the remaining network after the localized attack is (see supplementary information) where p is the fraction of unremoved nodes and f ≡ G −1 0 (p). The critical probability p c where the network collapses and the size of the giant component P ∞ (p) for p > p c can be derived analytically from Eq. (1). The generating function of the cluster sizes in the remaining network is H p [27]. By combining Eq. (1) and the criterion for the network to collapse [2,3] The size of the giant component S(p) as a fraction of the remaining network satisfies where (1)). The relative size of the giant component as a fraction of the original network is P ∞ (p) = pS(p).
We apply the above mathematical framework to three types of complex networks: Erdős-Rényi (ER) networks, random-regular (RR) networks, and scale-free (SF) networks, and compare the results of a localized attack with those of a random attack.
For an ER network with an average degree k , the degree distribution follows a Poissonian distribution P (k) = e − k k k /k! and the corresponding generating function , which is the same as the generating function of the degree distribution for the remaining network after a random attack. Thus the effect of a localized attack is exactly the same as that of a random attack on an ER network (see Fig. 2(a)), and the critical threshold is p c = 1/ k . The size of the giant component P ∞ (p) satisfies P ∞ (p) = p(1−e − k P∞(p) ). In an RR network each node is connected to k 0 other nodes randomly and the generating function of the degree distribution is G 0 (x) = x k 0 . Using Eq. (2) we find that the critical threshold for a localized attack on an RR network is Note that for an RR network under random attack the critical threshold is p c = (k 0 − 1) −1 . Thus, for k 0 > 2, p c under localized attack is always smaller than p c under random attack (see Fig. 2(b)). This means that an RR network is more resilient against localized attack than against random attack. When k 0 ≫ 1, random and localized attacks have the same critical threshold (p c = 1/(k 0 − 1)), since in this limit every node is a neighbor of the root node and there is no difference between random and localized attacks. Since lim k 0 →2 p c = e −2 ≈ 0.135 and lim k 0 →∞ p c = 0, one can see that p c for a localized attack on an RR network is always within the range (0, e −2 ) for all k 0 > 2. For p > p c , from Eq. (3), the relative size of the giant component P ∞ (p) satisfies For a SF network the degree distribution is P (k) ∼ k −λ (m ≤ k ≤ M), where m and M are the lower and upper bound of the degree, respectively, and λ is the power exponent. The critical threshold p c and the size of the giant component P ∞ (p) are solved numerically by using the theoretical framework developed in Eq. (1) (see Fig. 3). We find that the degree heterogeneity plays an important role in the robustness of SF networks against localized attack. The critical threshold p c and the size of the giant component P ∞ (p) for the percolation transition of the SF network under localized attack depends on λ. We find that in a SF network there is a critical value λ c below which a localized attack is significantly more severe than a random attack, but when λ > λ c a random attack is more severe. Indeed, as seen in Fig. 3(a), for λ < λ c , p c for a localized attack is significantly higher than for a random attack. As λ increases and the network becomes less heterogeneous, p c decreases and the network becomes more robust against localized attacks. The specific value of λ c depends on other parameters, such as m, M, and k . In Fig. 3(b)−(d), we plot the size of the giant component P ∞ (p) as a function of p and compare the results of a localized attack with those of a random attack. One intuitive explanation for the dependence of network robustness on λ is that, on the one hand, there is a higher probability that higher degree nodes will be within the attacked hole, which accelerates the fragmentation of the SF network; on the other, only nodes on the surface of the attacked hole are connected to the remaining network and contribute to its breakdown, which mitigates the fragmentation process. The total impact of the localized attack is the result of the competition between these two effects. As λ increases and the SF network becomes less heterogeneous, the first effect becomes less dominant and the network becomes more robust. Our analytical analysis shows that for an ER network these two effects always compensate each other and yield equal effects from both localized attack and random attack. For an RR network, on the other hand, the degrees are all the same and therefore only the second effect exists, and the underlying network becomes more robust against localized attack than against random attack.
We also investigate the robustness of real-world networks against localized attack and random attack using a peer-to-peer computer network [32] and a global airline route network [33]. The real-world data proves the feasibility of our model, as shown in supplementary information.
To conclude, we have developed a mathematical framework for studying the percolation of localized attacks on complex networks with an arbitrary degree distribution. Using generating function methods, we have solved exactly for the percolation properties of random networks under localized node removal. Our results show that the effects of localized attack and random attack on an Erdős-Rényi network are identical. While a random-regular network is more robust against localized attack than against random attack, the robustness of a scale-free network depends on the heterogeneity of the degree distribution. When λ < λ c , the SF network is found to be significantly more vulnerable with respect to localized attack compared to random attack. When λ > λ c , the opposite is true. Our results can provide insight into understanding the robustness of complex systems and facilitate the design of resilient infrastructures.   k 0 −2 = 0.125, as confirmed in simulations. Note that for 2 < λ ≤ 3, p c → 0 in the thermodynamic limit (N → ∞) for random attack [2]. (b) When λ < λ c , the SF network is more vulnerable to localized attack compared to random attack. (c) When λ = λ c , p c for localized attack and for random attack are equal. (d) When λ > λ c , the SF network is more robust against localized attack compared to random attack.

REMAINING NETWORK AFTER LOCALIZED ATTACK
Consider a random network with arbitrary degree distribution P (k), which represents the probability of a node in the network to have k links. The corresponding generating function is defined as We separate the process of a localized attack into two stages: (i) at the first stage, we remove all the nodes belonging to the attacked area but keep the links connecting the removed nodes to the remaining nodes; (ii) at the second stage, we remove those links. Now consider the degree distribution P p (k) of the remaining nodes after the first stage. Following Ref. [3] and letting A p (k) be the number of nodes with degree k in the remaining network, we have With one more node being removed, A p (k) changes as where k(p) ≡ P p (k)k. In the limit N → ∞, Eq. (3) can be presented in terms of a derivative of A p (k) with respect to p, By differentiating Eq. (2) with respect to p and plugging it into Eq. (4), we have p dP p (k) dp The solution of Eq. (5) can be expressed as and the average degree of the remaining network is 2 where f ≡ G −1 0 (p). Thus the generating function of P p (k) is Now consider the second stage of removing the links of the remaining nodes which lead to the removed nodes. The number of links belonging to the nodes on the outer shell of the attacked hole, L(f ), can be expressed as [3], Since loops are allowed in the random network model, those links can be connected either to the remaining nodes or to other nodes on the same outer shell of the attacked hole. The number of links of the remaining nodes which lead to the removed nodes is ].
The probability that a link in the remaining network will end at an unremoved node is equal Because the network is randomly connected, removing the links that end at the removed nodes is equivalent to randomly removing a 1 −p fraction of links of the remaining network.
The generating function of the remaining network after the random removal of a 1 −p fraction of links is equal to [4] where f ≡ G −1 0 (p). Note that Eq. (12) is the generating function of the degree distribution of the remaining network after a localized attack.

II. ROBUSTNESS OF REAL-WORLD NETWORKS AGAINST LOCALIZED AT-TACK
We test and compare the robustness of real-world networks against localized attack and random attack using a peer-to-peer computer network [9] and a global airline route network [10]. The degree distributions of both networks approximately follow power law (see Figure 1). Figure 2 shows that, for both real-world networks, localized attack can collapse the network much more easily: a node failure of 30% in the global airline route network and 55% in the peer-to-peer computer network can disable the total network. When the attack is random, however, a node failure of 98% in the global airline route network and 90% in the peer-to-peer computer network must occur before the network collapses. This shows that a localized attack is significantly more harmful to real-world SF networks than a random attack, supporting our theoretical results for SF networks with λ < λ c .