Attack Vulnerability of Network with Duplication-Divergence Mechanism
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1 Commun. Theor. Phys. (Beijing, China) 48 (2007) pp c International Academic Publishers Vol. 48, No. 4, October 5, 2007 Attack Vulnerability of Network with Duplication-Divergence Mechanism WANG Li, YAN Jia-Ren, LIU Zi-Ran, and ZHANG Jian-Guo Department of Physics, Hunan Normal University, Changsha 4008, China (Received October 8, 2006; Revised December 2, 2006) Abstract We study the attack vulnerability of network with duplication-divergence mechanism. Numerical results have shown that the duplication-divergence network with larger retention probability σ is more robust against target attack relatively. Furthermore, duplication-divergence network is broken down more quickly than its counterpart BA network under target attack. Such result is consistent with the fact of WWW and Internet networks under target attack. So duplication-divergence model is a more realistic one for us to investigate the characteristics of the world wide web in future. We also observe that the exponent γ of degree distribution and average degree are important parameters of networks, reflecting the performance of networks under target attack. Our results are helpful to the research on the security of network. PACS numbers: Fb, a, a Key words: complex network systems, attack vulnerability, statistical dynamics Introduction Examples complex networks are abundant in many disciplines of science, ranging from financial markets to biological systems. [ 3] Security of complex networks in response to random attack or target attack has become a topic of recent interest. [4 6] Originated from studies of computer networks, attack vulnerability denotes the decrease performance due to target removal of vertices or edges. [7] The meaningful purpose for attack vulnerability studies is for the sake of protection. If one wants to protect the network by guarding or by a temporary isolation of some vertices, the most important vertices, breaking of which would make the whole network malfunction, should be identified. Furthermore, one can learn how to build attack-robust networks. Duplication-divergence model was investigated by Ispolatov et al. [8] It turns out that such the model well describes the degree distribution found in biological proteinprotein network. Such mechanism also plays a role in creation of new vertices and links in the world wide web, growth of various networks of human contacts by introduction of close acquaintance of existing members, and the evolution of many other nonbiological networks. In such a model, at each time step the amount of links added to the network depends on the degree of the original vertex and the link retention probability σ. A question is then how this retention probability affects the networks robustness against target attack. This problem is important, but to our knowledge, has not been studied. Previous studies [9 ] have shown that, the network with BA mechanism is sensitive to target attack. These mechanism s main principles are growth and preferential attachment. The duplication-divergence mechanism is a different role to generate network. So the issue of these two different networks attack vulnerabilities is interesting, however has not been considered. These are our motivations for this paper. In this paper, we investigate the attack vulnerability of duplication-divergence network. The measure of the network s response to target attack is the decrease in its average inverse geodesic length and the size of the giant component. Our study focuses on the exact value of critical fraction needed to remove for breaking down the network. We show that the duplication-divergence network with larger retention probability σ is more robust against target attack relatively. Comparing the duplication-divergence model with the BA model, we found that the former network is broken down more quickly than the latter one under target attack. WWW and Internet networks have been simulated under target attack. It is found that they break down more quickly than the BA model. Compared with the BA model, we think the duplication-divergence model describes the realistic WWW and Internet networks better and is promising to investigate characteristics of them. For these results, we give our comprehensions and consider that the exponent γ of degree distribution and average degree are important parameters of scale-free networks, reflecting the performance of the networks under target attack. 2 Model and Definition of Quantities and Attack Strategy 2. Model and Degree Distribution We first introduce duplication-divergence model as follows (Fig. ). The network generated by duplicationdivergence mechanism is an evolving graph with vertices The project supported by National Natural Science Foundation of China under Grant No Corresponding author, jiarenyan@hunnu.edu.cn
2 No. 4 Attack Vulnerability of Network with Duplication-Divergence Mechanism 755 correspond to single members and links mimics the interactions among them. Starting from a small and arbitrary initial network (we choose a connected initial network to guarantee a connected network). The graph consisting of N 0 vertices and E 0 links (in this paper, for all cases considered we choose N 0 = 0, E 0 = 30), suffers from the following modification in each discrete time step. [8] (i) A randomly chosen target vertex is duplicated including its links, that is, a new vertex is introduced and connected to each neighbor of the target vertex; (ii) each link of the new vertex is retented, with probability σ. The amount of new vertex links depends on the target one and the link retention probability σ. If at least one edge of the new vertex is established, the new vertex is preserved; otherwise, the attempt is considered as a failure and the network does not change. Step (i) mimics a duplication process, where the duplicates are linked to the same neighbors of the target one and step (ii) represents the divergence process, that is, some links between the new vertex and its neighbors disappear. Fig. Illustration of duplication-divergence event. The vertex i is the original one and j is the duplicated one. Links between the duplicated vertex j and vertices 3 and 4 disappeared as a result of divergence. Fig. 2 The degree distribution P k vs. k for different σ values. The size of the network is N = The results are averaged over 00 realizations. The analytical solution about degree distribution of this model is observed by Ispolatov et al. [8] Figure 2 shows the numerically computed degree distribution P (k) with N = 5000 for different σ values, where the open circles, the stars, and the squares denote σ = /4, σ = 2/5, and σ = /2, respectively. Ispolatov et al. pointed out that only when σ < /2 can we obtain a stationary scaling power-law degree distribution P (k) k γ. And the exponent γ varies with σ, decreasing as σ increases. So we consider all duplication-divergence networks with σ < /2 in our study. The mechanisms leading to power-law degree distribution of BA model are argued to be growth and preferential attachment, where the former means that the size of the network keeps increasing with time and the latter assumes that the relative probability for an already heavily connected vertex to get new links is proportionally large. In particular, at a given time the probability Π i for a vertex in the network with k links to acquire a new link is assumed to be [3] Π i k. In general, in duplicationdivergence model, the original vertex is randomly chosen; the new links added to the network at each time step are uncertain, which depends on the original vertex and the retention probability. These are the main differences between duplication-divergence model and BA model. 2.2 Definition of Quantities To quantify the attack vulnerability of a network against target attack, we use the average inverse geodesic length l as well as the size S of the giant component. [2] Another key quantity is the average length l. [9,2] If l is larger, the dynamics (of epidemics, information flow, etc.) is slow in the network. However, as the number of removed vertices increases, the network will eventually break into disconnected subgraphs. The average geodesic length become infinite for such a disconnected graph. So we choose the average inverse geodesic length, l d(υ, ω) N(N ) υ ν ω υ ν d(υ, ω), () where ν is the set of vertices, and d(υ, ω) is the length of the geodesic between ν and ω. l has a finite value for a disconnected graph since /d(υ, ω) = 0 if no path connects υ and ω. It should be noted that the notation l is not the reciprocal of l. The functionality of the network is measured by l : the larger l is, the better the network functions. For the calculation of d(υ, ω), we use the algorithm presented in Ref. [3]. Since subsequent attack will disintegrate the network, the size of the largest connected subgraph is also an interesting quantity for measuring the functionality of the networks. In social networks, the largest connected subgraph is known to have a size of the order of the entire network, and accordingly is called giant component. [4] Throughout the present paper, we denote the size of the giant component as S, which will be used together with l to study the attack vulnerability. We consider that the network is broken down when S/N 0. And f c is defined as a fraction of vertices removed when the network is broken down. [5,6]
3 756 WANG Li, YAN Jia-Ren, LIU Zi-Ran, and ZHANG Jian-Guo Vol Attack Strategy As for strategy of target attack, we choose the subsequent target attack. That is we remove vertices one by one starting from the vertex with the highest degree until the network is broken down. As more vertices are removed, the network structure changes, leading to different distribution of the degree. So we recalculate at every removal step, and remove the vertices with highest degree at that time. In any cases where two or more vertex should be equally chosen, the selection is done randomly. As the vertex attack proceeds, both the remaining number of vertices and the average inverse geodesic length decrease, which from the definition of l, suggesting that l can be both increasing and decreasing, depending on how much damage is made by the removals. If l and S decrease quickly it means that such network is very sensitive to target attack. 3 Results Now we present results concerning the attack vulnerability of duplication-divergence networks with same size and different retention probabilities σ. Figure 3 shows that for networks consisting of 4000 vertices, its average inverse geodesic length l and the size S of giant component versus the fraction of removed vertices N rm /N, respectively. Apparently, as N rm is increased, the l and S decrease rapidly for all σ considered. For instance, from Fig. 3(b) we can see that the network with σ = 0.25 is broken down only when N rm /N 0.0. That is, f c 0.0. This indicates that the network with duplication-divergence mechanism is very sensitive to target attack, regardless of what value σ is. This can be explained from the large variation in the importance (measured by degree) of the vertices. The degree distribution is P (k) k γ, that is to say, there exist very important vertices, or center vertices, which play a very important role in the network functionality. If we attack on these vertices, there would be a significant impact on the performance of the network. So the duplication-divergence network is broken down quickly, when facing subsequent target attack. Fig. 3 Attack vulnerability of networks constructed according to duplicated-divergence mechanism with different retention probabilities at a time when there are 4000 vertices in the network. The results of attack vulnerability are measured by (a) the average inverse geodesic length l and (b) the S size of giant component S versus the fraction of removed vertices N rm/n, normalized by the values for initial network. The different symbols represent different retention probabilities σ, which is interpreted in the top right corner of figure. As N rm /N is increased, the average inverse geodesic length l and the size S of giant component all decrease rapidly. However, we can see clearly that the duplicationdivergence networks with different σ behave differently under target attack. The larger the σ is, the larger the f c will be. For instance, f c 0.07 for the network with σ = 0.20 while about f c 0.30 for network with σ = It means that the duplication-divergence network with smaller retention probability σ is found to be even more vulnerable against subsequent target attack relatively. To investigate the reason for such results, we now study the average degree k of networks with different σ. We show the numerical results in Fig. 4. For the same size of the network, the larger the σ is, the larger the average degree k will be. The average degree k is larger in network with larger σ. This can be seen from the comparison between Fig. 3 and Fig. 4. These results are in agreement with our comprehension. Naturally, vertices in the network with larger average degree have more interactions with other vertices. In such networks, the vertices contact with each other more closely. The increase of the average degree reduces the damage of the subsequent attack. So the network with larger σ (also with larger average degree k ) is broken down more slowly. That is to say, the network with larger σ is more robust. The practical implications of this result is the following. Suppose one wishes to design a network with the duplication-divergence mechanism, which should also be robust against target attack relatively. We should
4 No. 4 Attack Vulnerability of Network with Duplication-Divergence Mechanism 757 make great effort to obtain larger retention probability. because such network with larger retention probability is more robust against target attack relatively. Fig. 4 The average degree versus N in duplication-divergence networks. The retention probability is σ = 0.25, σ = 3/8, σ = 0.45, from bottom to top. Fig. 5 The degree distribution of duplication-divergence model (σ = 0.25) and BA model. The value of exponent γ is about 2.92 in the BA network, while it is about 2.54 in duplication-divergence network. Next we compare the attack vulnerability of duplication-divergence network with that of BA network. These two networks are all expected to be fragile under target attack. Do they response similarly under target attack? To make a meaningful comparison, we present the attack results for these two network models with the same size (number of vertices N = 4000) and the same average degree k ( k = 4). These two networks, whose degree distributions are P (k) k γ, have different exponent γ, as shown in Fig. 5. The value of exponent γ is about 2.92 in the BA network, while it is about 2.54 in duplication-divergence network. Fig. 6 The comparison results of duplication-divergence model and BA model for vulnerability to subsequent target attack. The notations are the same as the ones in Fig. 3. The two networks have the same number of vertices N = 4000, and the same average degree k = 4. Figure 6 displays the results of networks response under subsequent target attack. The two networks behave differently. As N rm /N becomes larger, the l and S decrease rapidly for both networks, indicating both networks are easily to crash into small fragments under target attack, which is in agreement with our prediction. The very interesting and also important result is that, duplication-divergence network (σ = 0.25) is broken down more quickly than its counterpart BA network. WWW and Internet networks have been simulated under attack. It was found that they broken down more quickly than BA model. [3] And the duplication-divergence mechanism plays a role in a creation of new vertices and links in the world wide web. This result is consistent with real fact. So we think this model is a more realistic one for us to study the world wide web in future. These two networks contain the same size vertices and the same average degree. Why do they behave differently when facing subsequent target attack? In order to investigate the reason of their diverse behaviors, we do research on the networks exponent γ values. Figure 5 shows that the exponent γ of duplication-divergence network (γ 2.54) is
5 758 WANG Li, YAN Jia-Ren, LIU Zi-Ran, and ZHANG Jian-Guo Vol. 48 smaller than BA network (γ 2.92). For instance, the value of p k is larger in duplication-divergence network when k is equal to 00. For these two networks, the same number of largest degree vertices possesses more links with other vertices in the duplication-divergence network. When the same number of largest vertices removed, the vertices will lose more interactions with other vertices the damage to duplication-divergence network is more serious. So that the duplication-divergence network appears more fragile crash into fragments more quickly. 4 Summary and Discussion In summary, we have investigated the attack vulnerability of duplication-divergence network. Due to the rule used to create network, the structure and characteristic are affected by the link retention probability σ. So we focus on the relationship between σ and the networks response to target attack qualitatively. And duplication-divergence networks are all sensitive to target attack for all σ considered, and as σ increased, the critical point f c becomes larger; Also we found that the duplication-divergence network is more fragile than the BA network. As mentioned above, this result is consistent with previous simulation result about Internet network. So the investigation of the networks robust and other dynamics will reveal important aspects of networks. As for these results, we give our comprehension in some degree. However, there are some aspects we did not considered. For the duplication-divergence networks we investigated, the values of exponent γ and average degree k are all different. We only consider the likely effect of the average degree k. What will the combination influence of the two parameters be? This is our further preparation. We also hope for other researchers interpretations for these results. In recent years, various complex systems have been under both theoretical and empirical scrutiny, and the studies are abundant. Yet, besides the large-scale degree distribution, there are still many aspects that we cannot explain and many new methods and techniques requires theoretical description. We hope the networks attack vulnerability studied in this article may offer important input into understanding of complex networks and contribute to designing robust and effective networks. Acknowledgment We thank Prof. Tang Yi for helpful discussions. References [] D.J. Watts and S.H. Strogatz, Nature (London) 393 (998) 440. [2] S.N. Dorogovtsev and J.F.F. Mendes, Phys. Rev. E 63 (200) [3] R. Albert and A.L. Barabási, Rev. Mod. Phys. 74 (2002) 47. [4] B.A. Huberman and L.A. Adamic, Nature (London) 40 (999) 3. [5] M. Faloutsos, P. Faloutsos, and C. Faloutsos, Comput. Commu. Rev. 29 (999)5. [6] P.L. Krapivsky and S. Redner, Phys. Rev. E 7 (2005) [7] A.L. Barabási, R. Albert, and H. Jeong, Physica A 272 (999) 73. [8] I. Ispolatov, P.L. Krapivsky, and A. Yuryev, Phys. Rev. E 7 (2005) 069. [9] R. Albert, H. Jeong, and A.L. Barabási, Nature (London) 406 (2000) 378. [0] R. Cohen, K. Erez, D. Avrahan, and S. Havlin, Phys. Rev. Lett. 86 (200) [] L.K. Gallos, R. Cohen, and P. Argyrakis, Phys. Rev. Lett. 94 (2005) [2] P. Holme and B.J. Kim, Phys. Rev. E 65 (2002) [3] M.E.J. Newman, Phys. Rev. E 64 (200) 063. [4] E.M. Adison and Y.C. Lai, Phys. Rev. E 66 (2002) 06502(R). [5] G.J. Lin, X. Cheng, and Y.Q. Ou, Chin. Phys. Lett. 20 (2003) 22. [6] T. Tanizawa, G. Paul, R. Cohen, S. Havlin, and H.E. Stanley, Phys. Rev. E 7 (2005) 0470.
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