Error and Attack Tolerance of Scale-Free Networks: Effects of Geometry

Transcription

1 Error and Attack Tolerance of Scale-Free Networks: Effects of Geometry Advisor: Prof. Hawoong Jeong A thesis submitted to the faculty of the KAIST in partial fulfillment of the requirements for the degree Bachelor of Science in Department of Physics Korea Advanced Institute of Science and Technology by Ji Hyun Bak 2010

2 Abstract Tolerance of a scale-free network against errors and various schemes of attacks is reported, focusing on the effect of network-embedding geometry and the method of node assignment. Network tolerance is assessed by two global measures, the topological diameter D =< d ij > and the largest cluster size S, and is recorded versus the fraction of the number of attacked nodes. Maintaining the scale-free statistics, networks are embedded on three types of geometries: topological (no geometry), the two-dimensional regular lattice, and the two-dimensional fractal geometry known as the Sierpinski carpet. In embedding a topologically constructed network onto a geometry, the node assignment may be done either randomly or under a Coulomb-like repulsion between the nodes.

3 Contents List of Figures Acknowledgements Vita iii iv v 1 Introduction Motivation Aims Structure of the Paper The Topological Scale-Free Network The Scale-Free Network Parameters of Network Efficiency Network Diameter D Largest Cluster Size S Attack Schemes Error and Attack Tolerance The Geometry Two-Dimensional Regular Lattice Two-Dimensional Fractal Geometry i

4 4 Node Assignment and Link Reconstruction Node Assignment Random Assignment Coulomb-like Repulsive Assignment Link Reconstruction Error and Attack Tolerance on Geometry 18 6 Conclusion 22 Bibliography 24 ii

5 List of Figures 2.1 The Scale-Free Network The Power-Law Connectivity Distribution Error and Attack Tolerance of the Scale-Free Network Error and Attack Tolerance of the Scale-Free Network when the Network is Kept Pruned to the Largest Cluster The Fractal Geometry The Coulomb-like Repulsive Assignments of Nodes Error and Attack Tolerance on Regular Lattice Error and Attack Tolerance on Fractal Geometry iii

6 Acknowledgements My undergraduate studies at KAIST have been supported by the Presidential Science Scholarship [M ]. In doing most of the calculations for this paper I have gratefully used gmul, a network analyzing software by Dong-Hee Kim, an ex-member of Complex Systems and Statistical Physics Lab., KAIST. To produce the images for this paper I have been benefited by the free network analyzing software pajek, along with Microsoft Excel and Windows Paint. iv

7 Vita Education Undergraduate Program (B.S.) KAIST, Feb Current. Secondary Education, Korea Science Academy, s.c.l., Mar Feb Employment Student Research Associate, Computational Nano-Bio Laboratory, KAIST Graduate School of NanoScience and Technology, Aug Current. v

8 Chapter 1 Introduction 1.1 Motivation Communication is the everlasting topic throughout the history of human civilization, and it is crucial that we understand the properties of networks on which any mass communication may take place. However networks cannot be perfectly durable: they may suffer from random or intended failures (errors and attacks). Network tolerance against random and intended failures under various settings is thus an urgent topic for establishing stable networks. 1.2 Aims The tolerance of networks has been extensively studied during the last few decades, but the interests have been limited to the topological networks. In real situations there is always some metric involved: be it the geographical distance, the social distance, or the transportation cost. Although two nodes of a network are topologically close (that is, connected by small number of links) they may be far apart in terms of, for instance, the geographic locations. I will consider the simplest geometry of two dimensional regular lattice, as well as the two dimensional fractal case which more closely 1

9 resembles the natural pattern of human residence on the physical globe. The presence of a specific geometry gives to a pair of nodes a metric distance as well as their topological distance in the network. Since this intertwinning of the topological and the geographical space is an essential reality of any physical network, the problem of connecting the two spaces must naturally be considered as the key issue. The method of assigning the topological nodes onto the geometrical lattice points may affect the global tolerance of the network significantly. As first reported by Albert, Jeong and Barabási (AJB) [1] the scale-free network is remarkably robust against errors (random failures) but relatively vulnerable to attacks (targeted failures). Knowing that many complex networks in the real world have the scale-free structure, we aim to see how the introduction of geometry affects the error and attack tolerance of the scale-free network. 1.3 Structure of the Paper In Chapter 2 I will first introduce the concept of the scale-free network and the relavant parameter to assess network tolerance. Section 2.3 introduces the basic attack schemes we use throughout this study. And in Section 2.4 we investigate the error and attack tolerance of the topological scale-free networks. The following two chapters, Chapter 3 and Chapter 4, are the major concerns of this paper, introduction of the network-embedding structure and the mapping between topology and geometry. Finally I will show in Chapter 5 how the presence of metric (geometrical structure) affects the error and attack tolerance of the scale-free networks. 2

10 Chapter 2 The Topological Scale-Free Network 2.1 The Scale-Free Network Scale-free networks are characterized by their power-law tail in the connectivity distribution. Most of the links are concentrated to a very small fraction of nodes in the network, while the vast majority of nodes have nearly zero nodes attached to them. A typical topology of the scale-free network is shown in Figure 2.1. The name scale-free comes from that while the exponential (Erdös-Rényi) network or the small-world network displays a peak in the distribution and thus the characteristic value of connectivity (the scale), this type of network is free of a characteristic scale in the connectivity distribution. [3] The Static Model [4] suggested by Goh, Kahng and Kim is employed to implement the scale-free network. Initially we have N nodes, designated by i = 1,, N and a weight w i = i α for each node. Given the weighted probability we can choose a pair of two nodes and form a link between them. We repeat the linking process until the total number of link becomes L = mn, where m is a given integer. [2] Every network created using this model has N nodes, L links and the power-law connectivity distribution 3

11 Figure 2.1: The Scale-Free Network A scale-free network with N = 1000, m = 2, α = 0.5(γ = 3.0). This image is produced by the drawing tool of pajek. characterized by the parameter α. P (k) = 1 N N p i (k) ck γ (2.1) i=1 The degree distribution can also be characterized by the so-called connectivity exponent γ = 1 + 1/α. In this work all networks have α = 0.5 and γ = 3.0. Figure 2.2 shows the power-law connectivity distribution of the scale-free network created by this model. 2.2 Parameters of Network Efficiency Since we want to investigate how the network maintains its efficiency against failures, appropriate choice of parameters characterizing the network efficiency may be the most crucial part of all. What is an efficient network? I shall follow the answer of AJB. [5] [1] 4

12 Figure 2.2: The Power-Law Connectivity Distribution The characteristic power-law distribution of the scale-free network. N = 1000, m = 2, α = 0.5(γ = 3.0). Averaged over an ensemble of 6 realizations Network Diameter D We want a network to be able to transport, in short time and/or low cost, information or materials from one place to another. This interconnectedness is described by the diameter D =< d ij >, defined as the average distance between any two nodes in the network. The distance d ij is defined to be the length of the shortest path between nodes i and j. A smaller value of diameter indicates the network can connect its two nodes in shorter time or lower cost, in other words, that the network is closely interconnceted. We shall see the diameter increasing as the network is destroyed by failures. In this paper I have used the harmonic average of the distances. Taking the harmonic average has an advantage that it evades the problem of having infinite distance between two isolated nodes. In this way we can deal with the connected pairs of nodes only. The calculation is largely based on the network analysis software gmul. 5

13 2.2.2 Largest Cluster Size S We also want a network to be compact. That is, one large cluster of 100 nodes is preferred to ten fractured clusters of 10 nodes each. This property can be traced by keeping the relative size (number of nodes) of the largest cluster S = N l.c. /N total, 0 S 1 with respect to the total size of the network. We may regard a network as fragmented if the largest cluster size S falls below a certain value, for example 0.5. This calculation was also based on gmul. 2.3 Attack Schemes A failure of a node may result from either an error or an attack. An error is a random failure, while an attack is a intended failure preferentially targeted to damage the network. Technically a failure the removal of a nodes from the network. An informed agent will attack the most powerful node of the network, to reduce the network performance by the largest extent. The agent would establish a scheme to determine the target, which is done by identifying a parameter representing the significance of a node in the network. There are many such parameters identified in the preceding studies. For the topological network I tried three parameters for the node power: large connectivity, the total number of links attached to the node; large betweenness centrality, the accumulated total number of data packets passing through the node when every pair of nodes sends and receives a data packet along the shortest path connecting the pair [4]; and small closing centrality, the average distance from the node to all other nodes in the network, and performed a sequence of preferentially targeted attacks. There are two ways to determine the target given the information: 6

14 sort the nodes in the initial configuration and attack in the fixed order, from the most powerful node; or sort the nodes after each node removal, and attack the most powerful node following the updated order. A failure in the topological network is implemented by removing a single target (of either error or attack) from the network, that is, removing all links attached to the node. In the geometry-embedded networks, to see the effect of geometry, failure accompanies removals of all nodes in a finite region centered at the target. Hence the geometrical distance comes into significance. This process of destroying the neighboring region I call bombing. In this work, a bomb blows off a 3 3 region on the lattice centered at the target. The bombimg range may be increased to see a more dramatic effect of geometry. 2.4 Error and Attack Tolerance The network simulation results for the error and attack tolerance are shown in Figure 2.3 and Figure 2.4. The network diameter D and the largest cluster size S are plotted versus the fraction of failed nodes i, under various settings. In Figure 2.3 the diameter D mostly appear to be linear, with slight bends in the case of attacks by connectivity, connectivity updated, and betweenness centrality. As we will see in the later chapters, that the graph is linear means that it is not yet falling: it is not yet severely damaged. Yet we confirm the significant difference between the error tolerance and the attack tolerances. The scale-free network is much more vulnerable to attacks than to errors. The largest cluster sizes S do not fall abruptly but stay close to S = 1 up to i = 0.1, our range of investigation. This again means the network has not yet severely damaged. The curves for the attacks by connectivity, connectivity updated and betweenness centrality are the fastest in falling. 7

15 Figure 2.4 shows the situation when the network is pruned each time to leave the largest cluster only. The largest cluster size S is naturally kept at unity. The diameter D turns out to be larger than in the non-pruned case, which is obvious since we are effectively eliminating more nodes at a single failure. As the network is further destroyed a single attack eliminates yet more nodes, because more nodes (or groups of nodes) will be holding only one link to be conneted to the largest cluster of the network, and when the link breaks down they become isolated. Hence D shows an exponentiallike increase. 8

16 Figure 2.3: Error and Attack Tolerance of the Scale-Free Network A scale-free network with N = 1000, m = 2, α = 0.5(γ = 3.0). Tolerance against errors are marked with cross symbols. Empty and filled symbols indicate the fixed-order and the updated-order attack scheme respectively. Squares, circles and triangles stand for attacks by connectivity, betweenness centrality, and closing centrality. Each averaged over an ensemble of 10 realizations. 9

17 Figure 2.4: Error and Attack Tolerance of the Scale-Free Network when the Network is Kept Pruned to the Largest Cluster A scale-free network with N = 1000, m = 2, α = 0.5(γ = 3.0), pruned to the largest cluster. Tolerance against errors are marked with cross symbols. Empty and filled symbols indicate the fixed- and updated-order attack respectively. Squares, circles and triangles stand for attacks by connectivity, b.c., and c.c. Each averaged over an ensemble of 8 realizations. 10

18 Chapter 3 The Geometry The mathematical model of network consists of only two components: nodes and links. They dwell in the topological space. The only relevant quantity we can draw out of a pair of nodes is the distance d ij. From here on we shall call this quantity the topological distance. Topological networks considered in this paper are the scale-free networks with the weight parameter α = 0.5 ( the connectivity exponent γ = 3.0 ) the total number of nodes N = 10000, and the total number of links L = A topological failure affects no more than the targeted node(s). An attack on the network embedded on certain geometry, however, may affect not only the target, but also the neighboring nodes on the lattice. 3.1 Two-Dimensional Regular Lattice As introduced before, for any type of real network there may be certain metrics involved. The metric can be geographical length, (inverse) social intimacy, transportation cost, or any other quantity that comes in the actual performance of the network. Whatever it is, we can think of a space whose Euclidean metric represents our metric of interest. This space we call the geometry as opposed to the topology. Now the space consists of lattice points and the metric. The metric is now called the geometric distance. 11

19 We consider the regular (Euclidean) lattice in the geometric space. Periodic boundary condition is imposed. This geometric space need not be two dimensional: there may be a network system whose metric can only be properly embedded in higher dimensions. Nevertheless I proceed by assuming a two dimensional geometric space. The geographic distance best fits to this rather abstractized model. To investigate the effect of geometry, a topological scale-free network of N = nodes is first generated. From the network we extract the connectivity (the number of links attached) of each node, which exhibits the power-law distribution. Then we assign the nodes onto the lattice points on the lattice, and realize the connectivity by forming links. The method of node assignment and link reconstruction is an important issue and will be discussed in Chapter Two-Dimensional Fractal Geometry The previous discussions of regular lattice geometry has been derived from any metric involved in the network performance. Now we shall focus on the geographic space. The geographic space is special that it is itself the corresponding geometric space, so that one can directly perceive the metric space which is in general complicated. In this special case we can talk about the position of a point in the space as well as the distance between two points. It has been observed that the geographic distribution of human residence on the globe exhibits a fractal pattern. Most human-related networks are subjected to the geography of human residence, and it is worth considering the effect of allowing a fractal-structured available sites on the geometry. The Internet is a good example of a human-residence-dependent network, while the World-Wide Web is independent of the geography of human residence. A site on the previous Euclidean lattice is set either available or prohibited to accommodate a node of the network. The available sites are deliberately fixed in a two dimensional fractal structure, namely the Sier- 12

20 Figure 3.1: The Fractal Geometry The Sierpinski carpet is employed to implement the fractal geometry on two dimensions. A carpet is repeated in a lattice frame, to give the total number of available sites N a = A square on the left picture represent a smaller Sierpinski carpet of 9 9 lattice points, as shown on the right. pinski carpet. [6] The original Sierpinski carpet is constructed by dividing a square into nine subsquares, removing the one at the center and repeating the process infinitely many times on the remaining subsquares. Since our space is a discrete lattice the repetition is stopped when the subsquare size reaches 1 1, or a single lattice point. The size of the underlying lattice and initial carpet is carefully chosen to maintain the scale independence upon periodic repetition. As shown in Figure 3.1, a carpet is repeated in a lattice frame, leaving the total number of available sites N a =

21 Chapter 4 Node Assignment and Link Reconstruction Node assignment and link reconstruction establishes the mapping between topology and geometry. These two are the key steps in introducing geometries to networks. In this chapter we seek how to map a scale-free network to a geometry so that it becomes most tolerant against failures. My scheme is as follows: A topological network is created, and the nodes are assigned to the lattice points along with their respective connectivities. Then the connectivities are realized by reconstructing the links. This procedure ensures that the geometry-embedded network maintains the characteristic scale-free property. In the literal sense we are not embedding an existing network onto a geometry; rather, we are creating a network displaying certain connetivity distribution with both the topological and the geometrical structure. 4.1 Node Assignment A scale-free network is characterized to have a hub: a topological site around which the powerful nodes are concentrated. This property leads to the marked robustness against errors and the vulnerability to attacks. It is 14

22 then obvious that if we concentrate the powerful nodes in a small region of the geometric space, the network will be increasingly vulnerable to attacks. Putting aside this trivial case, I shall investigate the two other interesting assignment schemes, the random and the repulsive assignment Random Assignment The easiest way is simply to assign to each node a lattice point at random choice. A targeted bombing in this case is just equivalent to a targeted attack and eight random failures, since the bombing destroys the 3 3 region Coulomb-like Repulsive Assignment As an attempt to enhance the attack tolerance of scale-free networks, I have tried an assignment scheme in which powerful nodes are placed as far apart as possible. A Coulomb-like repulsive potential is introduced, with mathematical form the electrostatic potential proportional to the charge (the connectivity) and inversely proportional to the length (geometrical distance) squared. Nodes, along with respective connectivities, are placed one by one onto the lattice plane in order of decreasing connectivity. Once placed on the lattice, the nodes form the Coulomb-like potential which determines the position of the next node. Hence if the node of largest connectivity is placed at the lattice center (the starting position is arbitrary), the node with next largest connectivity will settle on the lattice corner, and the next one on the midpoint of lattice side, and so forth. From the fourth node the settling point will depend on the connectivities of the preceding nodes. The process has been implemented by solving the familiar Poisson differential equation 2 φ = k, (4.1) where φ is the corresponding potential and k is the connectivity of the nodes already assigned on the lattice points. Finite dimensional method 15

23 was used to solve the equation numerically. The actual process of repulsive assignment is illustrated in Figure Link Reconstruction Once the nodes are completely assigned onto the lattice, the connectivity distribution is to be realized to construct a network. I adopt the link construction method of Rozenfeld et al.[7], who imposed the natural restriction that the total length of links in the system be minimal. This minimal-length construction selects at random a lattice point with a node embedded, and connects it to its closest neighbors until its previously assigned connectivity k is realized. The neighboring nodes can accept links only if their assigned connectivity is not already filled. The link formation process may explore the neighboring nodes up to a distance r(k) = Ak 1/d (4.2) where A is an external parameter and d is the dimension of the lattice. I call quantity r(k) the exploration threshold. This concept of threshold can be understood with the real-world constraint of costs. According to Rozenfeld et al., networks considered in this paper (γ = 3) can be successfully embedded to the lattice place (the characteristic distance ξ 100) upon an appropriate choice of the value of A. In this paper A = 2 has been used. 16

24 Figure 4.1: The Coulomb-like Repulsive Assignments of Nodes Nodes are placed one by one onto the lattice plane in order of decreasing connectivity. Once placed on the lattice, the nodes form the Coulomb-like potential which determines the position of the next node. The left and the right column show the potential φ and the connectivity of the assigned nodes. The situations are shown when 1, 2, 5, 10 and 100 nodes (from the top) out of are assigned by the mechanism. 17

25 Chapter 5 Error and Attack Tolerance on Geometry Figure 5.1 and Figure 5.2 shows the error and attack tolerance of scalefree networks when embedded on certain geometries, the regular lattice and the Sierpinski fractal carpet. Again, the network diameter D and the largest cluster size S are plotted versus the fraction of failed nodes i, under various settings. In both cases the diameter D first rise: this indicates the network getting less and less dense, which is exactly the same mechanism as in the topological network case. By removing a node we are destroying the shortcuts conneting other pairs of nodes, the distances between the pairs increase, and the diameter increases. Then there is the characteristic fall, or a bent. Now that the network is so loosely connected, that removing a new node has more effect in reducing the absolute size of the network (the farthest pairs of nodes to be connected) than in destroying the shortcuts. After the fall the diameter shows another increase. At this point the system shall no more be called a network; rather, it is a collection of small clusters. Further breaking of these clusters increase the diameter D as it is defined, but the quantity no more conveys a physical significance. 18

26 The largest cluster size S shows a peculiarly abrupt drop. The flat region before the drop corresponds to the linear-rise region of the diameter D, where the network is relatively dense and compact into one large cluster. The narrow region of drop corresponds to the decreasing curve of D, where the breakdown of the network is most critical. The region bottomed out after the drop indicates that the network is now completely broken, corresponding to the second rise of D. Hence we may understand the decreasing region in the graph of D (and the drop in the graph of S) to be the critical breakdown stage of the network. In both cases of the regular and the fractal geometry, performing attacks by the updated order turned out to destroy the network a little faster. 19

27 Figure 5.1: Error and Attack Tolerance on Regular Lattice A scale-free network, N = 1000, m = 2, α = 0.5(γ = 3.0), embedded on a two dimensional regular lattice. Empty and filled symbols indicate the random and the Coulomb-like repulsive node assignment schemes respectively. Diamonds, squares and triangles stand for errors, attacks by connectivity in fixed order, and attacks by connectivity in updated order. Each averaged over an ensemble of 10 realizations. 20

28 Figure 5.2: Error and Attack Tolerance on Fractal Geometry A scale-free network, N = 1000, m = 2, α = 0.5(γ = 3.0), embedded on the Sierpinski fractal carpet. Empty and filled symbols indicate the random and the Coulomb-like repulsive node assignment schemes respectively. Diamonds, squares and triangles stand for errors, attacks by connectivity in fixed order, and attacks by connectivity in updated order. Each averaged over an ensemble of 10 realizations. 21

29 Chapter 6 Conclusion I have observed the error and attack tolerance of the topological scalefree network against various attack schemes, when the smaller clusters are taken along or discarded. The attacks are intendedly performed from the information of the connectivity, the betweenness centrality, and the closing centrality of the individual nodes. The target node for attack can be either chosen in the order fixed from the initial configuration, or in the order updated each time a node is eliminated from the network. The network tolerance has been assessed by two parameters: the network diameter D and the largest cluster size S. And I have discussed the significance of introducing geometrical structures to the study of networks. The topological networks may be embedded onto the two dimensional regular lattice or the two dimensional fractal structure known as the Sierpinski carpet. In introducing geometry to networks, the mapping between topology and geometry is the most crucial step. This is done as node assignment and link reconstruction. I suggest two mechanisms, random and the Coulomb-like repulsive, to assign topological nodes to the geometrical lattice points. I adopt the minimal-length link reconstruction scheme from [7]. Finally I have investigated the error and attack tolerance of the scalefreen network on geometry. The network breaks down completely within 22

30 our number range of attacks, that is, i < 0.1. I could identify the three stages of network damage: the dense state, the critical breakdown and the scattered state. In both regular and fractal geometries, performing attacks by the updated order destroyed the network faster. Also it turned out that the earlier result of topological network could be considered as the limiting case (i 0.1) of this: the dense state. No significant difference was found between the failure tolerances of the regular lattice and the fractal geometry case. 23

31 Bibliography [1] Albert, R., Jeong, H. & Barabasi, A. L., Error and attack tolerance of complex networks, Nature 406, (2000) [2] Kahng, B. et al., Complex Networks: Structure and Dynamics, Sae Mulli (The Korean Physical Society) 48, (2004) [3] Barabasi, A. L., Albert, R. & Jeong H., Mean-field theory for scale-free random networks, Physica A 272, (1999) [4] Goh, K.-I., Kahng, B. & Kim, D., Universal Behavior of Load Distribution in Scale-Free Networks, Physical Review Letters 87, (2001) [5] Albert, R., Jeong, H. & Barabasi, A. L., Diameter of the World-Wide Web Nature 401, (1999) [6] Sierpiński, W., Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donne, Comptes Rendus 162, (1916) [7] Rozenfeld et al., Scale-free networks on Latticies, Physical Review Letters 89, (2002) 24