Error and attack tolerance of complex networks

TL;DR

Problem addressed: why some complex networks remain functional despite local component failures and how error tolerance differs from attack tolerance. Approach: focus on scale-free, inhomogeneously wired networks to analyze robustness to failures and vulnerability to targeted removals. Findings: scale-free networks exhibit surprising robustness to high rates of random node failures, maintaining communication, but are extremely vulnerable to attacks that remove a few highly important nodes. Significance: reveals a trade-off between error tolerance and attack susceptibility in scale-free topologies, with implications for protecting and designing real-world networks such as the WWW, Internet, social networks, and cellular systems.

Abstract

Many complex systems, such as communication networks, display a surprising degree of robustness: while key components regularly malfunction, local failures rarely lead to the loss of the global information-carrying ability of the network. The stability of these complex systems is often attributed to the redundant wiring of the functional web defined by the systems' components. In this paper we demonstrate that error tolerance is not shared by all redundant systems, but it is displayed only by a class of inhomogeneously wired networks, called scale-free networks. We find that scale-free networks, describing a number of systems, such as the World Wide Web, Internet, social networks or a cell, display an unexpected degree of robustness, the ability of their nodes to communicate being unaffected by even unrealistically high failure rates. However, error tolerance comes at a high price: these networks are extremely vulnerable to attacks, i.e. to the selection and removal of a few nodes that play the most important role in assuring the network's connectivity.

Figures (4)

• Figure 1: Visual illustration of the difference between an exponential and a scale-free network. The exponential network a is rather homogeneous, i.e. most nodes have approximately the same number of links. In contrast, the scale-free network b is extremely inhomogeneous: while the majority of the nodes have one or two links, a few nodes have a large number of links, guaranteeing that the system is fully connected. We colored with red the five nodes with the highest number of links, and with green their first neighbors. While in the exponential network only $27\%$ of the nodes are reached by the five most connected nodes, in the scale-free network more than $60\%$ are, demonstrating the key role the connected nodes play in the scale-free network. Note that both networks contain $130$ nodes and $215$ links ($\langle k\rangle=3.3$). The network visualization was done using the Pajek program for large network analysis $<$http://vlado.fmf.uni-lj.si/pub/networks/pajek/pajekman.htm$>$.
• Figure 2: Changes in the diameter of the network as a function of the fraction of the removed nodes. a, Comparison between the exponential (E) and scale-free (SF) network models, each containing $N=10,000$ nodes and $20,000$ links (i.e. $\langle k\rangle=4$). The blue symbols correspond to the diameter of the exponential (triangles) and the scale-free (squares) network when a fraction $f$ of the nodes are removed randomly (error tolerance). Red symbols show the response of the exponential (diamonds) and the scale-free (circles) networks to attacks, when the most connected nodes are removed. We determined the $f$ dependence of the diameter for different system sizes ($N=1,000$, $5,000$, $20,000$) and found that the obtained curves, apart from a logarithmic size correction, overlap with those shown in a, indicating that the results are independent of the size of the system. Note that the diameter of the unperturbed ($f=0$) scale-free network is smaller than that of the exponential network, indicating that scale-free networks use more efficiently the links available to them, generating a more interconnected web. b, The changes in the diameter of the Internet under random failures (squares) or attacks (circles). We used the topological map of the Internet, containing $6,209$ nodes and $12,200$ links ($\langle k\rangle=3.4$), collected by the National Laboratory for Applied Network Research $<$http://moat.nlanr.net/Routing/rawdata/$>$. c, Error (squares) and attack (circles) survivability of the world-wide web, measured on a sample containing $325,729$ nodes and $1,498,353$ linksdiam, such that $\langle k\rangle=4.59$.
• Figure 3: Network fragmentation under random failures and attacks. The relative size of the largest cluster $S$ (open symbols) and the average size of the isolated clusters $\langle s \rangle$ (filled symbols) in function of the fraction of removed nodes $f$ for the same systems as in Fig.$\,$2. The size $S$ is defined as the fraction of nodes contained in the largest cluster (i.e. $S=1$ for $f=0$). a, Fragmentation of the exponential network under random failures (squares) and attacks (circles). b, Fragmentation of the scale-free network under random failures (blue squares) and attacks (red circles). The inset shows the error tolerance curves for the whole range of $f$, indicating that the main cluster falls apart only after it has been completely deflated. Note that the behavior of the scale-free network under errors is consistent with an extremely delayed percolation transition: at unrealistically high error rates ($f_{max}\simeq 0.75$) we do observe a very small peak in $\langle s\rangle$ ($\langle s_{max}\rangle\simeq 1.06$) even in the case of random failures, indicating the existence of a critical point. For a and b we repeated the analysis for systems of sizes $N=1,000$, $5,000$, and $20,000$, finding that the obtained $S$ and $\langle s\rangle$ curves overlap with the one shown here, indicating that the overall clustering scenario and the value of the critical point is independent of the size of the system. Fragmentation of the Internet ( c) and www ( d), using the topological data described in Fig.$\,$2. The symbols are the same as in b. Note that $\langle s\rangle$ in d in the case of attack is shown on a different scale, drawn in the right side of the frame. While for small $f$ we have $\langle s\rangle\simeq 1.5$, at $f_c^w=0.067$ the average fragment size abruptly increases, peaking at $\langle s_{max}\rangle\simeq 60$, then decays rapidly. For the attack curve in d we ordered the nodes in function of the number of outgoing links, $k_{out}$. Note that while the three studied networks, the scale-free model, the Internet and the www have different $\gamma$, $\langle k\rangle$ and clustering coefficientsmall_world, their response to attacks and errors is identical. Indeed, we find that the difference between these quantities changes only $f_c$ and the magnitude of $d$, $S$ and $\langle s\rangle$, but not the nature of the response of these networks to perturbations.
• Figure 4: Summary of the response of a network to failures or attacks. The insets show the cluster size distribution for various values of $f$ when a scale-free network of parameters given in Fig.$\,$3b is subject to random failures ( a- c) or attacks ( d- f). Upper panel: Exponential networks under random failures and attacks and scale-free networks under attacks behave similarly: for small $f$ clusters of different sizes break down, while there is still a large cluster. This is supported by the cluster size distribution: while we see a few fragments of sizes between $1$ and $16$, there is a large cluster of size $9,000$ (the size of the original system being $10,000$). At a critical $f_c$ (see Fig.$\,$3) the network breaks into small fragments between sizes $1$ and $100$ ( b) and the large cluster disappears. At even higher $f$ ( c) the clusters are further fragmented into single nodes or clusters of size two. Lower panel: Scale-free networks follow a different scenario under random failures: The size of the largest cluster decreases slowly as first single nodes, then small clusters break off. Indeed, at $f=0.05$ only single and double nodes break off ( d). At $f=0.18$, when under attack the network is fragmented ( b), under failures the large cluster of size $8,000$ coexists with isolated clusters of size $1$ through $5$ ( e). Even for unrealistically high error rate of $f=0.45$ the large cluster persists, the size of the broken-off fragments not exceeding $11$ ( f).