Recent years have seen the world become a closely connected society with the emergence of different types of social networks. Online social networks have provided a way to bridge long distances and establish numerous communication channels which were not possible earlier. These networks exhibit interesting behavior under intentional attacks and random failures where different structural properties influence the resilience in different ways. In this paper, we perform two sets of experiments and draw conclusions from the results pertaining to the resilience of social networks. The first experiment performs a comparative analysis of four different classes of networks namely small world networks, scale free networks, small world-scale free networks and random networks with four semantically different social networks under different attack strategies. The second experiment compares the resilience of these semantically different social networks under different attack strategies. Empirical analysis reveals interesting behavior of different classes of networks with different attack strategies.
One of the earliest studies to demonstrate that scale free networks are more robust against random failures was conducted by @cite_6 . The authors also discuss the vulnerability of scale free networks to targeted attacks. Cohen @cite_8 @cite_16 study the resilience of internet under random and targeted attacks on nodes. For the case of random attacks, they conclude that even after $100 Focusing on the random failure of nodes and edges, although previous researchers had predicted completely different behavior for Poisson and power law networks, in practice the differences, are vital but not huge. Our results re-enforce these results specially for the case of social networks. The authors also invalidate the explanation that targeted attacks are very efficient on power-law networks because they remove many links, random removal of as many links also result in breakdown of the network. Networks with Poisson degree distribution behave similarly in case of random node failures and targeted attacks, it must be noted that their threshold is significantly lower in the second case. This goes against the often claimed assumption that, because all nodes have almost the same degree in a Poisson network, there is little difference between random node failures and targeted attacks.
@cite_16 @cite_6 @cite_8
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It has appeared recently that the underlying degree distribution of networks may play a crucial role concerning their robustness. Previous work insisted on the fact that power-law degree distributions induce high resilience to random failures but high sensitivity to attack strategies, while Poisson degree distributions are quite sensitive in both cases. Then much work has been done to extend these results. We aim here at studying in depth these results, their origin, and limitations. We review in detail previous contributions in a unified framework, and identify the approximations on which these results rely. We then present new results aimed at clarifying some important aspects. We also provide extensive rigorous experiments which help evaluate the relevance of the analytic results. We reach the conclusion that, even if the basic results are clearly important, they are in practice much less striking than generally thought. The differences between random failures and attacks are not so huge and can be explained with simple facts. Likewise, the differences in the behaviors induced by power-law and Poisson distributions are not as striking as often claimed. The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology. We study the stability of random scale-free networks to degree-dependent attacks. We present analytical and numerical results to compute the critical fraction @math of nodes that need to be removed for destroying the network under this attack for different attack parameters. We study the effect of different defense strategies, based on the addition of a constant number of links on network robustness. We test defense strategies based on adding links to either low degree, middegree or high degree nodes. We find using analytical results and simulations that the middegree nodes defense strategy leads to the largest improvement to the network robustness against degree-based attacks. We also test these defense strategies on an internet autonomous systems map and obtain similar results. The spread of epidemics and malware is commonly modeled by diffusion processes on networks. Protective interventions such as vaccinations or installing anti-virus software are used to contain their spread. Typically, each node in the network has to decide its own strategy of securing itself, and its benefit depends on which other nodes are secure, making this a natural game-theoretic setting. There has been a lot of work on network security game models, but most of the focus has been either on simplified epidemic models or homogeneous network structure. We develop a new formulation for an epidemic containment game, which relies on the characterization of the SIS model in terms of the spectral radius of the network. We show in this model that pure Nash equilibria (NE) always exist, and can be found by a best response strategy. We analyze the complexity of finding NE, and derive rigorous bounds on their costs and the Price of Anarchy or PoA (the ratio of the cost of the worst NE to the optimum social cost) in general graphs as well as in random graph models. In particular, for arbitrary power-law graphs with exponent β > 2, we show that the PoA is bounded by O(T2(β -1))), where T = γ α is the ratio of the recovery rate to the transmission rate in the SIS model. We prove that this bound is tight up to a constant factor for the Chung-Lu random power-law graph model. We study the characteristics of Nash equilibria empirically in different real communication and infrastructure networks, and find that our analytical results can help explain some of the empirical observations.