The sensitivity of networks to the removal of vertices has been studied extensively over the last 15 years. A common approach to measuring this sensitivity is (i) successively removing vertices following a specific removal strategy and (ii) comparing the original and the modified network using a specific comparison method. In this paper we apply a wide range of removal strategies and comparison methods in order to study the sensitivity of medium-sized networks from the real world and randomly generated networks. In the first part of our study we observe that social networks and web graphs differ in sensitivity. When removing vertices, social networks are robust, web graphs are not. This effect is consistent with the work of who analyzed very large social networks and web graphs. For randomly generated networks we find that their sensitivity depends significantly on the comparison method. The choice of removal strategy has surprisingly marginal impact on the sensitivity for removal strategies derived from common centrality measures. However, the removal strategy has a strong impact when removing the vertices in random order.
@cite_26: Complex networks can model the structure and dynamics of different types of systems. It has been shown that they are characterized by a set of measures. In this work, we evaluate the variability of complex network measures face to perturbations and, for this purpose, we impose controlled perturbations and quantify their effect. We analyze theoretical models (random, small-world and scale-free) and real networks (a collaboration network and a metabolic networks) along with the shortest path length, vertex degree, local cluster coefficient and betweenness centrality measures.
@cite_7: We study the response of complex networks subject to attacks on vertices and edges. Several existing complex network models as well as real-world networks of scientific collaborations and Internet traffic are numerically investigated, and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph. For each case of attacks on vertices and edges, four different attacking strategies are used: removals by the descending order of the degree and the betweenness centrality, calculated for either the initial network or the current network during the removal procedure. It is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network, suggesting that the network structure changes as important vertices or edges are removed. Furthermore, the correlation between the betweenness centrality and the degree in complex networks is studied.
@cite_12: 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.
@cite_6: Given a social network, which of its nodes have a stronger impact in determining its structure? More precisely, which node-removal order has the greatest impact on the network structure? We approach this well-known problem for the first time in a setting that combines both web graphs and social networks. Our experiments are performed on datasets that are of orders of magnitude larger than those appearing in the previous literature: this is possible, thanks to some recently developed algorithms and software tools that approximate accurately the number of reachable pairs and the distribution of distances in large graphs. Our experiments highlight deep differences in the structure of social networks and web graphs, show significant limitations of previous experimental results; at the same time, they reveal clustering by label propagation as a new and very effective way of locating nodes that are important from a structural viewpoint.
Albert and Barabási @cite_12 examined the error and attack tolerance of random graphs with respect to node removal. Based on centrality measures, @cite_7 removed nodes and edges from real-world networks and random graphs to investigate the attack vulnerability regarding the average inverse of the geodesic length and the size of the largest connected subgraph. The behavior of very large social networks and web graphs with respect to vertex removal based on various removal strategies and comparison methods based on the neighborhood function has been studied by @cite_6 . @cite_26 measured the impact of random errors to real-world networks and generated graphs by means of stochastic quantifiers.