We propose an algorithm to locate the most critical nodes to network robustness. Such critical nodes may be thought of as those most related to the notion of network centrality. Our proposal relies only on a localized spectral analysis of a limited subnetwork centered at each node in the network. We also present a procedure allowing the navigation from any node towards a critical node following only local information computed by the proposed algorithm. Experimental results confirm the effectiveness of our proposal considering networks of different scales and topological characteristics.
Recent work on the Internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes or links. Such deletions include, for example, the failure of Internet routers or power transmission lines. Percolation models on random graphs provide a simple representation of this process but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real-world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience. Assessing network vulnerability before potential disruptive events such as natural disasters or malicious attacks is vital for network planning and risk management. It enables us to seek and safeguard against most destructive scenarios in which the overall network connectivity falls dramatically. Existing vulnerability assessments mainly focus on investigating the inhomogeneous properties of graph elements, node degree for example, however, these measures and the corresponding heuristic solutions can provide neither an accurate evaluation over general network topologies, nor performance guarantees to large scale networks. To this end, in this paper, we investigate a measure called pairwise connectivity and formulate this vulnerability assessment problem as a new graph-theoretical optimization problem called &#946;-disruptor, which aims to discover the set of critical node edges, whose removal results in the maximum decline of the global pairwise connectivity. Our results consist of the NP-Completeness and inapproximability proof of this problem, an O(log n loglog n) pseudo-approximation algorithm for detecting the set of critical nodes and an O(log^1.5 n) pseudo-approximation algorithm for detecting the set of critical edges. In addition, we devise an efficient heuristic algorithm and validate the performance of the our model and algorithms through extensive simulations. We consider the issue of protection in very large networks displaying randomness in topology. We employ random graph models to describe such networks, and obtain probabilistic bounds on several parameters related to reliability. In particular, we take the case of random regular networks for simplicity and consider the length of primary and backup paths in terms of the number of hops. First, for a randomly picked pair of nodes, we derive a lower bound on the average distance between the pair and discuss the tightness of the bound. In addition, noting that primary and protection paths form cycles, we obtain a lower bound on the average length of the shortest cycle around the pair. Finally, we show that the protected connections of a given maximum finite length are rare. We then generalize our network model so that different degrees are allowed according to some arbitrary distribution, and show that the second moment of degree over the first moment is an important shorthand for behavior of a network. Notably, we show that most of the results in regular networks carry over with minor modifications, which significantly broadens the scope of networks to which our approach applies. We present as an example the case of networks with a power-law degree distribution. Many complex systems display a surprising degree of tolerance against errors. For example, relatively simple organisms grow, persist and reproduce despite drastic pharmaceutical or environmental interventions, an error tolerance attributed to the robustness of the underlying metabolic network1. Complex communication networks2 display a surprising degree of robustness: although key components regularly malfunction, local failures rarely lead to the loss of the global information-carrying ability of the network. The stability of these and other complex systems is often attributed to the redundant wiring of the functional web defined by the systems' components. Here we demonstrate that error tolerance is not shared by all redundant systems: it is displayed only by a class of inhomogeneously wired networks, called scale-free networks, which include the World-Wide Web3,4,5, the Internet6, social networks7 and cells8. We find that such networks display an unexpected degree of robustness, the ability of their nodes to communicate being unaffected even by unrealistically high failure rates. However, error tolerance comes at a high price in that these networks are extremely vulnerable to attacks (that is, to the selection and removal of a few nodes that play a vital role in maintaining the network's connectivity). Such error tolerance and attack vulnerability are generic properties of communication networks. Centrality is a concept often used in social network analysis to study different properties of networks that are modeled as graphs. We present a new centrality metric called localized bridging centrality (LBC). LBC is based on the bridging centrality (BC) metric that recently introduced. Bridging nodes are nodes that are strategically located in between highly connected regions. LBC is capable of identifying bridging nodes with an accuracy comparable to that of the BC metric for most networks. As the name suggests, we use only local information from surrounding nodes to compute the LBC metric, whereas, global knowledge is required to calculate the BC metric. The main difference between LBC and BC is that LBC uses the egocentric definition of betweenness centrality to identify bridging nodes, while BC uses the sociocentric definition of betweenness centrality. Thus, our LBC metric is suitable for distributed or parallel computation and has the benefit of being an order of magnitude faster to calculate in computational complexity. We compare the results produced by BC and LBC in three examples. We applied our LBC metric for network analysis of a real wireless mesh network. Our results indicate that the LBC metric is as powerful as the BC metric at identifying bridging nodes. The LBC metric is thus an important tool that can help network administrators identify critical nodes that are important for the robustness of the network in a distributed manner. A complex network can be modeled as a graph representing the ''who knows who'' relationship. In the context of graph theory for social networks, the notion of centrality is used to assess the relative importance of nodes in a given network topology. For example, in a network composed of large dense clusters connected through only a few links, the nodes involved in those links are particularly critical as far as the network survivability is concerned. This may also impact any application running on top of it. Such information can be exploited for various topological maintenance issues to prevent congestion and disruption. This can also be used offline to identify the most important actors in large social interaction graphs. Several forms of centrality have been proposed so far. Yet, they suffer from imperfections: initially designed for small social graphs, they are either of limited use (degree centrality), either incompatible in a distributed setting (e.g. random walk betweenness centrality). In this paper we introduce a novel form of centrality: the second order centrality which can be computed in a distributed manner. This provides locally each node with a value reflecting its relative criticity and relies on a random walk visiting the network in an unbiased fashion. To this end, each node records the time elapsed between visits of that random walk (called return time in the sequel) and computes the standard deviation (or second order moment) of such return times. The key point is that central nodes see regularly the random walk compared to other topology nodes. Both through theoretical analysis and simulation, we show that the standard deviation can be used to accurately identify critical nodes as well as to globally characterize graphs topology in a distributed way. We finally compare our proposal to well-known centralities to assess its competitivity.