In this article we present the Hayastan Shakarian (HS), a robustness index for complex networks. HS measures the impact of a network disconnection (edge) while comparing the sizes of the remaining connected components. Strictly speaking, the Hayastan Shakarian index is defined as edge removal that produces the maximal inverse of the size of the largest connected component divided by the sum of the sizes of the remaining ones. We tested our index in attack strategies where the nodes are disconnected in decreasing order of a specified metric. We considered using the Hayastan Shakarian cut (disconnecting the edge with max HS) and other well-known strategies as the higher betweenness centrality disconnection. All strategies were compared regarding the behavior of the robustness (R-index) during the attacks. In an attempt to simulate the internet backbone, the attacks were performed in complex networks with power-law degree distributions (scale-free networks). Preliminary results show that attacks based on disconnecting using the Hayastan Shakarian cut are more dangerous (decreasing the robustness) than the same attacks based on other centrality measures. We believe that the Hayastan Shakarian cut, as well as other measures based on the size of the largest connected component, provides a good addition to other robustness metrics for complex networks.
Another interesting edge-based metrics are: the increase of distances in a network caused by the deletion of vertices and edges @cite_14 ; the mean connectivity @cite_12 , that is, the probability that a network is disconnected by the random deletion of edges; the average connected distance (the average length of the shortest paths between connected pairs of nodes in the network) and the fragmentation (the decay of a network in terms of the size of its connected components after random edge disconnections) @cite_20 ; the balance-cut resilience, that is, the capacity of a minimum cut such that the two resulting vertex sets contain approximately the same number (similar to the HS-index but aiming to divide the network only in halves @cite_4 , not in equal sized connected components); the effective diameter ( '' @cite_9 ); and the Dynamic Network Robustness (DYNER) @cite_1 , where a backup path between nodes after a node disconnection and its length is used as a robustness metric in communication networks.
@cite_14 @cite_4 @cite_9 @cite_1 @cite_12 @cite_20
2768318902 1967385589 2067935583 2100256240
Abstract As network analysis methods prevail, more metrics are applied to co-word networks to reveal hot topics in a field. However, few studies have examined the relationships among these metrics. To bridge this gap, this study explores the relationships among different ranking metrics, including one frequency-based and six network-based metrics, in order to understand the impact of network structural features on ranking themes on co-word networks. We collected bibliographic data from three disciplines from Web of Science (WoS), and generated 40 simulation networks following the preferential attachment assumption. Correlation analysis on the empirical and simulated networks shows strong relationships among the metrics. Their relationships are consistent across disciplines. The metrics can be categorized into three groups according to the strength of their correlations, where Degree Centrality, H-index, and Coreness are in one group, Betweenness Centrality, Clustering Coefficient, and frequency in another, and Weighted PageRank by itself. Regression analysis on the simulation networks reveals that network topology properties, such as connectivity, sparsity, and aggregation, influence the relationships among selected metrics. In addition, when comparing the top keywords ranked by the metrics in the three disciplines, we found the metrics exhibit different discriminative capacity. Coreness and H-index may be better suited for categorizing keywords rather than ranking keywords. Findings from this study contribute to a better understanding of the relationships among different metrics and provide guidance for using them effectively in different contexts. 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. In practical military or first responder deployment scenarios, information flows need to adhere to specified policies regardless of the physical connectivity of nodes. Nodes in such networks are associated with various levels in a command-and-control hierarchy, and therefore typically form a logical hierarchical tree network that is used to route both command and data traffic. Associated with this logical hierarchical network is a communication network that represents the connectivity of these nodes in the deployed scenario. Such composite networks introduce constraints that can result in information flows having to traverse much longer paths in the underlying communication network. In this paper, we look at the problem of adding edges to a logical hierarchical network (or any other social network) so as to minimize the number of hops required to route data traffic in the underlying communication network from a node to other specified nodes. The edges added are a subset of all possible edges in the complementary logical hierarchical graph and have to satisfy specified hierarchical constraints. First, we consider the general problem of minimizing the eccentricity of a source node 's' (where eccentricity of 's' is the maximum of the shortest paths from 's' to all other nodes) in a metric graph on adding upto 'B' unequal cost metric edges from the set of all edges in the complementary graph. We develop an efficient constant factor approximation algorithm for this case that outperforms existing constant factor algorithms for eccentricity minimization. Here the added edge metric cost as well as the graph edge metric cost correspond to the number of hops in the shortest path required to route traffic in the actual deployed topology (i.e., underlying communication network). Next, we consider the case where the set of possible added edges is a specified subset of the edges in the complementary graph and the set of destinations is a subset of the graph nodes. For this case, we develop heuristic algorithms based on the previous eccentricity minimizing algorithm that show good performance. We validate our algorithms using two realistic military deployment scenarios. We find that adding even a low number of hierarchically constrained edges (of the order of 10) can cause a significant decrease (around 50 ) in the eccentricity of a node in the logical hierarchical network and thus can reduce the number of hops required for data traffic traversal. Researchers have proposed a variety of metrics to measure important graph properties, for instance, in social, biological, and computer networks. Values for a particular graph metric may capture a graph's resilience to failure or its routing efficiency. Knowledge of appropriate metric values may influence the engineering of future topologies, repair strategies in the face of failure, and understanding of fundamental properties of existing networks. Unfortunately, there are typically no algorithms to generate graphs matching one or more proposed metrics and there is little understanding of the relationships among individual metrics or their applicability to different settings. We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dK-series implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d=0, 1, 2, 3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d=2 case is sufficient for most practical purposes, while d=3 essentially reconstructs the Internet AS-and router-level topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.