Graph measures and network robustness

Network robustness research aims at finding a measure to quantify network robustness. Once such a measure has been established, we will be able to compare networks, to improve existing networks and to design new networks that are able to continue to perform well when it is subject to failures or attacks. In this paper we survey a large amount of robustness measures on simple, undirected and unweighted graphs, in order to offer a tool for network administrators to evaluate and improve the robustness of their network. The measures discussed in this paper are based on the concepts of connectivity (including reliability polynomials), distance, betweenness and clustering. Some other measures are notions from spectral graph theory, more precisely, they are functions of the Laplacian eigenvalues. In addition to surveying these graph measures, the paper also contains a discussion of their functionality as a measure for topological network robustness

A measure of centrality based on the spectrum of the Laplacian

We introduce a family of new centralities, the k-spectral centralities. k-Spectral centrality is a measurement of importance with respect to the deformation of the graph Laplacian associated with the graph. Due to this connection, k-spectral centralities have various interpretations in terms of spectrally determined information. We explore this centrality in the context of several examples. While for sparse unweighted networks 1-spectral centrality behaves similarly to other standard centralities, for dense weighted networks they show different properties. In summary, the k-spectral centralities provide a novel and useful measurement of relevance (for single network elements as well as whole subnetworks) distinct from other known measures.Comment: 12 pages, 6 figures, 2 table

A Topological Investigation of Phase Transitions of Cascading Failures in Power Grids

Cascading failures are one of the main reasons for blackouts in electric power transmission grids. The economic cost of such failures is in the order of tens of billion dollars annually. The loading level of power system is a key aspect to determine the amount of the damage caused by cascading failures. Existing studies show that the blackout size exhibits phase transitions as the loading level increases. This paper investigates the impact of the topology of a power grid on phase transitions in its robustness. Three spectral graph metrics are considered: spectral radius, effective graph resistance and algebraic connectivity. Experimental results from a model of cascading failures in power grids on the IEEE power systems demonstrate the applicability of these metrics to design/optimize a power grid topology for an enhanced phase transition behavior of the system

A Bag-of-Paths Node Criticality Measure

This work compares several node (and network) criticality measures quantifying to which extend each node is critical with respect to the communication flow between nodes of the network, and introduces a new measure based on the Bag-of-Paths (BoP) framework. Network disconnection simulation experiments show that the new BoP measure outperforms all the other measures on a sample of Erdos-Renyi and Albert-Barabasi graphs. Furthermore, a faster (still O(n^3)), approximate, BoP criticality relying on the Sherman-Morrison rank-one update of a matrix is introduced for tackling larger networks. This approximate measure shows similar performances as the original, exact, one

Synchronization in Networks of Identical Systems via Pinning: Application to Distributed Secondary Control of Microgrids

Motivated by the need for fast synchronized operation of power microgrids, we analyze the problem of single and multiple pinning in networked systems. We derive lower and upper bounds on the algebraic connectivity of the network with respect to the reference signal. These bounds are utilized to devise a suboptimal algorithm with polynomial complexity to find a suitable set of nodes to pin the network effectively and efficiently. The results are applied to secondary voltage pinning control design for a microgrid in islanded operation mode. Comparisons with existing single and multiple pinning strategies clearly demonstrates the efficacy of the obtained results.Comment: 11 pages, 9 figures, submitted to Transactions on Control Systems Technolog

Using network centrality measures to manage landscape connectivity

We use a graph-theoretical landscape modeling approach to investigate how to identify central patches in the landscape as well as how these central patches influence (1) organism movement within the local neighborhood, and (2) the dispersal of organisms beyond the local neighborhood. Organism movements were theoretically estimated based on the spatial configuration of the habitat patches in the studied landscape. We find that centrality depends on the way the graph-theoretical model of habitat patches is constructed, although even the simplest network representation, not taking strength and directionality of potential organisms flows into account, still provides a coarse-grained assessment of the most important patches according to their contribution to landscape connectivity. Moreover, we identify (at least) two general classes of centrality. One accounts for the local flow of organisms in the neighborhood of a patch and the other for the ability to maintain connectivity beyond the scale of the local neighborhood. Finally, we study how habitat patches with high scores on different network centrality measures are distributed in a fragmented agricultural landscape in Madagascar. Results show that patches with high degree-, and betweenness centrality are widely spread, while patches with high subgraph- and closeness centrality are clumped together in dense clusters. This finding may enable multi-species analyses of single-species network models

A network approach for power grid robustness against cascading failures

Cascading failures are one of the main reasons for blackouts in electrical power grids. Stable power supply requires a robust design of the power grid topology. Currently, the impact of the grid structure on the grid robustness is mainly assessed by purely topological metrics, that fail to capture the fundamental properties of the electrical power grids such as power flow allocation according to Kirchhoff's laws. This paper deploys the effective graph resistance as a metric to relate the topology of a grid to its robustness against cascading failures. Specifically, the effective graph resistance is deployed as a metric for network expansions (by means of transmission line additions) of an existing power grid. Four strategies based on network properties are investigated to optimize the effective graph resistance, accordingly to improve the robustness, of a given power grid at a low computational complexity. Experimental results suggest the existence of Braess's paradox in power grids: bringing an additional line into the system occasionally results in decrease of the grid robustness. This paper further investigates the impact of the topology on the Braess's paradox, and identifies specific sub-structures whose existence results in Braess's paradox. Careful assessment of the design and expansion choices of grid topologies incorporating the insights provided by this paper optimizes the robustness of a power grid, while avoiding the Braess's paradox in the system.Comment: 7 pages, 13 figures conferenc