Global Robustness vs. Local Vulnerabilities in Complex Synchronous Networks

In complex network-coupled dynamical systems, two questions of central importance are how to identify the most vulnerable components and how to devise a network making the overall system more robust to external perturbations. To address these two questions, we investigate the response of complex networks of coupled oscillators to local perturbations. We quantify the magnitude of the resulting excursion away from the unperturbed synchronous state through quadratic performance measures in the angle or frequency deviations. We find that the most fragile oscillators in a given network are identified by centralities constructed from network resistance distances. Further defining the global robustness of the system from the average response over ensembles of homogeneously distributed perturbations, we find that it is given by a family of topological indices known as generalized Kirchhoff indices. Both resistance centralities and Kirchhoff indices are obtained from a spectral decomposition of the stability matrix of the unperturbed dynamics and can be expressed in terms of resistance distances. We investigate the properties of these topological indices in small-world and regular networks. In the case of oscillators with homogeneous inertia and damping coefficients, we find that inertia only has small effects on robustness of coupled oscillators. Numerical results illustrate the validity of the theory.Comment: 11 pages, 9 figure

The Walk Distances in Graphs

The walk distances in graphs are defined as the result of appropriate transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic; moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. We also show that the logarithmic forest distances which are known to generalize the resistance distance and the shortest path distance are a subclass of walk distances. On the other hand, the long walk distance is equal to the resistance distance in a transformed graph.Comment: Accepted for publication in Discrete Applied Mathematics. 26 pages, 3 figure

Robustness of Synchrony in Complex Networks and Generalized Kirchhoff Indices

In network theory, a question of prime importance is how to assess network vulnerability in a fast and reliable manner. With this issue in mind, we investigate the response to parameter changes of coupled dynamical systems on complex networks. We find that for specific, non-averaged perturbations, the response of synchronous states critically depends on the overlap between the perturbation vector and the eigenmodes of the stability matrix of the unperturbed dynamics. Once averaged over properly defined ensembles of such perturbations, the response is given by new graph topological indices, which we introduce as generalized Kirchhoff indices. These findings allow for a fast and reliable method for assessing the specific or average vulnerability of a network against changing operational conditions, faults or external attacks.Comment: 5 pages + supplemental material Fina

Adjustable reach in a network centrality based on current flows

Centrality, which quantifies the "importance" of individual nodes, is among the most essential concepts in modern network theory. Most prominent centrality measures can be expressed as an aggregation of influence flows between pairs of nodes. As there are many ways in which influence can be defined, many different centrality measures are in use. Parametrized centralities allow further flexibility and utility by tuning the centrality calculation to the regime most appropriate for a given network. Here, we identify two categories of centrality parameters. Reach parameters control the attenuation of influence flows between distant nodes. Grasp parameters control the centrality's potential to send influence flows along multiple, often nongeodesic paths. Combining these categories with Borgatti's centrality types [S. P. Borgatti, Social Networks 27, 55-71 (2005)], we arrive at a novel classification system for parametrized centralities. Using this classification, we identify the notable absence of any centrality measures that are radial, reach parametrized, and based on acyclic, conservative flows of influence. We therefore introduce the ground-current centrality, which is a measure of precisely this type. Because of its unique position in the taxonomy, the ground-current centrality has significant advantages over similar centralities. We demonstrate that, compared to other conserved-flow centralities, it has a simpler mathematical description. Compared to other reach centralities, it robustly preserves an intuitive rank ordering across a wide range of network architectures. We also show that it produces a consistent distribution of centrality values among the nodes, neither trivially equally spread (delocalization), nor overly focused on a few nodes (localization). Other reach centralities exhibit both of these behaviors on regular networks and hub networks, respectively

Weighted Spectral Embedding of Graphs

We present a novel spectral embedding of graphs that incorporates weights assigned to the nodes, quantifying their relative importance. This spectral embedding is based on the first eigenvectors of some properly normalized version of the Laplacian. We prove that these eigenvectors correspond to the configurations of lowest energy of an equivalent physical system, either mechanical or electrical, in which the weight of each node can be interpreted as its mass or its capacitance, respectively. Experiments on a real dataset illustrate the impact of weighting on the embedding