Enrico Bozzo

Resistance distance, closeness, and betweenness

Abstract In a seminal paper Stephenson and Zelen (1989) rethought centrality in networks proposing an information-theoretic distance measure among nodes in a network. The suggested information distance diverges from the classical geodesic metric since it is sensible to all paths (not just to the shortest ones) and it diminishes as soon as there are more routes between a pair of nodes. Interestingly, information distance has a clear interpretation in electrical network theory that was missed by the proposing authors. When a fixed resistor is imagined on each edge of the graph, information distance, known as resistance distance in this context, corresponds to the effective resistance between two nodes when a battery is connected across them. Here, we review resistance distance, showing once again, with a simple proof, that it matches information distance. Hence, we interpret both current-flow closeness and current-flow betweenness centrality in terms of resistance distance. We show that this interpretation has semantic, theoretical, and computational benefits.

Approximations of the Generalized Inverse of the Graph Laplacian Matrix

We devise methods for finding approximations of the generalized inverse of the graph Laplacian matrix, which arises in many graph-theoretic applications. Finding this matrix in its entirety involves solving a matrix inversion problem, which is resource-demanding in terms of consumed time and memory and hence impractical whenever the graph is relatively large. Our approximations use only a few eigenpairs of the Laplacian matrix and are parametric with respect to this number, so that the user can compromise between effectiveness and efficiency of the approximate solution. We apply the devised approximations to the problem of computing current-flow betweenness centrality on a graph. However, given the generality of the Laplacian matrix, many other applications can be sought. We experimentally demonstrate that the approximations are effective already with a constant number of eigenpairs. These few eigenpairs can be stored with a linear amount of memory in the number of nodes of the graph, and in the realistic case of sparse networks, they can be efficiently computed using one of the many methods for retrieving a few eigenpairs of sparse matrices that abound in the literature.

Vulnerability and power on networks

Inspired by socio-political scenarios, like dictatorships, in which a minority of people exercise control over a majority of weakly interconnected individuals, we propose vulnerability and power measures defined on groups of actors of networks. We establish an unexpected connection between network vulnerability and graph regularizability. We use the Shapley value of coalition games to introduce fresh notions of vulnerability and power at node level defined in terms of the corresponding measures at group level. We investigate the computational complexity of computing the defined measures, both at group and node levels, and provide effective methods to quantify them. Finally we test vulnerability and power on both artificial and real networks.

A theory on power in networks

Actors linked to central others in networks are generally central, even as actors linked to powerful others are powerless.

Rate Switching Filters: Model and Efficient Approximation

A filter model is proposed, allowing for the realization of a digital structure that computes a decimated version of the output signal. Each time the sampling rate is switched, pre-calculated coefficients are loaded by the processor in parallel to computing a filter state that fits the new rate. Sufficient conditions for the existence of the new state are given: holding these conditions, the sampling rate can be varied at runtime without introducing spurious transients in the output signal. The equivalence between the proposed filter model and existing polyphase networks for the efficient computation of decimated signals is discussed. If the input is null, the rate-switching structure performs a fraction of the computations that equals the decimation factor. Otherwise, the same efficiency can be achieved by linearly interpolating in between decimated input values, at the cost of introducing an error in the output signal. Particularly in the second-order case, an efficient rate-switching structure can be figured out capable of producing an error-free output also in presence of an input which is not null.

Decimation in Time and Space of Finite-Difference Time-Domain Schemes: Standard Isotropic Lossless Model

Finite-difference time-domain (FDTD) schemes permit changes in the grid density on selected regions of the wave propagation domain, which can reduce the computational load of the simulations. One possible alternative to varying the spatial density is to change the simulation temporal rate. This idea looks attractive when the wave signals exhibit pronounced bandwidth fluctuations across time. This is particularly true in sound synthesis, where a physically based acoustic resonator can be conveniently modeled using such schemes. To overcome the computational constraints that must be met by real-time distributed resonator models, this paper deals with the decimation in time and space of isotropic lossless finite-difference time-domain schemes holding conventional Nyquist-Shannon limits on the bandwidth of the wave signals. Formulas for the reconstruction of these signals at runtime over the interpolated grid are provided for both the 1D and 2D orthogonal case, depending on the ideal boundary conditions (either Neumann or Dirichlet) holding at each side of the grid in connection with the domain side lengths (either even or odd). Together, the boundaries and size determine the type of Discrete Cosine Transform used in the corresponding interpolation formula. Numerical artifacts arising as a consequence of decimating in space in 2D are discussed in terms of dispersion error and aliasing. Considerations concerning the temporal reconstruction of components lying at the decimated Nyquist frequency are addressed in the conclusion.

Decimation of finite-difference time-domain schemes in 1D and 2D boundary-absorbing acoustic model simulations

A decimated version of a broad family of Finite-Difference Time-Domain schemes is derived by an algebraic rearrangement of their matrix formulation, allowing for computing the grid nodes at half the temporal steps compared to the original scheme. This rearrangement can ask for solving a generalized matrix inversion problem. The decimated scheme generates solutions having comparable accuracy to that exhibited by the original simulations. However, the broader applicability of the proposed technique requires to solve currently unanswered theoretical issues of spatial grid decimation, as well as to make extensive tests using large matrices.

A Priori Estimates on the Structured Conditioning of Cauchy and Vandermonde Matrices

We analyze the componentwise and normwise sensitivity of inverses of Cauchy, Vandermonde, and Cauchy-Vandermonde matrices, with respect to relative componentwise perturbations in the nodes defining these matrices. We obtain a priori, easily computable upper bounds for these condition numbers. In particular, we improve known estimates for Vandermonde matrices with generic real nodes; twe consider in detail Vandermonde matrices with nonnegative or symmetric nodes; and we extend the analysis to the class of complex Cauchy-Vandermonde matrices.

Arbitrarily regularizable graphs

A graph is regularizable if it is possible to assign weights to its edges so that all nodes have the same degree. Weights can be positive, nonnegative or arbitrary as soon as the regularization degree is not null. Positive and nonnegative regularizable graphs have been thoroughly investigated in the literature. In this work, we propose and study arbitrarily regularizable graphs. In particular, we investigate necessary and sufficient regularization conditions on the topology of the graph and of the corresponding adjacency matrix. Moreover, we study the computational complexity of the regularization problem and characterize it as a linear programming model.