The Deformed Graph Laplacian and Its Applications to Network Centrality Analysis

Abstract

We introduce and study a new network centrality measure based on the concept of nonbacktracking walks; that is, walks not containing subsequences of the form uvu where u and v are any distinct connected vertices of the underlying graph. We argue that this feature can yield more meaningful rankings than traditional walk-based centrality measures. We show that the resulting Katz-style centrality measure may be computed via the so-called deformed graph Laplacian-a quadratic matrix polynomial that can be associated with any graph. By proving a range of new results about this matrix polynomial, we gain insights into the behavior of the algorithm with respect to its Katz-like parameter. The results also inform implementation issues. In particular we show that, in an appropriate limit, the new measure coincides with the nonbacktracking version of eigenvector centrality introduced by Martin, Zhang and Newman in 2014. Rigorous analysis on star and star-like networks illustrates the benefits of the new approach, and further results are given on real networks.

Non-backtracking walk centrality for directed networks

“…A more accessible treatment from the perspective of graph theory and linear algebra may be found in [33]. We also note that in the case of an undirected network, the recurrence collapses to a two-term expression that was discovered independently in the theory of zeta functions of graphs [32] and exploited in [14] from a network science perspective. Proof.…”

“…Moreover, quantities defined using the concept of nonbacktracking walks have been proved to be useful in the framework of undirected networks when tackling decycling problems, as well as dismantling problems and the problem of optimal percolation (see, e.g., [25,26] and references therein). A nonbacktracking analogue of eigenvector centrality was developed in [24] for undirected networks, and a Katz version was proposed in [14] and studied from a matrix polynomial perspective. However, none of those references handle directed edges.…”

“…For example, in Figure 1, 717171 has the same influence on the centrality measure as 712756, even though the latter, nonbacktracking, walk, traverses the network more widely and is likely to be more relevant in terms of message passing or disease spreading. This type of argument motivated the authors in [14] to propose and study a nonbacktracking analogue of Katz centrality, restricted to the case of undirected networks, where A = A T . Here, the node centrality vector is defined as…”

“…It was shown in [14] that, for undirected networks, the NBTW centrality vector (2.2a) can be computed as cheaply as the Katz version in (2.1a). Further, in an appropriate sense, the limiting value of b as a function of t in (2.2a) agrees with the nonbacktracking eigenvector centrality measure proposed in [24] for undirected networks.…”

“…In [14] it was shown in the undirected case that the Katz-style NBTW measure recovers the eigenvector version of [24] as the attenuation parameter approaches its upper limit. Here we show that in the more general case of directed edges, a related eigenvector limit arises.…”