Compatibility, embedding and regularization of non-local random walks on graphs

Abstract

Several variants of the graph Laplacian have been introduced to model non-local diffusion processes, which allow a random walker to "jump" to non-neighborhood nodes, most notably the transformed path graph Laplacians and the fractional graph Laplacian. From a rigorous point of view, this new dynamics is made possible by having replaced the original graph G with a weighted complete graph G on the same node-set, that depends on G and wherein the presence of new edges allows a direct passage between nodes that were not neighbors in G.We show that, in general, the graph G is not compatible with the dynamics characterizing the original model graph G: the random walks on G subjected to move on the edges of G are not stochastically equivalent, in the wide sense, to the random walks on G. From a purely analytical point of view, the incompatibility of G with G means that the normalized graph Ĝ can not be embedded into the normalized graph Ĝ . Eventually, we provide a regularization method to guarantee such compatibility and preserving at the same time all the nice properties granted by G .

Decay bounds for Bernstein functions of Hermitian matrices with applications to the fractional graph Laplacian

“…where C is a constant, showing a power-law decay, as already observed in [3,10], but with the improved exponent −2α instead of −α. Thus, our new bounds show that the strength of connection between far apart nodes in G α must actually drop off faster than known so far.…”

“…Recently, interest in the fractional graph Laplacian has emerged, which allows to model nonlocal diffusion processes on graphs or use nonlocal random walks for the exploration of large networks [3,7,9,10,14]. The fractional graph Laplacian is simply defined by taking a fractional power of the ordinary Laplacian, i.e., by L α G .…”

“…Power law decay in the fractional Laplacian. In [3,10] it was observed that the entries of the fractional graph Laplacian L α G exhibit a power-law decay away from the sparsity pattern of L G . In particular, e.g., the following result was shown, which is based on Jackson's theorem [30].…”

“…As the condition (1.1) means that f is a completely monotonic function, the class of Bernstein functions is intimately related to the completely monotonic function classes of Laplace-Stieltjes and Cauchy-Stieltjes transforms. While the latter classes have received considerable interest in the analysis of matrix functions in recent years, see, e.g., [1,16,22,23,29] and the references therein, the class of Bernstein functions has not been investigated as thoroughly, although it also frequently occurs in applications: Our present study is motivated by the fact that fractional powers L α G , α ∈ (0, 1), where L G is the Laplacian of an undirected graph G, have recently emerged as a useful tool in modeling non-local diffusion processes on graphs and in the efficient exploration of large networks; see, e.g., [3,10,33,34]. Clearly, z α , α ∈ (0, 1) is nonnegative on (0, ∞) and fulfills the condition (1.1), so that it is a Bernstein function, whereas it is neither a Laplace-Stieltjes nor a Cauchy-Stieltjes function.…”

“…A priori knowledge of the decay in matrix functions has many different applications, e.g., the efficient construction of sparse approximations [21,36], the design of linearly scaling algorithms for certain linear algebra problems [5,11] and the analysis of probing methods for trace estimation [20]. In the context of fractional powers of the graph Laplacian L G , decay estimates can give insight into transition probabilities of non-local random walks on G; see [3,10].…”