Abstract
Wilson (Proceedings of the twenty-eight annual acm symposium on the theory of computing, pp. 296–303, 1996) in the 1990s described a simple and efficient algorithm based on loop-erased random walks to sample uniform spanning trees and more generally weighted trees or forests spanning a given graph. This algorithm provides a powerful tool in analyzing structures on networks and along this line of thinking, in recent works (Avena and Gaudillière in A proof of the transfer-current theorem in absence of reversibility, in Stat. Probab. Lett. 142, 17–22 (2018); Avena and Gaudillière in J Theor Probab, 2017. https://doi.org/10.1007/s10959-017-0771-3; Avena et al. in Approximate and exact solutions of intertwining equations though random spanning forests, 2017. arXiv:1702.05992v1; Avena et al. in Intertwining wavelets or multiresolution analysis on graphs through random forests, 2017. arXiv:1707.04616, to appear in ACHA (2018)) we focused on applications of spanning rooted forests on finite graphs. The resulting main conclusions are reviewed in this paper by collecting related theorems, algorithms, heuristics and numerical experiments. A first foundational part on determinantal structures and efficient sampling procedures is followed by four main applications: (1) a random-walk-based notion of well-distributed points in a graph, (2) a framework to describe metastable-like dynamics in finite settings by means of Markov intertwining dualities, (3) coarse graining schemes for networks and associated processes, (4) wavelets-like pyramidal algorithms for graph signals.
Highlights
Networks, Trees and ForestsThe aim of this paper is to survey some recent results [2,3,4,5] on a certain measure on spanning forests of a given graph and its applications within the context of networks analysis
G = (V, E, w), where V denotes a finite vertex set of size |V| = n, E stands for a directed edge set seen as a prescribed collection of ordered pairs of vertices {(x, y) ∈ V × V}, and w : V × V → R+ is a weight function, which associates to each ordered pair (x, y) ∈ E a strictly positive weight w(x, y)
A rooted spanning forest φ is a subgraph of G without cycle, with V as set of vertices and such that, for each x ∈ V, there is at most one y ∈ V
Summary
The aim of this paper is to survey some recent results [2,3,4,5] on a certain measure on spanning forests of a given graph and its applications within the context of networks analysis. A rooted spanning forest φ is a subgraph of G without cycle, with V as set of vertices and such that, for each x ∈ V, there is at most one y ∈ V such that (x, y) is an edge of φ. Let us notice that the random forest q induces a partition of the graph into trees, and the measure in (1) can be seen on the one hand as a clustering measure similar in spirit to the well-known FK-percolation [19]. The forest q is rooted and the set of roots R( q ) forms an interesting random subset of vertices whose distribution can be explicitly characterized. The presence of the tuning parameter q, controlling size and number of trees, and related efficient sampling algorithms make this measure flexible and suitable for applications
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