Random Forests and Networks Analysis

Avena, Luca; Castell, Fabienne; Gaudillière, Alexandre; Mélot, Clothilde

doi:10.1007/s10955-018-2124-8

Cited by 16 publications

(14 citation statements)

References 41 publications

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“…The transition path theory [54,101,33,98] has been designed for this set-up, as well as the intertwining method [11,8,9,10]. See also the results by Bianchi and Gaudillière [24] Remark 2.8.…”

Section: Remark 25mentioning

confidence: 99%

Metastable Markov chains

Landim¹

2019

Probab. Surveys

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Section: Remark 25mentioning

confidence: 99%

Metastable Markov chains

Landim¹

2019

Probab. Surveys

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“…In this sense the loop-erased partition has the tendency to capture metastable-like regions (blocks), namely, regions of points from which is difficult for the SRW to escape on time scale 1/q. This makes the probability µ q an interesting measure for randomized clustering procedures, see in this direction [2] and [3,Sec. 5].…”

Section: Z(q)mentioning

confidence: 99%

“…Several properties of the forest measure associated to the loop-erased partitioning have been derived in the recent [5,6]. Based on these results, in [3,Prop. 6] and [4,Sect.…”

Section: Leq Xmentioning

confidence: 99%

Loop-erased partitioning of a graph: mean-field analysis

Avena,

Gaudilliere,

Milanesi

et al. 2019

Preprint

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We consider a random partition of the vertex set of an arbitrary graph that can be efficiently sampled using loop-erased random walks stopped at a random independent exponential time of parameter q > 0. The related random blocks tend to cluster nodes visited by the random walk-with generator minus the graph Laplacian-on time scale 1/q, with q being the tuning parameter. We explore the emerging macroscopic structure by analyzing 2-point correlations. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block of the random partition. This interaction potential can be seen as an affinity measure for "densely connected nodes" and capture well-separated regions in network models presenting non-homogeneous landscapes. In this spirit, we compute the potential and its scaling limits on a complete graph and on a non-homogeneous weighted version with community structures. For the latter we show a phase-transition for detectability as a function of the tuning parameter and the edge weights.

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“…As such, it naturally leads to dynamic approaches(see [4,Thm.2]), and new structures and questions which do not make sense within the more restrictive global and static UST context. Applications of rooted forest measure and LEP: In a series of recent works [1,4,5] we started to explore these new features. For example, the roots [4, Prop.2.2] in Φq form a determinantal point process with kernel given by the RW Green's function, that is: for any A ⊂ V…”

mentioning

confidence: 99%

“…Further, these roots turn out to be well-distributed in the given network [4, Thm.1] and, conditional on the induced partition, their law is as the stationary measures of the random walk X restricted to each block [4, Prop.2.3] of the underlying partition. These and other features of the LEP have been recently exploited to build novel algorithms for the following different applications in data science: wavelets basis and filters for signal processing on graphs [3,33,34], estimate traces of discrete Laplacians and other diagonally dominant matrices [8], network renormalization [1,2], centrality measures [16] and statistical learning [7]. These applications give further motivations to explore in more details this LEP.…”

mentioning

confidence: 99%

Loop-erased partitioning of a network: monotonicities & analysis of cycle-free graphs

Avena¹,

Jannetje²,

Koperberg³

2021

Preprint

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We consider random partitions of the vertex set of a given finite graph that can be sampled by means of loop-erased random walks stopped at a random independent exponential time of parameter q > 0. The related random blocks tend to cluster nodes visited by the random walk associated to the graph on time scale 1/q. This random partitioning is induced by a measure of spanning rooted forest of the graph which naturally generalize the classical uniform spanning tree measure and which can be obtained as a "zero-limit" of FK-percolation with an external cemetery state. General properties of this rooted forest measure and related determinantal observables, along with a number of applications in data analysis have been recently explored. We are here interested in understanding the emergent partitioning, referred to as loop-erased partitioning, as the intensity parameter q scales with the graph size. Some first results in this direction have been investigated in the recent [6] for dense geometries. In this work we instead look at very sparse geometries. We start by characterising monotone events in q by deriving a general Russo-like formula for this rooted arboreal gas measure. We then explore the resulting clusters on simple insightful tree-like graphs by looking at the probability that two given vertices do not belong to the same block. In particular, a detailed analysis of the resulting integer partitioning on line segments is pursued. We finally look at simple trees and other almost tree-like geometries, without and with implanted modular structures. For the latter, we characterize the emergence of giants and asymptotic detection of these implanted modules as the scale parameter q varies.

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Random Forests and Networks Analysis

Cited by 16 publications

References 41 publications

Metastable Markov chains

Metastable Markov chains

Loop-erased partitioning of a graph: mean-field analysis

Loop-erased partitioning of a network: monotonicities & analysis of cycle-free graphs

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