# A simple spectral algorithm for recovering planted partitions

Sam Cole; Shmuel Friedland; Lev Reyzin

Special Matrices (2017)

- Volume: 5, Issue: 1, page 139-157
- ISSN: 2300-7451

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topSam Cole, Shmuel Friedland, and Lev Reyzin. "A simple spectral algorithm for recovering planted partitions." Special Matrices 5.1 (2017): 139-157. <http://eudml.org/doc/288505>.

@article{SamCole2017,

abstract = {In this paper, we consider the planted partition model, in which n = ks vertices of a random graph are partitioned into k “clusters,” each of size s. Edges between vertices in the same cluster and different clusters are included with constant probability p and q, respectively (where 0 ≤ q < p ≤ 1). We give an efficient algorithm that, with high probability, recovers the clusters as long as the cluster sizes are are least (√n). Informally, our algorithm constructs the projection operator onto the dominant k-dimensional eigenspace of the graph’s adjacency matrix and uses it to recover one cluster at a time. To our knowledge, our algorithm is the first purely spectral algorithm which runs in polynomial time and works even when s = Θ (√n), though there have been several non-spectral algorithms which accomplish this. Our algorithm is also among the simplest of these spectral algorithms, and its proof of correctness illustrates the usefulness of the Cauchy integral formula in this domain.},

author = {Sam Cole, Shmuel Friedland, Lev Reyzin},

journal = {Special Matrices},

language = {eng},

number = {1},

pages = {139-157},

title = {A simple spectral algorithm for recovering planted partitions},

url = {http://eudml.org/doc/288505},

volume = {5},

year = {2017},

}

TY - JOUR

AU - Sam Cole

AU - Shmuel Friedland

AU - Lev Reyzin

TI - A simple spectral algorithm for recovering planted partitions

JO - Special Matrices

PY - 2017

VL - 5

IS - 1

SP - 139

EP - 157

AB - In this paper, we consider the planted partition model, in which n = ks vertices of a random graph are partitioned into k “clusters,” each of size s. Edges between vertices in the same cluster and different clusters are included with constant probability p and q, respectively (where 0 ≤ q < p ≤ 1). We give an efficient algorithm that, with high probability, recovers the clusters as long as the cluster sizes are are least (√n). Informally, our algorithm constructs the projection operator onto the dominant k-dimensional eigenspace of the graph’s adjacency matrix and uses it to recover one cluster at a time. To our knowledge, our algorithm is the first purely spectral algorithm which runs in polynomial time and works even when s = Θ (√n), though there have been several non-spectral algorithms which accomplish this. Our algorithm is also among the simplest of these spectral algorithms, and its proof of correctness illustrates the usefulness of the Cauchy integral formula in this domain.

LA - eng

UR - http://eudml.org/doc/288505

ER -

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