Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2211.03939

Detecting Hidden Communities by Power Iterations with Connections to Vanilla Spectral Algorithms

Chandra Sekhar Mukherjee·Jiapeng Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Community detection in the stochastic block model is one of the central problems of graph clustering. Since its introduction, many subsequent papers have made great strides in solving and understanding this model. In this setup, spectral algorithms have been one of the most widely used frameworks. However, despite the long history of study, there are still unsolved challenges. One of the main open problems is the design and analysis of "simple"(vanilla) spectral algorithms, especially when the number of communities is large. In this paper, we provide two algorithms. The first one is based on the power-iteration method. It is a simple algorithm which only compares the rows of the powered adjacency matrix. Our algorithm performs optimally (up to logarithmic factors) compared to the best known bounds in the dense graph regime by Van Vu (Combinatorics Probability and Computing, 2018). Furthermore, our algorithm is also robust to the "small cluster barrier", recovering large clusters in the presence of an arbitrary number of small clusters. Then based on a connection between the powered adjacency matrix and eigenvectors, we provide a vanilla spectral algorithm for large number of communities in the balanced case. This answers an open question by Van Vu (Combinatorics Probability and Computing, 2018) in the balanced case. Our methods also partially solve technical barriers discussed by Abbe, Fan, Wang and Zhong (Annals of Statistics, 2020). In the technical side, we introduce a random partition method to analyze each entry of a powered random matrix. This method can be viewed as an eigenvector version of Wigner's trace method. Recall that Wigner's trace method links the trace of powered matrix to eigenvalues. Our method links the whole powered matrix to the span of eigenvectors. We expect our method to have more applications in random matrix theory.

Explore related subjects

math.CO

Keep this discovery

BibTeXRIS

Chandra Sekhar Mukherjee, Jiapeng Zhang. 2022-11-08. Detecting Hidden Communities by Power Iterations with Connections to Vanilla Spectral Algorithms. https://arxiv.org/abs/2211.03939

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Balanced even cycles in signed graphs:Tur\'an bounds, double covers, and parity obstructions

We study Tur\'an problems for balanced even cycles in simple signed graphs, where signed subgraphs are considered up to switching. For every balanced bipartite signed graph, the signed and ordinary Tur\'an numbers differ by at most a factor of two. Our main structural results concern the underlying graphs that admit a signing in which every $2k$-cycle is unbalanced. We characterize these graphs by the absence of an odd dependence among their $2k$-cycle incidence vectors, give a cohomological formulation, and construct subgraph-minimal obstructions of arbitrarily large order. In particular, there is no finite forbidden-subgraph characterization. We also give an exact closed-walk criterion for cycles in double covers and derive a direct signed breadth-first-search upper bound. As applications, we prove \[ \hex(n,C_{+4})=\left(\frac{\sqrt2}{2}+o(1)\right)n^{3/2} \] and study the signed hexagon number $R_6(n)=\hex(n,\{C_{-3},C_{+6}\})$. We characterize the underlying graphs counted by $R_6$ and express it as an extremal problem for ordinary $C_6$-free graphs with a prescribed involution. For every sufficiently large $n$, we construct examples with $\Omega(n^{4/3})$ edges, and we give an equivariant construction attaining the coefficient obtained from the F\"uredi--Naor--Verstra\"ete lower bound by double-cover transfer. Finally, we give $n$-vertex $C_{+10}$-free signed graphs with $\Omega(n^{6/5})$ edges and use octagon examples to illustrate the limitations of theta-freeness as a signing criterion.

math.CO↗
arXiv2026

Fractional DP-colorings of $d$-degenerate locally sparse graphs

Bernshteyn, Kostochka, and Zhu (2020) introduced the notion of fractional DP-coloring, which generalizes both fractional coloring and fractional list coloring. Among several foundational results, they proved that every $d$-degenerate bipartite graph $G$ satisfies $\chi_f^{\mathrm{DP}} \le (1 + o(1))\frac{d}{\log d}$, and that this bound is optimal---a stark contrast to ordinary fractional coloring. In this paper, we extend this upper bound to all $d$-degenerate triangle-free graphs, proving that $\chi_f^{\mathrm{DP}} \le (4 + o(1))\frac{d}{\log d}$. This generalizes a recent result of Martinsson and Steiner (2025) for ordinary fractional coloring. We derive this result as a corollary of a more general upper bound concerning locally sparse graph orderings. Specifically, a $d$-degenerate graph $G$ is left $k$-locally-sparse if it admits a degeneracy ordering in which, for every vertex $v$, the subgraph induced by its back-neighbors contains at most $k$ edges. We show that if a $d$-degenerate graph $G$ is left $\frac{d^2}{f}$-locally-sparse, then \[ \chi_f^{\mathrm{DP}}(G) \le (8 + o(1))\frac{d}{\log f}. \] This immediately yields an identical upper bound on the ordinary fractional chromatic number $\chi_f(G)$, improving upon the leading constants of previously known bounds. Additionally, we establish the asymptotic sharpness of this result up to the leading constant. For any $1 \ll f \le d^2$, we construct $d$-degenerate graphs that are left $\frac{d^2}{f}$-locally-sparse and satisfy $\chi_f(G) \ge (1 - o(1))\frac{d}{\log f}$. Finally, as applications of our main theorem, we obtain improved upper bounds on the fractional DP-chromatic number of $d$-degenerate $K_{1,t,t}$-free graphs, as well as $K_{t,t,t}$-free graphs with maximum degree $\Delta$. Notably, these bounds improve upon existing results even in the setting of ordinary fractional coloring.

math.CO↗
arXiv2026

Erd\H{o}s-S\'{o}s for digraphs

It is shown that every Eulerian digraph on $n$ vertices with more than $(t-1)n$ arcs contains every oriented tree with $t$ edges. The digraphs have no loops or repeated arcs, but opposite arcs are permitted. The bound is sharp for each fixed oriented tree, as witnessed by disjoint unions of complete bidirected graphs. Previously, such tight bounds were not known, even just for directed paths. This can be considered as a directed analog of the recently proved Erd\H{o}s-S\'os conjecture. The result was proved by GPT-6 Astra.

math.CO↗