Cheeger constants, structural balance, and spectral clustering analysis for signed graphs

  title={Cheeger constants, structural balance, and spectral clustering analysis for signed graphs},
  author={Fatihcan M. Atay and Shiping Liu},
  journal={Discret. Math.},
Sublinear-Time Clustering Oracle for Signed Graphs
This work provides a local clustering oracle for signed graphs with such a clear community structure, that can answer membership queries, i.e., “ Given a vertex v, which community does v belong to? ”, in sublinear time by reading only a small portion of the graph.
Symmetric Matrices, Signed Graphs, and Nodal Domain Theorems
In 2001, Davies, Gladwell, Leydold, and Stadler proved discrete nodal domain theorems for eigenfunctions of generalized Laplacians, i.e., symmetric matrices with non-positive off-diagonal entries. In
A View of Exact Inference in Graphs from the Degree-4 Sum-of-Squares Hierarchy
This work applies a powerful hierarchy of relaxations, known as the sum-of-squares (SoS) hierarchy, to the combinatorial problem of exactly recovering an unknown ground-truth binary labeling of the nodes from a single corrupted observation of each edge.
SP ] 2 1 O ct 2 01 8 Hamiltonian surgery : Cheeger-type inequalities for nonpositive ( stoquastic ) , real , and Hermitian matrices
Cheeger inequalities bound the spectral gap γ of a space by isoperimetric properties of that space and vice versa. In this paper, I derive Cheeger-type inequalities for nonpositive matrices (aka
Searching for polarization in signed graphs: a local spectral approach
This paper forms the problem of finding local polarized communities in signed graphs as a locally-biased eigen-problem and shows that the locally- biased vector can be used to find communities with approximation guarantee with respect to a local analogue of the Cheeger constant on signed graphs.
Hamiltonian surgery: Cheeger-type gap inequalities for nonpositive (stoquastic), real, and Hermitian matrices
This paper derives Cheeger-type inequalities for nonpositive matrices (aka stoquastic Hamiltonians), real matrices, and Hermitian matrices and sketches a bashful adiabatic algorithm that aborts the adiABatic process early, uses the resulting state to approximate the weighted Cheeger constant, and restarts the process using the updated information.
Curvature and Higher Order Buser Inequalities for the Graph Connection Laplacian
In this process, the concepts of Cheeger type constants and a discrete Ricci curvature for connection Laplacians are discussed and their properties systematically are studied.
Bipartite Communities via Spectral Partitioning
In this paper we are interested in finding communities with bipartite structure. A bipartite community is a pair of disjoint vertex sets S, \(S'\) such that the number of edges with one endpoint in S
Discrete-to-Continuous Extensions: Lov\'asz extension, optimizations and eigenvalue problems
In this paper, we use various versions of Lovász extension to systematically derive continuous formulations of problems from discrete mathematics. This will take place in the following context: For
Signatures, lifts, and eigenvalues of graphs
It is proved that the existence of an infinite tower of $3$-cyclic lifts, each of which is again Ramanujan, is possible.


Balancedness and the least eigenvalue of Laplacian of signed graphs
Signatures and signed switching classes
Balancing signed graphs
A Coding Approach to Signed Graphs
The cocycle code of an undirected graph $\Gamma$ is the linear span over F2 of the characteristic vectors of cutsets, and bounds on $D(T)$ are obtained in terms of the genus $\gamma$ and on the number of unlabeled even-degree subgraphs in terms.
Improved Cheeger's inequality: analysis of spectral partitioning algorithms through higher order spectral gap
It is proved that for any graph G and any k ≥ 2, [φ(G) = O(k) l<sub>2</sub>/√l <sub>k</sub>) and this performance guarantee is achieved by the spectral partitioning algorithm, and the bound is optimal up to a constant factor for any $k$.
Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees
The existence of infinite families of (c, d)-biregular bipartite graphs with all non-trivial eigenvalues bounded by √c-1+√d-1, for all c, d ≥ q 3 is proved.
Lifts, Discrepancy and Nearly Optimal Spectral Gap*
It is shown that every graph of maximal degree d has a 2-lift such that all “new” eigenvalues are in the range, leading to a deterministic polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue O(d/α)+1.
Correlation Clustering: maximizing agreements via semidefinite programming
This work gives a 0.7666-approximation algorithm for maximizing agreements on any graph even when the edges have non-negative weights (along with labels) and they want to maximize the weight of agreements.
Ollivier’s Ricci Curvature, Local Clustering and Curvature-Dimension Inequalities on Graphs
This paper employs a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau to derive lower RicCI curvature bounds on graphs in terms of such local clustering coefficients.