The minimum number of triangular edges and a symmetrization method for multiple graphs

We give an asymptotic formula for the minimum number of edges contained in triangles in a graph having n vertices and e edges. Our main tool is a generalization of Zykov's symmetrization method that can be applied for several graphs simultaneously.Comment: Same paper, better presentation. Now it has 10 pages with 7 figure

Quantum Hall Ground States, Binary Invariants, and Regular Graphs

Extracting meaningful physical information out of a many-body wavefunction is often impractical. The polynomial nature of fractional quantum Hall (FQH) wavefunctions, however, provides a rare opportunity for a study by virtue of ground states alone. In this article, we investigate the general properties of FQH ground state polynomials. It turns out that the data carried by an FQH ground state can be essentially that of a (small) directed graph/matrix. We establish a correspondence between FQH ground states, binary invariants and regular graphs and briefly introduce all the necessary concepts. Utilizing methods from invariant theory and graph theory, we will then take a fresh look on physical properties of interest, e.g. squeezing properties, clustering properties, etc. Our methodology allows us to `unify' almost all of the previously constructed FQH ground states in the literature as special cases of a graph-based class of model FQH ground states, which we call \emph{accordion} model FQH states

Detecting communities of triangles in complex networks using spectral optimization

The study of the sub-structure of complex networks is of major importance to relate topology and functionality. Many efforts have been devoted to the analysis of the modular structure of networks using the quality function known as modularity. However, generally speaking, the relation between topological modules and functional groups is still unknown, and depends on the semantic of the links. Sometimes, we know in advance that many connections are transitive and, as a consequence, triangles have a specific meaning. Here we propose the study of the modular structure of networks considering triangles as the building blocks of modules. The method generalizes the standard modularity and uses spectral optimization to find its maximum. We compare the partitions obtained with those resulting from the optimization of the standard modularity in several real networks. The results show that the information reported by the analysis of modules of triangles complements the information of the classical modularity analysis.Comment: Computer Communications (in press

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.

Edge Expansion and Spectral Gap of Nonnegative Matrices

The classic graphical Cheeger inequalities state that if $M$ is an $n\times n$ symmetric doubly stochastic matrix, then $$\frac{1-\lambda_{2}(M)}{2}\leq\phi(M)\leq\sqrt{2\cdot(1-\lambda_{2}(M))}$$ where $\phi(M)=\min_{S\subseteq[n],|S|\leq n/2}\left(\frac{1}{|S|}\sum_{i\in S,j\not\in S}M_{i,j}\right)$ is the edge expansion of $M$, and $\lambda_{2}(M)$ is the second largest eigenvalue of $M$. We study the relationship between $\phi(A)$ and the spectral gap $1-\text{Re}\lambda_{2}(A)$ for any doubly stochastic matrix $A$ (not necessarily symmetric), where $\lambda_{2}(A)$ is a nontrivial eigenvalue of $A$ with maximum real part. Fiedler showed that the upper bound on $\phi(A)$ is unaffected, i.e., $\phi(A)\leq\sqrt{2\cdot(1-\text{Re}\lambda_{2}(A))}$. With regards to the lower bound on $\phi(A)$, there are known constructions with $$\phi(A)\in\Theta\left(\frac{1-\text{Re}\lambda_{2}(A)}{\log n}\right),$$ indicating that at least a mild dependence on $n$ is necessary to lower bound $\phi(A)$. In our first result, we provide an exponentially better construction of $n\times n$ doubly stochastic matrices $A_{n}$, for which $$\phi(A_{n})\leq\frac{1-\text{Re}\lambda_{2}(A_{n})}{\sqrt{n}}.$$ In fact, all nontrivial eigenvalues of our matrices are $0$, even though the matrices are highly nonexpanding. We further show that this bound is in the correct range (up to the exponent of $n$), by showing that for any doubly stochastic matrix $A$, $$\phi(A)\geq\frac{1-\text{Re}\lambda_{2}(A)}{35\cdot n}.$$ Our second result extends these bounds to general nonnegative matrices $R$, obtaining a two-sided quantitative refinement of the Perron-Frobenius theorem in which the edge expansion $\phi(R)$ (appropriately defined), a quantitative measure of the irreducibility of $R$, controls the gap between the Perron-Frobenius eigenvalue and the next-largest real part of any eigenvalue

Edge Expansion and Spectral Gap of Nonnegative Matrices

The classic graphical Cheeger inequalities state that if M is an n × n symmetric doubly stochastic matrix, then 1−λ₂(M)/2 ≤ ϕ(M) ≤ √2⋅(1−λ₂(M)) where ϕ(M) = min_(S⊆[n],|S|≤n/2)(1|S|∑_(i∈S,j∉S)M_(i,j)) is the edge expansion of M, and λ₂(M) is the second largest eigenvalue of M. We study the relationship between φ(A) and the spectral gap 1 – Re λ₂(A) for any doubly stochastic matrix A (not necessarily symmetric), where λ₂(A) is a nontrivial eigenvalue of A with maximum real part. Fiedler showed that the upper bound on φ(A) is unaffected, i.e., ϕ(A) ≤ √2⋅(1−Reλ₂(A)). With regards to the lower bound on φ(A), there are known constructions with ϕ(A) ∈ Θ(1−Reλ₂(A)/log n) indicating that at least a mild dependence on n is necessary to lower bound φ(A). In our first result, we provide an exponentially better construction of n × n doubly stochastic matrices A_n, for which ϕ(An) ≤ 1−Reλ₂(A_n)/√n. In fact, all nontrivial eigenvalues of our matrices are 0, even though the matrices are highly nonexpanding. We further show that this bound is in the correct range (up to the exponent of n), by showing that for any doubly stochastic matrix A, ϕ(A) ≥ 1−Reλ₂(A)/35⋅n As a consequence, unlike the symmetric case, there is a (necessary) loss of a factor of n^α for ½ ≤ α ≤ 1 in lower bounding φ by the spectral gap in the nonsymmetric setting. Our second result extends these bounds to general matrices R with nonnegative entries, to obtain a two-sided gapped refinement of the Perron-Frobenius theorem. Recall from the Perron-Frobenius theorem that for such R, there is a nonnegative eigenvalue r such that all eigenvalues of R lie within the closed disk of radius r about 0. Further, if R is irreducible, which means φ(R) > 0 (for suitably defined φ), then r is positive and all other eigenvalues lie within the open disk, so (with eigenvalues sorted by real part), Re λ₂(R) < r. An extension of Fiedler's result provides an upper bound and our result provides the corresponding lower bound on φ(R) in terms of r – Re λ₂(R), obtaining a two-sided quantitative version of the Perron-Frobenius theorem

Solving Directed Laplacian Systems in Nearly-Linear Time through Sparse LU Factorizations

We show how to solve directed Laplacian systems in nearly-linear time. Given a linear system in an $n \times n$ Eulerian directed Laplacian with $m$ nonzero entries, we show how to compute an $\epsilon$-approximate solution in time $O(m \log^{O(1)} (n) \log (1/\epsilon))$. Through reductions from [Cohen et al. FOCS'16] , this gives the first nearly-linear time algorithms for computing $\epsilon$-approximate solutions to row or column diagonally dominant linear systems (including arbitrary directed Laplacians) and computing $\epsilon$-approximations to various properties of random walks on directed graphs, including stationary distributions, personalized PageRank vectors, hitting times, and escape probabilities. These bounds improve upon the recent almost-linear algorithms of [Cohen et al. STOC'17], which gave an algorithm to solve Eulerian Laplacian systems in time $O((m+n2^{O(\sqrt{\log n \log \log n})})\log^{O(1)}(n \epsilon^{-1}))$. To achieve our results, we provide a structural result that we believe is of independent interest. We show that Laplacians of all strongly connected directed graphs have sparse approximate LU-factorizations. That is, for every such directed Laplacian ${\mathbf{L}}$, there is a lower triangular matrix $\boldsymbol{\mathit{{\mathfrak{L}}}}$ and an upper triangular matrix $\boldsymbol{\mathit{{\mathfrak{U}}}}$, each with at most $\tilde{O}(n)$ nonzero entries, such that their product $\boldsymbol{\mathit{{\mathfrak{L}}}} \boldsymbol{\mathit{{\mathfrak{U}}}}$ spectrally approximates ${\mathbf{L}}$ in an appropriate norm. This claim can be viewed as an analogue of recent work on sparse Cholesky factorizations of Laplacians of undirected graphs. We show how to construct such factorizations in nearly-linear time and prove that, once constructed, they yield nearly-linear time algorithms for solving directed Laplacian systems.Comment: Appeared in FOCS 201