arXiv subjects

Cristopher Moore

Publications and source records attributed to Cristopher Moore.

At least 37 records · Page 2Linked to original sources

arXiv2020

Spectral Planting and the Hardness of Refuting Cuts, Colorability, and Communities in Random Graphs

We study the problem of efficiently refuting the k-colorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, showing optimality of a simple spectral certificate. This evidence takes the form of a computationally-quiet planting: we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishable by a large class of algorithms. We generalize our results to the more general problem of certifying an upper bound on the maximum k-cut. This quiet planting is achieved by minimizing the effect of the planted structure (e.g. colorings or cuts) on the graph spectrum. Specifically, the planted structure corresponds exactly to eigenvectors of the adjacency matrix. This avoids the pushout effect of random matrix theory, and delays the point at which the planting becomes visible in the spectrum or local statistics. To illustrate this further, we give similar results for a Gaussian analogue of this problem: a quiet version of the spiked model, where we plant an eigenspace rather than adding a generic low-rank perturbation. Our evidence for computational hardness of distinguishing two distributions is based on three different heuristics: stability of belief propagation, the local statistics hierarchy, and the low-degree likelihood ratio. Of independent interest, our results include general-purpose bounds on the low-degree likelihood ratio for multi-spiked matrix models, and an improved low-degree analysis of the stochastic block model.

cs.CC↗

arXiv2019

Percolation is Odd

We prove a remarkable combinatorial symmetry in the number of spanning configurations in site percolation: for a large class of lattices, the number of spanning configurations with an odd or even number of occupied sites differs by $\pm 1$. In particular, this symmetry implies that the total number of spanning configurations is always odd, independent of the size or shape of the lattice. The class of lattices that share this symmetry includes the square lattice and the hypercubic lattice in any dimension, with a wide variety of boundary conditions.

cond-mat.stat-mech↗

arXiv2019

Linear Consistency for Proof-of-Stake Blockchains

The blockchain data structure maintained via the longest-chain rule---popularized by Bitcoin---is a powerful algorithmic tool for consensus algorithms. Such algorithms achieve consistency for blocks in the chain as a function of their depth from the end of the chain. While the analysis of Bitcoin guarantees consistency with error $2^{-k}$ for blocks of depth $O(k)$, the state-of-the-art of proof-of-stake (PoS) blockchains suffers from a quadratic dependence on $k$: these protocols, exemplified by Ouroboros (Crypto 2017), Ouroboros Praos (Eurocrypt 2018) and Sleepy Consensus (Asiacrypt 2017), can only establish that depth $Θ(k^2)$ is sufficient. Whether this quadratic gap is an intrinsic limitation of PoS---due to issues such as the nothing-at-stake problem---has been an urgent open question, as deployed PoS blockchains further rely on consistency for protocol correctness. We give an axiomatic theory of blockchain dynamics that permits rigorous reasoning about the longest-chain rule and achieve, in broad generality, $Θ(k)$ dependence on depth in order to achieve consistency error $2^{-k}$. In particular, for the first time, we show that PoS protocols can match proof-of-work protocols for linear consistency. We analyze the associated stochastic process, give a recursive relation for the critical functionals of this process, and derive tail bounds in both i.i.d. and martingale settings via associated generating functions.

cs.CR↗

arXiv2019

Lecture Notes on Automata, Languages, and Grammars

These lecture notes are intended as a supplement to Moore and Mertens' The Nature of Computation or as a standalone resource, and are available to anyone who wants to use them. Comments are welcome, and please let me know if you use these notes in a course. There are 61 exercises. I emphasize that automata are elementary playgrounds where we can explore the issues of deterministic and nondeterministic computation. Unlike P vs. NP, we can prove that nondeterminism is equivalent to determinism, or strictly more powerful than determinism, in finite-state and push-down automata respectively. I also correct several historical and aesthetic injustices: in particular, the Myhill-Nerode theorem and the idea of building minimal DFAs from equivalence classes of prefixes is restored to its rightful place above the Pumping Lemma for regular languages. I also discuss the Pumping Lemma for context-free languages, and briefly discuss counter automata, queue automata, and the connection between unambiguous context-free languages and algebraic generating functions.

cs.CC↗

arXiv2018

Series Expansion of the Percolation Threshold on Hypercubic Lattices

We study proper lattice animals for bond- and site-percolation on the hypercubic lattice $\mathbb{Z}^d$ to derive asymptotic series of the percolation threshold $p_c$ in $1/d$, The first few terms of these series were computed in the 1970s, but the series have not been extended since then. We add two more terms to the series for $\pcsite$ and one more term to the series for $\pcbond$, using a combination of brute-force enumeration, combinatorial identities and an approach based on Padé approximants, which requires much fewer resources than the classical method. We discuss why it took 40 years to compute these terms, and what it would take to compute the next ones. En passant, we present new perimeter polynomials for site and bond percolation and numerical values for the growth rate of bond animals.

cond-mat.stat-mech↗

arXiv2018

Community detection, link prediction, and layer interdependence in multilayer networks

Complex systems are often characterized by distinct types of interactions between the same entities. These can be described as a multilayer network where each layer represents one type of interaction. These layers may be interdependent in complicated ways, revealing different kinds of structure in the network. In this work we present a generative model, and an efficient expectation-maximization algorithm, which allows us to perform inference tasks such as community detection and link prediction in this setting. Our model assumes overlapping communities that are common between the layers, while allowing these communities to affect each layer in a different way, including arbitrary mixtures of assortative, disassortative, or directed structure. It also gives us a mathematically principled way to define the interdependence between layers, by measuring how much information about one layer helps us predict links in another layer. In particular, this allows us to bundle layers together to compress redundant information, and identify small groups of layers which suffice to predict the remaining layers accurately. We illustrate these findings by analyzing synthetic data and two real multilayer networks, one representing social support relationships among villagers in South India and the other representing shared genetic substrings material between genes of the malaria parasite.

cs.SI↗

arXiv2018

Percolation Thresholds and Fisher Exponents in Hypercubic Lattices

We use invasion percolation to compute highly-accurate numerical values for bond and site percolation thresholds p_c on the hypercubic lattice Z^d for d = 4,,,,,13. We also compute the Fisher exponent tau governing the cluster size distribution at criticality. Our results support the claim that the mean-field value tau = 5/2 holds for d >= 6, with logarithmic corrections to power-law scaling at d=6.

cond-mat.stat-mech↗

arXiv2018

A physical model for efficient ranking in networks

We present a physically-inspired model and an efficient algorithm to infer hierarchical rankings of nodes in directed networks. It assigns real-valued ranks to nodes rather than simply ordinal ranks, and it formalizes the assumption that interactions are more likely to occur between individuals with similar ranks. It provides a natural statistical significance test for the inferred hierarchy, and it can be used to perform inference tasks such as predicting the existence or direction of edges. The ranking is obtained by solving a linear system of equations, which is sparse if the network is; thus the resulting algorithm is extremely efficient and scalable. We illustrate these findings by analyzing real and synthetic data, including datasets from animal behavior, faculty hiring, social support networks, and sports tournaments. We show that our method often outperforms a variety of others, in both speed and accuracy, in recovering the underlying ranks and predicting edge directions.

physics.soc-ph↗

arXiv2018

Graph Isomorphism and Circuit Size

It is well-known [KST93] that the complexity of the Graph Automorphism problem is characterized by a special case of Graph Isomorphism, where the input graphs satisfy the "promise" of being rigid (that is, having no nontrivial automorphisms). In this brief note, we observe that the reduction of Graph Automorphism to the Rigid Graph Ismorphism problem can be accomplished even using Grollman and Selman's notion of a "smart reduction".

cs.CC↗

arXiv2017

Minimum Circuit Size, Graph Isomorphism, and Related Problems

We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted as MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP / MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive proof system for the complement of Graph Isomorphism (GI). It yields a randomized reduction with zero-sided error from GI to MKTP. We generalize the result and show that GI can be replaced by any isomorphism problem for which the underlying group satisfies some elementary properties. Instantiations include Linear Code Equivalence, Permutation Group Conjugacy, and Matrix Subspace Conjugacy. Along the way we develop encodings of isomorphism classes that are efficiently decodable and achieve compression that is at or near the information-theoretic optimum; those encodings may be of independent interest.

cs.CC↗

arXiv2017

Percolation Thresholds in Hyperbolic Lattices

We use invasion percolation to compute numerical values for bond and site percolation thresholds $p_c$ (existence of an infinite cluster) and $p_u$ (uniqueness of the infinite cluster) of tesselations $\{P,Q\}$ of the hyperbolic plane, where $Q$ faces meet at each vertex and each face is a $P$-gon. Our values are accurate to six or seven decimal places, allowing us to explore their functional dependency on $P$ and $Q$ and to numerically compute critical exponents. We also prove rigorous upper and lower bounds for $p_c$ and $p_u$ that can be used to find the scaling of both thresholds as a function of $P$ and $Q$.

cond-mat.stat-mech↗

arXiv2017

Designing Strassen's algorithm

In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than $O(n^3)$. While the recursive construction in his algorithm is very clear, the key gain was made by showing that 2 x 2 matrix multiplication could be performed with only 7 multiplications instead of 8. The latter construction was arrived at by a process of elimination and appears to come out of thin air. Here, we give the simplest and most transparent proof of Strassen's algorithm that we are aware of, using only a simple unitary 2-design and a few easy lines of calculation. Moreover, using basic facts from the representation theory of finite groups, we use 2-designs coming from group orbits to generalize our construction to all n (although the resulting algorithms aren't optimal for n at least 3).

cs.DS↗

arXiv2017

The Lovász Theta Function for Random Regular Graphs and Community Detection in the Hard Regime

We derive upper and lower bounds on the degree $d$ for which the Lovász $\vartheta$ function, or equivalently sum-of-squares proofs with degree two, can refute the existence of a $k$-coloring in random regular graphs $G_{n,d}$. We show that this type of refutation fails well above the $k$-colorability transition, and in particular everywhere below the Kesten-Stigum threshold. This is consistent with the conjecture that refuting $k$-colorability, or distinguishing $G_{n,d}$ from the planted coloring model, is hard in this region. Our results also apply to the disassortative case of the stochastic block model, adding evidence to the conjecture that there is a regime where community detection is computationally hard even though it is information-theoretically possible. Using orthogonal polynomials, we also provide explicit upper bounds on $\vartheta(\overline{G})$ for regular graphs of a given girth, which may be of independent interest.

cs.CC↗

arXiv2017

The Computer Science and Physics of Community Detection: Landscapes, Phase Transitions, and Hardness

Community detection in graphs is the problem of finding groups of vertices which are more densely connected than they are to the rest of the graph. This problem has a long history, but it is undergoing a resurgence of interest due to the need to analyze social and biological networks. While there are many ways to formalize it, one of the most popular is as an inference problem, where there is a "ground truth" community structure built into the graph somehow. The task is then to recover the ground truth knowing only the graph. Recently it was discovered, first heuristically in physics and then rigorously in probability and computer science, that this problem has a phase transition at which it suddenly becomes impossible. Namely, if the graph is too sparse, or the probabilistic process that generates it is too noisy, then no algorithm can find a partition that is correlated with the planted one---or even tell if there are communities, i.e., distinguish the graph from a purely random one with high probability. Above this information-theoretic threshold, there is a second threshold beyond which polynomial-time algorithms are known to succeed; in between, there is a regime in which community detection is possible, but conjectured to require exponential time. For computer scientists, this field offers a wealth of new ideas and open questions, with connections to probability and combinatorics, message-passing algorithms, and random matrix theory. Perhaps more importantly, it provides a window into the cultures of statistical physics and statistical inference, and how those cultures think about distributions of instances, landscapes of solutions, and hardness.

cs.CC↗

arXiv2017

Information-theoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization

We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these "planted models" and null models where the signal matrix is zero. Our bounds differ by at most a factor of root two when the rank is large (in the clustering and submatrix localization problems, when the number of clusters or blocks is large) or the signal matrix is very sparse. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information- theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs.

math.ST↗

arXiv2016

Matrix multiplication algorithms from group orbits

We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of orbits under some finite group action. We show how to use the representation theory of the corresponding group to derive simple constraints on the decomposition, which we solve by hand for n=2,3,4,5, recovering Strassen's algorithm (in a particularly symmetric form) and new algorithms for larger n. While these new algorithms do not improve the known upper bounds on tensor rank or the matrix multiplication exponent, they are beautiful in their own right, and we point out modifications of this idea that could plausibly lead to further improvements. Our constructions also suggest further patterns that could be mined for new algorithms, including a tantalizing connection with lattices. In particular, using lattices we give the most transparent proof to date of Strassen's algorithm; the same proof works for all n, to yield a decomposition with $n^3 - n + 1$ terms.

cs.CC↗

arXiv2016

Codes, Lower Bounds, and Phase Transitions in the Symmetric Rendezvous Problem

In the rendezvous problem, two parties with different labelings of the vertices of a complete graph are trying to meet at some vertex at the same time. It is well-known that if the parties have predetermined roles, then the strategy where one of them waits at one vertex, while the other visits all $n$ vertices in random order is optimal, taking at most $n$ steps and averaging about $n/2$. Anderson and Weber considered the symmetric rendezvous problem, where both parties must use the same randomized strategy. They analyzed strategies where the parties repeatedly play the optimal asymmetric strategy, determining their role independently each time by a biased coin-flip. By tuning the bias, Anderson and Weber achieved an expected meeting time of about $0.829 n$, which they conjectured to be asymptotically optimal. We change perspective slightly: instead of minimizing the expected meeting time, we seek to maximize the probability of meeting within a specified time $T$. The Anderson-Weber strategy, which fails with constant probability when $T= Θ(n)$, is not asymptotically optimal for large $T$ in this setting. Specifically, we exhibit a symmetric strategy that succeeds with probability $1-o(1)$ in $T=4n$ steps. This is tight: for any $α< 4$, any symmetric strategy with $T = αn$ fails with constant probability. Our strategy uses a new combinatorial object that we dub a "rendezvous code," which may be of independent interest. When $T \le n$, we show that the probability of meeting within $T$ steps is indeed asymptotically maximized by the Anderson-Weber strategy. Our results imply new lower bounds, showing that the best symmetric strategy takes at least $0.638 n$ steps in expectation. We also present some partial results for the symmetric rendezvous problem on other vertex-transitive graphs.

math.CO↗

arXiv2016

Random graph models for dynamic networks

We propose generalizations of a number of standard network models, including the classic random graph, the configuration model, and the stochastic block model, to the case of time-varying networks. We assume that the presence and absence of edges are governed by continuous-time Markov processes with rate parameters that can depend on properties of the nodes. In addition to computing equilibrium properties of these models, we demonstrate their use in data analysis and statistical inference, giving efficient algorithms for fitting them to observed network data. This allows us, for instance, to estimate the time constants of network evolution or infer community structure from temporal network data using cues embedded both in the probabilities over time that node pairs are connected by edges and in the characteristic dynamics of edge appearance and disappearance. We illustrate our methods with a selection of applications, both to computer-generated test networks and real-world examples.

cs.SI↗