Random Regular Graphs Research Articles

Randomly Twisted Hypercubes: Between Structure and Randomness

ABSTRACTTwisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these graphs have optimal diameter. We study twisted hypercubes in the setting where the instances can have general dependence, and also in the particular case where they are identical. We show that the resultant graph shares properties with random regular graphs, including small diameter, large vertex expansion, a semicircle law for its eigenvalues and no non‐trivial automorphisms. However, in contrast to random regular graphs, twisted hypercubes allow for short routing schemes.

Counting and hardness-of-finding fixed points in cellular automata on random graphs

Abstract We study the fixed points of outer-totalistic cellular automata on sparse random regular graphs. These can be seen as constraint satisfaction problems, where each variable must adhere to the same local constraint, which depends solely on its state and the total number of its neighbors in each possible state. Examples of this setting include classical problems such as independent sets or assortative/dissasortative partitions. We analyse the existence and number of fixed points in the large system limit using the cavity method, under both the replica symmetric (RS) and one-step replica symmetry breaking (1RSB) assumption. This method allows us to characterize the structure of the space of solutions, in particular, if the solutions are clustered and whether the clusters contain frozen variables. This last property is conjectured to be linked to the typical algorithmic hardness of the problem. We bring experimental evidence for this claim by studying the performance of the belief-propagation reinforcement algorithm, a message-passing-based solver for these constraint satisfaction problems.

Quantum-like states on complex synchronized networks

Recent work has exposed the idea that interesting quantum-like (QL) probability laws, including interference effects, can be manifest in classical systems. Here, we propose a model for QL states and QL bits. We suggest a way that huge, complex systems can host robust states that can process information in a QL fashion. Axioms that such states should satisfy are proposed. Specifically, it is shown that building blocks suited for QL states are networks, possibly very complex, that we defined based on k -regular random graphs. These networks can dynamically encode a lot of information that is distilled into the emergent states we can use for QL processing. Although the emergent states are classical, they have properties analogous to quantum states. Concrete examples of how QL functions are possible are given. The possibility of a ‘QL advantage’ for computing-type operations and the potential relevance for new kinds of function in the brain are discussed and left as open questions.

Renormalization group analysis of the Anderson model on random regular graphs

We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.

Statistical physics of principal minors: Cavity approach.

Determinants are useful to represent the state of an interacting system of (effectively) repulsive and independent elements, like fermions in a quantum system and training samples in a learning problem. A computationally challenging problem is to compute the sum of powers of principal minors of a matrix which is relevant to the study of critical behaviors in quantum fermionic systems and finding a subset of maximally informative training data for a learning algorithm. Specifically, principal minors of positive square matrices can be considered as statistical weights of a random point process on the set of the matrix indices. The probability of each subset of the indices is in general proportional to a positive power of the determinant of the associated submatrix. We use Gaussian representation of the determinants for symmetric and positive matrices to estimate the partition function (or free energy) and the entropy of principal minors within the Bethe approximation. The results are expected to be asymptotically exact for diagonally dominant matrices with locally treelike structures. We consider the Laplacian matrix of random regular graphs of degree K=2,3,4 and exactly characterize the structure of the relevant minors in a mean-field model of such matrices. No (finite-temperature) phase transition is observed in this class of diagonally dominant matrices by increasing the positive power of the principal minors, which here plays the role of an inverse temperature.

Markov processes on quasi-random graphs

We study Markov population processes on large graphs, with the local state transition rates of a single vertex being a linear function of its neighborhood. A simple way to approximate such processes is by a system of ODEs called the homogeneous mean-field approximation (HMFA). Our main result is showing that HMFA is guaranteed to be the large graph limit of the stochastic dynamics on a finite time horizon if and only if the graph-sequence is quasi-random. An explicit error bound is given and it is 1N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{\\sqrt{N}}$$\\end{document} plus the largest discrepancy of the graph. For Erdős–Rényi and random regular graphs we show an error bound of order the inverse square root of the average degree. In general, diverging average degrees is shown to be a necessary condition for the HMFA to be accurate. Under special conditions, some of these results also apply to more detailed type of approximations like the inhomogenous mean field approximation (IHMFA). We pay special attention to epidemic applications such as the SIS process.

Shared-Memory Parallel Edmonds Blossom Algorithm for Maximum Cardinality Matching in General Graphs.

The Edmonds Blossom algorithm is implemented here using depth-first search, which is intrinsically serial. By streamlining the code, our serial implementation is consistently three to five times faster than the previously fastest general graph matching code. By extracting parallelism across iterations of the algorithm, with coarse-grain locking, we are able to further reduce the run time on random regular graphs four-fold and obtain a two-fold reduction of run time on real-world graphs with similar topology. Solving very sparse graphs (average degree less than four) exhibiting community structure with eight threads led to a slow down of three-fold, but this slow down is replaced by marginal speed up once the average degree is greater than four. We conclude that our parallel coarse-grain locking implementation performs well when extracting parallelism from this augmenting-path-based algorithm and may work well for similar algorithms.

On minimum vertex bisection of random d-regular graphs

Minimum vertex bisection is a graph partitioning problem in which the aim is to find a partition of the vertices into two equal parts that minimizes the number of vertices in one partition set that have a neighbor in the other set. In this work we are interested in providing asymptotically almost surely upper bounds on the minimum vertex bisection of random d-regular graphs, for constant values of d. Our approach is based on analyzing a greedy algorithm by using the differential equation method. In this way, we obtain the first known non-trivial upper bounds for the vertex bisection number in random regular graphs. The numerical approximations of these theoretical bounds are compared with the emprical ones, and with the lower bounds from Kolesnik and Wormald (2014) [30].

Ant Mill: an adversarial traffic pattern for low-diameter direct networks

Since today’s HPC and data center systems can comprise hundreds of thousands of servers and beyond, it is crucial to equip them with a network that provides high performance. New topologies proposed to achieve such performance need to be evaluated under different traffic conditions, aiming to closely replicate real-world scenarios. While most optimizations should be guided by common traffic patterns, it is essential to ensure that no pathological traffic pattern can compromise the entire system. Determining synthetic adversarial traffic patterns for a network typically relies on a thorough understanding of its topology and routing. In this paper, we address the problem of identifying a generic adversarial traffic pattern for low-diameter direct interconnection networks. We first focus on Random Regular Graphs (RRGs), which represent a typical case for these networks. Moreover, RRGs have been proposed as topologies for interconnection networks due to their superior scalability and expandability, among other advantages. We introduce Ant Mill, an adversarial traffic pattern for RRGs when using routes of minimal length. Secondly, we demonstrate that the Ant Mill traffic pattern is also adversarial in other low-diameter direct interconnection networks such as Slimfly, Dragonfly, and Projective networks. Ant Mill is thoroughly motivated and evaluated, enabling future studies of low-diameter direct interconnection networks to leverage its findings.

Correlated volumes for extended wave functions on a random-regular graph

Correlated volumes for extended wave functions on a random-regular graph

Full degree spanning trees in random regular graphs

We study the problem of maximizing the number of full degree vertices in a spanning tree T of a graph G; that is, the number of vertices whose degree in T equals its degree in G. In cubic graphs, this problem is equivalent to maximizing the number of leaves in T and minimizing the size of a connected dominating set of G. We provide an algorithm that produces (w.h.p.) a tree with at least 0.4591n vertices of full degree (and also, leaves) when run on a random cubic graph. This improves the previously best known lower bound of 0.4146n. We also provide lower bounds on the number of full degree vertices in the random regular graph G(n,r) for r≤10.

Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry.

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal, and right localized states depending on both parameters and graph structure properties such as node degree d. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around W∼20 for a random regular graph d=4. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around W∼5, localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.

Robust extended states in Anderson model on partially disordered random regular graphs

In this work we analytically explain the origin of the mobility edge in the partially disordered random regular graphs of degree d, i.e., with a fraction \betaβ of the sites being disordered, while the rest remain clean. It is shown that the mobility edge in the spectrum survives in a certain range of parameters (d,\betaβ) at infinitely large uniformly distributed disorder. The critical curve separating extended and localized states is derived analytically and confirmed numerically. The duality in the localization properties between the sparse and extremely dense RRG has been found and understood.

Large-scale quantum approximate optimization on nonplanar graphs with machine learning noise mitigation

Quantum computers are increasing in size and quality but are still very noisy. Error mitigation extends the size of the quantum circuits that noisy devices can meaningfully execute. However, state-of-the-art error mitigation methods are hard to implement and the limited qubit connectivity in superconducting qubit devices restricts most applications to the hardware's native topology. Here we show a quantum approximate optimization algorithm (QAOA) on nonplanar random regular graphs with up to 40 nodes enabled by a machine learning-based error mitigation. We use a swap network with careful decision-variable-to-qubit mapping and a feed-forward neural network to optimize a depth-two QAOA on up to 40 qubits. We observe a meaningful parameter optimization for the largest graph which requires running quantum circuits with 958 two-qubit gates. Our paper emphasizes the need to mitigate samples, and not only expectation values, in quantum approximate optimization. These results are a step towards executing quantum approximate optimization at a scale that is not classically simulable. Reaching such system sizes is key to properly understanding the true potential of heuristic algorithms like QAOA. Published by the American Physical Society 2024

Learning interpretable collective variables for spreading processes on networks.

Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this Letter, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barabási model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.

Hyperoptimized Approximate Contraction of Tensor Networks with Arbitrary Geometry

Tensor network contraction is central to problems ranging from many-body physics to computer science. We describe how to approximate tensor network contraction through bond compression on arbitrary graphs. In particular, we introduce a hyperoptimization over the compression and contraction strategy itself to minimize error and cost. We demonstrate that our protocol outperforms both handcrafted contraction strategies in the literature as well as recently proposed general contraction algorithms on a variety of synthetic and physical problems on regular lattices and random regular graphs. We further showcase the power of the approach by demonstrating approximate contraction of tensor networks for frustrated three-dimensional lattice partition functions, dimer counting on random regular graphs, and to access the hardness transition of random tensor network models, in graphs with many thousands of tensors. Published by the American Physical Society 2024

Triangles and Subgraph Probabilities in Random Regular Graphs

We improve the estimates of the subgraph probabilities in a random regular graph. Using the improved results, we further improve the limiting distribution of the number of triangles in random regular graphs.

Discordant edges for the voter model on regular random graphs

Discordant edges for the voter model on regular random graphs

Majority vote in social networks

Consider a graph G and suppose that initially each node is colored either black or white. In the majority model, in each round all nodes simultaneously update their color to the most frequent color among their neighbors.Experiments on the graph data from the real world social networks (SNs) suggest that if an extremely small set of high-degree nodes, often referred to as the elites, all agree on a color, that color becomes the dominant color at the end of the process. We propose two countermeasures that can be adopted by individual nodes relatively easily and guarantee that the elites will not have this disproportionate power to engineer the dominant output color. The first countermeasure essentially requires each node to make some new connections at random, while the second one demands the nodes to be more reluctant towards changing their color. We verify their effectiveness and correctness both theoretically and experimentally.We also investigate the majority model and a variant of it when the initial coloring is random on the real world SNs and several random graph models. In particular, our results on the Erdős-Rényi and regular random graphs confirm or support several theoretical findings or conjectures by the prior work regarding the threshold behavior of the process.

Heat-Bath and Metropolis Dynamics in Ising-like Models on Directed Regular Random Graphs.

Using a single-site mean-field approximation (MFA) and Monte Carlo simulations, we examine Ising-like models on directed regular random graphs. The models are directed-network implementations of the Ising model, Ising model with absorbing states, and majority voter models. When these nonequilibrium models are driven by the heat-bath dynamics, their stationary characteristics, such as magnetization, are correctly reproduced by MFA as confirmed by Monte Carlo simulations. It turns out that MFA reproduces the same result as the generating functional analysis that is expected to provide the exact description of such models. We argue that on directed regular random graphs, the neighbors of a given vertex are typically uncorrelated, and that is why MFA for models with heat-bath dynamics provides their exact description. For models with Metropolis dynamics, certain additional correlations become relevant, and MFA, which neglects these correlations, is less accurate. Models with heat-bath dynamics undergo continuous phase transition, and at the critical point, the power-law time decay of the order parameter exhibits the behavior of the Ising mean-field universality class. Analogous phase transitions for models with Metropolis dynamics are discontinuous.