Ensemble Of Graphs Research Articles

An uncertainty-aware strategy for plasma mechanism reduction with directed weighted graphs

In this work, we present a framework for the analysis and reduction of plasma mechanisms by means of weighted directed graphs, in which reactions and species are both treated as nodes. The methodology consists of two distinct analyses. The first, which is qualitative, relies on graph spatializations via force-directed algorithms to discover the predominant global patterns in the chemical model. The second ranks the reactions based on their shortest paths' lengths from/to the species of interest and their relative contributions to the power balance. This quantitative investigation enables a strategy for mechanism reduction that is fully automatized, as it does not require any expert knowledge, highly effective, as it generates reduced mechanisms that are highly accurate while relying on a small number of processes, and easily interpretable, as the algorithm justifies the importance of the retained reactions by outputting their related chemical pathways. Additionally, the work proposes a methodology extension that employs ensembles of graphs to improve the robustness of the reduced mechanism to reaction parameter uncertainties. The approach, here tested for steady-state predictions of a plasma system characterizing negative hydrogen ion sources, is general and can be used in a wide variety of applications outside the particular nuclear fusion context demonstrated in this work.

Improving the adversarial transferability with relational graphs ensemble adversarial attack.

In transferable black-box attacks, adversarial samples remain adversarial across multiple models and are more likely to attack unknown models. From this view, acquiring and exploiting multiple models is the key to improving transferability. For exploiting multiple models, existing approaches concentrate on differences among models but ignore the underlying complex dependencies. This exacerbates the issue of unbalanced and inadequate attacks on multiple models. To this problem, this paper proposes a novel approach, called Relational Graph Ensemble Attack (RGEA), to exploit the dependencies among multiple models. Specifically, we redefine the multi-model ensemble attack as a multi-objective optimization and create a sub-optimization problem to compute the optimal attack direction, but there are serious time-consuming problems. For this time-consuming problem, we define the vector representation of the model, extract the dependency matrix, and then equivalently simplify the sub-optimization problem by utilizing the dependency matrix. Finaly, we theoretically extend to investigate the connection between RGEA and the traditional multiple gradient descent algorithm (MGDA). Notably, combining RGEA further enhances the transferability of existing gradient-based attacks. The experiments using ten normal training models and ten defensive models on the labeled face in the wild (LFW) dataset demonstrate that RGEA improves the success rate of white-box attacks and further boosts the transferability of black-box attacks.

Revan Sombor indices: Analytical and statistical study

In this paper, we perform analytical and statistical studies of Revan indices on graphs $ G $: $ R(G) = \sum_{uv \in E(G)} F(r_u, r_v) $, where $ uv $ denotes the edge of $ G $ connecting the vertices $ u $ and $ v $, $ r_u $ is the Revan degree of the vertex $ u $, and $ F $ is a function of the Revan vertex degrees. Here, $ r_u = \Delta + \delta - d_u $ with $ \Delta $ and $ \delta $ the maximum and minimum degrees among the vertices of $ G $ and $ d_u $ is the degree of the vertex $ u $. We concentrate on Revan indices of the Sombor family, i.e., the Revan Sombor index and the first and second Revan $ (a, b) $-$ KA $ indices. First, we present new relations to provide bounds on Revan Sombor indices which also relate them with other Revan indices (such as the Revan versions of the first and second Zagreb indices) and with standard degree-based indices (such as the Sombor index, the first and second $ (a, b) $-$ KA $ indices, the first Zagreb index and the Harmonic index). Then, we extend some relations to index average values, so they can be effectively used for the statistical study of ensembles of random graphs.

Efficient Deployment of UWB Anchors for Wide-Area Positioning Coverage Using Bipartite Graph

Indoor mobile robots have been developed rigorously, with more diversified functions and wide applications. The research of indoor robot positioning systems is increasingly critical, and UWB is one of the most accurate methods. However, due to its short reception range, current UWB systems are limited to small activity spaces. Two important questions must be answered to enable UWB systems to be used in a wider area. How many anchors does it need? Where to deploy these anchors? For the first time, this study converts the problem of deploying anchors into finding an ensemble of bipartite graphs. The problem is an np-hard problem, which will be addressed in the study. Nevertheless, the study presents a systematic approach to efficiently solving the number and placement of UWB anchors. The generalized method applies to all places where UWB anchors can be deployed, regardless of spatial shape or size constraints. Moreover, a systematic approach can be used when the total number of UWB anchors is limited; the process can find where to place the anchors so that the robot can be positioned within the maximum range. This method enables the UWB indoor positioning system to be widely and efficiently applied in more fields. Experiments were also conducted to verify the feasibility of this method.

Eigenfunction and eigenmode-spacing statistics in chaotic photonic crystal graphs.

The statistical properties of wave chaotic systems of varying dimensionalities and realizations have been studied extensively. These systems are commonly characterized by the statistics of the eigenmode spacings and the statistics of the eigenfunctions. Here, we propose photonic crystal (PC) defect waveguide graphs as a physical setting for chaotic graph studies. Photonic crystal waveguides possess a dispersion relation for the propagating modes, which is engineerable. Graphs constructed by joining these waveguides possess junctions and bends with distinct scattering properties. We present numerically determined statistical properties of an ensemble of such PC graphs including both eigenfunction amplitude and eigenmode-spacing studies. Our proposed system is compatible with silicon nanophotonic technology and opens chaotic graph studies to a new community of researchers.

Cutting-plane algorithms and solution whitening for the vertex-cover problem.

The phase-transition behavior of the NP-hard vertex-cover (VC) combinatorial optimization problem is studied numerically by linear programming (LP) on ensembles of random graphs. As the basic Simplex (SX) algorithm suitable for such LPs may produce incomplete solutions for sufficiently complex graphs, the application of cutting-plane (CP) methods is sought. We consider Gomory and {0,1/2} cuts. We measure the probability of obtaining complete solutions with these approaches as a function of the average node degree c and observe transition between typically complete and incomplete phase regions. While not generally complete solutions are obtained for graphs of arbitrarily high complexity, the CP approaches still advance the boundary in comparison to the pure SX algorithm, beyond the known replica-symmetry breaking (RSB) transition at c=e≈2.718. In fact, our results provide evidence for another algorithmic transition at c≈2.90(2). Besides this, we quantify the transition between easy and hard solvability of the VC problem also in terms of numerical effort. Further we study the so-called whitening of the solution, which is a measure for the degree of freedom that single vertices experience with respect to degenerate solutions. Inspection of the quantities related to clusters of white vertices reveals that whitening is affected, only slightly but measurably, by the RSB transition.

Destruction of localization by thermal inclusions: Anomalous transport and Griffiths effects in the Anderson and André-Aubry-Harper models

We discuss and compare two recently proposed toy models for anomalous transport and Griffiths effects in random systems near the Many-Body Localization transitions: the random dephasing model, which adds thermal inclusions in an Anderson Insulator as local Markovian dephasing channels that heat up the system, and the random Gaussian Orthogonal Ensemble (GOE) approach which models them in terms of ensembles of random regular graphs. For these two settings we discuss and compare transport and dissipative properties and their statistics. We show that both types of dissipation lead to similar Griffiths-like phenomenology, with the GOE bath being less effective in thermalising the system due to its finite bandwidth. We then extend these models to the case of a quasi-periodic potential as described by the André-Aubry-Harper model coupled to random thermal inclusions, that we show to display, for large strength of the quasiperiodic potential, a similar phenomenology to the one of the purely random case. In particular, we show the emergence of subdiffusive transport and broad statistics of the local density of states, suggestive of Griffiths like effects arising from the interplay between quasiperiodic localization and random coupling to the baths.

Robust Graph Neural Networks via Ensemble Learning

Graph neural networks (GNNs) have demonstrated a remarkable ability in the task of semi-supervised node classification. However, most existing GNNs suffer from the nonrobustness issues, which poses a great challenge for applying GNNs into sensitive scenarios. Some researchers concentrate on constructing an ensemble model to mitigate the nonrobustness issues. Nevertheless, these methods ignore the interaction among base models, leading to similar graph representations. Moreover, due to the deterministic propagation applied in most existing GNNs, each node highly relies on its neighbors, leaving the nodes to be sensitive to perturbations. Therefore, in this paper, we propose a novel framework of graph ensemble learning based on knowledge passing (called GEL) to address the above issues. In order to achieve interaction, we consider the predictions of prior models as knowledge to obtain more reliable predictions. Moreover, we design a multilayer DropNode propagation strategy to reduce each node’s dependence on particular neighbors. This strategy also empowers each node to aggregate information from diverse neighbors, alleviating oversmoothing issues. We conduct experiments on three benchmark datasets, including Cora, Citeseer, and Pubmed. GEL outperforms GCN by more than 5% in terms of accuracy across all three datasets and also performs better than other state-of-the-art baselines. Extensive experimental results also show that the GEL alleviates the nonrobustness and oversmoothing issues.

Emergent time, cosmological constant and boundary dimension at infinity in combinatorial quantum gravity

Combinatorial quantum gravity is governed by a discrete Einstein-Hilbert action formulated on an ensemble of random graphs. There is strong evidence for a second-order quantum phase transition separating a random phase at strong coupling from an ordered, geometric phase at weak coupling. Here we derive the picture of space-time that emerges in the geometric phase, given such a continuous phase transition. In the geometric phase, ground-state graphs are discretizations of Riemannian, negative-curvature Cartan-Hadamard manifolds. On such manifolds, diffusion is ballistic. Asymptotically, diffusion time is soldered with a manifold coordinate and, consequently, the probability distribution is governed by the wave equation on the corresponding Lorentzian manifold of positive curvature, de Sitter space-time. With this asymptotic Lorentzian picture, the original negative curvature of the Riemannian manifold turns into a positive cosmological constant. The Lorentzian picture, however, is valid only asymptotically and cannot be extrapolated back in coordinate time. Before a certain epoch, coordinate time looses its meaning and the universe is a negative-curvature Riemannian “shuttlecock” with ballistic diffusion, thereby avoiding a big bang singularity. The emerging coordinate time leads to a de Sitter version of the holographic principle relating the bulk isometries with boundary conformal transformations. While the topological boundary dimension is (D − 1), the so-called “dimension at infinity” of negative curvature manifolds, i.e. the large-scale spectral dimension seen by diffusion processes with no spectral gap, those that can probe the geometry at infinity, is always three.

Motif estimation via subgraph sampling: The fourth-moment phenomenon

Network sampling is an indispensable tool for understanding features of large complex networks where it is practically impossible to search over the entire graph. In this paper, we develop a framework for statistical inference for counting network motifs, such as edges, triangles and wedges, in the widely used subgraph sampling model, where each vertex is sampled independently, and the subgraph induced by the sampled vertices is observed. We derive necessary and sufficient conditions for the consistency and the asymptotic normality of the natural Horvitz–Thompson (HT) estimator, which can be used for constructing confidence intervals and hypothesis testing for the motif counts based on the sampled graph. In particular, we show that the asymptotic normality of the HT estimator exhibits an interesting fourth-moment phenomenon, which asserts that the HT estimator (appropriately centered and rescaled) converges in distribution to the standard normal whenever its fourth-moment converges to 3 (the fourth-moment of the standard normal distribution). As a consequence, we derive the exact thresholds for consistency and asymptotic normality of the HT estimator in various natural graph ensembles, such as sparse graphs with bounded degree, Erdős–Rényi random graphs, random regular graphs and dense graphons.

Nonuniform random graphs on the plane: A scaling study.

We consider random geometric graphs on the plane characterized by a nonuniform density of vertices. In particular, we introduce a graph model where n vertices are independently distributed in the unit disk with positions, in polar coordinates (l,θ), obeying the probability density functions ρ(l) and ρ(θ). Here we choose ρ(l) as a normal distribution with zero mean and variance σ∈(0,∞) and ρ(θ) as a uniform distribution in the interval θ∈[0,2π). Then, two vertices are connected by an edge if their Euclidean distance is less than or equal to the connection radius ℓ. We characterize the topological properties of this random graph model, which depends on the parameter set (n,σ,ℓ), by the use of the average degree 〈k〉 and the number of nonisolated vertices V_{×}, while we approach their spectral properties with two measures on the graph adjacency matrix: the ratio of consecutive eigenvalue spacings r and the Shannon entropy S of eigenvectors. First we propose a heuristic expression for 〈k(n,σ,ℓ)〉. Then, we look for the scaling properties of the normalized average measure 〈X[over ¯]〉 (where X stands for V_{×}, r, and S) over graph ensembles. We demonstrate that the scaling parameter of 〈V_{×}[over ¯]〉=〈V_{×}〉/n is indeed 〈k〉, with 〈V_{×}[over ¯]〉≈1-exp(-〈k〉). Meanwhile, the scaling parameter of both 〈r[over ¯]〉 and 〈S[over ¯]〉 is proportional to n^{-γ}〈k〉 with γ≈0.16.

Semi-Supervised anchor graph ensemble for large-scale hyperspectral image classification

ABSTRACT Existing graph-based, semi-supervised hyperspectral image (HSI) classification models often suffer from prolonged execution time due to high computational complexity. In this work, we first propose a fast anchor graph regularization (FAGR) model for large scale, HSI classification. FAGR employs a simple anchor-based graph construction procedure and a new adjacency matrix among anchors to dramatically reduce the computational complexity while attaining good classification performance. In order to further improve the classification accuracy of hyperspectral images, we propose a novel semi-supervised anchor graph ensemble (SAGE) model. SAGE is an ensemble realization of multiple FAGR with each component FAGR operating on a randomly selected subset of features. Ameta classifier is applied to aggregate the outputs of component classifiers to yield an ensemble classification result. We performed extensive experimentations using three real-world HSI datasets, to compare the performance of FAGR and SAGE against several existing graph-based HSI classifiers. The experiment results show that the proposed SAGE achieves 95.78% classification accuracy on the Indian Pines dataset using limited labeled samples, out-performing existing models in terms of shorter execution time and better classification accuracy.

The Random Plots Graph Generation Model for Studying Systems with Unknown Connection Structures.

We consider the problem of modeling complex systems where little or nothing is known about the structure of the connections between the elements. In particular, when such systems are to be modeled by graphs, it is unclear what vertex degree distributions these graphs should have. We propose that, instead of attempting to guess the appropriate degree distribution for a poorly understood system, one should model the system via a set of sample graphs whose degree distributions cover a representative range of possibilities and account for a variety of possible connection structures. To construct such a representative set of graphs, we propose a new random graph generator, Random Plots, in which we (1) generate a diversified set of vertex degree distributions and (2) target a graph generator at each of the constructed distributions, one-by-one, to obtain the ensemble of graphs. To assess the diversity of the resulting ensembles, we (1) substantialize the vague notion of diversity in a graph ensemble as the diversity of the numeral characteristics of the graphs within this ensemble and (2) compare such formalized diversity for the proposed model with that of three other common models (Erdős–Rényi–Gilbert (ERG), scale-free, and small-world). Computational experiments show that, in most cases, our approach produces more diverse sets of graphs compared with the three other models, including the entropy-maximizing ERG. The corresponding Python code is available at GitHub.

Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm

The K-core of a graph is the maximal subgraph within which each vertex is connected to at least K other vertices. It is a fundamental network concept for understanding threshold cascading processes with a discontinuous percolation transition. A minimum attack set contains the smallest number of vertices whose removal induces complete collapse of the K-core. Here we tackle this prototypical optimal initial-condition problem from the spin-glass perspective of cycle-tree maximum packing and propose a cycle-tree guided attack (CTGA) message-passing algorithm. The good performance and time efficiency of CTGA are verified on the regular random and Erd\"os-R\'enyi random graph ensembles. Our central idea of transforming a long-range correlated dynamical process to static structural patterns may also be instructive to other hard optimization and control problems.

Phase transition in the bipartite z-matching

We study numerically the maximum z-matching problems on ensembles of bipartite random graphs. The z-matching problems describes the matching between two types of nodes, users and servers, where each server may serve up to z users at the same time. Using a mapping to standard maximum-cardinality matching, and because for the latter there exists a polynomial-time exact algorithm, we can study large system sizes of up to 10^6 nodes. We measure the capacity and the energy of the resulting optimum matchings. First, we confirm previous analytical results for bipartite regular graphs. Next, we study the finite-size behaviour of the matching capacity and find the same scaling behaviour as before for standard matching, which indicates the universality of the problem. Finally, we investigate for bipartite Erdős–Rényi random graphs the saturability as a function of the average degree, i.e. whether the network allows as many customers as possible to be served, i.e. exploiting the servers in an optimal way. We find phase transitions between unsaturable and saturable phases. These coincide with a strong change of the running time of the exact matching algorithm, as well with the point where a minimum-degree heuristic algorithm starts to fail.Graphical

Fixed-angle conjectures for the quantum approximate optimization algorithm on regular MaxCut graphs

The quantum approximate optimization algorithm (QAOA) is a near-term combinatorial optimization algorithm suitable for noisy quantum devices. However, little is known about performance guarantees for $p>2$. A recent work computing MaxCut performance guarantees for 3-regular graphs conjectures that any $d$-regular graph evaluated at particular fixed angles has an approximation ratio greater than some worst-case guarantee. In this work, we provide numerical evidence for this fixed angle conjecture for $p<12$. We compute and provide these angles via numerical optimization and tensor networks. These fixed angles serve for an optimization-free version of QAOA and have universally good performance on any 3-regular graph. Heuristic evidence is presented for the fixed angle conjecture on graph ensembles, which suggests that these fixed angles are ``close'' to global optimum. Under the fixed angle conjecture, QAOA has a larger performance guarantee than the Goemans Williamson algorithm on 3-regular graphs for $p\ensuremath{\ge}11$.

Statistics of Complex Wigner Time Delays as a Counter of S-Matrix Poles: Theory and Experiment.

We study the statistical properties of the complex generalization of Wigner time delay τ_{W} for subunitary wave-chaotic scattering systems. We first demonstrate theoretically that the mean value of the Re[τ_{W}] distribution function for a system with uniform absorption strength η is equal to the fraction of scattering matrix poles with imaginary parts exceeding η. The theory is tested experimentally with an ensemble of microwave graphs with either one or two scattering channels and showing broken time-reversal invariance and variable uniform attenuation. The experimental results are in excellent agreement with the developed theory. The tails of the distributions of both real and imaginary time delay are measured and are also found to agree with theory. The results are applicable to any practical realization of a wave-chaotic scattering system in the short-wavelength limit, including quantum wires and dots, acoustic and electromagnetic resonators, and quantum graphs.

Local topological moves determine global diffusion properties of hyperbolic higher-order networks

From social interactions to the human brain, higher-order networks are key to describe the underlying network geometry and topology of many complex systems. While it is well known that network structure strongly affects its function, the role that network topology and geometry has on the emerging dynamical properties of higher-order networks is yet to be clarified. In this perspective, the spectral dimension plays a key role since it determines the effective dimension for diffusion processes on a network. Despite its relevance, a theoretical understanding of which mechanisms lead to a finite spectral dimension, and how this can be controlled, still represents a challenge and is the object of intense research. Here, we introduce two nonequilibrium models of hyperbolic higher-order networks and we characterize their network topology and geometry by investigating the intertwined appearance of small-world behavior, δ-hyperbolicity, and community structure. We show that different topological moves, determining the nonequilibrium growth of the higher-order hyperbolic network models, induce tuneable values of the spectral dimension, showing a rich phenomenology which is not displayed in random graph ensembles. In particular, we observe that, if the topological moves used to construct the higher-order network increase the area/volume ratio, then the spectral dimension continuously decreases, while the opposite effect is observed if the topological moves decrease the area/volume ratio. Our work reveals a new link between the geometry of a network and its diffusion properties, contributing to a better understanding of the complex interplay between network structure and dynamics.

Polygon-Based Hierarchical Planar Networks Based on Generalized Apollonian Construction

Experimentally observed complex networks are often scale-free, small-world and have an unexpectedly large number of small cycles. An Apollonian network is one notable example of a model network simultaneously having all three of these properties. This network is constructed by a deterministic procedure of consequentially splitting a triangle into smaller and smaller triangles. In this paper, a similar construction based on the consequential splitting of tetragons and other polygons with an even number of edges is presented. The suggested procedure is stochastic and results in the ensemble of planar scale-free graphs. In the limit of a large number of splittings, the degree distribution of the graph converges to a true power law with an exponent, which is smaller than three in the case of tetragons and larger than three for polygons with a larger number of edges. It is shown that it is possible to stochastically mix tetragon-based and hexagon-based constructions to obtain an ensemble of graphs with a tunable exponent of degree distribution. Other possible planar generalizations of the Apollonian procedure are also briefly discussed.

Ensemble of correlation, parenclitic and synolitic graphs as a tool to detect universal changes in complex biological systems: Comment on "Dynamic and thermodynamic models of adaptation" by A.N. Gorban et al.

Ensemble of correlation, parenclitic and synolitic graphs as a tool to detect universal changes in complex biological systems: Comment on "Dynamic and thermodynamic models of adaptation" by A.N. Gorban et al.