Random Multigraphs Research Articles

Hyperspectral Imagery Classification via Random Multigraphs Ensemble Learning

Hyperspectral imagery (HSI) classification, which attempts to assign hyperspectral pixels with proper labels, has drawn significant attention in various applications. Recently, the graph-based semisupervised learning (SSL) approaches have shown the outstanding ability to handle the situation of limited labeled data for classification task. However, it is hard to construct the pairwise adjacent graph due to the high dimensionality of hyperspectral data. Besides, the graph-based SSL models are usually decided by a single classifier, which fail to effectively learn the complex structures and intrinsic properties of HSI. To address these problems, we propose a novel graph-based SSL classification model for HSI, which is based on random multigraphs construction and ensemble strategy (RMGE). Specifically, the anchor graph (AG) is constructed with spatial–spectral features, which integrates the spatial characteristics extracted by local binary pattern on each selected spectrum, preserving fine structures of local region. In order to enhance the discriminative capability of the classifier and avoid the trivial solution, the maximum entropy regularization is added into adjacent AG model. In addition, to capture the diversity of HSI data effectively, we design the ensemble framework by employing multiple AGs to learn HSI features. Experiments conducted on real hyperspectral datasets indicate that the proposed RMGE shows better performance than that of state-of-the-art approaches.

Spectral-Spatial Hyperspectral Image Semisupervised Classification by Fusing Maximum Noise Fraction and Adaptive Random Multigraphs

Hyperspectral images (HSIs) contain large amounts of spectral and spatial information, and this provides the possibility for ground object classification. However, when using the traditional method, achieving a satisfactory classification result is difficult because of the insufficient labeling of samples in the training set. In addition, parameter adjustment during HSI classification is time-consuming. This paper proposes a novel fusion method based on the maximum noise fraction (MNF) and adaptive random multigraphs for HSI classification. Considering the overall spectrum of the object and the correlation of adjacent bands, the MNF was utilized to reduce the spectral dimension. Next, a multiscale local binary pattern (LBP) analysis was performed on the MNF dimension-reduced data to extract the spatial features of different scales. The obtained multiscale spatial features were then stacked with the MNF dimension-reduced spectral features to form multiscale spectral-spatial features (SSFs), which were sent into the RMG for HSI classification. Optimal performance was obtained by fusion. For all three real datasets, our method achieved competitive results with only 10 training samples. More importantly, the classification parameters corresponding to different hyperspectral data can be automatically optimized using our method.

Moments of Uniform Random Multigraphs with Fixed Degree Sequences

We study the expected adjacency matrix of a uniformly random multigraph with fixed degree sequence $\mathbf{d} \in \mathbb{Z}_+^n$. This matrix arises in a variety of analyses of networked data sets, including modularity-maximization and mean-field theories of spreading processes. Its structure is well understood for large, sparse, simple graphs: the expected number of edges between nodes $i$ and $j$ is roughly $\frac{d_id_j}{\sum_\ell{d_\ell}}$. Many network data sets are neither large, sparse, nor simple, and in these cases the standard approximation no longer applies. We derive a novel estimator using a dynamical approach: the estimator emerges from the stationarity conditions of a class of Markov Chain Monte Carlo algorithms for graph sampling. We derive error bounds for this estimator and provide an efficient scheme with which to compute it. We test the estimator on synthetic and empirical degree sequences, finding that it enjoys relative error against ground truth a full order of magnitude smaller than the standard approximation. We then compare modularity maximization techniques using both the standard and novel estimators, finding that the qualitative structure of the optimization landscape depends significantly on the estimator choice. Our results emphasize the importance of using carefully specified random graph models in data scientific applications.

Goldberg's conjecture is true for random multigraphs

In the 70s, Goldberg, and independently Seymour, conjectured that for any multigraph G, the chromatic index χ′(G) satisfies χ′(G)≤max⁡{Δ(G)+1,⌈ρ(G)⌉}, where ρ(G)=max⁡{e(G[S])⌊|S|/2⌋|S⊆V}. We show that their conjecture (in a stronger form) is true for random multigraphs. Let M(n,m) be the probability space consisting of all loopless multigraphs with n vertices and m edges, in which m pairs from [n] are chosen independently at random with repetitions. Our result states that, for a given m:=m(n), M∼M(n,m) typically satisfies χ′(G)=max⁡{Δ(G),⌈ρ(G)⌉}. In particular, we show that if n is even and m:=m(n), then χ′(M)=Δ(M) for a typical M∼M(n,m). Furthermore, for a fixed ε>0, if n is odd, then a typical M∼M(n,m) has χ′(M)=Δ(M) for m≤(1−ε)n3log⁡n, and χ′(M)=⌈ρ(M)⌉ for m≥(1+ε)n3log⁡n. To prove this result, we develop a new structural characterization of multigraphs with chromatic index larger than the maximum degree.

Spectral and Spatial Classification of Hyperspectral Images Based on Random Multi-Graphs

Hyperspectral image classification has been acknowledged as the fundamental and challenging task of hyperspectral data processing. The abundance of spectral and spatial information has provided great opportunities to effectively characterize and identify ground materials. In this paper, we propose a spectral and spatial classification framework for hyperspectral images based on Random Multi-Graphs (RMGs). The RMG is a graph-based ensemble learning method, which is rarely considered in hyperspectral image classification. It is empirically verified that the semi-supervised RMG deals well with small sample setting problems. This kind of problem is very common in hyperspectral image applications. In the proposed method, spatial features are extracted based on linear prediction error analysis and local binary patterns; spatial features and spectral features are then stacked into high dimensional vectors. The high dimensional vectors are fed into the RMG for classification. By randomly selecting a subset of features to create a graph, the proposed method can achieve excellent classification performance. The experiments on three real hyperspectral datasets have demonstrated that the proposed method exhibits better performance than several closely related methods.

Synthetic aperture radar image change detection based on frequency-domain analysis and random multigraphs

With the development of earth observation programs, many multitemporal synthetic aperture radar (SAR) images over the same geographical area are available. It is demanding to develop automatic change detection techniques to take advantage of these images. Most existing techniques directly analyze the difference image (DI), and therefore, they are easily affected by the speckle noise. We proposed an SAR image change detection method based on frequency-domain analysis and random multigraphs. The proposed method follows a coarse-to-fine procedure: in the coarse changed regions localization stage, frequency-domain analysis is utilized to select distinctive and salient regions from the DI. Therefore, nonsalient regions are neglected, and noisy unchanged regions incurred by the speckle noise are suppressed. In the fine changed regions classification stage, random multigraphs are employed as the classification model. By selecting a subset of neighborhood features to create graphs, the proposed method can efficiently exploit the nonlinear relations between multitemporal SAR images. The experimental results on two real SAR datasets and one simulated dataset have demonstrated the effectiveness of the proposed method.

Random multigraphs and aggregated triads with fixed degrees

AbstractRandom multigraphs with fixed degrees are obtained by the configuration model or by so called random stub matching. New combinatorial results are given for the global probability distribution of edge multiplicities and its marginal local distributions of loops and edges. The number of multigraphs on triads is determined for arbitrary degrees, and aggregated triads are shown to be useful for analyzing regular and almost regular multigraphs. Relationships between entropy and complexity are given and numerically illustrated for multigraphs with different number of vertices and specified average and variance for the degrees.

Push is Fast on Sparse Random Graphs

We consider the classical push broadcast process on a large class of sparse random multigraphs that includes random power law graphs and multigraphs. Our analysis shows that for every $\varepsilon>0$, whp $O(\log n)$ rounds are sufficient to inform all but an $\varepsilon$-fraction of the vertices. It is not hard to see that, e.g. for random power law graphs, the push process needs whp $n^{\Omega(1)}$ rounds to inform all vertices. Fountoulakis, Panagiotou and Sauerwald proved that for random graphs that have power law degree sequences with $\beta>3$, the push-pull protocol needs $\Omega(\log n)$ to inform all but $\varepsilon n$ vertices whp. Our result demonstrates that, for such random graphs, the pull mechanism does not (asymptotically) improve the running time. This is surprising as it is known that, on random power law graphs with $2<\beta<3$, push-pull is exponentially faster than pull.

Random Multi-Graphs: A semi-supervised learning framework for classification of high dimensional data

Currently, high dimensional data processing confronts two main difficulties: inefficient similarity measure and high computational complexity in both time and memory space. Common methods to deal with these two difficulties are based on dimensionality reduction and feature selection. In this paper, we present a different way to solve high dimensional data problems by combining the ideas of Random Forests and Anchor Graph semi-supervised learning. We randomly select a subset of features and use the Anchor Graph method to construct a graph. This process is repeated many times to obtain multiple graphs, a process which can be implemented in parallel to ensure runtime efficiency. Then the multiple graphs vote to determine the labels for the unlabeled data. We argue that the randomness can be viewed as a kind of regularization. We evaluate the proposed method on eight real-world data sets by comparing it with two traditional graph-based methods and one state-of-the-art semi-supervised learning method based on Anchor Graph to show its effectiveness. We also apply the proposed method to the subject of face recognition.

Analytic description of the phase transition of inhomogeneous multigraphs

We introduce a new model of random multigraphs with colored vertices and weighted edges. It is similar to the inhomogeneous random graph model of Söderberg (2002), extended by Bollobás, Janson and Riordan (2007). By means of analytic combinatorics, we then analyze the birth of complex components, which are components with at least two cycles. We apply those results to give a complete picture of the finite size scaling and the critical exponents associated to a rather broad family of decision problems. As applications, we derive new proofs of known results on the 2-colorability problem (Pittel and Yeum, 2010) and on the enumeration of properly q-colored multigraphs (Wright, 1972), and new results on the phase transition of the satisfiability of quantified 2-Xor-formulas (Daudé and Ravelomanana, 2011, Creignou et al., 2007).

Periodic Partial Words and Random Bipartite Graphs

The interaction property or the Fine--Wilf property for periodic partial words is studied. Partial words with two periods are represented by bipartite graphs and the interaction property is related to the edge connectedness property of these graphs. Threshold functions for the edge connectedness of random bipartite graphs and multigraphs are found. As a corollary, the interaction property is described in probabilistic terms.

Multigraph models for causal quantum gravity and scale dependent spectral dimension

We study random walks on ensembles of a specific class of random multigraphs which provide an ‘effective graph ensemble’ for the causal dynamical triangulation (CDT) model of quantum gravity. In particular, we investigate the spectral dimension of the multigraph ensemble for recurrent as well as transient walks. We investigate the circumstances in which the spectral dimension and Hausdorff dimension are equal and show that this occurs when ρ, the exponent for anomalous behaviour of the resistance to infinity, is zero. The concept of scale dependent spectral dimension in these models is introduced. We apply this notion to a multigraph ensemble with a measure induced by a size biased critical Galton–Watson process which has a scale dependent spectral dimension of two at large scales and one at small scales. We conclude by discussing a specific model related to four dimensional CDT which has a spectral dimension of four at large scales and two at small scales.

The degree distribution of the random multigraphs

National Natural Science Fund of China [10831001, 10871046, 10971027]; Science and Technology of Science Fund of Fujian Province [A0950059]; Fuzhou University [2009-XQ-27]

Bubblestorm

Peer-to-peer systems promise inexpensive scalability, adaptability, and robustness. Thus, they are an attractive platform for file sharing, distributed wikis, and search engines. These applications often store weakly structured data, requiring sophisticated search algorithms. To simplify the search problem, most scalable algorithms introduce structure to the network. However, churn or violent disruption may break this structure, compromising search guarantees. This paper proposes a simple probabilistic search system, BubbleStorm, built on random multigraphs. Our primary contribution is a flexible and reliable strategy for performing exhaustive search. BubbleStorm also exploits the heterogeneous bandwidth of peers. However, we sacrifice some of this bandwidth for high parallelism and low latency. The provided search guarantees are tunable, with success probability adjustable well into the realm of reliable systems. For validation, we simulate a network with one million low-end peers and show BubbleStorm handles up to 90% simultaneous peer departure and 50% simultaneous crash.

Appearance of complex components in a random bigraph

AbstractThis is an investigation into the limiting distribution of the sparse connected components in an n1 × n2 bipartite multigraph with approximately ½n edges, where n1, n2 ≈︁ ½n. We will show that the probability of finding no complex components in a random n1 × n2 bigraph, with approximately ½n edges, is asymptotically the same as for random graphs with n vertices and approximately ½n edges, namely √2/3. In addition, we will show that, for n1 × n2 multi‐bigraphs, the probability distribution for finitely many bicyclic components, tricyclic components, etc., with no components having more than a given number of cycles, asymptotically equals the corresponding distribution for random multigraphs with n vertices and approximately half as many edges. As an application of this analysis we present a method for estimating the efficiency of memory access in a distributed, replicated data base.

Triad count statistics

The triad counts of a graph G are the numbers of various distinct induced subgraphs of order 3. If G is an undirected graph, there are 4 triad counts, and if G is a digraph, there are 16 triad counts. A multigraph is a sequence G = (G1,..., Gr) of graphs and digraphs defined on a common vertex set. The concept of triad counts is generalized to multigraphs with colored vertices, edges, and arcs. It is shown how triad counts in multigraphs can be used in various kinds of statistical analyses of graph data. In particular, probability distributions are investigated of the triad counts in random multigraphs.