Undirected Bipartite Graph Research Articles

Extraction of logical networks during decompiling transistor-level CMOS circuit descriptions

Objectives. The problem of restoring the functional description of digital VLSI devices presented at the transistor level is considered. The objective of the work is to develop means for extraction of blocks representing logical networks from two-level descriptions of CMOS circuits at the transistor level, which were obtained as a result of recognition (extraction) of subcircuits that implement logic elements.Methods. Graph based methods and software tools are proposed for extracting a connected blocks representing a logical network from two-level descriptions of a transistor circuits in SPICE format. In the graph interpretation, the task is reduced to constructing a labeled directed graph of a logical network based on a labeled undirected bipartite graph specifying a two-level description of the transistor circuit.Results. The proposed method makes it possible to identify lexicographically ranked logical networks, from which a transition is made to logical equations that specify the functions implemented at the outputs of the resulting networks. Software tools have been developed that provide the generation of a hierarchical description in SPICE format that implements the original circuit at the transistor level, as well as descriptions of found logical networks in the SF language of hierarchical structural and functional descriptions of discrete devices and in high-level languages (VHDL and Verilog).Conclusion. The developed methods are implemented in C++, included in the program for decompiling transistor CMOS circuits and tested within it on practical examples of transistor-level circuits. The paper provides examples of reverse engineering of some practical transistor circuits.

Extending Undirected Graph Techniques to Directed Graphs via Category Theory

We use Category Theory to construct a ‘bridge’ relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call ‘prime graphs category’; this has as objects labeled undirected bipartite graphs (which we call prime graphs), and as morphisms undirected graph morphisms that preserve the labeling (which we call prime graph morphisms). This theoretical bridge allows us to extend undirected graph techniques to directed graphs by converting the directed graphs into prime graphs. To give a proof of concept, we show that our construction preserves topological features when applied to the problems of network alignment and spectral graph clustering.

Disentangled Partial Label Learning

Partial label learning (PLL) induces a multi-class classifier from training examples each associated with a set of candidate labels, among which only one is valid. The formation of real-world data typically arises from heterogeneous entanglement of series latent explanatory factors, which are considered intrinsic properties for discriminating between different patterns. Though learning disentangled representation is expected to facilitate label disambiguation for partial-label (PL) examples, few existing works were dedicated to addressing this issue. In this paper, we make the first attempt towards disentangled PLL and propose a novel approach named TERIAL, which makes predictions according to derived disentangled representation of instances and label embeddings. The TERIAL approach formulates the PL examples as an undirected bipartite graph where instances are only connected with their candidate labels, and employs a tailored neighborhood routing mechanism to yield disentangled representation of nodes in the graph. Specifically, the proposed routing mechanism progressively infers the explanatory factors that contribute to the edge between adjacent nodes and augments the representation of the central node with factor-aware embedding information propagated from specific neighbors simultaneously via iteratively analyzing the promising subspace clusters formed by the node and its neighbors. The estimated labeling confidence matrix is also introduced to accommodate unreliable links owing to the inherent ambiguity of PLL. Moreover, we theoretically prove that the neighborhood routing mechanism will converge to the point estimate that maximizes the marginal likelihood of observed PL training examples. Comprehensive experiments over various datasets demonstrate that our approach outperforms the state-of-the-art counterparts.

On Hamiltonian property of bi-Cayley graphs

ABSTRACT For a finite group G, and a subset S of G (possibly, contains the identity element), the bi-Cayley graph Bcay(G, S) of G with respect to S is defined as the undirected bipartite graph with vertex set and edge set . In this paper, we give some sufficient conditions for the existence of hamiltonian cycle in , where is a semiproduct of Zm by a subgroup H of G. In addition, we introduce a new graph operation, which produces some classes of bi-Cayley graphs having hamiltonian cycle.

Brownian-Huygens Propagation: Modeling Wave Functions with Discrete Particle-Antiparticle Random Walks

We present a simple method of discretely modeling solutions of the classical wave and Klein-Gordon equations using variations of random walks on a graph. Consider a collection of particles executing random walks on an undirected bipartite graph embedded in mathbb {R}^{D} at discrete times mathbb {Z}, and assume those walks are “heat-like”, in the sense that a (conserved) density of particles obeys the heat equation in the continuum limit. If the particles possess a binary degree of freedom (so that they may be said to be either particles or “antiparticles”, i.e., “positive” or “negative), then there exist closely related branching random walks on the same graph that are “wave-like”, in the sense that their (also conserved) net density obeys the D-dimensional classical wave equation. Such wave-like paths can be generated even on random graphs. The transformation by which the heat-like random walks become wave-like branching random walks is as follows: at every time step, any “incoming” particle arriving at any node X of the graph along some edge eX creates a particle-antiparticle pair at that node, with the stipulation that the newly created particle with the opposite sign of the incoming particle must initially step (in “Huygens” back-propagation fashion) along eX in the reverse direction, while the other two take a step (in “Brownian” fashion) along any edge of X (including possibly eX) with equal likelihood, and chosen independently. An additional degree of freedom (resulting in “bra” and “ket” particles) leads to quasi-probability densities proportional to the square of the wave functions.

A Bipartite Graph Partition-Based Coclustering Approach With Graph Nonnegative Matrix Factorization for Large Hyperspectral Images

Clustering large hyperspectral images (HSIs) is a very challenging problem because large HSIs have high dimensionality, large spectral variability, and large computational and memory consumption. Recently, sparse subspace clustering (SSC) has achieved remarkable success in HSI clustering. However, most SSC-based methods suffer from the following bottlenecks for large HSIs: 1) high computational consumption and memory space during the construction of the similarity matrix and decomposition of the graph Laplacian matrix and 2) failure to capture the relationships among dictionary atoms, sparse coefficients, and hyperspectral pixels. To address these challenges, we propose a novel algorithm that extends SSC to cocluster large HSIs, called bipartite graph partition with graph nonnegative matrix factorization (BGP-GNMF). Specifically, to fully explore the characteristics of the spectral and spatial contexts in HSIs, we propose a novel superpixel and pixel coclustering framework with bipartite graph partitioning in the joint sparse representation domain, where superpixel-based dictionary atoms are defined as disjoint vertex sets of the bipartite graph and the joint sparsity representation is mapped into the adjacency matrix of the undirected bipartite graph. To overcome the challenges of high computational consumption and large memory space for large HSIs, the bipartite graph partition with orthonormal constrained nonnegative matrix factorization is proposed to simultaneously cluster the structured dictionary atoms and hyperspectral pixels with an indicator matrix. Finally, to exploit the intrinsic geometry of HSIs, we incorporate manifold regularization into the bipartite graph partition to improve final clustering accuracy. The effectiveness and efficiency of the proposed method are verified on three classical HSIs, and the experimental results illustrate the superiority of the proposed method compared with other state-of-the-art HSI clustering methods.

Efficient Performance Modeling for Automated CMOS Analog Circuit Synthesis

Fast and accurate performance estimation can significantly enhance the efficiency of automated analog circuit synthesis. This article presents a novel performance modeling method that can efficiently estimate circuit performance with ignorable model building overhead for variant circuit topologies. The proposed method starts with accurate transistor modeling by taking advantage of the advanced neural network (NN) fitting technique. It then utilizes the established transistor models and topology information from a circuit netlist to precisely discover the circuit dc operating point. Specialized deterministic schemes have been developed with the aid of an undirected bipartite graph converted from the circuit netlist. Moreover, the accurate NN transistor models help directly derive the small-signal model parameter values, which can be further applied to conduct symbolic analysis to evaluate circuit performances. Our experimental results not only compare various deterministic dc operating point computation schemes but also demonstrate the efficient model development, general applicability, speedy execution, and fair prediction of our proposed performance modeling method.

Automorphisms of linear functional graphs over vector spaces

Let be a finite field with q elements, a positive integer, a n-dimensional vector space over and the set of all linear functionals from to . Let and . The linear functional graph of dented by , is an undirected bipartite graph, whose vertex set V is partitioned into two sets as and two vertices and are adjacent if and only if f sends v to the zero element of (i.e. ). In this paper, the structure of all automorphisms of this graph is characterized and formalized. Also the cardinal number of automorphisms group for this graph is determined.

Moving Vehicle Tracking Optimization Method Based on SPF

In the intelligent transportation system, the license information can be automatically recognized by the computer and the vehicle can be tracked. Red light running, illegal change of lanes, vehicle retrograde, and other illegal driving events are reasonably recorded. This is undoubtedly an effective help for the traffic police to relieve the huge work pressure. However, in China, a considerable number of vehicle tracking methods have certain limitations in resisting complex external environmental influences. The external environmental factors include but not limited to variable factors such as camera movement, jitter, and severe rain and snow. These factors cannot be controlled well, so the tracking accuracy is greatly reduced. In regard to this, this paper proposes an optimization method for moving vehicle tracking based on SPF. First, according to the size of the overlapping area of the motion area between the two images, the researcher can construct and simplify the vertex adjacency matrix that reflects the characteristics of the undirected bipartite graph. Then according to the corresponding relationship between the vertex adjacency matrix and the regional behavior and vehicle behavior, the researcher completes the regional behavior analysis and vehicle behavior analysis. On this basis, a particle filter vehicle tracking algorithm based on segmentation compensation is introduced, and the vector sum of the tracked segmentation area is used as the final position of the target vehicle. In this way, as many scattered particles fall on the target area as possible, which will greatly improve the efficiency of particle utilization, enhance tracking accuracy, and avoid the problem of tracking failure caused by too fast vehicle movement. Through experimental simulation, it can be seen that the method proposed in this paper can greatly enhance the vehicle tracking ability when tracking vehicles in “complex environments.”

A Modified Decomposition Algorithm for Maximum Weight Bipartite Matching and Its Experimental Evaluation

Let G be an undirected bipartite graph with positive integer weights on the edges. We refine the existing decomposition theorem originally proposed by Kao et al., for computing maximum weight bipartite matching. We apply it to design an efficient version of the decomposition algorithm to compute the weight of a maximum weight bipartite matching of G in O(|V |W /k(|V |, W /N))-time by employing an algorithm designed by Feder and Motwani as a subroutine, where |V | and N denote the number of nodes and the maximum edge weight of G, respectively and k(x, y) = log x/ log(x 2 /y). The parameter W is smaller than the total edge weight W, essentially when the largest edge weight differs by more than one from the second largest edge weight in the current working graph in any decomposition step of the algorithm. In best case W = O(|E|) where |E| be the number of edges of G and in worst case W = W, that is, |E| ≤ W ≤ W. In addition, we talk about a scaling property of the algorithm and research a better bound of the parameter W. An experimental evaluation on randomly generated data shows that the proposed improvement is significant in general.

Bipartite graphs as polynomials and polynomials as bipartite graphs

The aim of this paper is to show that any finite undirected bipartite graph can be considered as a polynomial [Formula: see text], and any directed finite bipartite graph can be considered as a polynomial [Formula: see text], and vise verse. We also show that the multiplication in the semirings [Formula: see text], [Formula: see text] corresponds to an operation of the corresponding graphs. This operation is exactly the product of Petri nets in the sense of Winskel [G. Winskel and M. Nielsen, Models of concurrency, in Handbook of Logic in Computer Science, Vol. 4, eds. Abamsky, Gabbay and Maibaum (Oxford University Press, 1995), pp. 1–148]. As an application, we give an approach to dividing in the semirings [Formula: see text], [Formula: see text], and a criteria for parallalization of Petri nets. Finally, we endow the set of all bipartite graphs with the Zariski topology.

The non-negative spectrum of a digraph

Abstract Given the adjacency matrix A of a digraph, the eigenvalues of the matrix AAT constitute the so-called non-negative spectrum of this digraph. We investigate the relation between the structure of digraphs and their non-negative spectra and associated eigenvectors. In particular, it turns out that the non-negative spectrum of a digraph can be derived from the traditional (adjacency) spectrum of certain undirected bipartite graphs.

M-Channel Graph Filter Banks: Polyphase Analysis and Structures

Two channel critically sampled filter banks for signal over graph domains were first proposed for undirected bipartite graphs by Narang and Ortega. Extension to the M-channel critically sampled case for balanced M-block cyclic graphs was then proposed by Teke and Vaidynathan but the filter bank does not achieve strict perfect reconstruction (PR), only generalized PR. In this letter, we consider the more general case of filter banks on unbalanced M-block cyclic graphs where strict PR is achieved. A polyphase analysis to derive the implementation structures in the downsampled domain is presented here. The relevant system/filter matrices have interesting cyclic properties and projection operators are needed to map signals between subgraphs.

Bipartite divisor graph for the set of irreducible character degrees

‎Let $G$ be a finite group‎. ‎We consider the set of the irreducible complex characters of $G$‎, ‎namely $Irr(G)$‎, ‎and the related degree set $cd(G)={chi(1)‎ : ‎chiin Irr(G)}$‎. ‎Let $rho(G)$ be the set of all primes which divide some character degree of $G$‎. ‎In this paper we introduce the bipartite divisor graph for $cd(G)$ as an undirected bipartite graph with vertex set $rho(G)cup (cd(G)setminus{1})$‎, ‎such that an element $p$ of $rho(G)$ is adjacent to an element $m$ of $cd(G)setminus{1}$ if and only if $p$ divides $m$‎. ‎We denote this graph simply by $B(G)$‎. ‎Then by means of combinatorial properties of this graph‎, ‎we discuss the structure of the group $G$‎. ‎In particular‎, ‎we consider the cases where $B(G)$ is a path or a cycle‎.

Matchings with Lower Quotas: Algorithms and Complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G= (A, dot{cup }, P, E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between textsf {NP}-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota u_{max } as basis, and we prove that this dependence is necessary unless textsf {FPT}= textsf {W}[1]. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee u_{max }+1, which is asymptotically best possible unless textsf {P}= textsf {NP}. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas.

Efficient detection of anomaly patterns through global search in remotely sensed big data

In order to leverage computational complexity and avoid information losses, “big data” analysis requires a new class of algorithms and methods to be designed and implemented. In this sense, information theory-based techniques can play a key role to effectively unveil change and anomaly patterns within big data sets. A framework that aims at detecting the anomaly patterns of a given dataset is introduced. The proposed method, namely PROMODE, relies on a representation of the given dataset performed by means of undirected bipartite graphs. Then the anomalies are searched and detected by progressively spanning the graph. The proposed architecture delivers a computational load that is less than that carried by typical frameworks in literature, so that PROMODE can be considered as a valid algorithm for efficient detection of change patterns in remotely sensed big data.

Graph-induced restricted Boltzmann machines for document modeling

Discovering knowledge from unstructured texts is a central theme in data mining and machine learning. We focus on fast discovery of thematic structures from a corpus. Our approach is based on a versatile probabilistic formulation – the restricted Boltzmann machine (RBM) – where the underlying graphical model is an undirected bipartite graph. Inference is efficient – document representation can be computed with a single matrix projection, making RBMs suitable for massive text corpora available today. Standard RBMs, however, operate on bag-of-words assumption, ignoring the inherent underlying relational structures among words. This results in less coherent word thematic grouping. We introduce graph-based regularization schemes that exploit the linguistic structures, which in turn can be constructed from either corpus statistics or domain knowledge. We demonstrate that the proposed technique improves the group coherence, facilitates visualization, provides means for estimation of intrinsic dimensionality, reduces overfitting, and possibly leads to better classification accuracy.

Design of Near Orthogonal Graph Filter Banks

The processing of signal on graphs is becoming an important emerging area that has great potential in a wide variety of applications, e.g. social network. The work by Narang and Ortega (2012) laid the framework for critically sampled orthogonal two-channel filter bank for signal on undirected bipartite graphs. The design method presented by Narang and Ortega (2012) does not allow the tailoring of the filters spectral response and the control of the reconstruction error of the filter bank. This work present a method to design spectral filters that is based on Bernstein polynomial approximation and constrained optimization. The method allows the trade-off between transition band sharpness, ripple magnitude and reconstruction error.

Stationary vectors of stochastic matrices subject to combinatorial constraints

Given a strongly connected directed graph D, let S_D denote the set of all stochastic matrices whose directed graph is a spanning subgraph of D. We consider the problem of completely describing the set of stationary vectors of irreducible members of S_D. Results from the area of convex polytopes and an association of each matrix with an undirected bipartite graph are used to derive conditions which must be satisfied by a positive probability vector x in order for it to be admissible as a stationary vector of some matrix in S_D. Given some admissible vector x, the set of matrices in S_D that possess x as a stationary vector is also characterised.

Spectral–spatial co-clustering of hyperspectral image data based on bipartite graph

The high dimensionality of hyperspectral images are usually coupled with limited data available, which degenerates the performances of clustering techniques based only on pixel spectral. To improve the performances of clustering, incorporation of spectral and spatial is needed. As an attempt in this direction, in this paper, we propose an unsupervised co-clustering framework to address both the pixel spectral and spatial constraints, in which the relationship among pixels is formulated using an undirected bipartite graph. The optimal partitions are obtained by spectral clustering on the bipartite graph. Experiments on four hyperspectral data sets are performed to evaluate the effectiveness of the proposed framework. Results also show our method achieves similar or better performance when compared to the other clustering methods.