Graph Partitioning and Continuous Quadratic Programming

Automatic detection of salient image regions is a useful tool with applications in intelligent camera control, virtual cinematography, video summarization and editing, evaluation of viewer preferences, and many others. This paper presents an effective method for detecting potentially salient foreground regions. Salient regions are identified by eigenvalue analysis of a graph Laplacian that is defined over the color similarity of image superpixels, under the assumption that the majority of pixels on the image boundary show non-salient background. In contrast to previous methods based on graph-cuts or graph partitioning, our method provides continuously-valued saliency estimates with complementary properties to recently proposed color contrast-based approaches. Moreover, exploiting discriminative properties of the Fiedler vector, we devise an SVM-based classifier that allows us to determine whether an image contains any salient objects at all, a problem that has been largely neglected in previous works. We also describe how the per-frame saliency detection can be extended to improve its spatiotemporal coherence when computed on video sequences. Extensive evaluation on several datasets demonstrates and validates the state-of-the-art performance of the proposed method.

The Fiedler vector of a graph, namely the eigenvector corresponding to the second smallest eigenvalue of a graph Laplacian matrix, plays an important role in spectral graph theory with applications in problems such as graph bi-partitioning and envelope reduction. Algorithms designed to estimate this quantity usually rely on a priori knowledge of the entire graph, and employ techniques such as graph sparsification and power iterations, which have obvious shortcomings in cases where the graph is unknown, or changing dynamically. In this paper, we develop a framework in which we construct a stochastic process based on a set of interacting random walks on a graph and show that a suitably scaled version of our stochastic process converges to the Fiedler vector for a sufficiently large number of walks. Like other techniques based on exploratory random walks and on-the-fly computations, such as Markov Chain Monte Carlo (MCMC), our algorithm overcomes challenges typically faced by power iteration based approaches. But, unlike any existing random walk based method such as MCMCs where the focus is on the leading eigenvector, our framework with interacting random walks converges to the Fiedler vector (second eigenvector). We also provide numerical results to confirm our theoretical findings on different graphs, and show that our algorithm performs well over a wide range of parameters and the number of random walks. Simulations results over time varying dynamic graphs are also provided to show the efficacy of our random walk based technique in such settings. As an important contribution, we extend our results and show that our framework is applicable for approximating not just the Fiedler vector of graph Laplacians, but also the second eigenvector of any time reversible Markov Chain kernel via interacting random walks.

It is well known that the algebraic connectivity of a graph is the second small eigenvalue of its Laplacian matrix. In this paper, we mainly research the relationships between the algebraic connectivity and the disjoint vertex subsets of graphs, which are presented through some upper bounds on algebraic connectivity.

In this paper, we demonstrate a phase transition phenomenon in algebraic connectivity of small-world networks. Algebraic connectivity of a graph is the second smallest eigenvalue of its Laplacian matrix and a measure of speed of solving consensus problems in networks. We demonstrate that it is possible to dramatically increase the algebraic connectivity of a regular complex network by 1000 times or more without adding new links or nodes to the network. This implies that a consensus problem can be solved incredibly fast on certain small-world networks giving rise to a network design algorithm for ultra fast information networks. Our study relies on a procedure called "random rewiring" due to Watts & Strogatz (Nature, 1998). Extensive numerical results are provided to support our claims and conjectures. We prove that the mean of the bulk Laplacian spectrum of a complex network remains invariant under random rewiring. The same property only asymptotically holds for scale-free networks. A relationship between increasing the algebraic connectivity of complex networks and robustness to link and node failures is also shown. This is an alternative approach to the use of percolation theory for analysis of network robustness. We also show some connections between our conjectures and certain open problems in the theory of random matrices.

This paper presents a complex network and graph spectral approach to calculate the resiliency of complex engineered systems. Resiliency is a key driver in how systems are developed to operate in an unexpected operating environment, and how systems change and respond to the environments in which they operate. This paper deduces resiliency properties of complex engineered systems based on graph spectra calculated from their adjacency matrix representations, which describes the physical connections between components in a complex engineered systems. In conjunction with the adjacency matrix, the degree and Laplacian matrices also have eigenvalue and eigenspectrum properties that can be used to calculate the resiliency of the complex engineered system. One such property of the Laplacian matrix is the algebraic connectivity. The algebraic connectivity is defined as the second smallest eigenvalue of the Laplacian matrix and is proven to be directly related to the resiliency of a complex network. Our motivation in the present work is to calculate the algebraic connectivity and other graph spectra properties to predict the resiliency of the system under design.

The algebraic connectivity of a graph is the second smallest eigenvalue of the graph Laplacian, and is a measure of how well-connected the graph is. We study the problem of adding edges (from a set of candidate edges) to a graph so as to maximize its algebraic connectivity. This is a difficult combinatorial optimization, so we seek a heuristic for approximately solving the problem. The standard convex relaxation of the problem can be expressed as a semidefinite program (SDP); for modest sized problems, this yields a cheaply computable upper bound on the optimal value, as well as a heuristic for choosing the edges to be added. We describe a new greedy heuristic for the problem. The heuristic is based on the Fiedler vector, and therefore can be applied to very large graphs

An important application of graph partitioning is data clustering using a graph model - the pairwise similarities between all data objects form a weighted graph adjacency matrix that contains all necessary information for clustering. In this paper, we propose a new algorithm for graph partitioning with an objective function that follows the min-max clustering principle. The relaxed version of the optimization of the min-max cut objective function leads to the Fiedler vector in spectral graph partitioning. Theoretical analyses of min-max cut indicate that it leads to balanced partitions, and lower bounds are derived. The min-max cut algorithm is tested on newsgroup data sets and is found to out-perform other current popular partitioning/clustering methods. The linkage-based refinements to the algorithm further improve the quality of clustering substantially. We also demonstrate that a linearized search order based on linkage differential is better than that based on the Fiedler vector, providing another effective partitioning method.

An important challenge in the study of complex systems is to identify appropriate effective variables at different times. In this paper, we explain why structures that are persistent with respect to changes in length and time scales are proper effective variables, and illustrate how persistent structures can be identified from the spectra and Fiedler vector of the graph Laplacian at different stages of the topological data analysis (TDA) filtration process for twelve toy models. We then investigated four market crashes, three of which were related to the COVID-19 pandemic. In all four crashes, a persistent gap opens up in the Laplacian spectra when we go from a normal phase to a crash phase. In the crash phase, the persistent structure associated with the gap remains distinguishable up to a characteristic length scale where the first non-zero Laplacian eigenvalue changes most rapidly. Before , the distribution of components in the Fiedler vector is predominantly bi-modal, and this distribution becomes uni-modal after Our findings hint at the possibility of understanding market crashs in terms of both continuous and discontinuous changes. Beyond the graph Laplacian, we can also employ Hodge Laplacians of higher order for future research.

PurposeThe purpose of this paper is to introduce a general equation for eigensolution. Eigenvalues and eigenvectors of graphs have many applications in combinatorial optimization and structural mechanics. Some important applications of graph products consist of nodal ordering and graph partitioning for structuring the structural matrices and finite element subdomaining, respectively.Design/methodology/approachIn the existing methods for the eigensolution of Laplacian matrices, members have been added to the model of a graph product such that for its Laplacian matrix an algebraic relation between blocks become possible. These methods are categorized as topological approaches. Here, using concepts of linear algebra a general algebraic method is developed.FindingsA new algebraic method is introduced for calculating the eigenvalues of Laplacian matrices in graph products.Originality/valueThe present method provides a simple tool for calculating the eigenvalues of the Laplacian matrices without using the configurational model and merely by using the Laplacian matrices. The developed formula for calculating the eigenvalues contains approximate terms which can be managed by the analyst.

Algebraic connectivity maximization of an air transportation network: The flight routes’ addition/deletion problem

This paper proposes a new scheme for very fast detection of a sensor going offline or loss of connectivity of sensor in Wireless Mesh Networks (WMN) in order to avoid any packet loss. A sensor is declared offline by a gateway if it does not receive any packet from the sensor for a specific time interval termed as keep-alive time. In mesh networks, this keep-alive window is kept for a very long duration; sometimes it is as high as 2.5 to 3 hours. The time duration is kept high so that gateway does not declare a sensor that is in deep sleep to conserve battery, as offline. However, keeping a large keep-alive time window leads to the very likely possibility of missing the timely detection of a critical event like a smoke sensor or gas leak sensor going offline. In the proposed method, the gateway maintains its table of neighbourhood matrix in order to facilitate fast and efficient tracking as well as tracing of the nodes for the detection of the loss of connectivity of a sensor. A Laplacian matrix formulation proposed by Fiedler is invoked for tracking and tracing lost nodes of WMN. Through the Fiedler value, this paper substantiates the efficacy of monitoring a large WMN not only for its algebraic connectivity but also to compute the number of disconnected partitions in it. This is achieved through the number of null values of eigenvalues of the Laplacian matrix of the graph of WMN. Through simulation studies, this paper demonstrates achieving an 88 % success rate in the detection of disconnection with an average latency of 39.806 ms. On the contrary, a conventional process for the time duration after which a sensor can be declared offline might be an hour or even more depending upon the pre-set duration of the keep-alive time window.

Let G be an undirected unweighted graph on n vertices and let L be its Laplacian matrix. It is known that if is the group inverse of L, then for is a lower bound on the algebraic connectivity μ(G of G Merris has introduced and characterized the class of all maximal graphs of all orders. These are graphs whose degree sequence is not majorized by the degree sequence of any other graph. Here we show that if Ç is a maximal graph and L is its Laplacian, then 1/(L #)=μ(Ç). We provide an example to show that the converse of this result is not valid.

The communication links shared among a team of Unmanned Aerial Vehicles (UAVs) can be represented as a Laplacian matrix. The second smallest eigenvalue of the matrix, known as algebraic connectivity, is a metric typically used to measure the connectivity of UAVs. Algebraic connectivity represents the robustness of the communication network. For an increasing number of UAV applications, varying levels of connectivity at different points throughout the duration of a mission are needed to meet operational requirements. Thus, the ability to track and obtain a desired connectivity profile over the course of a mission is critical. This work compares four different methods for increasing or decreasing the connectivity for a team of UAVs by adding or removing communication links between UAVs. A connectivity tracking algorithm was developed using these methods to track a profile of varying connectivity over time and an analysis of the results is provided.

Synchronous neural oscillations within the beta frequency range are observed across the parkinsonian basal ganglia network, including within the subthalamic nucleus (STN) - globus pallidus (GPe) subcircuit. The emergence of pathological synchrony in Parkinson's disease is often attributed to changes in neural properties or connection strength, and less often to the network topology, i.e. the structural arrangement of connections between neurons. This study investigates the relationship between network structure and neural synchrony in a model of the STN-GPe circuit comprised of conductance-based spiking neurons. Changes in net synaptic input were controlled for through a synaptic scaling rule, which facilitated separation of the effects of network structure from net synaptic input. Five topologies were examined as structures for the STN-GPe circuit: Watts-Strogatz, preferential attachment, spatial, stochastic block, k-regular random. Beta band synchrony generally increased as the number of connections increased, however the exact relationship was topology specific. Varying the wiring pattern while maintaining a constant number of connections caused network synchrony to be enhanced or suppressed, demonstrating the ability of purely structural changes to alter synchrony. This relationship was well-captured by the algebraic connectivity of the network, the second smallest eigenvalue of the network's Laplacian matrix. The structure-synchrony relationship was further investigated in a network model designed to emulate the action selection role of the STN-GPe circuit. It was found that increasing the number of connections and/or the overlap of action selection channels could lead to a rapid transition to synchrony, which was also predicted by the algebraic connectivity.

Effective communication is crucial for the performance and collaboration within cooperative networked multi-agent systems. However, existing literature lacks comprehensive solutions for dynamically monitoring and adjusting communication topologies to balance connectivity and energy efficiency. This study addresses this gap by proposing a distributed approach for estimating and controlling system connectivity over time. We introduce a modified consensus protocol where agents exchange local assessments of communication link quality, enabling the estimation of a global weighted adjacency matrix without requiring centralized information. The system's connectivity is measured using the second smallest eigenvalue of the communication graph Laplacian, commonly referred to as algebraic connectivity. Additionally, we enhance the consensus protocol with an adaptive mechanism to expedite convergence, irrespective of system size or structure. Furthermore, we present an analytical method for connectivity control based on the Fiedler vector approximation, facilitating the addition or removal of communication links. This method adjusts control parameters to accommodate minor variations in link quality while reconfiguring the network in response to significant changes. Notably, it identifies and eliminates energy-consuming yet non-contributory links, improving long-term connectivity efficiency. Simulation experiments across diverse scenarios and the number of agents validate the efficacy of our proposed algebraic connectivity estimation and tracking strategy. Results demonstrate robust connectivity maintenance against external disturbances and agent failures, underscoring the practical utility of our approach for real-world multi-agent systems.