Resilient Design of Complex Engineered Systems

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.

Water distribution systems are regarded as large sparse graphs with complex network characteristics. Topological aspects of system resilience, viewed as the antonym to structural vulnerability, are assessed in connection with the network architecture, robustness and loop redundancy. Deterministic techniques from complex networks and spectral graph theory are utilized to quantify well-connectedness and estimate loop redundancy in the studied benchmark networks. By using graph connectivity and expansion properties, system robustness against node/link failures and isolation of the demand nodes from the source(s) are assessed and network tolerance against random failures and targeted attacks on their bridges and cut sets are analyzed. Among other measurements, two metrics of meshed-ness and algebraic connectivity are proposed as candidates for quantification of redundancy and robustness, respectively, in optimization design models. A brief discussion on the scope and relevance of the provided measurements in the analysis of resilience of water distribution networks is presented.

Algebraic connectivity metric for spare capacity allocation problem in survivable networks

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.

For network graphs, numerous graph features are intimately linked to eigenvalues of the Laplacian matrix, such as connectivity and diameter. Thus, it is very important to solve eigenvalues of the Laplacian matrix for graphs. Similarly, for higher-order networks, eigenvalues of combinatorial Laplacian matrices are also important for invariants of graphs. However, for large-scale networks, it is difficult to calculate eigenvalues of the Laplacian matrix directly because it is either very difficult to obtain the whole network structure or requires a lot of computing resources. Therefore, this article makes the following contributions. Firstly, this paper proposes a random walk approach for estimating the bounds of the greatest eigenvalues of Laplacian matrices for large-scale networks. Considering the relationship between the spectral moments of the adjacency matrix and the closed paths in the network, we utilize the relationship between the adjacency matrix and the Laplacian matrix to establish the relationship between the Laplacian matrix and the closed paths. Then, we employ equiprobable random walks to sample the large graph to obtain the small graph. Through algebraic topology knowledge, we obtain the bounds of the largest eigenvalue of the Laplacian matrix of the large graph by using Laplacian spectral moments of the small graph. Secondly, for high-order networks, this paper proposes a method based on random walks and graph transformations. The graph transformation we propose mainly converts graphs with second-order simplices into ordinary weighted graphs, thereby transforming the problem of solving the spectral moments of the second-order combined Laplacian matrix into solving the spectral moments of the adjacency matrix. Then, we use the aforementioned random walk method to solve bounds of the greatest eigenvalue of the second-order combinatorial Laplacian matrix. Finally, by comparing the proposed method with existing algorithms in synthetic and real networks, its accuracy and superiority are demonstrated.

Robustness and Optimization of Complex Networks: Reconstructability, Algorithms and Modeling

Optimizing algebraic connectivity by edge rewiring

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.

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.

On graphs whose signless Laplacian index does not exceed 4.5

Graphs whose signless Laplacian spectral radius does not exceed the Hoffman limit value

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.

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.

A multi-level finite element nodal ordering using algebraic graph theory

Graph theory plays a fundamental role in various fields of science and engineering, providing powerful tools for modeling and analyzing relationships among entities. One of the most effective ways to study graphs is through matrix representation. This paper explores the three primary matrix representations of graphs: the adjacency matrix, The adjacency matrix provides direct insight into vertex connectivity and the incidence matrix, the incidence matrix reflects the relationship between edges and vertices. and the Laplacian matrix defined as the difference between the degree matrix and the adjacency matrix, plays a central role in spectral graph theory. Matrix representations enable efficient storage, computation, and analysis of graphs using linear algebraic techniques. They form the basis for many modern algorithms in graph theory, This paper discusses the mathematical foundations, construction methods, and practical applications of these matrix forms, highlighting their essential role in both theoretical and applied graph analysis.