Structural engineering of evolving complex dynamical networks

Networks are ubiquitous in nature and many natural and man-made systems can be modelled as networked systems. Complex networks, systems comprising a number of nodes that are connected through edges, have been frequently used to model large-scale systems from various disciplines such as biology, ecology, and engineering. Dynamical systems interacting through a network may exhibit collective behaviours such as synchronisation, consensus, opinion formation, flocking and unusual phase transitions. Evolution of such collective behaviours is highly dependent on the structure of the interaction network. Optimisation of network topology to improve collective behaviours and network robustness can be achieved by intelligently modifying the network structure. Here, it is referred to as "Engineering of the Network". Although coupled dynamical systems can develop spontaneous synchronous patterns if their coupling strength lies in an appropriate range, in some applications one needs to control a fraction of nodes, known as driver nodes, in order to facilitate the synchrony. This thesis addresses the problem of identifying the set of best drivers, leading to the best pinning control performance. The eigen-ratio of the augmented Laplacian matrix, that is the largest eigenvalue divided by the second smallest one, is chosen as the controllability metric. The approach introduced in this thesis is to obtain the set of optimal drivers based on sensitivity analysis of the eigen-ratio, which requires only a single computation of the eigenvector associated with the largest eigenvalue, and thus is applicable for large-scale networks. This leads to a new "controllability centrality" metric for each subset of nodes. Simulation results reveal the effectiveness of the proposed metric in predicting the most important driver(s) correctly.     Interactions in complex networks might also facilitate the propagation of undesired effects, such as node/edge failure, which may crucially affect the performance of collective behaviours. In order to study the effect of node failure on network synchronisation, an analytical metric is proposed that measures the effect of a node removal on any desired eigenvalue of the Laplacian matrix. Using this metric, which is based on the local multiplicity of each eigenvalue at each node, one can approximate the impact of any node removal on the spectrum of a graph. The metric is computationally efficient as it only needs a single eigen-decomposition of the Laplacian matrix. It also provides a reliable approximation for the "Laplacian energy" of a network. Simulation results verify the accuracy of this metric in networks with different topologies. This thesis also considers formation control as an application of network synchronisation and studies the "rigidity maintenance" problem, which is one of the major challenges in this field. This problem is to preserve the rigidity of the sensing graph in a formation during motion, taking into consideration constraints such as line-of-sight requirements, sensing ranges and power limitations. By introducing a "Lattice of Configurations" for each node, a distributed rigidity maintenance algorithm is proposed to preserve the rigidity of the sensing network when failure in a sensing link would result in loss of rigidity. The proposed algorithm recovers rigidity by activating, almost always, the minimum number of new sensing links and considers real-time constraints of practical formations. A sufficient condition for this problem is proved and tested via numerical simulations. Based on the above results, a number of other areas and applications of network dynamics are studied and expounded upon in this thesis

Markov Tensor Theory and Cascade, Reachability, and Routing in Complex Networks

University of Minnesota Ph.D. dissertation. 2017. Major: Electrical Engineering. Advisor: Zhi-Li Zhang. 1 computer file (PDF); 180 pages.In this dissertation, we study and characterize the networks as the medium and substrate for communications, interactions, and flows by addressing various crucial problems under the general topics of cascade, reachability, and routing. These are general problem domains common in several applications and from a variety of networks. We address these problems in a unified way by a theoretical platform that we have developed in this research, which we call Markov Tensor Theory. How does a phenomena, influence, or a failure cascade in a network and what are the key factors in this cascade? We study the influence cascade in social networks and introduce the Heat Conduction (HC) Model which captures both social influence and non-social influence, and extends many of the existing non-progressive models. We then prove that selecting the optimal seed set of influential nodes for maximizing the influence spread is NP-hard for HC, however, by establishing the submodularity of influence spread, we tackle the influence maximization problem with a scalable and provably near-optimal greedy algorithm. We also study failure cascade in inter-dependent networks where we considered the effects of cascading failures both within and across different layers. In this study, we investigate how different couplings (i.e., inter-dependencies) between network elements across layers affect the cascading failure dynamics. How failures or disruptions affect the network in terms of reachability of entities from each other, how to identify the reachabilities efficiently after failures, and who are the pivotal players in the reachabilities? We develop an oracle to answer dynamic reachabilities efficiently for failure-prone networks with frequent reachability query requirement. Founded on the concept of reachability, we also introduce and provide a formulation for finding articulation points, measuring network load balancing, and computing pivotality ranking of nodes. Once the reachabilities are determined, how to quickly and robustly route a flow from a part of the network to the other part of a network under the failures? To avoid solely relying on the shortest path and generate alternative paths on one hand, and to correct the degeneracy of hitting time distance, on the other hand, we develop a novel routing continuum method from shortest-path routing to all-path routing which provides both a closed form formulation for computing the continuum distances and an efficient routing strategy. We also devise an oracle for efficiently answering to single-source shortest path queries as well as finding the replacement paths in the case of multiple failures. For these studies, we develop Markov Tensor Theory as a platform of powerful theories and tools founded on Markov chain theory and random walk methods which supports the general weighted and directed networks

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum