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Register a new userQuantum walks on complex networks with connection instabilities and community structure
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Dimitris Tsomokos Dr
Publication date
2010
fields
Physics
and research's language is
English
Authors
Dimitris I. Tsomokos
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A continuous-time quantum walk is investigated on complex networks with the characteristic property of community structure, which is shared by most real-world networks. Motivated by the prospect of viable quantum networks, I focus on the effects of network instabilities in the form of broken links, and examine the response of the quantum walk to such failures. It is shown that the reconfiguration of the quantum walk is determined by the community structure of the network. In this context, quantum walks based on the adjacency and Laplacian matrices of the network are compared, and their responses to link failures is analyzed.
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In this theoretical study, we analyze quantum walks on complex networks, which model network-based processes ranging from quantum computing to biology and even sociology. Specifically, we analytically relate the average long time probability distribution for the location of a unitary quantum walker to that of a corresponding classical walker. The distribution of the classical walker is proportional to the distribution of degrees, which measures the connectivity of the network nodes and underlies many methods for analyzing classical networks including website ranking. The quantum distribution becomes exactly equal to the classical distribution when the walk has zero energy and at higher energies the difference, the so-called quantumness, is bounded by the energy of the initial state. We give an example for which the quantumness equals a Renyi entropy of the normalized weighted degrees, guiding us to regimes for which the classical degree-dependent result is recovered and others for which quantum effects dominate.
Determining community structure is a central topic in the study of complex networks, be it technological, social, biological or chemical, in static or interacting systems. In this paper, we extend the concept of community detection from classical to quantum systems---a crucial missing component of a theory of complex networks based on quantum mechanics. We demonstrate that certain quantum mechanical effects cannot be captured using current classical complex network tools and provide new methods that overcome these problems. Our approaches are based on defining closeness measures between nodes, and then maximizing modularity with hierarchical clustering. Our closeness functions are based on quantum transport probability and state fidelity, two important quantities in quantum information theory. To illustrate the effectiveness of our approach in detecting community structure in quantum systems, we provide several examples, including a naturally occurring light-harvesting complex, LHCII. The prediction of our simplest algorithm, semiclassical in nature, mostly agrees with a proposed partitioning for the LHCII found in quantum chemistry literature, whereas our fully quantum treatment of the problem uncovers a new, consistent, and appropriately quantum community structure.
Community structure is one of the most relevant features encountered in numerous real-world applications of networked systems. Despite the tremendous effort of scientists working on this subject over the past few decades to characterize, model, and analyze communities, more investigations are needed to better understand the impact of community structure and its dynamics on networked systems. Here, we first focus on generative models of communities in complex networks and their role in developing strong foundation for community detection algorithms. We discuss modularity and the use of modularity maximization as the basis for community detection. Then, we overview the Stochastic Block Model, its different variants, and inference of community structures from such models. Next, we focus on time evolving networks, where existing nodes and links can disappear and/or new nodes and links may be introduced. The extraction of communities under such circumstances poses an interesting and non-trivial problem that has gained considerable interest over the last decade. We briefly discuss considerable advances made in this field recently. Finally, we focus on immunization strategies essential for targeting the influential spreaders of epidemics in modular networks. Their main goal is to select and immunize a small proportion of individuals from the whole network to control the diffusion process. Various strategies have emerged over the years suggesting different ways to immunize nodes in networks with overlapping and non-overlapping community structure. We first discuss stochastic strategies that require little or no information about the network topology at the expense of their performance. Then, we introduce deterministic strategies that have proven to be very efficient in controlling the epidemic outbreaks, but require complete knowledge of the network.
Crystals arise as the result of the breaking of a spatial translation symmetry. Similarly, translation symmetries can also be broken in time so that discrete time crystals appear. Here, we introduce a method to describe, characterize, and explore the physical phenomena related to this phase of matter using tools from graph theory. The analysis of the graphs allows to visualizing time-crystalline order and to analyze features of the quantum system. For example, we explore in detail the melting process of a minimal model of a period-2 discrete time crystal and describe it in terms of the evolution of the associated graph structure. We show that during the melting process, the network evolution exhibits an emergent preferential attachment mechanism, directly associated with the existence of scale-free networks. Thus, our strategy allows us to propose a previously unexplored far-reaching application of time crystals as a quantum simulator of complex quantum networks.
Controlling complex networks is of paramount importance in science and engineering. Despite the recent development of structural-controllability theory, we continue to lack a framework to control undirected complex networks, especially given link weights. Here we introduce an exact-controllability paradigm based on the maximum multiplicity to identify the minimum set of driver nodes required to achieve full control of networks with arbitrary structures and link-weight distributions. The framework reproduces the structural controllability of directed networks characterized by structural matrices. We explore the controllability of a large number of real and model networks, finding that dense networks with identical weights are difficult to be controlled. An efficient and accurate tool is offered to assess the controllability of large sparse and dense networks. The exact-controllability framework enables a comprehensive understanding of the impact of network properties on controllability, a fundamental problem towards our ultimate control of complex systems.
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