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Understanding the localization properties of eigenvectors of complex networks is important to get insight into various structural and dynamical properties of the corresponding systems. Here, we analytically develop a scheme to construct a highly localized network for a given set of networks parameters that is the number of nodes and the number of interactions. We find that the localization behavior of the principal eigenvector (PEV) of such a network is sensitive against a single edge rewiring. We find evidences for eigenvalue crossing phenomena as a consequence of the single edge rewiring, in turn providing an origin to the sensitive behavior of the PEV localization. These insights were then used to analytically construct the highly localized network for a given set of networks parameters. The analysis provides fundamental insight into relationships between the structural and the spectral properties of networks for PEV localized networks. Further, we substantiate the existence of the eigenvalue crossing phenomenon by considering a linear-dynamical process, namely the ribonucleic acid (RNA) neutral network population dynamical model. The analysis presented here on model networks aids in understanding the steady-state behavior of a broad range of linear-dynamical processes, from epidemic spreading to biochemical dynamics associated with the adjacency matrices.
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We study the effect of localized attacks on a multiplex spatial network, where each layer is a network of communities. The system is considered functional when the nodes belong to the giant component in all the multiplex layers. The communities are of linear size $zeta$, such that within them any pair of nodes are linked with same probability, and additionally nodes in nearby communities are linked with a different (typically smaller) probability. This model can represent an interdependent infrastructure system of cities where within the city there are many links while between cities there are fewer links. We develop an analytical method, similar to the finite element method applied to a network with communities, and verify our analytical results by simulations. We find, both by simulation and theory, that for different parameters of connectivity and spatiality --- there is a critical localized size of damage above which it will spread and the entire system will collapse.
Much of neuroscience aims at reverse engineering the brain, but we only record a small number of neurons at a time. We do not currently know if reverse engineering the brain requires us to simultaneously record most neurons or if multiple recordings from smaller subsets suffice. This is made even more important by the development of novel techniques that allow recording from selected subsets of neurons, e.g. using optical techniques. To get at this question, we analyze a neural network, trained on the MNIST dataset, using only partial recordings and characterize the dependency of the quality of our reverse engineering on the number of simultaneously recorded neurons. We find that reverse engineering of the nonlinear neural network is meaningfully possible if a sufficiently large number of neurons is simultaneously recorded but that this number can be considerably smaller than the number of neurons. Moreover, recording many times from small random subsets of neurons yields surprisingly good performance. Application in neuroscience suggests to approximate the I/O function of an actual neural system, we need to record from a much larger number of neurons. The kind of scaling analysis we perform here can, and arguably should be used to calibrate approaches that can dramatically scale up the size of recorded data sets in neuroscience.
It has been widely assumed that a neural network cannot be recovered from its outputs, as the network depends on its parameters in a highly nonlinear way. Here, we prove that in fact it is often possible to identify the architecture, weights, and biases of an unknown deep ReLU network by observing only its output. Every ReLU network defines a piecewise linear function, where the boundaries between linear regions correspond to inputs for which some neuron in the network switches between inactive and active ReLU states. By dissecting the set of region boundaries into components associated with particular neurons, we show both theoretically and empirically that it is possible to recover the weights of neurons and their arrangement within the network, up to isomorphism.
Sustainable urban design or planning is not a LEGO-like assembly of prefabricated elements, but an embryo-like growth with persistent differentiation and adaptation towards a coherent whole. The coherent whole has a striking character - called living structure - that consists of far more small substructures than large ones. To detect the living structure, natural streets or axial lines have been previously adopted to be topologically represent an urban environment as a coherent whole. This paper develops a new approach to detecting the underlying living structure of urban environments. The approach takes an urban environment as a whole and recursively decomposes it into meaningful subwholes at different levels of hierarchy or scale ranging from the largest to the smallest. We compared the new approach to natural street and axial line approaches and demonstrated, through four case studies, that the new approach is better and more powerful. Based on the study, we further discuss how the new approach can be used not only for understanding, but also for effectively designing or planning the living structure of an urban environment to be more living or more livable. Keywords: Urban design or planning, structural beauty, space syntax, natural streets, life, wholeness
Networks are universally considered as complex structures of interactions of large multi-component systems. In order to determine the role that each node has inside a complex network, several centrality measures have been developed. Such topological features are also important for their role in the dynamical processes occurring in networked systems. In this paper, we argue that the dynamical activity of the nodes may strongly reshape their relevance inside the network making centrality measures in many cases misleading. We show that when the dynamics taking place at the local level of the node is slower than the global one between the nodes, then the system may lose track of the structural features. On the contrary, when that ratio is reversed only global properties such as the shortest distances can be recovered. From the perspective of networks inference, this constitutes an uncertainty principle, in the sense that it limits the extraction of multi-resolution information about the structure, particularly in the presence of noise. For illustration purposes, we show that for networks with different time-scale structures such as strong modularity, the existence of fast global dynamics can imply that precise inference of the community structure is impossible.
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