Complex Networks

Introduction to the Special Issue on Complex Networks, Artificial Life journal.Comment: 7 pages, in pres

Complex Networks

An outline of recent work on complex networks is given from the point of view of a physicist. Motivation, achievements and goals are discussed with some of the typical applications from a wide range of academic fields. An introduction to the relevant literature and useful resources is also given.Comment: Review for Contemporary Physics, 31 page

Sznajd Complex Networks

The Sznajd cellular automata corresponds to one of the simplest and yet most interesting models of complex systems. While the traditional two-dimensional Sznajd model tends to a consensus state (pro or cons), the assignment of the contrary to the dominant opinion to some of its cells during the system evolution is known to provide stabilizing feedback implying the overall system state to oscillate around null magnetization. The current article presents a novel type of geographic complex network model whose connections follow an associated feedbacked Sznajd model, i.e. the Sznajd dynamics is run over the network edges. Only connections not exceeding a maximum Euclidean distance $D$ are considered, and any two nodes within such a distance are randomly selected and, in case they are connected, all network nodes which are no further than $D$ are connected to them. In case they are not connected, all nodes within that distance are disconnected from them. Pairs of nodes are then randomly selected and assigned to the contrary of the dominant connectivity. The topology of the complex networks obtained by such a simple growth scheme, which are typically characterized by patches of connected communities, is analyzed both at global and individual levels in terms of a set of hierarchical measurements introduced recently. A series of interesting properties are identified and discussed comparatively to random and scale-free models with the same number of nodes and similar connectivity.Comment: 10 pages, 4 figure

Deep Complex Networks

At present, the vast majority of building blocks, techniques, and architectures for deep learning are based on real-valued operations and representations. However, recent work on recurrent neural networks and older fundamental theoretical analysis suggests that complex numbers could have a richer representational capacity and could also facilitate noise-robust memory retrieval mechanisms. Despite their attractive properties and potential for opening up entirely new neural architectures, complex-valued deep neural networks have been marginalized due to the absence of the building blocks required to design such models. In this work, we provide the key atomic components for complex-valued deep neural networks and apply them to convolutional feed-forward networks and convolutional LSTMs. More precisely, we rely on complex convolutions and present algorithms for complex batch-normalization, complex weight initialization strategies for complex-valued neural nets and we use them in experiments with end-to-end training schemes. We demonstrate that such complex-valued models are competitive with their real-valued counterparts. We test deep complex models on several computer vision tasks, on music transcription using the MusicNet dataset and on Speech Spectrum Prediction using the TIMIT dataset. We achieve state-of-the-art performance on these audio-related tasks

Portraits of Complex Networks

We propose a method for characterizing large complex networks by introducing a new matrix structure, unique for a given network, which encodes structural information; provides useful visualization, even for very large networks; and allows for rigorous statistical comparison between networks. Dynamic processes such as percolation can be visualized using animations. Applications to graph theory are discussed, as are generalizations to weighted networks, real-world network similarity testing, and applicability to the graph isomorphism problem.Comment: 6 pages, 9 figure

Immunization of complex networks

Complex networks such as the sexual partnership web or the Internet often show a high degree of redundancy and heterogeneity in their connectivity properties. This peculiar connectivity provides an ideal environment for the spreading of infective agents. Here we show that the random uniform immunization of individuals does not lead to the eradication of infections in all complex networks. Namely, networks with scale-free properties do not acquire global immunity from major epidemic outbreaks even in the presence of unrealistically high densities of randomly immunized individuals. The absence of any critical immunization threshold is due to the unbounded connectivity fluctuations of scale-free networks. Successful immunization strategies can be developed only by taking into account the inhomogeneous connectivity properties of scale-free networks. In particular, targeted immunization schemes, based on the nodes' connectivity hierarchy, sharply lower the network's vulnerability to epidemic attacks