Complex Networks: New Concepts and Tools for Real-Time Imaging and Vision

This article discusses how concepts and methods of complex networks can be applied to real-time imaging and computer vision. After a brief introduction of complex networks basic concepts, their use as means to represent and characterize images, as well as for modeling visual saliency, are briefly described. The possibility to apply complex networks in order to model and simulate the performance of parallel and distributed computing systems for performance of visual methods is also proposed.Comment: 3 page

Exploring Complex Networks through Random Walks

Most real complex networks -- such as protein interactions, social contacts, the internet -- are only partially known and available to us. While the process of exploring such networks in many cases resembles a random walk, it becomes a key issue to investigate and characterize how effectively the nodes and edges of such networks can be covered by different strategies. At the same time, it is critically important to infer how well can topological measurements such as the average node degree and average clustering coefficient be estimated during such network explorations. The present article addresses these problems by considering random and Barab\'asi-Albert (BA) network models with varying connectivity explored by three types of random walks: traditional, preferential to untracked edges, and preferential to unvisited nodes. A series of relevant results are obtained, including the fact that random and BA models with the same size and average node degree allow similar node and edge coverage efficiency, the identification of linear scaling with the size of the network of the random walk step at which a given percentage of the nodes/edges is covered, and the critical result that the estimation of the averaged node degree and clustering coefficient by random walks on BA networks often leads to heavily biased results. Many are the theoretical and practical implications of such results.Comment: 5 pages, 5 figure

The Path-Star Transformation and its Effects on Complex Networks

A good deal of the connectivity of complex networks can be characterized in terms of their constituent paths and hubs. For instance, the Barab\'asi-Albert model is known to incorporate a significative number of hubs and relatively short paths. On the other hand, the Watts-Strogatz model is underlain by a long path and almost complete absence of hubs. The present work investigates how the topology of complex networks changes when a path is transformed into a star (or, for long paths, a hub). Such a transformation keeps the number of nodes and does not increase the number of edges in the network, but has potential for greatly changing the network topology. Several interesting results are reported with respect to Erdos-R\'enyi, Barab\'asi-Albert and Watts-Strogats models, including the unexpected finding that the diameter and average shortest path length of the former type of networks are little affected by the path-star transformation. In addition to providing insight about the organization of complex networks, such transformations are also potentially useful for improving specific aspects of the network connectivity, e.g. average shortest path length as required for expedite communication between nodes.Comment: 8 pages, 2 figures, 1 table. A working manuscript, comments welcome

On the Separability of Attractors in Grandmother Dynamic Systems with Structured Connectivity

The combination of complex networks and dynamic systems research is poised to yield some of the most interesting theoretic and applied scientific results along the forthcoming decades. The present work addresses a particularly important related aspect, namely the quantification of how well separated can the attractors be in dynamic systems underlined by four types of complex networks (Erd\H{o}s-R\'enyi, Barab\'asi-Albert, Watts-Strogatz and as well as a geographic model). Attention is focused on grandmother dynamic systems, where each state variable (associated to each node) is used to represent a specific prototype pattern (attractor). By assuming that the attractors spread their influence among its neighboring nodes through a diffusive process, it is possible to overlook the specific details of specific dynamics and focus attention on the separability among such attractors. This property is defined in terms of two separation indices (one individual to each prototype and the other considering also the immediate neighborhood) reflecting the balance and proximity to attractors revealed by the activation of the network after a diffusive process. The separation index considering also the neighborhood was found to be much more informative, while the best separability was observed for the Watts-Strogatz and the geographic models. The effects of the involved parameters on the separability were investigated by correlation and path analyses. The obtained results suggest the special importance of some measurements in underlying the relationship between topology and dynamics.Comment: 23 pages, 12 figures, 3 table

Knitted Complex Networks

To a considerable extent, the continuing importance and popularity of complex networks as models of real-world structures has been motivated by scale free degree distributions as well as the respectively implied hubs. Being related to sequential connections of edges in networks, paths represent another important, dual pattern of connectivity (or motif) in complex networks (e.g., paths are related to important concepts such as betweeness centrality). The present work proposes a new supercategory of complex networks which are organized and/or constructed in terms of paths. Two specific network classes are proposed and characterized: (i) PA networks, obtained by star-path transforming Barabasi-Albert networks; and (ii) PN networks, built by performing progressive paths involving all nodes without repetition. Such new networks are important not only from their potential to provide theoretical insights, but also as putative models of real-world structures. The connectivity structure of these two models is investigated comparatively to four traditional complex networks models (Erdos-Renyi, Barabasi-Albert, Watts-Strogatz and a geographical model). A series of interesting results are described, including the corroboration of the distinct nature of the two proposed models and the importance of considering a comprehensive set of measurements and multivariated statistical methods for the characterization of complex networks.Comment: 10 pages, 5 figures, 1 table. A working manuscript, comments and suggestions welcome

Trajectory Networks and Their Topological Changes Induced by Geographical Infiltration

In this article we investigate the topological changes undergone by trajectory networks as a consequence of progressive geographical infiltration. Trajectory networks, a type of knitted network, are obtained by establishing paths between geographically distributed nodes while following an associated vector field. For instance, the nodes could correspond to neurons along the cortical surface and the vector field could correspond to the gradient of neurotrophic factors, or the nodes could represent towns while the vector fields would be given by economical and/or geographical gradients. Therefore trajectory networks are natural models of a large number of geographical structures. The geographical infiltrations correspond to the addition of new local connections between nearby existing nodes. As such, these infiltrations could be related to several real-world processes such as contaminations, diseases, attacks, parasites, etc. The way in which progressive geographical infiltrations affect trajectory networks is investigated in terms of the degree, clustering coefficient, size of the largest component and the lengths of the existing chains measured along the infiltrations. It is shown that the maximum infiltration distance plays a critical role in the intensity of the induced topological changes. For large enough values of this parameter, the chains intrinsic to the trajectory networks undergo a collapse which is shown not to be related to the percolation of the network also implied by the infiltrations.Comment: 10 pages, 8 figures. A working manuscript: suggestions and collaborations welcome

Diffusion of Time-Varying Signals in Complex Networks: A Structure-Dynamics Investigation Focusing the Distance to the Source of Activation

The way in which different types of dynamics unfold in complex networks is intrinsically related to the propagation of activation along nodes, which is strongly affected by the network connectivity. In this work we investigate to which extent a time-varying signal emanating from a specific node is modified as it diffuses, at the equilibrium regime, along uniformly random (Erd\H{o}s-R\'enyi) and scale-free (Barab\'asi-Albert) networks. The degree of preservation is quantified in terms of the Pearson cross-correlation between the original signal and the derivative of the signals appearing at each node along time. Several interesting results are reported. First, the fact that quite distinct signals are typically obtained at different nodes in the considered networks implies mean-field approaches to be completely inadequate. It has also been found that the peak and lag of the similarity time-signatures obtained for each specific node are strongly related to the respective distance between that node and the source node. Such a relationship tends to decrease with the average degree of the networks. Also, in the case of the lag, a less intense relationship is verified for scale-free networks. No relationship was found between the dispersion of the similarity signature and the distance to the source.Comment: 9 pages, 5 figure

Random and Longest Paths: Unnoticed Motifs of Complex Networks

Paths are important structural elements in complex networks because they are finite (unlike walks), related to effective node coverage (minimum spanning trees), and can be understood as being dual to star connectivity. This article introduces the concept of random path applies it for the investigation of structural properties of complex networks and as the means to estimate the longest path. Random paths are obtained by selecting one of the network nodes at random and performing a random self-avoiding walk (here called path-walk) until its termination. It is shown that the distribution of random paths are markedly different for diverse complex network models (i.e. Erdos-Renyi, Barabasi-Albert, Watts-Strogatz, a geographical model, as well as two recently introduced path-based network types), with the BA structures yielding the shortest random walks, while the longest paths are produced by WS networks. Random paths are also explored as the means to estimate the longest paths (i.e. several random paths are obtained and the longest taken). The convergence to the longest path and its properties ire characterized with respect to several networks models. Several results are reported and discussed, including the markedly distinct lengths of the longest paths obtained for the different network models.Comment: 13 pages, 8 figures. A working manuscript: comments welcome

The Hierarchical Backbone of Complex Networks

Given any complex directed network, a set of acyclic subgraphs - the hierarchical backbone of the network - can be extracted that will provide valuable information about its hierarchical structure. The current paper presents how the interpretation of the network weight matrix as a transition matrix allows the hierarchical backbone to be identified and characterized in terms of the concepts of hierarchical degree, which expresses the total number of virtual edges established along successive transitions, and of hierarchical successors, namely the number of nodes accessible from a specific node while moving successive hierarchical levels. The potential of the proposed approach is illustrated with respect to word associations and gene sequencing data.Comment: 4 pages, 5 figure

The Heart of Protein-Protein Interaction Networks

Recent developments in complex networks have paved the way to a series of important biological insights, such as the fact that many of the essential proteins of S. cerevisae corresponds to the so-called hubs of the respective protein-protein interaction networks. Despite the special importance of hubs, other types of nodes such as those corresponding to the network border, as well as the innermost nodes, also deserve special attention. This work reports on how the application of the concept of distance transform to networks showed that a great deal of the innermost nodes correspond to essential proteins, with interesting biological implications.Comment: 4 pages, 1 figur