Static and dynamic properties of small-world connection topologies based on transit-stub networks

Many real complex networks are believed to belong to a class called small-world (SW) networks. SW networks are graphs with high local clustering and small distances between nodes. A standard approach to constructing SW networks consists of varying the probability of rewiring each edge on a regular graph. As the initial substrate for the regular graph some specific topologies are usually selected such as ring-lattices or grids. However, these regular graphs are not suitable for modeling certain hierarchical topologies. A new regular substrate is proposed in this paper. The proposed substrate resembles topologies with certain hierarchical propertiesmore accurately. Then, different dynamics inspired by networking protocols are used to characterize dynamical properties of a network. Measuring transmission times and error rates lead us to consider networks with SW features as the most reliable and fastest, regardless of the routing policies.We thank the MCyT (BFI 2000-015). (RH) was also funded by DE-FG03-96ER14092 and (CA) was supported by ARO-MURI grant DAA655-98-1-0249 during a four month stay at UCSD. We also thank Lev Trimsing for useful discussion

Normality-based validation for crisp clustering

This is the author’s version of a work that was accepted for publication in Pattern Recognition. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Pattern Recognition, 43, 36, (2010) DOI 10.1016/j.patcog.2009.09.018We introduce a new validity index for crisp clustering that is based on the average normality of the clusters. Unlike methods based on inter-cluster and intra-cluster distances, this index emphasizes the cluster shape by using a high order characterization of its probability distribution. The normality of a cluster is characterized by its negentropy, a standard measure of the distance to normality which evaluates the difference between the cluster's entropy and the entropy of a normal distribution with the same covariance matrix. The definition of the negentropy involves the distribution's differential entropy. However, we show that it is possible to avoid its explicit computation by considering only negentropy increments with respect to the initial data distribution, where all the points are assumed to belong to the same cluster. The resulting negentropy increment validity index only requires the computation of covariance matrices. We have applied the new index to an extensive set of artificial and real problems where it provides, in general, better results than other indices, both with respect to the prediction of the correct number of clusters and to the similarity among the real clusters and those inferred.This work has been partially supported with funds from MEC BFU2006-07902/BFI, CAM S-SEM-0255-2006 and CAM/UAM CCG08-UAM/TIC-442

A realistic substrate for Small-world networks modeling

Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. C. Aguirre, F. Corbacho, and R. Huerta, "A realistic substrate for Small-world networks modeling", Proceeding of 12th International Workshop on Database and Expert Systems Applications, Munich, Germany, 2001, pp. 649 - 653Small-World networks are networks with high local clustering and small distance between the nodes. In order to study the properties of these kinds of networks, Watts and Strogatz developed a method based on varying the probability of rewiring each edge on a regular graph. As initial substrate for the regular graph, some specific topologies are usually selected, such as for example, ring-lattice or grids. These regular graphs are not suitable for modeling of certain hierarchical topologies such as for example, holonic systems and Internet. We present a new regular substrate that models more accurately topologies with certain hierarchical properties. We also investigate the dynamics of the diffusion of information packages over the network for different types of network substratesWe thank the Ministerio de Ciencia y Tecnologia (BFI 2000-15). This work has been partially supported by the project ”The Science of Complexity” (ZiF, Bielefeld Universitat, German

The effect of low number of points in clustering validation via the negentropy increment

This is the author’s version of a work that was accepted for publication in Neurocomputing. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Neurocomputing, 74, 16, (2011) DOI: 10.1016/j.neucom.2011.03.023Selected papers of the 10th International Work-Conference on Artificial Neural Networks (IWANN2009)We recently introduced the negentropy increment, a validity index for crisp clustering that quantifies the average normality of the clustering partitions using the negentropy. This index can satisfactorily deal with clusters with heterogeneous orientations, scales and densities. One of the main advantages of the index is the simplicity of its calculation, which only requires the computation of the log-determinants of the covariance matrices and the prior probabilities of each cluster. The negentropy increment provides validation results which are in general better than those from other classic cluster validity indices. However, when the number of data points in a partition region is small, the quality in the estimation of the log-determinant of the covariance matrix can be very poor. This affects the proper quantification of the index and therefore the quality of the clustering, so additional requirements such as limitations on the minimum number of points in each region are needed. Although this kind of constraints can provide good results, they need to be adjusted depending on parameters such as the dimension of the data space. In this article we investigate how the estimation of the negentropy increment of a clustering partition is affected by the presence of regions with small number of points. We find that the error in this estimation depends on the number of points in each region, but not on the scale or orientation of their distribution, and show how to correct this error in order to obtain an unbiased estimator of the negentropy increment. We also quantify the amount of uncertainty in the estimation. As we show, both for 2D synthetic problems and multidimensional real benchmark problems, these results can be used to validate clustering partitions with a substantial improvement.This work has been funded by DGUI-CAM/UAM (Project CCG10-UAM/TIC-5864

Fast response and temporal coherent oscillations in small-world networks

We have investigated the role that different connectivity regimes play in the dynamics of a network of Hodgkin-Huxley neurons by computer simulations. The different connectivity topologies exhibit the following features: random topologies give rise to fast system response yet are unable to produce coherent oscillations in the average activity of the network; on the other hand, regular topologies give rise to coherent oscillations, but in a temporal scale that is not in accordance with fast signal processing. Finally, small-world topologies, which fall between random and regular ones, take advantage of the best features of both, giving rise to fast system response with coherent oscillations.We acknowledge G. Laurent, A. Bäcker, M. Bazhenov, M. Rabinovich, and H. Abarbanel for insightful discussions. We thank the Dirección General de Enseñanza Superior e Investigación Científica for financial support (PB97-1448), the CAM for financial support to L. F. L., and the CCCFC (UAM) for the use of computation resources

An autonomous robot that learns approach-avoidance behaviors: lessons from the brain to the robot

This is an electronic version of the paper presented at the I Jornadas Técnicas de la ETS de Informática, held in Madrid on 200

Fast response and coherent oscillations in small-world Hodgkin-Huxley neural networks

This is an electronic version of the paper presented at the I Jornadas Técnicas de la ETS de Informática, held in Madrid on 200

Small-world topology for multi-agent collaboration

Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. C. Aguirre, J. Martínez-Muñoz, F. Corbacho, and R. Huerta, "Small-world topology for multi-agent collaboration", 11th International Workshop on Database and Expert Systems Applications, London (United Kingdom), 2000, pp. 231 - 235This paper studies a specific methodology for the design of different topologies in multi-agent networks with the central objective of maximizing agent collaboration. In order to obtain this feature we rely on the use of a recently discovered type of topology, namely the “small world” (SW) topology. This topology has been shown to present several advantages such as enhancement of signal-propagation speed computational power, and synchronizability. We have extended the analysis to multi-agent networks searching for the topologies that maximize the flow of entities (data, energy, goods, etc.) with different complexities in the behaviour of each agent in the networkWe thank the Dirección General de Enseñanza Superior e Investigación Científica for financial support (PB97- 1448

Analysis of biologically inspired small-world networks

The final publication is available at Springer via http://dx.doi.org/10.1007/3-540-46084-5_5Proceedings of International Conference Madrid, Spain, August 28–30, 2002Small-World networks are highly clusterized networks with small distances between their nodes. There are some well known biological networks that present this kind of connectivity. On the other hand, the usual models of Small-World networks make use of undirected and unweighted graphs in order to represent the connectivity between the nodes of the network. These kind of graphs cannot model some essential characteristics of neural networks as, for example, the direction or the weight of the synaptic connections. In this paper we analyze different kinds of directed graphs and show that they can also present a Small-World topology when they are shifted from regular to random. Also analytical expressions are given for the cluster coefficient and the characteristic path of these graphs.We thank the Ministerio de Ciencia y Tecnología (BFI 2000-015). (RH) was also funded by DE-FG03-96ER14092, (CA) was partially supported by ARO-MURI grant DAA655-98-1-0249 during a four month stay in UCSD. (PP) and (CA) are partially supported by PB98-085

Feature discovery for data mining

This is an electronic version of the paper presented at the III Taller de Minería de Datos y Aprendizaje, held in Granada on 2005In most problems of Knowledge Discovery the human analyst previously constructs a new set of features, derived from the initial problem input attributes, based on a priori knowledge of the problem structure. These different features are constructed from different transformations which must be selected by the analyst. This paper provides a first step towards a methodology that allows the search for near-optimal representations in classification problems by allowing the automatic selection and composition of feature transformations from an initial set of basis functions. In many cases, the original representation for the problem data is not the most appropriate, and the search for a new representation space that is closer to the structure of the problem to be solved is critical for the successful solution of the problem. On the other hand, once this optimal representation is found, most of the problems may be solved by a linear classification method. As a proof of concept we present two classification problems where the class distributions have a very intricate overlap on the space of original attributes. For these problems, the proposed methodology is able to construct representations based on function compositions from the trigonometric and polynomial bases that provide a solution where some of the classical learning methods, e.g. multilayer perceptrons and decision trees, fail. The methodology consists of a discrete search within the space of compositions of the basis functions and a linear mapping performed by a Fisher discriminant. We play special emphasis on the first part. Finding the optimal composition of basis functions is a difficult problem because of its nongradient nature and the large number of possible combinations. We rely on the global search capabilities of a genetic algorithm to scan the space of function compositions