Mean-field theory for clustering coefficients in Barabasi-Albert networks

We applied a mean field approach to study clustering coefficients in Barabasi-Albert networks. We found that the local clustering in BA networks depends on the node degree. Analytic results have been compared to extensive numerical simulations finding a very good agreement for nodes with low degrees. Clustering coefficient of a whole network calculated from our approach perfectly fits numerical data.Comment: 8 pages, 3 figure

Measuring the Generalized Friendship Paradox in Networks with Quality-dependent Connectivity

The friendship paradox is a sociological phenomenon stating that most people have fewer friends than their friends do. The generalized friendship paradox refers to the same observation for attributes other than degree, and it has been observed in Twitter and scientific collaboration networks. This paper takes an analytical approach to model this phenomenon. We consider a preferential attachment-like network growth mechanism governed by both node degrees and `qualities'. We introduce measures to quantify paradoxes, and contrast the results obtained in our model to those obtained for an uncorrelated network, where the degrees and qualities of adjacent nodes are uncorrelated. We shed light on the effect of the distribution of node qualities on the friendship paradox. We consider both the mean and the median to measure paradoxes, and compare the results obtained by using these two statistics

Evolution of the social network of scientific collaborations

The co-authorship network of scientists represents a prototype of complex evolving networks. By mapping the electronic database containing all relevant journals in mathematics and neuro-science for an eight-year period (1991-98), we infer the dynamic and the structural mechanisms that govern the evolution and topology of this complex system. First, empirical measurements allow us to uncover the topological measures that characterize the network at a given moment, as well as the time evolution of these quantities. The results indicate that the network is scale-free, and that the network evolution is governed by preferential attachment, affecting both internal and external links. However, in contrast with most model predictions the average degree increases in time, and the node separation decreases. Second, we propose a simple model that captures the network's time evolution. Third, numerical simulations are used to uncover the behavior of quantities that could not be predicted analytically.Comment: 14 pages, 15 figure

Surface Scaling Analysis of a Frustrated Spring-network Model for Surfactant-templated Hydrogels

We propose and study a simplified model for the surface and bulk structures of crosslinked polymer gels, into which voids are introduced through templating by surfactant micelles. Such systems were recently studied by Atomic Force Microscopy [M. Chakrapani et al., e-print cond-mat/0112255]. The gel is represented by a frustrated, triangular network of nodes connected by springs of random equilibrium lengths. The nodes represent crosslinkers, and the springs correspond to polymer chains. The boundaries are fixed at the bottom, free at the top, and periodic in the lateral direction. Voids are introduced by deleting a proportion of the nodes and their associated springs. The model is numerically relaxed to a representative local energy minimum, resulting in an inhomogeneous, ``clumpy'' bulk structure. The free top surface is defined at evenly spaced points in the lateral (x) direction by the height of the topmost spring, measured from the bottom layer, h(x). Its scaling properties are studied by calculating the root-mean-square surface width and the generalized increment correlation functions C_q(x)= . The surface is found to have a nontrivial scaling behavior on small length scales, with a crossover to scale-independent behavior on large scales. As the vacancy concentration approaches the site-percolation limit, both the crossover length and the saturation value of the surface width diverge in a manner that appears to be proportional to the bulk connectivity length. This suggests that a percolation transition in the bulk also drives a similar divergence observed in surfactant templated polyacrylamide gels at high surfactant concentrations.Comment: 17 pages RevTex4, 10 imbedded eps figures. Expanded discussion of multi-affinit

Properties of a random attachment growing network

In this study we introduce and analyze the statistical structural properties of a model of growing networks which may be relevant to social networks. At each step a new node is added which selects 'k' possible partners from the existing network and joins them with probability delta by undirected edges. The 'activity' of the node ends here; it will get new partners only if it is selected by a newcomer. The model produces an infinite-order phase transition when a giant component appears at a specific value of delta, which depends on k. The average component size is discontinuous at the transition. In contrast, the network behaves significantly different for k=1. There is no giant component formed for any delta and thus in this sense there is no phase transition. However, the average component size diverges for delta greater or equal than one half.Comment: LaTeX, 19 pages, 6 figures. Discussion section, comments, a new figure and a new reference are added. Equations simplifie

Finite-time fluctuations in the degree statistics of growing networks

This paper presents a comprehensive analysis of the degree statistics in models for growing networks where new nodes enter one at a time and attach to one earlier node according to a stochastic rule. The models with uniform attachment, linear attachment (the Barab\'asi-Albert model), and generalized preferential attachment with initial attractiveness are successively considered. The main emphasis is on finite-size (i.e., finite-time) effects, which are shown to exhibit different behaviors in three regimes of the size-degree plane: stationary, finite-size scaling, large deviations.Comment: 33 pages, 7 figures, 1 tabl

Degree-distribution Stability of Growing Networks

In this paper, we abstract a kind of stochastic processes from evolving processes of growing networks, this process is called growing network Markov chains. Thus the existence and the formulas of degree distribution are transformed to the corresponding problems of growing network Markov chains. First we investigate the growing network Markov chains, and obtain the condition in which the steady degree distribution exists and get its exact formulas. Then we apply it to various growing networks. With this method, we get a rigorous, exact and unified solution of the steady degree distribution for growing networks.Comment: 12 page

Signatures of small-world and scale-free properties in large computer programs

A large computer program is typically divided into many hundreds or even thousands of smaller units, whose logical connections define a network in a natural way. This network reflects the internal structure of the program, and defines the ``information flow'' within the program. We show that, (1) due to its growth in time this network displays a scale-free feature in that the probability of the number of links at a node obeys a power-law distribution, and (2) as a result of performance optimization of the program the network has a small-world structure. We believe that these features are generic for large computer programs. Our work extends the previous studies on growing networks, which have mostly been for physical networks, to the domain of computer software.Comment: 4 pages, 1 figure, to appear in Phys. Rev.

Application of semidefinite programming to maximize the spectral gap produced by node removal

The smallest positive eigenvalue of the Laplacian of a network is called the spectral gap and characterizes various dynamics on networks. We propose mathematical programming methods to maximize the spectral gap of a given network by removing a fixed number of nodes. We formulate relaxed versions of the original problem using semidefinite programming and apply them to example networks.Comment: 1 figure. Short paper presented in CompleNet, Berlin, March 13-15 (2013

Visibility graphs of fractional Wu-Baleanu time series

[EN] We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos.J.A. Conejero is supported Ministerio de Economia y Competitividad Grant Project MTM2016-75963-P. Carlos Lizama is supported by CONICYT, under Fondecyt Grant number 1180041. Cristobal Rodero-Gomez is funded by European Commission H2020 research and Innovation programme under the Marie Sklodowska-Curie grant agreement No. 764738.Conejero, JA.; Lizama, C.; Mira-Iglesias, A.; Rodero-Gómez, C. (2019). Visibility graphs of fractional Wu-Baleanu time series. The Journal of Difference Equations and Applications. 25(9-10):1321-1331. https://doi.org/10.1080/10236198.2019.1619714S13211331259-10Anand, K., & Bianconi, G. (2009). Entropy measures for networks: Toward an information theory of complex topologies. Physical Review E, 80(4). doi:10.1103/physreve.80.045102Barabási, A.-L., & Albert, R. (1999). Emergence of Scaling in Random Networks. Science, 286(5439), 509-512. doi:10.1126/science.286.5439.509Brzeziński, D. W. (2017). Comparison of Fractional Order Derivatives Computational Accuracy - Right Hand vs Left Hand Definition. Applied Mathematics and Nonlinear Sciences, 2(1), 237-248. doi:10.21042/amns.2017.1.00020Brzeziński, D. W. (2018). 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Mathematische Nachrichten, 288(17-18), 2079-2092. doi:10.1002/mana.201400326Lizama, C. (2017). The Poisson distribution, abstract fractional difference equations, and stability. Proceedings of the American Mathematical Society, 145(9), 3809-3827. doi:10.1090/proc/12895Luque, B., Lacasa, L., Ballesteros, F., & Luque, J. (2009). Horizontal visibility graphs: Exact results for random time series. Physical Review E, 80(4). doi:10.1103/physreve.80.046103Luque, B., Lacasa, L., Ballesteros, F. J., & Robledo, A. (2011). Feigenbaum Graphs: A Complex Network Perspective of Chaos. PLoS ONE, 6(9), e22411. doi:10.1371/journal.pone.0022411Luque, B., Lacasa, L., & Robledo, A. (2012). Feigenbaum graphs at the onset of chaos. Physics Letters A, 376(47-48), 3625-3629. doi:10.1016/j.physleta.2012.10.050May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261(5560), 459-467. doi:10.1038/261459a0Núñez, Á. M., Luque, B., Lacasa, L., Gómez, J. P., & Robledo, A. (2013). Horizontal visibility graphs generated by type-I intermittency. Physical Review E, 87(5). doi:10.1103/physreve.87.052801Ravetti, M. G., Carpi, L. C., Gonçalves, B. A., Frery, A. C., & Rosso, O. A. (2014). Distinguishing Noise from Chaos: Objective versus Subjective Criteria Using Horizontal Visibility Graph. PLoS ONE, 9(9), e108004. doi:10.1371/journal.pone.0108004Robledo, A. (2013). Generalized Statistical Mechanics at the Onset of Chaos. Entropy, 15(12), 5178-5222. doi:10.3390/e15125178Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379-423. doi:10.1002/j.1538-7305.1948.tb01338.xSong, C., Havlin, S., & Makse, H. A. (2006). Origins of fractality in the growth of complex networks. Nature Physics, 2(4), 275-281. doi:10.1038/nphys266West, J., Lacasa, L., Severini, S., & Teschendorff, A. (2012). Approximate entropy of network parameters. Physical Review E, 85(4). doi:10.1103/physreve.85.046111Wu, G.-C., & Baleanu, D. (2013). Discrete fractional logistic map and its chaos. Nonlinear Dynamics, 75(1-2), 283-287. doi:10.1007/s11071-013-1065-7Wu, G.-C., & Baleanu, D. (2014). Discrete chaos in fractional delayed logistic maps. Nonlinear Dynamics, 80(4), 1697-1703. doi:10.1007/s11071-014-1250-3Zhang, J., & Small, M. (2006). Complex Network from Pseudoperiodic Time Series: Topology versus Dynamics. Physical Review Letters, 96(23). doi:10.1103/physrevlett.96.23870