arXiv subjects

Alexei Vazquez

Publications and source records attributed to Alexei Vazquez.

At least 55 records · Page 3Linked to original sources

arXiv2005

Exact results for the Barabasi model of human dynamics

Human activity patterns display a bursty dynamics, with interevent times following a heavy tailed distribution. This behavior has been recently shown to be rooted in the fact that humans assign their active tasks different priorities, a process that can be modeled as a priority queueing system [A.-L. Barabasi, Nature 435, 207 (2005)]. In this work we obtain exact results for the Barabasi model with two tasks, calculating the priority and waiting time distribution of active tasks. We demonstrate that the model has a singular behavior in the extremal dynamics limit, when the highest priority task is selected first. We find that independently of the selection protocol, the average waiting time is smaller or equal to the number of active tasks, and discuss the asymptotic behavior of the waiting time distribution. These results have important implications for understanding complex systems with extremal dynamics.

physics.soc-ph↗

arXiv2005

Polynomial growth in age-dependent branching processes with diverging reproductive number

We study the spreading dynamics on graphs with a power law degree distribution p_k ~ k^-gamma with 2<gamma<3, as an example of a branching process with diverging reproductive number. We provide evidence that the divergence of the second moment of the degree distribution carries as a consequence a qualitative change in the growth pattern, deviating from the standard exponential growth. First, the population growth is extensive, meaning that the average number of vertices reached by the spreading process becomes of the order of the graph size in a time scale that vanishes in the large graph size limit. Second, the temporal evolution is governed by a polynomial growth, with a degree determined by the characteristic distance between vertices in the graph. These results open a path to further investigation on the dynamics on networks.

cond-mat.dis-nn↗

arXiv2005

Distance-d covering problems in scale-free networks with degree correlations

A number of problems in communication systems demand the distributed allocation of network resources in order to provide better services, sampling and distribution methods. The solution to these issues is becoming more challenging due to the increasing size and complexity of communication networks. We report here on an heuristic method that allows to find near-optimal solutions to the covering problem in real communication networks. The results show that the optimal allocation of resources strongly depends on the topology of the underlying networks. Whether a centralized or distributed design is to be used relies upon the degree correlations between connected vertices.

cond-mat.stat-mech↗

arXiv2005

The inhomogeneous evolution of subgraphs and cycles in complex networks

Subgraphs and cycles are often used to characterize the local properties of complex networks. Here we show that the subgraph structure of real networks is highly time dependent: as the network grows, the density of some subgraphs remains unchanged, while the density of others increase at a rate that is determined by the network's degree distribution and clustering properties. This inhomogeneous evolution process, supported by direct measurements on several real networks, leads to systematic shifts in the overall subgraph spectrum and to an inevitable overrepresentation of some subgraphs and cycles.

cond-mat.dis-nn↗

arXiv2004

Exploring networks with traceroute-like probes: theory and simulations

Mapping the Internet generally consists in sampling the network from a limited set of sources by using traceroute-like probes. This methodology, akin to the merging of different spanning trees to a set of destination, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones. In this paper we explore these biases and provide a statistical analysis of their origin. We derive an analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph. In particular, we find that the edge and vertex detection probability depends on the betweenness centrality of each element. This allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with broad distributions of connectivity. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in network models with different topologies. We show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters. Moreover, we characterize the level of redundancy and completeness of the exploration process as a function of the topological properties of the network. Finally, we study numerically how the fraction of vertices and edges discovered in the sampled graph depends on the particular deployements of probing sources. The results might hint the steps toward more efficient mapping strategies.

cs.NI↗

arXiv2004

Clustering coefficient without degree correlations biases

The clustering coefficient quantifies how well connected are the neighbors of a vertex in a graph. In real networks it decreases with the vertex degree, which has been taken as a signature of the network hierarchical structure. Here we show that this signature of hierarchical structure is a consequence of degree correlation biases in the clustering coefficient definition. We introduce a new definition in which the degree correlation biases are filtered out, and provide evidence that in real networks the clustering coefficient is constant or decays logarithmically with vertex degree.

cond-mat.stat-mech↗

arXiv2004

A statistical approach to the traceroute-like exploration of networks: theory and simulations

Mapping the Internet generally consists in sampling the network from a limited set of sources by using "traceroute"-like probes. This methodology, akin to the merging of different spanning trees to a set of destinations, has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones. Here we explore these biases and provide a statistical analysis of their origin. We derive a mean-field analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph. In particular we find that the edge and vertex detection probability is depending on the betweenness centrality of each element. This allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with scale-free topology. We complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in different network models. We show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters. The numerical study also allows the identification of intervals of the exploration parameters that optimize the fraction of nodes and edges discovered in the sampled graph. This finding might hint the steps toward more efficient mapping strategies.

cond-mat.stat-mech↗

arXiv2003

Growing networks with local rules: preferential attachment, clustering hierarchy and degree correlations

The linear preferential attachment hypothesis has been shown to be quite successful to explain the existence of networks with power-law degree distributions. It is then quite important to determine if this mechanism is the consequence of a general principle based on local rules. In this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules. It is also shown that the local models offer an explanation to other properties like the clustering hierarchy and degree correlations recently observed in complex networks. These conclusions are based on both analytical and numerical results of different local rules, including some models already proposed in the literature.

cond-mat.stat-mech↗

arXiv2003

Resilience to damage of graphs with degree correlations

The existence or not of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking. In this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena. We obtain some general expressions that allow the computation of the transition thresholds or their bounds. Using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks.

cond-mat.dis-nn↗

arXiv2002

Computational complexity arising from degree correlations in networks

We apply a Bethe-Peierls approach to statistical-mechanics models defined on random networks of arbitrary degree distribution and arbitrary correlations between the degrees of neighboring vertices. Using the NP-hard optimization problem of finding minimal vertex covers on these graphs, we show that such correlations may lead to a qualitatively different solution structure as compared to uncorrelated networks. This results in a higher complexity of the network in a computational sense: Simple heuristic algorithms fail to find a minimal vertex cover in the highly correlated case, whereas uncorrelated networks seem to be simple from the point of view of combinatorial optimization.

cond-mat.dis-nn↗

arXiv2002

Disease Spreading in Structured Scale-Free Networks

We study the spreading of a disease on top of structured scale-free networks recently introduced. By means of numerical simulations we analyze the SIS and the SIR models. Our results show that when the connectivity fluctuations of the network are unbounded whether the epidemic threshold exists strongly depends on the initial density of infected individuals and the type of epidemiological model considered. Analytical arguments are provided in order to account for the observed behavior. We conclude that the peculiar topological features of this network and the absence of small-world properties determine the dynamics of epidemic spreading.

cond-mat.stat-mech↗

arXiv2002

Topology and correlations in structured scale-free networks

We study a recently introduced class of scale-free networks showing a high clustering coefficient and non-trivial connectivity correlations. We find that the connectivity probability distribution strongly depends on the fine details of the model. We solve exactly the case of low average connectivity, providing also exact expressions for the clustering and degree correlation functions. The model also exhibits a lack of small world properties in the whole parameters range. We discuss the physical properties of these networks in the light of the present detailed analysis.

cond-mat.stat-mech↗

arXiv2002

The Bak-Sneppen Model on Scale-Free Networks

We investigate by numerical simulations and analytical calculations the Bak-Sneppen model for biological evolution in scale-free networks. By using large scale numerical simulations, we study the avalanche size distribution and the activity time behavior at nodes with different connectivities. We argue the absence of a critical barrier and its associated critical behavior for infinite size systems. These findings are supported by a single site mean-field analytic treatment of the model.

cond-mat.stat-mech↗

arXiv2002

Dynamical and correlation properties of the Internet

The description of the Internet topology is an important open problem, recently tackled with the introduction of scale-free networks. In this paper we focus on the topological and dynamical properties of real Internet maps in a three years time interval. We study higher order correlation functions as well as the dynamics of several quantities. We find that the Internet is characterized by non-trivial correlations among nodes and different dynamical regimes. We point out the importance of node hierarchy and aging in the Internet structure and growth. Our results provide hints towards the realistic modeling of the Internet evolution.

cond-mat.stat-mech↗

arXiv2001

Large-scale topological and dynamical properties of Internet

We study the large-scale topological and dynamical properties of real Internet maps at the autonomous system level, collected in a three years time interval. We find that the connectivity structure of the Internet presents average quantities and statistical distributions settled in a well-defined stationary state. The large-scale properties are characterized by a scale-free topology consistent with previous observations. Correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the Internet. The study of the Internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models. We compare the properties of growing network models with the present real Internet data analysis.

cond-mat.stat-mech↗

arXiv2001

Statistics of citation networks

The out-degree distribution of citation networks is investigated. Statistical data of the number of papers cited within a paper (out-degree) for different journals in the period 1991-1999 is reported. The out-degree distribution is characterized by a maximum at intermediate out-degrees. At the left of the maximum there are strong fluctuations from journal to journal while is quite universal at the right, with two classes of journals. These two classes are associated with the existence or not of a restriction in the maximum number of pages per paper. The shape of the out-degree distribution does not change appreciable from period to period, but the average out-degree is observed to increase logarithmically with the number of published papers. These features are modeled using a recursive search model.

cond-mat.stat-mech↗

arXiv2000

Non conservative Abelian sandpile model with BTW toppling rule

A non conservative Abelian sandpile model with BTW toppling rule introduced in [Tsuchiya and Katori, Phys. Rev. E {\bf 61}, 1183 (2000)] is studied. Using a scaling analysis of the different energy scales involved in the model and numerical simulations it is shown that this model belong to a universality class different from that of previous models considered in the literature.

cond-mat.stat-mech↗

arXiv2000

Knowing a network by walking on it: emergence of scaling

A model for growing networks is introduced, having as a main ingredient that new nodes are attached to the network through one existing node and then explore the network through the links of the visited nodes. From exact calculations of two limiting cases and numerical simulations the phase diagram of the model is obtained. In the stationary limit, large network sizes, a phase transition from a network with finite average connectivity to a network with a power law distribution of connectivities, with no finite average, is found. Results are compared with measurements on real networks.

cond-mat.stat-mech↗