Raul J. Mondragon

The rich-club phenomenon in the Internet topology

We show that the Internet topology at the autonomous system (AS) level has a rich-club phenomenon. The rich nodes, which are a small number of nodes with large numbers of links, are very well connected to each other. The rich-club is a core tier that we measured using the rich-club connectivity and the node-node link distribution. We obtained this core tier without any heuristic assumption between the ASs. The rich-club phenomenon is a simple qualitative way to differentiate between power law topologies and provides a criterion for new network models. To show this, we compared the measured rich-club of the AS graph with networks obtained using the Baraba/spl acute/si-Albert (BA) scale-free network model, the Fitness BA model and the Inet-3.0 model.

Accurately modeling the internet topology

Based on measurements of the internet topology data, we found that there are two mechanisms which are necessary for the correct modeling of the internet topology at the autonomous systems (AS) level: the interactive growth of new nodes and new internal links, and a nonlinear preferential attachment, where the preference probability is described by a positive-feedback mechanism. Based on the above mechanisms, we introduce the positive-feedback preference (PFP) model which accurately reproduces many topological properties of the AS-level internet, including degree distribution, rich-club connectivity, the maximum degree, shortest path length, short cycles, disassortative mixing, and betweenness centrality. The PFP model is a phenomenological model which provides an insight into the evolutionary dynamics of real complex networks.

Neutrino Billiards: Time-Reversal Symmetry-Breaking Without Magnetic Fields

A Dirac hamiltonian describing massless spin-half particles (‘neutrinos’) moving in the plane r = (x, y) under the action of a 4-scalar (not electric) potential V(r) is, in position representation, H^=−ihcσ^⋅∇+V(r)σ^z,, where σ̂ = (σ̂x, σ̂y) and σ̂z are the Pauli matrices; Ĥ acts on two-component column spinor wavefunctions ψ(r) = (ψ1, ψ2) and has eigenvalues ћckn. Ĥ does not possess time-reversal symmetry (T). If V(r) describes a hard wall bounding a finite domain D (‘billiards’), this is equivalent to a novel boundary condition for ψ2/ψ1. T-breaking is interpreted semiclassically as a difference of π between the phases accumulated by waves travelling in opposite senses round closed geodesics in D with odd numbers of reflections. The semiclassical (large-k) asymptotics of the eigenvalue counting function (spectral staircase) N(k) are shown to have the ‘Weyl’ leading term Ak2/4π, where A is the area of D, but zero perimeter correction. The Dirac equation is transformed to an integral equation round the boundary of D, and forms the basis of a numerical method for computing the kn. When D is the unit disc, geodesics are integrable and the eigenvalues, which satisfy Jl(kn) = Jl+1(kn), are (locally) Poisson-distributed. When D is an ‘Africa’ shape (cubic conformal map of the unit disc), the eigenvalues are (locally) distributed according to the statistics of the gaussian unitary ensemble of random-matrix theory, as predicted on the basis of T-breaking and lack of geometric symmetry.

Weighted multiplex networks.

One of the most important challenges in network science is to quantify the information encoded in complex network structures. Disentangling randomness from organizational principles is even more demanding when networks have a multiplex nature. Multiplex networks are multilayer systems of nodes that can be linked in multiple interacting and co-evolving layers. In these networks, relevant information might not be captured if the single layers were analyzed separately. Here we demonstrate that such partial analysis of layers fails to capture significant correlations between weights and topology of complex multiplex networks. To this end, we study two weighted multiplex co-authorship and citation networks involving the authors included in the American Physical Society. We show that in these networks weights are strongly correlated with multiplex structure, and provide empirical evidence in favor of the advantage of studying weighted measures of multiplex networks, such as multistrength and the inverse multiparticipation ratio. Finally, we introduce a theoretical framework based on the entropy of multiplex ensembles to quantify the information stored in multiplex networks that would remain undetected if the single layers were analyzed in isolation.

Structural constraints in complex networks

We present a link rewiring mechanism to produce surrogates of a network where both the degree distribution and the rich-club connectivity are preserved. We consider three real networks, the autonomous system (AS)-Internet, protein interaction and scientific collaboration. We show that for a given degree distribution, the rich-club connectivity is sensitive to the degree–degree correlation, and on the other hand the degree–degree correlation is constrained by the rich-club connectivity. In particular, in the case of the Internet, the assortative coefficient is always negative and a minor change in its value can reverse the networks rich-club structure completely; while fixing the degree distribution and the rich-club connectivity restricts the assortative coefficient to such a narrow range, that a reasonable model of the Internet can be produced by considering mainly the degree distribution and the rich-club connectivity. We also comment on the suitability of using the maximal random network as a null model to assess the rich-club connectivity in real networks.

Towards Modelling The Internet Topology - The Interactive Growth Model

The Internet topology at the Autonomous Systems level (AS graph) has a power-law degree distribution and a tier structure. In this paper, we introduce the Interactive Growth (IG) model based on the joint growth of new nodes and new links. This simple and dynamic model compares favorable with other Internet power-law topology generators because it not only closely resembles the degree distribution of the AS graph, but also accurately matches the hierarchical structure, which is measured by the recently reported rich-club phenomenon.

Anatomy of funded research in science

Significance The study of scientific collaborations has been predominately focused on characterizing publication coauthorships. Here, we study instead collaboration networks by looking at project partnerships funded over the years and show clearly the substantial impact of funding shifts on the pattern of interactions. We find that the leading universities form a cohesive clique among themselves and occupy brokerage positions between otherwise disconnected entities, and as the inequality in the distribution of funding grows over time, so does the degree of brokerage. Specifically, the elites overattract resources but they also reward in variety of research and quality. We are the first to our knowledge to systematically quantify the far-reaching effects of external forces on the complex interactions in team research that underpin the production and evolution of science. Seeking research funding is an essential part of academic life. Funded projects are primarily collaborative in nature through internal and external partnerships, but what role does funding play in the formulation of these partnerships? Here, by examining over 43,000 scientific projects funded over the past three decades by one of the major government research agencies in the world, we characterize how the funding landscape has changed and its impacts on the underlying collaboration networks across different scales. We observed rising inequality in the distribution of funding and that its effect was most noticeable at the institutional level—the leading universities diversified their collaborations and increasingly became the knowledge brokers in the collaboration network. Furthermore, it emerged that these leading universities formed a rich club (i.e., a cohesive core through their close ties) and this reliance among them seemed to be a determining factor for their research success, with the elites in the core overattracting resources but also rewarding in terms of both research breadth and depth. Our results reveal how collaboration networks organize in response to external driving forces, which can have major ramifications on future research strategy and government policy.

Redundancy and robustness of AS-level Internet topology and its models

A comparison between the topological properties of the measured Internet topology, at the autonomous system level (AS graph), and the equivalent graphs generated by two different power law topology generators are presented. Only one of the synthetic generators reproduces the tier connectivity of the AS graph.

Rich-Cores in Networks

A core comprises of a group of central and densely connected nodes which governs the overall behaviour of a network. It is recognised as one of the key meso-scale structures in complex networks. Profiling this meso-scale structure currently relies on a limited number of methods which are often complex and parameter dependent or require a null model. As a result, scalability issues are likely to arise when dealing with very large networks together with the need for subjective adjustment of parameters. The notion of a rich-club describes nodes which are essentially the hub of a network, as they play a dominating role in structural and functional properties. The definition of a rich-club naturally emphasises high degree nodes and divides a network into two subgroups. Here, we develop a method to characterise a rich-core in networks by theoretically coupling the underlying principle of a rich-club with the escape time of a random walker. The method is fast, scalable to large networks and completely parameter free. In particular, we show that the evolution of the core in World Trade and C. elegans networks correspond to responses to historical events and key stages in their physical development, respectively.

Chaotic maps for traffic modelling and queueing performance analysis

Abstract In this paper we present an overview of the progress made using chaotic maps to model individual and aggregated self-similar traffic streams and in particular their impact on queue performance. Our findings show that the asymptotic behaviour of the queue is a function only of the tail of the ON active periods, and that the Hurst parameter is not a good parameter to achieve traffic control: two different self-similar traffic traces can have the same Hurst parameter but have a very different effect on the queue statistics. These results are part of a framework for developing chaotic control of networks.