Global epidemic spreading processes in coupled networks

Projecte final de Màster Oficial fet en col.laborció amb Universitat de Barcelona. Departament de Física Fonamental i Departament de Química Física.English: We study the effect of coupling two random networks where an epidemic process propagates. A theoretical SIS model is applied and a new critical threshold for the existence of an endemic state is analytically calculated for the two coupled networks, under the assumption that there is total correlation between the outer degree and the inner degree of each node. Our main result is that a global endemic state can exist in the coupled networks, even though the epidemics does not propagate on each network separately. Finally, we checked these results by running large scale computer simulations

Mesures de regularitat per a polígons convexos

Al llarg d'aquesta memòria, hem plantejat possibles mesures de regularitat, totes elles justificades, per a un n-gon convex qualsevol que han donat lloc a problemes de geometria discreta i computacional. A més, hem estat capaços d'oferir algorismes per al seu càlcul de complexitat baixa (i, en alguns casos, òptima) i, per tant, realitzables. Els mètodes proposats són, en algunes ocasions, l'aplicació de resultats més generals, en d'altres, algorismes ad hoc, i, en d'altres, un estudi acurat permet transformar el problema que s'ha de resoldre en un altre problema d'optimització geomètrica que té una solució eficient coneguda

Mesures de regularitat per a polígons convexos

Al llarg d'aquesta memòria, hem plantejat possibles mesures de regularitat, totes elles justificades, per a un n-gon convex qualsevol que han donat lloc a problemes de geometria discreta i computacional. A més, hem estat capaços d'oferir algorismes per al seu càlcul de complexitat baixa (i, en alguns casos, òptima) i, per tant, realitzables. Els mètodes proposats són, en algunes ocasions, l'aplicació de resultats més generals, en d'altres, algorismes ad hoc, i, en d'altres, un estudi acurat permet transformar el problema que s'ha de resoldre en un altre problema d'optimització geomètrica que té una solució eficient coneguda

Epidemic spreading on interconnected networks

Many real networks are not isolated from each other but form networks of networks, often interrelated in non trivial ways. Here, we analyze an epidemic spreading process taking place on top of two interconnected complex networks. We develop a heterogeneous mean field approach that allows us to calculate the conditions for the emergence of an endemic state. Interestingly, a global endemic state may arise in the coupled system even though the epidemics is not able to propagate on each network separately, and even when the number of coupling connections is small. Our analytic results are successfully confronted against large-scale numerical simulations

Terrain prickliness: theoretical grounds for high complexity viewsheds

An important task when working with terrain models is computing viewsheds: the parts of the terrain visible from a given viewpoint. When the terrain is modeled as a polyhedral terrain, the viewshed is composed of the union of all the triangle parts that are visible from the viewpoint. The complexity of a viewshed can vary significantly, from constant to quadratic in the number of terrain vertices, depending on the terrain topography and the viewpoint position. In this work we study a new topographic attribute, the prickliness, that measures the number of local maxima in a terrain from all possible perspectives. We show that the prickliness effectively captures the potential of 2.5D terrains to have high complexity viewsheds, and we present near-optimal algorithms to compute the prickliness of 1.5D and 2.5D terrains. We also report on some experiments relating the prickliness of real word 2.5D terrains to the size of the terrains and to their viewshed complexity.Peer ReviewedPostprint (author's final draft

FastSIR Algorithm: A Fast Algorithm for simulation of epidemic spread in large networks by using SIR compartment model

The epidemic spreading on arbitrary complex networks is studied in SIR (Susceptible Infected Recovered) compartment model. We propose our implementation of a Naive SIR algorithm for epidemic simulation spreading on networks that uses data structures efficiently to reduce running time. The Naive SIR algorithm models full epidemic dynamics and can be easily upgraded to parallel version. We also propose novel algorithm for epidemic simulation spreading on networks called the FastSIR algorithm that has better average case running time than the Naive SIR algorithm. The FastSIR algorithm uses novel approach to reduce average case running time by constant factor by using probability distributions of the number of infected nodes. Moreover, the FastSIR algorithm does not follow epidemic dynamics in time, but still captures all infection transfers. Furthermore, we also propose an efficient recursive method for calculating probability distributions of the number of infected nodes. Average case running time of both algorithms has also been derived and experimental analysis was made on five different empirical complex networks.Comment: 8 figure

Flips in combinatorial pointed pseudo-triangulations with face degree at most four

In this paper we consider the flip operation for combinatorial pointed pseudo-triangulations where faces have size 3 or 4, so-called combinatorial 4-PPTs. We show that every combinatorial 4-PPT is stretchable to a geometric pseudo-triangulation, which in general is not the case if faces may have size larger than 4. Moreover, we prove that the flip graph of combinatorial 4-PPTs with triangular outer face is connected and has diameter O(n2).European Science FoundationAustrian Science FundMinisterio de Ciencia e InnovaciónJunta de Castilla y Leó

Improving shortest paths in the Delaunay triangulation

We study a problem about shortest paths in Delaunay triangulations. Given two nodes s; t in the Delaunay triangulation of a point set P, we look for a new point p that can be added, such that the shortest path from s to t in the Delaunay triangulation of P u{p} improves as much as possible. We study properties of the problem and give efficient algorithms to find such a point when the graph-distance used is Euclidean and for the link-distance. Several other variations of the problem are also discussed

Epidemics in partially overlapped multiplex networks

Many real networks exhibit a layered structure in which links in each layer reflect the function of nodes on different environments. These multiple types of links are usually represented by a multiplex network in which each layer has a different topology. In real-world networks, however, not all nodes are present on every layer. To generate a more realistic scenario, we use a generalized multiplex network and assume that only a fraction $q$ of the nodes are shared by the layers. We develop a theoretical framework for a branching process to describe the spread of an epidemic on these partially overlapped multiplex networks. This allows us to obtain the fraction of infected individuals as a function of the effective probability that the disease will be transmitted $T$. We also theoretically determine the dependence of the epidemic threshold on the fraction $q > 0$ of shared nodes in a system composed of two layers. We find that in the limit of $q \to 0$ the threshold is dominated by the layer with the smaller isolated threshold. Although a system of two completely isolated networks is nearly indistinguishable from a system of two networks that share just a few nodes, we find that the presence of these few shared nodes causes the epidemic threshold of the isolated network with the lower propagating capacity to change discontinuously and to acquire the threshold of the other network.Comment: 13 pages, 4 figure

Multiplex PageRank

(15 pages, 6 figures