Emmanuel Jacob - Academia.edu (original) (raw)

Papers by Emmanuel Jacob

Le Centre pour la Communication Scientifique Directe - HAL - Université Paris Descartes, Jun 2, 2022

We investigate the contact process on four different types of scale-free inhomogeneous random gra... more We investigate the contact process on four different types of scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where each potential edge is updated with a rate depending on the strength of the adjacent vertices. Depending on the type of graph, the tail exponent of the degree distribution and the updating rate, we find parameter regimes of fast and slow extinction and in the latter case identify metastable exponents that undergo first order phase transitions. Résumé: Nousétudions le processus de contact sur quatre types différents de graphes aléatoires inhomogènes invariants d'échelleévoluant selon une dynamique stationnaire, où chaque arête potentielle est rafraîchieà un taux dépendant de la force des sommets adjacents. En fonction du type de graphe, de l'exposant de la queue de distribution des degrés et du taux de rafraîchissement, nous trouvons des régimes d'extinction rapide ou lente et, dans ce dernier cas, nous identifions des exposants métastables qui subissent des transitions de phase de premier ordre.

We prove that a uniform rooted plane map with n edges converges in distribution after a suitable ... more We prove that a uniform rooted plane map with n edges converges in distribution after a suitable normalization to the Brownian map for the Gromov-Hausdorff topology. A recent bijection due to Ambjørn and Budd allows to derive this result by a direct coupling with a uniform random quadrangulation with n faces.

The Annals of Applied Probability, 2019

We study the contact process in the regime of small infection rates on finite scale-free networks... more We study the contact process in the regime of small infection rates on finite scale-free networks with stationary dynamics based on simultaneous updating of all connections of a vertex. We allow the update rates of individual vertices to increase with the strength of a vertex, leading to a fast evolution of the network. We first develop an approach for inhomogeneous networks with general kernel and then focus on two canonical cases, the factor kernel and the preferential attachment kernel. For these specific networks, we identify and analyse four possible strategies how the infection can survive for a long time. We show that there is fast extinction of the infection when neither of the strategies is successful, otherwise there is slow extinction and the most successful strategy determines the asymptotics of the metastable density as the infection rate goes to zero. We identify the domains in which these strategies dominate in terms of phase diagrams for the exponent describing the decay of the metastable density.

Royal Society open science, 2017

We study the contact process on a class of evolving scale-free networks, where each node updates ... more We study the contact process on a class of evolving scale-free networks, where each node updates its connections at independent random times. We give a rigorous mathematical proof that there is a transition between a phase where for all infection rates the infection survives for a long time, at least exponential in the network size, and a phase where for sufficiently small infection rates extinction occurs quickly, at most polynomially in the network size. The phase transition occurs when the power-law exponent crosses the value four. This behaviour is in contrast with that of the contact process on the corresponding static model, where there is no phase transition, as well as that of a classical mean-field approximation, which has a phase transition at power-law exponent three. The new observation behind our result is that temporal variability of networks can simultaneously increase the rate at which the infection spreads in the network, and decrease the time at which the infection...

The Annals of Probability, 2017

A growing family of random graphs is called robust if it retains a giant component after percolat... more A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the d-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering, we can independently tune the power law exponent τ of the degree distribution and the rate −δd at which the connection probability decreases with the distance of two vertices. We show that the network is robust if τ < 2 + 1 δ , but fails to be robust if τ > 3. In the case of one-dimensional space, we also show that the network is not robust if τ > 2 + 1 δ−1. This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks, our model is not locally treelike, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality.

Lecture Notes in Computer Science, 2015

We study robustness under random attack for a class of networks, in which new nodes are given a s... more We study robustness under random attack for a class of networks, in which new nodes are given a spatial position and connect to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering one can independently tune the power law exponent τ > 2 of the degree distribution and a parameter δ > 1 determining the decay rate of the probability of long edges. We argue that the network is robust if τ < 2 + 1 δ , but fails to be robust if τ > 2 + 1 δ−1. Hence robustness depends not only on the power-law exponent but also on the clustering features of the network.

The Annals of Applied Probability, 2015

We define a class of growing networks in which new nodes are given a spatial position and are con... more We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent τ > 2. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value τ = 3. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.

Electronic Journal of Probability, 2014

We prove that a uniform rooted plane map with n edges converges in distribution after a suitable ... more We prove that a uniform rooted plane map with n edges converges in distribution after a suitable normalization to the Brownian map for the Gromov-Hausdorff topology. A recent bijection due to Ambjørn and Budd allows us to derive this result by a direct coupling with a uniform random quadrangulation with n faces.

Lecture Notes in Computer Science, 2013

A class of growing networks is introduced in which new nodes are given a spatial position and are... more A class of growing networks is introduced in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favouring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Most notably, empirical degree distributions converge to a limit law, which can be a power law with any exponent τ > 2, and the average clustering coefficient converges to a positive limit. Our main tool to show these and other results is a weak law of large numbers in the spirit of Penrose and Yukich, which can be applied thanks to a novel rescaling idea. We also conjecture that the networks have a robust giant component if τ is sufficiently small.

International Review of Mission, 2002

If someone had asked me 25 years ago what the relevance of the message of Matthew was to Christia... more If someone had asked me 25 years ago what the relevance of the message of Matthew was to Christian mission in my own context I would have pointed directly to Matthew 25:31-46, and not to Matthew 28:19-20. This is because 25 years ago I was living in apartheid South Africa, my home country, where the contrast between rich and poor, and the powerful and powerless, was a stark and challenging reality. The question, “Who is Jesus Christ?” needed an appropriate, credible and relevant answer, and the need to identify the gospel and to live it in the face of a “different gospel” called for discernment, courage, and faithfulness. For South Africans the text which summed up the gospel in a way that made a real difference was, “For as much as you do it unto the least of these ...y ou do it to me.”

The Journal of Ecclesiastical History, 2000