Introductory Lectures on Stochastic Population Systems

Dawson, Donald A.

doi:10.48550/arxiv.1705.03781

Cited by 6 publications

(8 citation statements)

References 195 publications

(270 reference statements)

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“…Consider any node v which gets infected, say at time t 0 ≥ 1. Recall that G t denotes the infected subgraph at time t and for any t ≥ t 0 , (G t , u ) v denotes the subtree rooted at v and consisting of v and the infected descendants of v. It is easy to see that {(G t , u ) v } t≥t 0 forms a Galton-Watson (GW) branching process [3], which grows independent of the rest of the infection tree. A GW branching process is defined as follows.…”

Section: Jordan Center In the Ic Modelmentioning

confidence: 99%

“…We say that a branching process is 'dead' if ∃ n 0 such that Z n = 0 ∀n ≥ n 0 . For pd > 1, there is a positive probability, say p, that the GW branching process will not die [3]. We assume pd > 1 which implies that for any infected node v, there is a positive probability p that the rooted growing subtree {(G t , u ) v } t≥t 0 will not die and grow forever.…”

Section: Jordan Center In the Ic Modelmentioning

confidence: 99%

“…where (a) follows from Lemma 3.2, which dictates that since the deepest subtree 3 does not grow, the Jordan center will not move from time t 1 to t 1 + 1, and furthermore, the centrality will remain the same; and (b) follows from the fact that the Jordan center does not move from time t 1 to t 1 + 1, while the depth of the B 1 process increases by 1. Following t 1 + 1, we continue the process and again wait for the next timestep when the deepest subtree does not grow in depth.…”

Section: Jordan Center In the Discrete Si Modelmentioning

confidence: 99%

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Persistence of the Jordan Center in Random Growing Trees

Pattathil

Karamchandani

Shah

2018

2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM)

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The Jordan center of a graph is defined as a vertex whose maximum distance to other nodes in the graph is minimal, and it finds applications in facility location and source detection problems. We study properties of the Jordan center in the case of random growing trees. In particular, we consider a regular tree graph on which an infection starts from a root node and then spreads along the edges of the graph according to various random spread models. For the Independent Cascade (IC) model and the discrete Susceptible Infected (SI) model, both of which are discrete time models, we show that as the infected subgraph grows with time, the Jordan center persists on a single vertex after a finite number of timesteps. Finally, we also study the continuous time version of the SI model and bound the maximum distance between the Jordan center and the root node at any time.

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Section: Jordan Center In the Ic Modelmentioning

confidence: 99%

Section: Jordan Center In the Ic Modelmentioning

confidence: 99%

Section: Jordan Center In the Discrete Si Modelmentioning

confidence: 99%

See 1 more Smart Citation

Persistence of the Jordan Center in Random Growing Trees

Pattathil

Karamchandani

Shah

2018

2018 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM)

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“…Since McKean's seminal paper McKean (1966a), the mean-field theory has been widely used for the study of large stochastic interacting particle systems arising from various domains such as statistical physics McKean (1966a,b); Dawson (1983); Gärtner (1988), biological systems Dawson (2017); Méléard & Bansaye (2015), communication networks Graham & Méléard (1993, 1995; Benaïm & Boudec (2008); Graham (2000), mathematical finance Kley et al (2015); Giesecke et al (2015), etc. This theory, first initiated in connection with a mathematical foundation of the Boltzmann equation, aims for a mathematically rigorous treatment of the time evolution of stochastic systems with long-range weak interaction where the interaction between particles is realized via the empirical measure of the particle configuration.…”

Section: Introductionmentioning

confidence: 99%

Propagation of chaos and large deviations in mean-field models with jumps on block-structured networks

Dawson¹,

Sid-Ali²,

Zhao³

2020

Preprint

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A system of interacting multiclass finite-state jump processes is analyzed. The model under consideration consists of a block-structured network with dynamically changing multicolors nodes. The interaction is local and described through local empirical measures. Two levels of heterogeneity are considered: between and within the blocks where the nodes are labeled into two types. The central nodes are those connected only to the nodes of the same block whereas the peripheral nodes are connected to both the nodes of the same block and to some nodes from other blocks. The limits of such systems as the number of particles tends to infinity are investigated. Under regularity conditions on the peripheral nodes, propagation of chaos and law of large numbers are established in a multi-population setting. In particular, it is shown that, as the number of nodes goes to infinity, the behavior of the different classes of nodes can be represented by the solution of a McKean-Vlasov system. Moreover, we prove large deviation principles for the vectors of empirical measures and the empirical processes. Our large deviations results extend classical results of Gärtner (1987) andLéonard (1995).

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“…More recently, because of their relation to non-linear PDEs, they also have been helpful in some problems in mathematical finance as in [33] or [23]. For thorough introductions to the topic, we refer to the notes of Dawson [10] and Perkins [31] and for an introductory reading to [19] and [11].…”

Section: Introductionmentioning

confidence: 99%

A Functional Ito-Formula for Dawson-Watanabe Superprocesses

Mandler¹,

Overbeck²

2020

Preprint

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We derive an Itō-formula for the Dawson-Watanabe superprocess, a wellknown class of measure-valued processes, extending the classical Itō-formula with respect to two aspects. Firstly, we extend the state-space of the underlying process (X(t)) t∈[0,T ] to an infinite-dimensional one -the space of finite measure. Secondly, we extend the formula to functions F (t, Xt) depending on the entire paths Xt = (X(s ∧ t)) s∈[0,T ] up to times t. This later extension is usually called functional Itō-formula. Finally we remark on the application to predictable representation for martingales associated with superprocesses.

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Introductory Lectures on Stochastic Population Systems

Cited by 6 publications

References 195 publications

Persistence of the Jordan Center in Random Growing Trees

Persistence of the Jordan Center in Random Growing Trees

Propagation of chaos and large deviations in mean-field models with jumps on block-structured networks

A Functional Ito-Formula for Dawson-Watanabe Superprocesses

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