Weakly reinforced Pólya urns on countable networks

Couzinié, Yannick; Hirsch, Christian

doi:10.1214/21-ecp404

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…Iterating this argument, we see that it also holds for q > F (k) (1) for any number of iterations k, where F(q) = g −1 L (g R (q)). Now note that F (5) (1) < 1 2 , hence proving (29) on all the interval ( 1 2 , 1) and thus concluding the proof of the lemma. The graphs of functions y = ( q 1−q ) 1/7 (solid) and y = ( 0.5+q 2−q ) (dashed), plotted over q ∈ ( 1 2 , 1), cut off at y = 2.…”

Section: Proposition 10supporting

confidence: 56%

“…Part (0) of theorem 4 in fact holds for graphs that are 'uniformly disconnectable' 5 in the sense that there exists a constant C > 0 such that any finite A ⊂ G can be disconnected from infinity by removing at most C edges from E. Such graphs include graphs that are quasiisometric to Z, graphs with linear volume growth, etc.…”

Section: Definition 1 (Removable Edge Setmentioning

confidence: 99%

“…The cases α = 1 and α < 1 are rather different, see e.g. [5,11]). However, in the present paper, we focus on the natural generalisation of such models to infinite connected graphs (with countably many vertices).…”

Section: Introductionmentioning

confidence: 99%

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Infinite WARM graphs III: strong reinforcement regime

2023

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We study a reinforcement process on graphs G of bounded degree. The model involves a parameter α > 0 governing the strength of reinforcement, and Poisson clock rates λ v at the vertices v of the graph. When the Poisson clock at a vertex v rings, one of the edges incident to it is reinforced, with edge e being chosen with probability proportional to its current count (counts start from 1) raised to the power α. The main problem in such models is to describe the (random) subgraph , consisting of edges that are reinforced infinitely often. In this paper, we focus on the finite connected components of in the strong reinforcement regime (α > 1) with clock rates that are uniformly bounded above. We show here that when α is sufficiently large, all connected components of are trees. When the firing rates λ v are constant, we show that all components are trees of diameter at most 3 when α is sufficiently large, and that there are infinitely many phase transitions as α ↓ 1 . For example, on the triangular lattice, increasingly large (odd) cycles appear as α ↓ 1 (while on the square lattice no finite component of contains a cycle for any α > 1). Increasingly long paths and other structures appear in both lattices when taking α ↓ 1 . In the special case where G = Z and α > 1, all connected components of are finite and we show that the possible cluster sizes are non-monotone in α. We also present several open problems.

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Section: Proposition 10supporting

confidence: 56%

Section: Definition 1 (Removable Edge Setmentioning

confidence: 99%

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Infinite WARM graphs III: strong reinforcement regime

2023

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“…A naive modelling using so-called (W, A)-reinforcement models (or 'WARM') suggests a negative answer. In this model involving Pólya urns with graph-based competition, there are only two regimes: In the strong reinforcement regime, the process is supported on small isolated islands [9,10,12], whereas in the weak reinforcement regime the support is on the entire graph [5,11]. There is thus no regime in which a subgraph with suitable properties emerges.…”

Section: Motivationmentioning

confidence: 99%

Extremal linkage networks

Heydenreich¹,

Hirsch²

2020

Preprint

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We demonstrate how sophisticated graph properties, such as small distances and scale-free degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an a-priori i.i.d. fitness with max-stable distribution. The fitness determines the node attractiveness w.r.t. incoming edges as well as the spatial range for outgoing edges. For max-stable fitness distributions, we thus obtain complex spatial network, which we coin extremal linkage network.

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Extremal linkage networks

Heydenreich

Hirsch

2021

Extremes

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We demonstrate how sophisticated graph properties, such as small distances and scale-free degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an a-priori i.i.d. fitness with max-stable distribution. The fitness determines the node attractiveness w.r.t. incoming edges as well as the spatial range for outgoing edges. For max-stable fitness distributions, we thus obtain a complex spatial network, which we coin extremal linkage network.

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Weakly reinforced Pólya urns on countable networks

Cited by 4 publications

References 9 publications

Infinite WARM graphs III: strong reinforcement regime

Infinite WARM graphs III: strong reinforcement regime

Extremal linkage networks

Extremal linkage networks

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