Synchrony-Breaking Bifurcation at a Simple Real Eigenvalue for Regular Networks 1: 1-Dimensional Cells

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.

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Joint Distribution of Distances in Large Random Regular Networks

Journal of Applied Probability ◽

10.1017/s002190020000989x ◽

2013 ◽

Vol 50 (03) ◽

pp. 861-870 ◽

Cited By ~ 1

Author(s):

Justin Salez

Keyword(s):

Joint Distribution ◽

Marginal Distribution ◽

Weak Sense ◽

Regular Graphs ◽

Random Array ◽

Point To Point ◽

Regular Networks ◽

Different Sources ◽

Individual Entry

We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to ∞. The marginal distribution of an individual entry is now well understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to an explicit infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously along the edges.

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An Algorithm for Discovery of New Families of Optimal Regular Networks

Discovery Science - Lecture Notes in Computer Science ◽

10.1007/978-3-540-39644-4_21 ◽

2003 ◽

pp. 245-255

Author(s):

Oleg Monakhov ◽

Emilia Monakhova

Keyword(s):

Regular Networks

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The Diagnosabilities of and Diagnosis Algorithms for Regular Networks under Two Three-Valued Models

Advances in Soft Computing - Fuzzy Information and Engineering Volume 2 ◽

10.1007/978-3-642-03664-4_36 ◽

2009 ◽

pp. 321-330

Author(s):

Yi Yang ◽

Xiao-fan Yang ◽

Hai-yan Tang

Keyword(s):

Regular Networks

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The Speed of Innovation Diffusion in Social Networks

Econometrica ◽

10.3982/ecta17007 ◽

2020 ◽

Vol 88 (2) ◽

pp. 569-594

Author(s):

Itai Arieli ◽

Yakov Babichenko ◽

Ron Peretz ◽

H. Peyton Young

Keyword(s):

Social Networks ◽

Social Network ◽

Waiting Time ◽

Network Topology ◽

Upper Bound ◽

Status Quo ◽

Arbitrary Size ◽

The Status ◽

Expected Waiting Time ◽

Regular Networks

New ways of doing things often get started through the actions of a few innovators, then diffuse rapidly as more and more people come into contact with prior adopters in their social network. Much of the literature focuses on the speed of diffusion as a function of the network topology. In practice, the topology may not be known with any precision, and it is constantly in flux as links are formed and severed. Here, we establish an upper bound on the expected waiting time until a given proportion of the population has adopted that holds independently of the network structure. Kreindler and Young (2014) demonstrated such a bound for regular networks when agents choose between two options: the innovation and the status quo. Our bound holds for directed and undirected networks of arbitrary size and degree distribution, and for multiple competing innovations with different payoffs.

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