We consider the voter model on the giant component of a hyperbolic random graph, which is a spatial scale-free network, in the sparse and linear-giant regime $\alpha\in(1/2,1)$. We find that the quenched expected consensus time has order $n^{2-1/\alpha}$, as the number of vertices $n\to\infty$, with probability arbitrarily close to one. This is generalised to the voter model where each vertex changes its opinion at rates $q(v)={\rm d}(v)^\varphi$, where we also establish the consensus time orders for all $\varphi\geq 0$. These orders have 3 regimes, with a phase transition at $\varphi=2-2\alpha$. For the upper bounds, our main proof idea is to connect the meeting set to some fixed target vertex of appropriate height in the product chain electrical network, to make rigorous an argument due to Durrett.
We consider the weight-dependent random connection model on a Poisson point process of intensity $\lambda$ on $\mathbb{R}\times(0,1)$ in which the vertices $(x,t)$ and $(y,s)$ are joined precisely when $(t\vee s)|x-y|\le\beta$. Points at distance $d$ are joined with probability $\min(1,\beta/d)^2$, the critical decay o...
Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics...
Simi Haber, Mostafa Mirabi, S. Shelah· 0 citations
Consider a simple undirected graph $G$ on $N$ vertices, with each vertex holding an opinion from one of two options. Starting from the initial opinions on day 1, we run majority dynamics: on each subsequent day, all vertices simultaneously adopt the majority opinion among their neighbors, retaining their current opinio...
Two star exponential random graph models (ERGMs) are an interesting special case of both general ERGMs and mean-field Ising models. In this paper, we study two star ERGMs conditioning on the edge density $p$. We begin with an analytic characterization of the replica symmetric region, where the conditional model is clos...
Inspect the vertices of a finite simple graph in uniformly random order, accepting each vertex if none of its neighbors has previously been accepted. Let $X_G$ be the number of accepted vertices. For every triangle-free graph with $n\ge2$ vertices and $e$ edges, we prove $\operatorname{Var}(X_G)\le e((n-2)/n)^2$, with...
Consider an infinite edge-weighted graph satisfying an isoperimetric inequality of the type $\|\partial A\|\geq C|A|^\alpha$ for some $\alpha,C>0$, where $\|\partial A\|$ denotes the weighted size of the edge boundary of $A$. We prove that, for $C$ large enough depending on $\alpha$, if each edge is open independently...
Ivailo Hartarsky, Franco Severo, Augusto Teixeira· 1 citation
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