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Register a new userGraph-based Polyas urn: completion of the linear case
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Yuri Lima
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Given a finite connected graph $G$, place a bin at each vertex. Two bins are called a pair if they share an edge of $G$. At discrete times, a ball is added to each pair of bins. In a pair of bins, one of the bins gets the ball with probability proportional to its current number of balls. Previous works proved that when $G$ is not balanced bipartite, the proportion of balls in the bins converges to a point $w(G)$ almost surely. We prove almost sure convergence for balanced bipartite graphs: the possible limit is either a single point $w(G)$ or a closed interval $mathcal J(G)$.
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In this paper we are concerned with a generalized $N$-urn Ehrenfest model, where balls keeps independent random walks between $N$ boxes uniformly laid on $[0, 1]$. After a proper scaling of the transition rates function of the aforesaid random walk, we derive the hydrodynamic limit of the model, i.e., the law of large numbers which the empirical measure of the model follows, under an assumption where the initial number of balls in each box independently follows a Poisson distribution. We show that the empirical measure of the model converges weakly to a deterministic measure with density driven by an integral equation. Furthermore, we derive non-equilibrium fluctuation of the model, i.e, the central limit theorem from the above hydrodynamic limit. We show that the non-equilibrium fluctuation of the model is driven by a measure-valued time-inhomogeneous generalized O-U process. At last, we prove a large deviation principle from the hydrodynamic limit under an assumption where the transition rates function from $[0, 1]times [0, 1]$ to $[0, +infty)$ of the aforesaid random walk is a product of two marginal functions from $[0, 1]$ to $[0, +infty)$.
In this paper, we consider a multi-drawing urn model with random addition. At each discrete time step, we draw a sample of m balls. According to the composition of the drawn colors, we return the balls together with a random number of balls depending on two discrete random variables X and Y with finite means and variances. Via the stochastic approximation algorithm, we give limit theorems describing the asymptotic behavior of white balls.
Knowledge graphs have been demonstrated to be an effective tool for numerous intelligent applications. However, a large amount of valuable knowledge still exists implicitly in the knowledge graphs. To enrich the existing knowledge graphs, recent years witness that many algorithms for link prediction and knowledge graphs embedding have been designed to infer new facts. But most of these studies focus on the static knowledge graphs and ignore the temporal information that reflects the validity of knowledge. Developing the model for temporal knowledge graphs completion is an increasingly important task. In this paper, we build a new tensor decomposition model for temporal knowledge graphs completion inspired by the Tucker decomposition of order 4 tensor. We demonstrate that the proposed model is fully expressive and report state-of-the-art results for several public benchmarks. Additionally, we present several regularization schemes to improve the strategy and study their impact on the proposed model. Experimental studies on three temporal datasets (i.e. ICEWS2014, ICEWS2005-15, GDELT) justify our design and demonstrate that our model outperforms baselines with an explicit margin on link prediction task.
This paper is a further investigation of the generalized $N$-urn Ehrenfest model introduced in cite{Xue2020}. A moderate deviation principle from the hydrodynamic limit of the model is derived. The proof of this main result follows a routine procedure introduced in cite{Kipnis1989}, where a replacement lemma plays the key role. To prove the replacement lemma, the large deviation principle of the model given in cite{Xue2020} is utilized.
Scale-free percolation is a spatial random graph model with vertex set $mathbb{Z}^d$. Vertices $x$ and $y$ are connected with probability depending on i.i.d. vertex weights and the Euclidean distance. Depending on the various parameters involved, we get a rich phase diagram. We study graph distances (in comparison to Euclidean distances). Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.
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