有向Erdos-Renyi随机图的多数投票模型

Approximating the cumulant generating function of triangles in the Erdös-Rényi random graph

We study the pressure of the "edge-triangle model", which is equivalent to the cumulant generating function of triangles in the Erdös-Rényi random graph. By analyzing finite graphs of increasing volume, as well as the graphon variational problem in the infinite volume limit, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to the one of an equi-bipartite graph.

Phase transition in the diffusion and bootstrap percolation models on regular random and Erdős-Rényi networks

The diffusion and bootstrap percolation models were studied in regular random and Erdős-Rényi networks using the modified Newman-Ziff algorithms. We calculated the percolation threshold and the order parameter of the percolation transition (strength of the giant cluster) and its derivatives. The percolation transitions are classified by the results. The diffusion percolation with a small $k$ has a double transition, and the bootstrap percolation with $m\geq3$ has the first-order percolation transition. The diffusion percolation with a large $k$ and the bootstrap percolation with a small $m$ sh…

Erdos-Renyi随机图的首次通过性质

First Passage Properties of the Erdos-Renyi Random Graph

研究了Erdos-Renyi随机图中随机游动到任意两个点之间的平均时间。我们开发了一个有效的介质近似，预测节点对之间的平均首次通过时间，以及此首次通过时间的所有时刻，是不敏感的分数P占用链接。这一预测定性同意远离逾渗阈值的数值模拟。在渗流阈值附近，统计学上有意义的量是平均通过率，即首次通过时间的倒数。该速率随渗流转变附近的p而非单调地变化。这种行为的大部分可以通过简单的启发式论证来理解。

Antiferromagnetic Potts model on the Erdos-Renyi random graph

We study the antiferromagnetic Potts model on the Poissonian Erdös-Rényi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of Derrida-Ruelle probability cascades)and from a positive entropy argument.

无向有向Barabási-Albert网络和Erdös-Rényi随机图上$Q$ -态Potts模型零温动力学的持久性

Persistence in the zero-temperature dynamics of the $Q$-states Potts model on undirected-directed Barabási-Albert networks and Erdös-Rényi random graphs

本文利用零温Glauber动力学研究了Potts模型中的持续概率$P(t)$，其中$Q=3,4,5,7,9,12,24,64, 128$，$256, 512, 1024,4096,16384 $，. {\it有向}和{ \it无向} Barabási-Albert网络和Erdös-Rényi随机图上的$2^{30}$状态.在这个模型中，我们发现对于{ \it无向}和{ \it有向} Erdös-Rényi随机图，$P(t)$在短时间内指数衰减到零.对于{ \it定向}和{ \it无向} Barabási-Albert网络，相反，它在很长一段时间内以指数方式衰减到一个恒定值，即对于所有来自$Q=3,4,5,..., 2^{30}$的$Q$值（本文研究），$P(\infty)$都不为零;这表明所有这些$Q$值都是“阻塞”的。除了在{ \it无向}情况下的$Q=2^{30}$，$P(t)$以指数方式趋向于零;这可能只是一个有限大小的效应，因为在其他“阻塞”情况下，你可能只有几个不变的自旋。

Erdys-Rényi随机图谱的Lifshitz尾

Lifshitz tails for spectra of Erdős--Rényi random graphs

研究了具有$N$顶点和边概率$p/N$的Erdens-Rényi随机图上的离散Laplace算子$Δ^{(N)}$.我们感兴趣的是在亚临界制度$0<p<1$没有巨型集群出现的限制光谱特性的$Δ^{(N)}$作为$N\to\infty$。我们证明了在这个极限下，$Δ^{(N)}$的积分态密度的期望值在谱下边缘E=0处表现出Lifshitz尾行为。

Finite size corrections to disordered systems on Erdös-Rényi random graphs

We study the finite size corrections to the free energy density in disorder spin systems on sparse random graphs, using both replica theory and cavity method. We derive an analytical expressions for the $O(1/N)$ corrections in the replica symmetric phase as a linear combination of the free energies of open and closed chains. We perform a numerical check of the formulae on the Random Field Ising Model at zero temperature, by computing finite size corrections to the ground state energy density.

Erdos-Renyi随机图的大偏差性质

On large deviation properties of Erdos-Renyi random graphs

我们证明了Erdös-Rényi随机图的大偏差性质可以从统计力学的$q$ -态Potts模型的自由能导出.更确切地说，Potts自由能相对于$\ln q$的勒让德变换与图系综的分量生成函数有关。这推广了随机图的典型性质和Potts自由能的$q\to 1$极限之间的著名映射。对于指数罕见的图，我们显式地计算组件的数量，巨组件的大小，巨组件内部和外部的度分布，以及小组件大小的分布。我们还进行了数值模拟，这是非常好的协议与我们的分析工作。最后，我们展示了如何通过研究插入新的顶点和边的随机图的演变，而不求助于热力学的Potts模型，可以得出相同的结果。

Modularity of Erdős-Rényi random graphs

For a given graph $G$, each partition of the vertices has a modularity score, with higher values indicating that the partition better captures community structure in $G$. The modularity $q^*(G)$ of the graph $G$ is defined to be the maximum over all vertex partitions of the modularity score, and satisfies $0\leq q^*(G) < 1$. Modularity is at the heart of the most popular algorithms for community detection. We investigate the behaviour of the modularity of the Erdős-Rényi random graph $G_{n,p}$ with $n$ vertices and edge-probability $p$. Two key findings are that the modularity is $1+o(1)$ with…