O. C. Martin, P. Sulc
We consider random walks on random graphs, focusing on return probabilities and hitting times for sparse Erdos-Renyi graphs. Using the tree approach which is expected to be exact in the large graph limit, we show how to solve for the distribution of these quantities and we find that these distributions exhibit a form of self-similarity.
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The critical point of k-clique percolation in the Erdos-Renyi graph
受k-团渗透方法在大型真实网络中识别重叠社区的成功启发，本文研究了Erdos-Renyi图中的k-团渗透问题。当两个节点连通的概率p大于某个阈值p_c（k）时，大小为k的完全子图（k-团）被组织成一个巨簇。通过作出一些假设，预计是有效的阈值以下，我们确定的k-团渗流集群的平均大小，使用生成函数形式主义。从这个平均尺寸的发散，我们然后导出临界连接概率p_c（k）的解析表达式。
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。
We present analytical results for the distribution of first hitting times of non-backtracking random walks on finite Erdős-Rényi networks of $N$ nodes. The walkers hop randomly between adjacent nodes on the network, without stepping back to the previous node, until they hit a node which they have already visited before or get trapped in a dead-end node. At this point, the path is terminated. The length, $d$, of the resulting path, is called the first hitting time. Using recursion equations, we obtain analytical results for the tail distribution of first hitting times, $P(d>\ell)$, $\ell=0,1,2,…
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.
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。
On certain perturbations of the Erdos-Renyi random graph
我们研究扰动的Erdos-Renyi模型的统计权重的图依赖于丰富的某些几何图案。利用与精确可解有效模型的形式对应，我们证明了在临界极限下渗流跃迁的存在性，并微扰地导出了临界值的表达式。自由能和度分布的时刻也计算微扰在该限制和渗流准则相比，Molloy-Reed标准。
We present analytical results for the distribution of first hitting times of random walkers (RWs) on directed Erdős-Rényi (ER) networks. Starting from a random initial node, a random walker hops randomly along directed edges between adjacent nodes in the network. The path terminates either by the retracing scenario, when the walker enters a node which it has already visited before, or by the trapping scenario, when it becomes trapped in a dead-end node from which it cannot exit. The path length, namely the number of steps, $d$, pursued by the random walker from the initial node up to its termi…
Phase transition in the maximum clique problem: the case of Erdos-Renyi graphs
在ER图的最大团问题上发现了一个相变，类似于Kirkpatrick和Selman在布尔可满足性问题上发现的相变。虽然数据点的数量是有限的，但这种转变似乎服从有限尺寸标度。转换还显示了需要特别大的CPU时间来求解的图实例的集中。
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.
In this work we prove that the giant component of the Erdös--Renyi random graph $G(n,c/n)$ for c a constant greater than 1 (sparse regime), is not Gromov $δ$-hyperbolic for any positive $δ$ with probability tending to one as $n\to\infty$. As a corollary we provide an alternative proof that the giant component of $G(n,c/n)$ when c>1 has zero spectral gap almost surely as $n\to\infty$.
From Scale-free to Erdos-Renyi Networks
我们分析了无标度网络和Erdos-Renyi网络之间的插值模型。引入的模型生成一个单参数的网络家庭，并允许分析结构异质性的作用。分析计算进行了比较，以描述这两个重要的网络类之间的过渡与广泛的数值模拟。最后，应用所提出的模型研究的逾渗过渡。
In this work, we investigate the temporal evolution of the degree of a given vertex in a network by mapping the dynamics into a random walk problem in degree space. We analyze when the degree approximates a pre-established value through a parallel with the first-passage problem of random walks. The method is illustrated on the time-dependent versions of the Erdős-Rényi and Watts-Strogatz models, which originally were formulated as static networks. We have succeeded in obtaining an analytic form for the first and the second moments of the first-passage time and showing how they depend on the si…
Majority-vote model on directed Erdos-Renyi random graphs
通过蒙特卡罗模拟，研究了有向随机图上带噪声的多数表决模型。为了完整地表征所观察到的有序-无序相变，临界噪声参数$q_c$，以及临界指数$beta/nu$，$gamma/nu$和$1/nu$已被计算为随机图的连接度$z$的函数。
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.
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。
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。
Analytical results for the distribution of first hitting times of random walks on Erdős-Rényi networks are presented. Starting from a random initial node, a random walker hops between adjacent nodes until it hits a node which it has already visited before. At this point, the path terminates. The path length, namely the number of steps, $d$, pursued by the random walker from the initial node up to its termination is called the first hitting time or the first intersection length. Using recursion equations, we obtain analytical results for the tail distribution of the path lengths, $P(d > \ell)$.…
The SIS process on Erdös-Rényi graphs: determining the infected fraction
针对稀疏图中SIS过程感染比例估计难以兼顾相关性与计算效率的问题，提出一种基于度对的新方法，在Erdős-Rényi图上实现了准确且高效的估计。
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…
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