Questions tagged [random-graphs]
The study of probability distributions over graphs. For example, the Erdős–Rényi model where each edge occurs independently with equal probability.
346 questions
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Positive-semidefiniteness of Laplacian of signed graph
Consider a signed complete graph $G(E,V)$ with adjacency $A_{ij}\in\{-1,+1\}$. Define the Laplacian matrix as $L:=D-A$ where $D$ is the degree matrix, $D_{ii}=\sum_{j\neq 1}A_{ij}$.
my question.
If $\...
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Example of random walk in a random environment (RWRE) saying things on the environment
I was wondering if anyone is aware of works/articles/examples where random walks in a random environment (RWRE) are actually used for obtaining information on the random environment.
To clarify a bit, ...
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Connected components in random regular graphs
Suppose we take a random regular graph $G_{2n, r}$, where $n$ is large. Let us also assume that $r$ is fixed, (not dependent on $n$). Let's say that half of the vertices of the graph are colored black ...
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Which infinite random graphs with percolation threshold $p_c=0$ are transient?
I am interested in long-range percolation models with heavy-tailed degree distributions such as DHH13, GLM21, Y6. The simplest example is scale-free percolation in which vertices are elements $\mathbb{...
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Is a random $(r+1,r)$-biregular bipartite graph $r$-edge connected w.h.p?
A uniformly random $r$-regular bipartite graph on $n$ vertices is known to be $r$-edge connected. That is, with high probability as $n$ grows large, the minimum size of a cut in a random $r$-regular ...
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Are the eigenvalues of the 1D lattice with random weights known?
Consider the adjacency matrix $\mathbf{A}$ of a one dimensional lattice of size $N$. That is, $A$ is a $N\times N$ matrix with $A_{ij}=1$ if vertex $i$ adjacent to vertex $j$ (there exists an edge ...
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Asymptotically tight concentration of norms of subgaussian random vectors with independent coordinates, as the dimension $n \to \infty?$
Let $X=(X_1 \dots X_n)\in \mathbb{R}^n,$ be a subgaussian random vector so that $X_i$'s are independent, $\mathbb{E}X_i = 0, \mathbb{E}X_i^2=1.$ Before we pose our question, let's state the following:
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Modularity in a graph -- derivation of modularity score
Background
I am currently reading "Modularity and community structure in networks" (2006) by Newman [1].
In it, he derives a score for the modularity of a graph ...
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Longest Path in a Directed Graph with Specified Number of Edges
Let $G$ be a directed graph that has $n$ nodes and is strongly connected. Define a random path as the following: Pick two vertices uniformly at random and find the shortest path going from one vertex ...
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On lower bounds for harmonic functions on $\mathbb{Z}^d$
Consider a non-constant harmonic function $f$ on $\mathbb{Z}^d$ (meaning this that $f(x)$ if the average of the $2d$ values $f(y)$ such that the distance between $x$ and $y$ is one). Let $M_n$ denote ...
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When an Erdos-Renyi graph is locally tree like?
I would like to know when an ER graph is locally treeing like. In this post.
I found this comment:
I think $N$ is $\log2|V|$, or something like that, in that paper.
They consider binary vectors ...
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Random Walk 2D with dependent weights [closed]
I have spent a lot of time trying to solve this problem but have had no luck so far! Any help would be highly appreciated!
Suppose I have a 3x3 grid as shown below.
(3,1) (3,2) (3,3)
(2,1) (2,2) (...
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Finding loops and double edges ASAP in configuration model random graph
A common approach (at least in theory) to generating a random $n$ vertex graph uniformly subject to having a given (feasible) degree sequence $(d_i)_{i = 1}^n$ is to use the configuration model, i.e. ...
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Subgraphs of random graphs with a given degree sequence
Let $\mathbf{d}=(d_1,\dots, d_n)$ be a given degree sequence with $3\leq d_i\leq \Delta$ for every $i$, where $\Delta$ is constant. Let $G(n,\mathbf{d})$ denote the random graph uniformly distributed ...
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Average number of cycles in a directed regular graph?
A directed random regular graph is a graph where all vertices have exactly $d_{\rm in}$ edges going in and $d_{\rm out}$ going out. If the graph is undirected, i.e. all vertices have degree $d$, then ...
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Do Fagin's zero-one laws hold on stochastic block model?
Let $n$ be a positive integer (the number of vertices), $k$ be a positive integer (the number of communities), $p = (p_1, . . . , p_k)$ be a probability vector on $[k] := \{1, . . . , k\}$ (the prior ...
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Graphon convergence of uniform weighted graphs
I have a question that I need at some point my research. Suppose that the upper-triangular entries of an $n\times n$ symmetric matrix $A$ are i.i.d. Uniform$(0,1)$. Does the weighted graph with ...
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Union of two copies of uniform spanning forest on $\mathbb{Z}^3$ is transient? [closed]
Let $G$ be the (random) graph which is the union of two independent copies of the uniform spanning forest on $\mathbb{Z}^3$.
Question: Is (the simple random walk on) $G$ transient almost surely?
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Is homomorphism density of partially labeled graph continuous with respect to cut metric
Let $F=(V, E)$ be a finite simple graph on $n$ vertices with two labelled vertices, say $x, y$. Let $W:[0, 1]^2\to [-1, 1]$ be symmetric function. Lov'asz's book (Large Networks and Graph Limits) ...
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Cycle statistics of random endomorphism
Let $S$ be a set with $n$ elements and let $f:S\to S$ be a random function, chosen uniformly among the $n^n$ possibilities. Considering $f$ as a directed graph of constant outdegree $1$, i. e. with ...
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Generating a random graph with bounds on degree and diameter
What would be a way to generate a random simple graph with diameter lesser than a given number, and in which there are given lower and upper bounds (bounds being uniform across vertices) on the degree ...
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Are two degree sequences compatible, for random simple graph generation?
Consider a set $V$ of $n$ vertices, and three degree sequences $a_i$, $b_i$ and $c_i$ such that $c_i = a_i+b_i$, $i=1..n$.
Assume these degree sequences are graphical: there exist simple graphs (no ...
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Expansion of random subgraphs of a bi-regular bipartite graph
Let $G = (L, R, E)$ be a bi-regular bipartite graph, with $|L|=n$ and $|R| = C \cdot n$, where $C$ is a large constant. Let $d$ be its (constant) right-degree.
We know $G$ is a good spectral expander. ...
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Zero-One law for Hamiltonian path subgraphs of Hamming Distance Graphs?
$(\alpha,\beta,d)$-Hamming Distance Graph $G_d(\alpha,\beta)$ for $\alpha,\beta\in(0,1]$ is a graph on $2^d$ vertices $v_0,\dots,v_{2^d-1}$ with edges $(v_i,v_j)\in\mathcal E(G_d)$ iff $0<\sum_{t=1}...
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Is there a way to generate a graph of specified treewidth
The treewidth is a parameter of the graph that describes its similarity to a tree. Treewidth is NP-hard to find. For the introduction please see wikipedia
The question is how to generate interesting ...
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Edgeweight-Conditions for "Statistically Self-similar" Complete Weighted Graphs
Given a complete symmetric weighted graph with $n$ vertices, for such a graph there always exists a minimum spanning tree and, under the assumption of the uniqueness of that tree, the vertex degrees ...
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Transversal deviation in first passage percolation
Take the lattice $\mathbb{L}^{2}=(\mathbb{Z}^{2},\mathbb{E}^{2})$ with i.i.d. $\text{U}[0,1]$ weights on the edges, and the random variable $D$ giving the maximal transversal deviation of the geodesic ...
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Size of the last non-empty $k$-core of a random graph
Given $n$ and $p$ for $G(n,p)$, how to find the distribution of the size of the non-empty $k$-core with largest $k$?
In particular, what is the probability (for any $n$ and $p$) that only $c$ ...
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Largest eigenvalue of two types of slightly different random matrices
Consider two types of slightly different $n \times n$ symmetric random matrices $X$. The diagonal elements of $X$ are fixed as $1$. Suppose $\frac{k}{n} \to \alpha$ for some constant $\alpha\in(0,1)$.
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Minimum dominating sets in tournaments
It is known that in any tournament with $n$ vertices, there is a dominating set of size no more than $\lceil \log_2 n\rceil$. (See Fact 2.5 here.)
What about when the tournament is chosen ...
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Erdös-Renyi Model with prescribed subgraph
In the Erdös-Rényi model for random graphs there is a lot of results stating sharp phase transitions for the probability of a random graph to contain a fixed prescribed ...
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mixing time of random walks on dense Erdos Renyi graphs
Is there anything known about the mixing time of a simple random walk on an Erdos-Renyi graph with parameter $\langle n,d \rangle$ where $d=n^a (0<a<1 )$. I know about Reed et al and Benjamini ...
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expected number of edges for fixed min cut
It is known that a graph $G=(V,E)$ with $n$ nodes and min cut $k$, must have at least $\frac{1}{2}nk$ edges.
Are there any tighter bounds or expectations I can place on $|E|$ if I assume that $G$ ...
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How are Polynomials of Toric ideals Studied with Exponents as ST-cuts?
Topic: Toric ideals on Expected value of Structure Functions in Random Graphs
Goal: to understand the toric ideal where the exponents $h_i$ and $s_j$ are st-vertex-cuts of a digraph
\begin{equation}
...
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What is the gap between the two defined distances from node to node in a random graph?
Give a sparse random graph $G=(V,E)$, every edge $(u,v)\in E$ is associated with a weight $w(u,v)$. We assume each $w(u,v)$ is geometrically distributed with parameter $p_{u,v}$. The weight of a path ...
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"drift" of a random graph $G(n,p)$ with $p=\alpha\ln{n}/n$
Suppose $G\sim G(n,p)$ with $p=\alpha\ln{n}/n$ for some large constant $\alpha$. I wish to show a certain "drift" property of $G$, which can roughly be phrased as follows: if $u$ is "not that far" ...
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Reachability in dynamic random graphs
There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...
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Can this application of the moment closure method to epidemics on networks be made rigorous?
Short version: Consider the SI model of infectious disease spread on a random graph $G$ with a given degree sequence. Let $j$ be a vertex and let $i$ and $k$ be two of its neighbors. If $G$ ...
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What is the exact definition of a sharp transition?
In "Sharp threshold phenomena in statistical physics", H. Duminil-Copin, Japanese J. of Math. 14, 2019, a sharp transition of a boolean function is defined as follows:
A sequence of ...
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Probability of having no cycles of fixed length in $d$-regular graphs
According to this paper, the probability that a random $d$-regular graph of order $n$ has no cycles of length $c_1,c_2,\ldots,c_t$ is $$P=\exp\left(-\sum_{i=1}^t\mu_i+o(1)\right)$$ as $n\rightarrow\...
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Proving a random bipartite graph contains a perfect matching
I have the following problem
consider a random bipartite with vertex classes $A$ and $B$ of size $|A|=|B|=\mathrm{log}^{2}(n)$ graph in which every possible edge is chosen independently with ...
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Can we estimate the probability $\mathbf{P}(a-k|a - b) $ on a random graph?
Let $G=(V,E)$ be an undirected random graph such that
$V$ is the set of nodes, and $E$ is the set of edges
Assume the ground graph $G$ is sparse enough, for example, $\frac{|E|}{|V|}= c \in [10, 40]$ ...
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Threshold for perfect Matchings in Bipartite graph
Consider the random bipartite graph with vertex classes of size $n$ and each edge being present independently with probability $p(n)$.
I know one way to prove the threshold of a perfect matching is ...
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how rare random bipartite graphs in all random regular graphs
i read a note talking about this fact, bipartite graphs are rare in regular graphs. but it do not state how rare it is? just curious about it. thank you very much.
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603
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Hammersley-Clifford theorem
The paper spatial interaction and the statistical analysis of lattice systems by Besag (1974) presents an alternative proof for the Hammersley-Clifford theorem.
In order to prove the HM theorem, Besag ...
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Does exponential degree distribution entail Log-normal distance distribution in large complex graphs?
We've been exploring the graph structure of a large genealogical data base (WikiTree) of which main connected component contains about 23 million nodes. The graph edges are defined by any direct ...
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382
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An attempt to find expected value of clique number of special random graph
Let $G(n)=(V,\mathcal{E})$ be a random graph definded as follows:
$V=[n]=\{1,2, ... ,n\}$ and for all $i,j\in V$ so that $i\ne j$ we have $\{i,j\}\in\mathcal{E}$ with probability $p$. Where $p\in[0,1]...
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How to use probability to find a matching in a family of graphs?
In a conference, I heard that we can use some probabilistic methods to find a matching in some kind of graphs. I would like to see some examples of such technics. Can someone provide some references ...
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Figuring out a consistent definition for the percolation backbone
In the context of percolation, e.g., bond/site percolation, random graph connectivity in 2-3 dimensions, etc., once the percolation threshold is reached, that is the system is spanned by an infinite ...
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Random graphs with boundary in a game (Tsuro)
Suppose we have an $n \times n$ board and we have $n^2 - 1$ square tiles. These tiles consist of a 8 vertices, two on each edge, and every vertex is connected to precisely one other vertex. These ...
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