# 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.

235
questions

**-2**

votes

**1**answer

36 views

### Ensemble averaging in a random graph (or network) in the large $N$ limit [closed]

I have a random graph/network described by the adjacency matrix $(a_{ij})_{N\times N}$ where $a_{ij}=1$ with probability $p$. Each node in the graph is associated with a continuous quantity $\eta_i=\...

**0**

votes

**0**answers

34 views

### Average number of edges of an induced graph by using the edge-based node selection technique on a graph with arbitrary degree distribution

Let $G(U,V,E)$ be a simple, undirected, bipartite graph and $U=\{u_1,u_2,{\cdots},u_n\}$ and $V=\{ v_{1},v_{2},\cdots,v_{n}\}$. Let $d_k^l$ be the number of vertex with degree $k$ in $l$, where $l \in ...

**2**

votes

**1**answer

61 views

### Critical probability for Erdos-Renyi digraphs to be strongly connected

Given $p \in [0,1]$, an Erdos-Renyi graph ${ER}(n,p)$ on $n$ vertices is constructed by defining, for each unordered couple of distinct vertices ${i,j}$ an edge between $i$ and $j$ with probability $p$...

**3**

votes

**0**answers

162 views

### Probability that a random multigraph is simple

Question.
Consider a given sequence of $n$ integers $d_1$, $d_2$, $\cdots$, $d_n$ with $\sum_i d_i$ even and $d_i\le n$ for all $i$. One may sample a random multi-graph having this degree sequence ...

**7**

votes

**0**answers

252 views

### Graph with path of length $\geq n$ along grid diagonals - a known result in graph theory?

Is the following lemma a well known result in graph theory?
I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph ...

**1**

vote

**1**answer

48 views

### Relation between expected values of eigenvalues of Laplacian matrix of a graph and eigenvalues of expected Laplacian matrix of that graph?

Particularly, I am dealing with Erdős–Rényi random 𝐺(𝑛,𝑝), so the expected Laplacian matrix of 𝐺(𝑛,𝑝) is 𝑝(𝐽𝑛−𝐼𝑛), where 𝐽𝑛 and 𝐼𝑛 are one and identity matrices, respectively.
In ...

**1**

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**0**answers

24 views

### Clustering number on ring lattice

I have seen in several places a useful formula that let us calculates the clustering number of regular ring lattice graphs with even degree but I have not found a convincing proof of it. Concretely, ...

**1**

vote

**0**answers

38 views

### Count shortest path with different lengths in random graph

Let $G(n,p)$ be an Erdos-Renyi random graph on $n$ vertices with probability $p$, i.e. for each pair of vertices, they are connected directly by an undirected edge with probability $p$. Suppose we are ...

**1**

vote

**1**answer

26 views

### 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]...

**1**

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38 views

### 4-cycles vs eigenvalue information on quasi-random graphs

My (philosophical) question arises from reading the wonderful paper of Chung-Graham-Wilson where the authors introduces the notion of quasi-random graphs.
The main purpose of the paper is to show ...

**12**

votes

**1**answer

313 views

### Distribution of degree in graphs: when is the friendship paradox the paradox it wants to be?

$\DeclareMathOperator\deg{deg}\DeclareMathOperator\ndeg{ndeg}\newcommand\abs[1]{\lvert#1\rvert}$The friendship paradox goes most people have fewer friends than their friends have on average. The ...

**5**

votes

**1**answer

95 views

### Attacking a network at minimum cost

A target system is modelled as a giant, undirected, simple graph $G$ (simple meaning no hyperedge) that can be scrutinized in adequate detail to budget and plan the attack: its topology changes slowly ...

**1**

vote

**1**answer

97 views

### 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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**0**answers

38 views

### Concentration of $2$-norms of random variables whose co-ordinates are not independent?

Let us consider the random vector $X=[X_1 \dots X_d] \in \mathbb{R}^d, E[X]= 0, cov[X]= \Sigma.$ Then the random vector $Z:= \Sigma^{-1/2} X=[Z_1 \dots Z_d]$ has $E[Z]=0, cov[Z]=I_d.$ I'm looking for ...

**4**

votes

**0**answers

54 views

### Expected number of bridges in a random subgraph

I am researching connectivity in random subgraphs and have come across the following problem.
A bridge between two vertices $a$ and $b$ of a graph $G$ is an edge $e$ such that removing $e$ from $G$ ...

**3**

votes

**1**answer

90 views

### Unique maximum degree in tournament

Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) Let $p(n)$ denote ...

**1**

vote

**1**answer

72 views

### Size of minimum cut in random graph

Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) The score of each ...

**0**

votes

**0**answers

39 views

### Giant bounded degree subgraph in supercritical Erdos-Renyi graph

We are studying an epidemic model and find ourselves needing criteria for when a giant component of bounded degree vertices is contained in the giant component.
Let $G\sim G(n, \mu/n)$ with $\mu >...

**2**

votes

**2**answers

100 views

### Difference between two largest degrees

Consider a uniform random tournament with $n$ vertices. (Between any two vertices $x,y$, with probability $0.5$ draw an edge from $x$ to $y$; otherwise draw an edge from $y$ to $x$.) Let $S$ be the ...

**1**

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**1**answer

78 views

### Limit of normalized sum of Dirac measures at first $\lfloor p/2\rfloor$ eigenvalues of the sample covariance matrix, with Marcenko-Pastur assumptions?

Let $\lfloor{*}\rfloor$ denotes the nearest integer $\le *$. I'm asking myself the question what's the limit of the part of the empirical spectral distribution corresponding to the first $\lfloor{p/2}...

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**0**answers

16 views

### Bound on the number of bridges between vertices in a sampled subgraph

I am researching connectivity in sampled subgraphs and have come across the following problem.
A bridge between two vertices $a$ and $b$ of a graph $G$ is an edge $e$ such that removing $e$ from $G$ ...

**10**

votes

**4**answers

296 views

### When is a large graph with a given degree sequence likely to be connected?

Are there any results on whether a large random graph with a given degree distribution is likely to be connected?
In Erdős-Rényi graphs with $n$ vertices and $m$ edges, we have two sudden transitions ...

**1**

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**1**answer

95 views

### Continuum percolation in 1d

What is known about continuum percolation in 1d?
By this, I mean, for $d \in \mathbb{N}$, the Poisson-Boolean model of disks of radius $r_0 \in \mathbb{R}$ with centres arranged randomly in $[0,1]^{d}...

**7**

votes

**1**answer

181 views

### Graphons and Graphs

The situation is as follows: assume we have a sequence of simple weighted graphs $(G_n)_{n\in\Bbb{N}}$. For the terminology that follows I refer to Limits of dense graph sequences by László Lovász and ...

**0**

votes

**1**answer

52 views

### Small subgraphs of the random graph

If I look at the distribution of the number of small subgraphs in the random graph isomorphic to a connected graph $H$, this is asymptotically Poisson.
What proportion of these small subgraphs ...

**4**

votes

**1**answer

161 views

### Probability of a vertex being a “degree-celebrity” in a random graph

If $G(n,p)$ is a random graph of the Erdös-Rényi model,
what is the probability that $\mathrm{deg}(v)\gt\mathrm{deg}(u)\ \forall u\in\mathrm{adj}(v)$
Please feel free to relate answers to other ...

**3**

votes

**0**answers

48 views

### Largest component and number of components of random mappings with bounded in-degree

Let $S$ be a finite set of size $n$ and consider all functions $f$ from $S$ to itself, such that the preimage $f^{-1} (k)$ obeys $0 \leq |f^{-1} (k)| \leq m$.
Let $F$ be chosen uniformly at random ...

**2**

votes

**1**answer

111 views

### Electrode assignment problem in resistive networks

Main question
In the context of resistor networks, and particularly purely from a graph theory point of view, is there a consistent way of assigning the two electrode nodes in order to compare the ...

**4**

votes

**1**answer

223 views

### Probability of a subset of Bernoulli's being all 1's

Suppose we have $n$ iid Bernoulli's $X_1,\ldots,X_n$ with mean $p$, and a family $\mathcal{F}$ of subsets of $[n]$. The question is how to lower bound the probability that there is a set in the family ...

**2**

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**0**answers

68 views

### 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}...

**2**

votes

**2**answers

159 views

### Model for random graphs where clique number remains bounded

In the Erdös-Rényi model for random graphs,the clique number is seen to go to infinity as the number of vertices grows. Is anyone aware of models for random graphs with bounded clique number?

**1**

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**1**answer

77 views

### Behaviour of global clustering for common random graph models

In order to develop some intuition for some of the commonly used random graph models, I've been looking at the global clustering coefficient as a means of comparing them. In particular, for the ...

**6**

votes

**0**answers

99 views

### The properties of almost all directed graphs

A mathematician on the forum previously requested a reference on human brains modelled as directed graphs. This makes sense as neurons are mostly unidirectional and I have been thinking about similar ...

**4**

votes

**1**answer

111 views

### Almost all simple graphs are small world networks

Two days ago it occurred to me that almost all simple graphs are small world networks in the sense that if $G_N$ is a simple graph with $N$ nodes sampled from the Erdös-Rényi random graph distribution ...

**2**

votes

**1**answer

180 views

### Eigenvalues of random graphs

At time $t=0$, let $G_n(V,E)$ be a graph with $n$ vertices and $m < n$ edges. Then there exists a unique symmetric adjacency matrix $A_n$ associated with $G_n(V,E)$, defined as follows: $a_{ij} = 1$...

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**0**answers

34 views

### Giant component in continuum random graph

Is there some similar statement to Giant component theorem for the infinite graph (particularly with continuum vertices)?

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**0**answers

76 views

### Random graphs - multiple giant components

For a given random graph, a connected component that contains a finite fraction of the entire graph’s vertices is called giant.
A well known result in random graphs is the existence and uniqueness of ...

**2**

votes

**2**answers

73 views

### 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 ...

**3**

votes

**1**answer

63 views

### Difference of pseudoinverse bound under assumptions

This seems like a standard problem, but unable to find a solution online.
Suppose we have two singular PSD matrices A and B with the following assumptions:
$ 0 < x \leq ||A|| \leq y$
$ 0 < ||...

**3**

votes

**1**answer

290 views

### Asymptotic formula for the number of connected graphs

It can be shown that the set of graphs with $N$ vertices $G_N$ has cardinality:
\begin{equation}
\lvert G_N \rvert = 2^{N \choose 2} \tag{1}
\end{equation}
Recently, I wondered how much bigger $\...

**2**

votes

**0**answers

17 views

### 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 ...

**1**

vote

**2**answers

104 views

### 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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votes

**2**answers

161 views

### 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 ...

**2**

votes

**0**answers

36 views

### 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 ...

**3**

votes

**1**answer

88 views

### Expected size of matchings in a cubic graph

Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$?
In other ...

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vote

**1**answer

89 views

### 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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votes

**1**answer

202 views

### Switching oriented paths in a graph

Consider an oriented graph (e.g. a finite part of the standard grid with some random orientations).
Each minute the following operation takes place: we choose uniformly randomly an ordered pair $(A,B)...

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vote

**1**answer

56 views

### What is the expected distance between the sides of a random subgraph of the grid?

Let $G$ be the $n \times n$ grid, in which each vertex is connected to the vertices above it, below it and on either side. Let $G_p$ be the random subgraph of $G$ obtained by keeping each edge with ...

**4**

votes

**1**answer

350 views

### Open Problems in Random Graphs [closed]

I am a PhD student in mathematics. I'm interested in probabilistic methods in combinatorics and especially random graphs. I am looking for an open problem in this area for my PhD proposal. I know that ...

**0**

votes

**2**answers

131 views

### Random Optimization on Graphs: Minimum Cut

Consider a complete graph on $n$ vertices. To each edge, $(i,j)$, we assign a weight, $W_{ij}$, which comes from some known distribution iid. Then, we ask the following question. Among all (weighted) ...