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Questions tagged [random-graphs]

28 questions
13
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1 answer

Number of clique in random graphs

There is a family of random graphs $G(n, p)$ with $n$ nodes (due to Gilbert). Each possible edge is independently inserted into $G(n, p)$ with probability $p$. Let $X_k$ be the number of cliques of size $k$ in $G(n, p)$. I know that…
user1374864
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11
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Min-eigenvalue bound for a random d-regular graph

I need help proving the following fact: Let $G$ be a random $d$-regular graph with adjacency matrix $A$. The smallest eigenvalue $\lambda_n$ of $A$ should satisfy $|\lambda_n| = o_d(d)$. (In particular, I think $|\lambda_n| = O(\sqrt{d})$.) Exercise…
Sam
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11
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4 answers

What is a good algorithm for generating random DFAs?

I am generating random DFAs to test a DFA reduction algorithm on them. The algorithm that I'm using right now is as follows: for each state $q$, for each symbol in the alphabet $c$, add $\delta (q, c)$ to some random state. Each state has the same…
Duncan
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7
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2 answers

Has this model of random directed graphs been studied?

Youtube recently added a feature called autoplay, where each clip is assigned a (presumably related) clip that follows it. This, in effect, defines a directed graph on the set of youtube clips, where each vertex has outdegree 1. The user starts at a…
Zur Luria
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6
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1 answer

Expected number of maximal cliques in $G(n,p)$

The $G(n,p)$ random graph model creates graphs with $n$ vertices and each possible edge exists independently with probability $p\in (0,1)$. Much is known about the (expected) size of a largest clique in these graphs, and it has been shown that the…
Austin Buchanan
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5
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2 answers

Generation of random binary trees

Given n, I want to randomly generate a binary tree (unlabelled) that has n end nodes. Could someone kindly provide a reference containing an algorithm for doing that? I attempted to do as follows: From a PRNG obtain n PRNs in [0.0, 1.0) as…
Mok-Kong Shen
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4
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Generating graphs with partially overlapping cliques

Currently, I am working on a research project where I will utilise reinforcement learning for the diversified top-$k$ clique search problem. To train the reinforcement learning algorithm, I need to generate graphs that have similar properties, such…
Jesse
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4
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1 answer

Marginal Probability of Generating a Tree

Fix some finite graph $G = (V, E)$, and some vertex $x$. Suppose I generate a random sub-tree of $G$ of size $N$, containing $x$, as follows: Let $T_0 = \{ x \}$. For $0 < n \leqslant N$ i. Let $B_n$ be the set of neighbors of $T_{n-1}$ outside of…
πr8
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3
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1 answer

What is the probability of friendship conditioned on the number of mutual friends

Let Alice and Bob be two users chosen uniformly at random from a social network (e.g. Facebook). What is the probability that they are friends assuming that they share $k$ mutual friends? I am interested both in the experimental values (or…
Kaveh
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3
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1 answer

Generating sparse connected Erdős–Rényi random graphs

Given a random graph $G(n, p)$, where $n$ is the number of nodes and $p$ is the probability of connecting any two edges, it is known that $t = \frac{\ln(n)}{n}$ is a threshold for the connectedness of the graph: if $p$ is greater than $t$ the graph…
Gilbes
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3
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1 answer

Efficient Algorithms for Complex Networks

Most standard works on random graphs focus on $G_{n,p}$ and random regular graphs. However, such models are far from a good abstraction to describe the types of networks that one typically encounters in the real world. There are several simple…
Zur Luria
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3
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1 answer

Reference asking: phase transition in SAT

This is not a technical question, I hope this community has a room for such questions, but I will delete it in case this is inappropriate. It has been experimentally observed (e.g. here) that when choosing a $3$-SAT formula by the following…
Dani
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3
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Random linear arrangement of a tree with constrained edge lengths

Let $T$ be a tree with $V$ and edges $E$. Let a linear arrangement $\pi$ of $T$ be a bijective mapping from nodes to integers in the range $\{1, \dots, |V|\}$. You can think of $\pi$ as defining the position of each node in an arrangement on a line.…
Richard Futrell
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2
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Eigenvalues of an induced subgraph of a random graph

Suppose $G$ is a random graph on $n$ vertices where each edge appears with probability half. Suppose someone looks at the resulting graph and chooses an arbitrary subset $W$ of vertices of size $k>\sqrt{n}$. How do the eigenvalues of the induced…
chyle
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2
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Probability of k-clique in a random graph

I need to find the order of the minimum k = k(n) such that the probability of having at least 1 k-clique in a random graph $G(n, \frac{1}{2}$) is $\mathcal{O}(\frac{1}{n})$. $X_k$ is the random variable which count the number of k-cliques in a…
Pratozoo
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