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0
votes
1answer
42 views

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 ...
9
votes
1answer
106 views

What are some useful invariants for distinguishing between random graph models?

Quite a few probabilistic algorithms for generating random graphs exist in the literature, such as: The Erdos-Renyi model The Stochastic Block model The Watts-Strogatz model The Barabasi-Alber model ...
3
votes
1answer
73 views

Good broad review of agent-based modeling? [closed]

Trying to find some good review of agent-based models and networks, specifically models that are defined by a graph of interacting nodes, that covers analysis of collective behavior based on model of ...
1
vote
1answer
199 views

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, ...
1
vote
0answers
35 views

Modularity in a graph - definition of the random component

This question concerns the definition of modularity in a graph. Consider a simple, undirected, unweighted graph $G$ with vertices in set $V$ and edges in set $E$ . Between any two vertices there is ...
5
votes
0answers
96 views

Does squaring a directed random graph more than double its out-degree?

As far as I know, it is an unsolved question whether or not this is true: If $G$ is a directed an oriented graph, $G^2$ always has some node whose outdegree is at least double that of its ...
1
vote
0answers
35 views

Weak law for component count of Erdos-Renyi random graphs

Penrose and Yukich derive a weak law for functionals of binomial point processes, which implies a law of large numbers for the component count of random geometric graphs. Do similar results exist for ...
5
votes
1answer
153 views

Modification of matching

Suppose i have an $n \times n$ random bipartite graph and suppose that i repeat the following process $n$ times. At the start (stage 1) each edge is selected independently with probability $p(n)$, and ...
0
votes
0answers
74 views

Gromov-Hausdorff distance measure between minimum spanning trees

I am trying to compare minimum spanning trees through time. I have two questions: 1-Is it possible to measure the similarity between two minimum spanning trees with Gromov-Hausdorff distance measure ...
0
votes
0answers
25 views

Cardinality of maximum independent set for a given degree distribution

Consider undirected graph $G(V,E)$. Assume that $f_n(k)$ be the probability mass function of degree of a vertex in $G$. Further, assume that $f_n(k)$ is an strictly decreasing function of $k$ with ...
2
votes
2answers
151 views

Probability of relations in network

Imagine, i have a predicate $\text{friends}(x_1, x_2)$ and I know that $p(\text{friends}(x_1, x_2)) = p_2$. If I generate a world of $n$ people ($x_1$ to $x_n$), I expect there to be $\binom{n}{2}p_2$ ...
1
vote
1answer
173 views

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 ...
8
votes
2answers
605 views

random category theory

This question is in some sense dual to the one asked in Is there an introduction to probability theory from a structuralist/categorical perspective? since contrary to the OP who asks for references ...
3
votes
1answer
105 views

Expected number of leaf nodes in some theoretical graph models

If a leaf node of a graph refers to a node having the degree of 1, how can one compute the expected number of leaf nodes of: (A) a random graph (e.g., Erdos-Renyi graph), (B) a small-world graph ...
7
votes
1answer
199 views

Probability of a graph procedure

We are going to build $K_n$ one edge at a time. Begin with the empty graph on $n$ vertices. Take a random permutation of the edges of $K_n$ and, one at a time, place the edges onto the graph (so, ...
4
votes
2answers
224 views

Statistics of strongly connected components in random directed graphs

I'm interested in the statistics of strongly connected components in random directed graphs. However, I'm unable to find any results on this, partly because I don't know the terminology to search for. ...
4
votes
1answer
81 views

Probability of a giant component existing in a $G(n,p)$ random graph with $p=\omega(\frac 1n)$

Consider a random $G(n,p)$ graph where $p=\omega(\frac 1n)$, and let $x$ denote the probability that the graph has a connected component of size linear in $n$. It is well known that $x$ tends to $1$ ...
1
vote
0answers
113 views

Multiple Bipartite graphs and matchings

I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...
1
vote
2answers
283 views

Expected matching in a bipartite graph

Consider a random bipartite graph constructed on vertex classes of size $n$ with each edge present independently with probability $p$. How could I go about calculating the size of the expected ...
1
vote
1answer
206 views

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 ...
1
vote
2answers
192 views

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 ...
0
votes
0answers
64 views

Computation on Random Bipartite graphs

I'm looking at a random bipartite graph $K_{\omega(n)}*K_{\omega(n)}$ where $\mathrm{log}(n)\leq \omega(n) \leq n^{1/2}$, in which each of the $\omega(n)^{2}$ edges is placed randomly with probability ...
1
vote
0answers
68 views

Matchings in random bipartite graphs

I was wondering if anyone could point me in the direction of a text or paper which would help deal with the following problem Suppose i am given a $K_{\mathrm{log}(n)} \times K_{\mathrm{log}(n)}$ ...
1
vote
1answer
157 views

Probability of each edge in K-clique [closed]

For $c \in R$ and $k \in N$, $k \geq 3$ let $p_{k,c} := n^{\frac{−2}{k+1}}log^c(n)$. I would like to prove that exists $c\in R$ such that every edge in the random graph $G(n,p_{k,c})$ lies in a copy ...
4
votes
0answers
195 views

Navigation in a graph

The problem Let $G=(V,E)$ be a graph. $k = O\left(\log(|V|)\right)$ distinct vertices are picked randomly from $V$. We call the set of chosen $k$ vertices $T$. Assumptions about the graph: You may ...
0
votes
1answer
217 views

Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
9
votes
5answers
357 views

Do there exist sparse graphs with large crossing number?

Does there exist a sequence of graphs $\{ G_n \}$ such that $G_n$ has $n$ vertices, the number of edges of $G_n$ is $O(n)$, and the crossing number of $G_n$ is $\Omega(n)$? In particular, do ...
4
votes
1answer
191 views

Expectation of ratio of functions of i.i.d. Bernoullis: a concentration question

Consider the following $n \times n$ symmetric matrix of i.i.d. Bernoulli random variables, $X_{ij}$. For $i=1,...,n$ and $i<j\le n$. Let $X_{ij} \sim \text{Bernoulli}(p)$ when $i \ne j$ ($p$ ...
2
votes
0answers
45 views

“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" ...
3
votes
2answers
168 views

Independent Sets in random geometric graphs

I was wondering if a lot is known about independent sets in Random Geometric graphs? Most google searches don't bring up much. In addition I'm interested in any algorithms used for finding independent ...
2
votes
1answer
216 views

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. ...
3
votes
1answer
77 views

Hamiltonicity of random graphs with high girth

We say that $G\sim G_{n,f}$ (for $f=f(n)$) if $G$ is chosen uniformly at random from all graphs on $n$ vertices with girth $g(G)\ge f(n)$. Is there any threshold function $F(n)$ such that when $f\ll ...
11
votes
0answers
361 views

First passage percolation on a random geometric graph in the large connectivity limit

Let $V_\rho\subset\mathbb{R}^2$ be a point set in the plane obtained from a Poisson process of density $\rho$. The random geometric graph $G_\rho$ is obtained from $V_\rho$ by connecting points that ...
3
votes
1answer
118 views

limiting empirical spectral distribution of the Laplacian matrix on an Erdos-Renyi graph?

Let $G$ be an Erdos-Renyi random graph (i.e. an edge ($ij$) exists with probability $0 < p < 1$ and all edges are independent). Let $L$ be the Laplacian matrix of this graph (i.e $L=D-A$, where ...
3
votes
0answers
96 views

Node covering in a random graph

Given $N$ nodes randomly placed in a $D\times D$ area, i.e., the position of each node is randomly chosen. Assume that both $N$ and $D$ are sufficiantly large. An agent can move in the area at ...
0
votes
2answers
91 views

different way of selecting a random graph

Consider having a 'base' graph $G=(V,E)$ and selecting each vertex with independent probability $p$ and having the induced subgraph of $G$ with all 'selected' points as your random graph. Has this ...
6
votes
2answers
152 views

Convergence on a random graph

Assume a directed graph $G = (V,E)$ is drawn from a random graph distribution, for instance Erdős–Rényi's $G(n,p)$ (but with directed edges). Let $S:V\rightarrow\mathcal{P}(V)$ be the direct ...
4
votes
1answer
265 views

Expected number of connected components in a random graph

For a random graph G(n,p) what is the expected number of connected components? What is the probability distribution of this value? I'm specially interested in what happens for small values of p, ...
1
vote
0answers
56 views

Simulation of disassortative random graphs

Recently I have been trying to find a succinct algorithm for generation of disassortative networks. The best I have found is the algorithm by Newman described in his paper "Mixing patterns in ...
7
votes
0answers
99 views

Longest induced cycles in random geometric graphs near criticality

We make a random geometric graph $X(n;r)$ as follows. Choose $n$ points uniformly, independently, in the unit square $[0,1]^2$, for vertices, and then connect a pair of vertices $\{ p,q \}$ by an edge ...
4
votes
0answers
59 views

Cycle removal process

Consider the following stochastic process for generating a forest: start from a complete graph on $n$ vertices and proceed to repeatedly remove the edges of uniformly chosen cycles. Formally, let ...
3
votes
0answers
177 views

Hitting edges in graphs at random and let them die with honor

Let $G=(V,E)$ be a finite simple 2-connected graph. Let $B\subseteq E$ be a set of bad edges of size $k:=|B|$. For each edge $e\in E$ we toss a fair coin ($p=1/2$) and if the outcome is head, we hit ...
7
votes
1answer
195 views

Erdős-Renyi graph restricted to largest connected component

Suppose we have an instance of Erdős-Renyi $G(n,p)$ graph with $p = d/n$. Thus the expected node degree is $d$ which we will fix, while letting $n \to \infty$. Then, there will be more than one ...
5
votes
6answers
455 views

Random planar, bipartite graphs

I have a need to generate random planar graphs none of which have an odd cycle, i.e., bipartite graphs. I know there is a substantial two-decade literature on random planar graphs, little with which I ...
3
votes
0answers
72 views

Using Crump-Mode-Jagers processes to get logarithmic bound on a random tree height

I am currently pursuing my PhD degree and in my research I came across a family of random trees. I need to prove a logarithmic asymptotic bound for the heights of such trees as their size grows. I ...
3
votes
1answer
351 views

Expected Value for a Connected Graph

Consider a connected graph of N nodes. Assign randomly to each node a distinct number from 1 to N. For each node consider the maximum adjacent value or itself if all adjacent values are smaller. ...
2
votes
3answers
365 views

random networks and prime numbers

I have been studying networks recently and accidentally came up with an heuristic approach towards the distribution of prime numbers. The prime number theorem states that $\pi(n)\sim n/log(n)$ for ...
4
votes
3answers
181 views

Tracking automorphism groups of graph processes

Start with an edgeless graph on $n$ labeled vertices, and note that the automorphism group is $\Sigma_n$, the symmetric group on $n$ elements. Now imagine that we randomly start throwing in all of the ...
0
votes
0answers
65 views

Maximum Independent set of sparse graphs with few triangles

Notations used $\alpha(G) = $ Max sized independent set of graph $G$. $n(G) = $ Number of vertex in graph $G$. Theorem (by Ajtai et al.): For a triangle-free graph $G$ and max degree being ...
3
votes
0answers
66 views

Quasi-isometry of giant components in Erdos-Renyi graphs

Take two independent random graphs $G_1, G_2$ in the $G(n,p)$ model for $p = \frac{c}{n}$, $c > 1$ (the question probably makes sense also for $c=1$). Each of them will have a unique giant ...