All Questions
8 questions
7
votes
2
answers
1k
views
Why is the spectrum of Erdős–Renyi random graph approximately symmetric?
I am recently self-learning random matrix theory and made some simulations about the spectrum of Erdős–Renyi random graph $G(n,p)$ when $np\to\infty$,
and $np\to c=2,3$.
The plots above are already ...
- 5,405
4
votes
0
answers
1k
views
Number of arrangements that contain at least 1 path from top to bottom of 2D matrix
I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white.
With that information, I can easily calculate the total number of black element arrangements that exist ...
- 47
2
votes
0
answers
90
views
Odd $k$-cycle counts in graph with adjacency matrix $A$ is leading term in $\operatorname{tr} A^k$?
In a recent paper of Neeman, Radin, and Sadun, Moderate Deviations in Cycle Count, in the first line of section 7.3 they wrote $\tau_k(A)=\frac{\operatorname{tr}A^k}{n^k}+O(\frac 1n)$, but I don't ...
- 12.9k
1
vote
0
answers
127
views
Delocalization of eigenvectors of graph Laplacians
Let $(V,E)$ be an undirected, connected graph with $n$ nodes. The graph Laplacian is defined as $L = D - A$, where $D$ is the degree matrix and $A$ is the adjacency matrix. Let $0 = \lambda_1 < \...
- 41
7
votes
1
answer
463
views
Boundedness of total current in electrical network
Consider the following symmetric matrix (adjacency matrix):
$$A=(a_{ij})_{1\leq i,j\leq n}$$
such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...
- 1,495
11
votes
2
answers
353
views
Exponential decay of voltage potential difference
Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
- 17.2k
5
votes
1
answer
1k
views
Intuition on Kronecker Product of a Transition Matrix
Let $T$ be a $N\times N$ transition matrix for a markov chain with $N$ states. Thus $T_{ij}$ is the probability of transition from state $i$ to state $j$ (and thus rows summing to one). Now consider ...
- 101
3
votes
1
answer
119
views
Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed
The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed:
I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n \...
