Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn Independent Process Approximation to Sparse Random Graphs with a Prescribed Number of Edges and Triangles
88
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove a $pre$-$asymptotic$ bound on the total variation distance between the uniform distribution over two types of undirected graphs with $n$ nodes. One distribution places a prescribed number of $k_T$ triangles and $k_S$ edges not involved in a triangle independently and uniformly over all possibilities, and the other is the uniform distribution over simple graphs with exactly $k_T$ triangles and $k_S$ edges not involved in a triangle. As a corollary, for $k_S = o(n)$ and $k_T = o(n)$ as $n$ tends to infinity, the total variation distance tends to $0$, at a rate that is given explicitly. Our main tool is Chen-Stein Poisson approximation, hence our bounds are explicit for all finite values of the parameters.
rate research
Read More
Let $[mathcal{P}]$ be the points of a Poisson process on $mathbb{R}^d$ and $F$ a probability distribution with support on the non-negative integers. Models are formulated for generating translation invariant random graphs with vertex set $[mathcal{P}]$ and iid vertex degrees with distribution $F$, and the length of the edges is analyzed. The main result is that finite mean for the total edge length per vertex is possible if and only if $F$ has finite moment of order $(d+1)/d$.
In this work we present a simple and efficient algorithm which, with high probability, provides an almost uniform sample from the set of proper k-colourings on an instance of a sparse random graph G(n,d/n), where k=k(d) is a sufficiently large constant. Our algorithm is not based on the Markov Chain Monte Carlo method (M.C.M.C.). Instead, we provide a novel proof of correctness of our Algorithm that is based on interesting spatial mixing properties of colourings of G(n,d/n). Our result improves upon previous results (based on M.C.M.C.) that required a number of colours growing unboundedly with n.
We study the spectrum of a random multigraph with a degree sequence ${bf D}_n=(D_i)_{i=1}^n$ and average degree $1 ll omega_n ll n$, generated by the configuration model, and also the spectrum of the analogous random simple graph. We show that, when the empirical spectral distribution (ESD) of $omega_n^{-1} {bf D}_n $ converges weakly to a limit $ u$, under mild moment assumptions (e.g., $D_i/omega_n$ are i.i.d. with a finite second moment), the ESD of the normalized adjacency matrix converges in probability to $ uboxtimes sigma_{rm sc}$, the free multiplicative convolution of $ u$ with the semicircle law. Relating this limit with a variant of the Marchenko--Pastur law yields the continuity of its density (away from zero), and an effective procedure for determining its support. Our proof of convergence is based on a coupling between the random simple graph and multigraph with the same degrees, which might be of independent interest. We further construct and rely on a coupling of the multigraph to an inhomogeneous ErdH{o}s-Renyi graph with the target ESD, using three intermediate random graphs, with a negligible fraction of edges modified in each step.
We study geometric random graphs defined on the points of a Poisson process in $d$-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centred in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance.
For a real constant $alpha$, let $pi_3^alpha(G)$ be the minimum of twice the number of $K_2$s plus $alpha$ times the number of $K_3$s over all edge decompositions of $G$ into copies of $K_2$ and $K_3$, where $K_r$ denotes the complete graph on $r$ vertices. Let $pi_3^alpha(n)$ be the maximum of $pi_3^alpha(G)$ over all graphs $G$ with $n$ vertices. The extremal function $pi_3^3(n)$ was first studied by GyH{o}ri and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315--320]. In a recent progress on this problem, Kral, Lidicky, Martins and Pehova [Decomposing graphs into edges and triangles, Combin. Prob. Comput. 28 (2019) 465--472] proved via flag algebras that $pi_3^3(n)le (1/2+o(1))n^2$. We extend their result by determining the exact value of $pi_3^alpha(n)$ and the set of extremal graphs for all $alpha$ and sufficiently large $n$. In particular, we show for $alpha=3$ that $K_n$ and the complete bipartite graph $K_{lfloor n/2rfloor,lceil n/2rceil}$ are the only possible extremal examples for large $n$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
