Related papers: Urns with Multiple Drawings and Graph-Based Intera…
For graphs $G$ and $H$, an {\em $H$-colouring} of $G$ (or {\em homomorphism} from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colourings generalize such graph theory notions as…
Distributed voting is a fundamental topic in distributed computing. In pull voting, in each step every vertex chooses a neighbour uniformly at random, and adopts its opinion. The voting is completed when all vertices hold the same opinion.…
An extended Polya urn Model with two colors, black and white, is studied with some SLLN and CLT on the proportion of white balls.
Consider the following two-player game on the edges of $K_n$, the complete graph with $n$ vertices: Starting with an empty graph $G$ on the vertex set of $K_n$, in each round the first player chooses $b \in \mathbb{N}$ edges from $K_n$…
This work considers new entropy-based proofs of some known, or otherwise refined, combinatorial bounds for bipartite graphs. These include upper bounds on the number of the independent sets, lower bounds on the minimal number of colors in…
We study majority dynamics on the binomial random graph $G(n,p)$ with $p = d/n$ and $d > \lambda n^{1/2}$, for some large $\lambda>0$. In this process, each vertex has a state in $\{-1,+1 \}$ and at each round every vertex adopts the state…
We consider a random graph in which vertices can have one of two possible colours. Each vertex switches its colour at a rate that is proportional to the number of vertices of the other colour to which it is connected by an edge. Each edge…
We prove almost sure convergence of the maximum degree in an evolving graph model combining a growing number of local choices with sublinear preferential attachment. At each step in the growth of the graph, a new vertex is introduced. Then…
We study first passage statistics of the Polya urn model. In this random process, the urn contains two types of balls. In each step, one ball is drawn randomly from the urn, and subsequently placed back into the urn together with an…
A cyclic urn is an urn model for balls of types $0,\ldots,m-1$ where in each draw the ball drawn, say of type $j$, is returned to the urn together with a new ball of type $j+1 \mod m$. The case $m=2$ is the well-known Friedman urn. The…
Pull voting is a random process in which vertices of a connected graph have initial opinions chosen from a set of $k$ distinct opinions, and at each step a random vertex alters its opinion to that of a randomly chosen neighbour. If the…
By designating vertices with variables, a simple undirected graph can be augmented to have an associated representing rational function in two variables taking the complex bi-upper halfplane to itself. We give relations between representing…
Graph-theoretic methods have seen wide use throughout the literature on multi-agent control and optimization. When communications are intermittent and unpredictable, such networks have been modeled using random communication graphs. When…
We study P\'olya urns on hypergraphs and prove that, when the incidence matrix of the hypergraph is injective, there exists a point $v=v(H)$ such that the random process converges to $v$ almost surely. We also provide a partial result when…
Given a graph $F$, the random Tur\'an problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete…
We introduce a class of stochastic processes with reinforcement consisting of a sequence of random partitions $\{\mathcal{P}_t\}_{t \ge 1}$, where $\mathcal{P}_t$ is a partition of $\{1,2,\dots, Rt\}$. At each time~$t$,~$R$ numbers are…
The unit ball random geometric graph $G=G^d_p(\lambda,n)$ has as its vertices $n$ points distributed independently and uniformly in the $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most…
We present the theory of multifunctions applied to graphs. Its interesting feature is that walks are recognized as iterations. We consider the graphs with arbitrary number of vertices which are determined by multifunctions. The mutually…
This paper explores the distribution of indistinguishable balls into distinct urns with varying capacity constraints, a foundational issue in combinatorial mathematics with applications across various disciplines. We present a comprehensive…
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind,…