Weak saturation numbers in random graphs

For two given graphs $G$ and $F$, a graph $H$ is said to be weakly $(G, F)$-saturated if $H$ is a spanning subgraph of $G$ which has no copy of $F$ as a subgraph and one can add all edges in $E(G)\setminus E(H)$ to $H$ in some order so that a new copy of $F$ is created at each step. The weak saturation number $wsat(G, F)$ is the minimum number of edges of a weakly $(G, F)$-saturated graph. In this paper, we deal with the relation between $wsat(G(n,p), F)$ and $wsat(K_n, F)$, where $G(n,p)$ denotes the Erd\H{o}s--R\'enyi random graph and $K_n$ denotes the complete graph on $n$ vertices. For every graph $F$ and constant $p$, we prove that $wsat( G(n,p),F)= wsat(K_n,F)(1+o(1))$ with high probability. Also, for some graphs $F$ including complete graphs, complete bipartite graphs, and connected graphs with minimum degree $1$ or $2$, it is shown that there exists an $\varepsilon(F)>0$ such that, for any $p\geqslant n^{-\varepsilon(F)}\log n$, $wsat( G(n,p),F)= wsat(K_n,F)$ with high probability

Tight concentration of star saturation number in random graphs

For given graphs $F$ and $G$, the minimum number of edges in an inclusion-maximal $F$-free subgraph of $G$ is called the $F$-saturation number and denoted $\mathrm{sat}(G, F)$. For the star $F=K_{1,r}$, the asymptotics of $\mathrm{sat}(G(n,p),F)$ is known. We prove a sharper result: whp $\mathrm{sat}(G(n,p), K_{1,r})$ is concentrated in a set of 2 consecutive points

Average path length in random networks

Analytic solution for the average path length in a large class of random graphs is found. We apply the approach to classical random graphs of Erd\"{o}s and R\'{e}nyi (ER) and to scale-free networks of Barab\'{a}si and Albert (BA). In both cases our results confirm previous observations: small world behavior in classical random graphs $l_{ER} \sim \ln N$ and ultra small world effect characterizing scale-free BA networks $l_{BA} \sim \ln N/\ln\ln N$. In the case of scale-free random graphs with power law degree distributions we observed the saturation of the average path length in the limit of $N\to\infty$ for systems with the scaling exponent $2< \alpha <3$ and the small-world behaviour for systems with $\alpha>3$.Comment: 4 pages, 2 figures, changed conten

Structure and randomness in extremal combinatorics

In this thesis we prove several results in extremal combinatorics from areas including Ramsey theory, random graphs and graph saturation. We give a random graph analogue of the classical Andr´asfai, Erd˝os and S´os theorem showing that in some ways subgraphs of sparse random graphs typically behave in a somewhat similar way to dense graphs. In graph saturation we explore a ‘partite’ version of the standard graph saturation question, determining the minimum number of edges in H-saturated graphs that in some way resemble H themselves. We determine these values for K4, paths, and stars and determine the order of magnitude for all graphs. In Ramsey theory we give a construction from a modified random graph to solve a question of Conlon, determining the order of magnitude of the size-Ramsey numbers of powers of paths. We show that these numbers are linear. Using models from statistical physics we study the expected size of random matchings and independent sets in d-regular graphs. From this we give a new proof of a result of Kahn determining which d-regular graphs have the most independent sets. We also give the equivalent result for matchings which was previously unknown and use this to prove the Asymptotic Upper Matching Conjecture of Friedland, Krop, Lundow and Markstrom. Using these methods we give an alternative proof of Shearer’s upper bound on off-diagonal Ramsey numbers

Deterministic Random Walk Model in NetLogo and the Identification of Asymmetric Saturation Time in Random Graph

Interactive programming environments are powerful tools for promoting innovative network thinking, teaching science of complexity, and exploring emergent phenomena. This paper reports on our recent development of the deterministic random walk model in NetLogo, a leading platform for computational thinking, eco-system thinking, and multi-agent cross-platform programming environment. The deterministic random walk is foundational to modeling dynamical processes on complex networks. Inspired by the temporal visualizations offered in NetLogo, we investigated the relationship between network topology and diffusion saturation time for the deterministic random walk model. Our analysis uncovers that in Erd\H{o}s-R\'{e}nyi graphs, the saturation time exhibits an asymmetric pattern with a considerable probability of occurrence. This behavior occurs when the hubs, defined as nodes with relatively higher number of connections, emerge in Erd\H{o}s-R\'{e}nyi graphs. Yet, our analysis yields that the hubs in Barab\'{a}si-Albert model stabilize the the convergence time of the deterministic random walk model. These findings strongly suggest that depending on the dynamical process running on complex networks, complementing characteristics other than the degree need to be taken into account for considering a node as a hub. We have made our development open-source, available to the public at no cost at https://github.com/bravandi/NetLogo-Dynamical-Processes