A Sharp Threshold for Network Reliability

2002 ◽  
Vol 11 (5) ◽  
pp. 465-474 ◽  
Author(s):  
MICHAEL KRIVELEVICH ◽  
BENNY SUDAKOV ◽  
VAN H. VU
Keyword(s):  
Classical Result ◽  
Network Reliability ◽  
Average Degree ◽  
Edge Connectivity ◽  
Sharp Threshold ◽  
Random Subgraph ◽  
Connectivity Threshold

Given a graph G on n vertices with average degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that, if the edge connectivity k(G) satisfies k(G) [Gt ] d/log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

scholarly journals Long Paths and Cycles in Random Subgraphs of $\mathcal{H}$-Free Graphs

10.37236/3198 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Krivelevich ◽  
Wojciech Samotij
Keyword(s):  
Random Graph ◽  
Classical Result ◽  
Minimum Degree ◽  
Average Degree ◽  
Free Graph ◽  
Connected Graphs ◽  
Random Subgraph ◽  
Minimum Number ◽  
Random Subgraphs ◽  
Long Paths

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random subgraph $G_p$ of $G$ by independently keeping each edge of $G$ with probability $p$. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive $\varepsilon$, there exists a positive $\delta$ (depending only on $\varepsilon$) such that the following holds: If $p \geq \frac{1+\varepsilon}{k}$, then with probability tending to $1$ as $k \to \infty$, the random graph $G_p$ contains a cycle of length at least $n_{\mathcal{H}}(\delta k)$, where $n_\mathcal{H}(k)>k$ is the minimum number of vertices in an $\mathcal{H}$-free graph of average degree at least $k$. Thus in particular $G_p$ as above typically contains a cycle of length at least linear in $k$.

scholarly journals Topology of random -dimensional cubical complexes

2021 ◽  
Vol 9 ◽  
Author(s):  
Matthew Kahle ◽  
Elliot Paquette ◽  
Érika Roldán
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
High Probability ◽  
Cubical Complex ◽  
Sharp Threshold ◽  
Cubical Complexes ◽  
Simple Connectivity ◽  
Dimensional Cube ◽  
Dimensional Face ◽  
Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

scholarly journals Ramsey Properties of Random Subgraphs of Pseudo-Random Graphs

10.37236/2468 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Jia Shen
Keyword(s):  
Random Graphs ◽  
Regular Graph ◽  
Threshold Function ◽  
Vertex Coloring ◽  
Similar Argument ◽  
Sharp Threshold ◽  
Absolute Value ◽  
Random Subgraph ◽  
Vertex Disjoint ◽  
Random Subgraphs

Let $G=(V,E)$ be a $d$-regular graph of order $n$. Let $G_p$ be the random subgraph of $G$ for which each edge is selected from $E(G)$ independently at random with probability $p$. For a fixed graph $H$, define $m(H):=$max$\{e(H')/(v(H')-1):H' \subseteq H\}$. We prove that $n^{(m(H)-1)/m(H)}/d$ is a threshold function for $G_p$ to satisfy Ramsey, induced Ramsey, and canonical Ramsey properties with respect to vertex coloring, respectively, provided the eigenvalue $\lambda$ of $G$ that is second largest in absolute value is significantly smaller than $d$.As a consequence, it is also shown that $\displaystyle n^{(m(H)-1)/m(H)}/d$ is a threshold function for $G_p$ to contain a family of vertex disjoint copies of $H$ (an $H$ packing) that covers $(1-o(1))n$ vertices of $G$. Using a similar argument, the sharp threshold function for $G_p$ to contain $H$ as a subgraph is obtained as well.

The sharp threshold for percolation on expander graphs

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Finite Graph ◽  
Giant Component ◽  
Expander Graphs ◽  
Expander Graph ◽  
Bounded Degree ◽  
Sharp Threshold ◽  
Random Subgraph

AbstractWe consider a random subgraph G n(p) of a finite graph family G n = (V n, E n) formed by retaining each edge of G n independently with probability p. We show that if G n is an expander graph with vertices of bounded degree, then for any c n ∈ (0, 1) satisfying $$c_n \gg {1 \mathord{\left/ {\vphantom {1 {\sqrt {\ln n} }}} \right. \kern-\nulldelimiterspace} {\sqrt {\ln n} }}$$ and $$\mathop {\lim \sup }\limits_{n \to \infty } c_n < 1$$, the property that the random subgraph contains a giant component of order c n|V n| has a sharp threshold.

Expected Value Expansions in Random Subgraphs with Applications to Network Reliability

1998 ◽  
Vol 7 (4) ◽  
pp. 465-483 ◽  
Author(s):  
KYLE SIEGRIST
Keyword(s):  
Performance Measures ◽  
General Result ◽  
Network Reliability ◽  
Random Variable ◽  
Expected Value ◽  
Complete Graphs ◽  
Number Of Components ◽  
Random Subgraph ◽  
Special Cases ◽  
Random Subgraphs

Subgraph expansions are commonly used in the analysis of reliability measures of a failure-prone graph. We show that these expansions are special cases of a general result on the expected value of a random variable defined on a partially ordered set; when applied to random subgraphs, the general result defines a natural association between graph functions. As applications, we consider several graph invariants that measure the connectivity of a graph: the number of connected vertex sets of size k, the number of components of size k, and the total number of components. The expected values of these invariants on a random subgraph are global performance measures that generalize the ones commonly studied. Explicit results are obtained for trees, cycles, and complete graphs. Graphs which optimize these performance measures over a given class of graphs are studied

Sufficient Conditions for Maximally k-Isoperimetric Edge Connectivity of Graphs

2015 ◽  
Vol 26 (05) ◽  
pp. 583-598
Author(s):  
Shiying Wang ◽  
Kai Feng ◽  
Yubao Guo
Keyword(s):  
Connected Graph ◽  
Reliability Index ◽  
Network Reliability ◽  
Sufficient Conditions ◽  
Edge Connectivity ◽  
Free Graph ◽  
Optimal Graph

The k-isoperimetric edge connectivity is a more refined network reliability index than edge connectivity. The k-isoperimetric edge connectivity of a connected graph G is defined as γk(G) = min{|[X, [Formula: see text]]| : X ⊆ V (G), |X| ≥ k, |[Formula: see text]| ≥ k}. Let βk(G) = min{|[X,[Formula: see text]]| : X ⊆ V (G), |X| = k}. A graph G is called a γk-optimal graph if γk(G) = βk(G). An edge cut S = [X,[Formula: see text]] is called a γk-cut if |S| = γk(G), X ⊆ V (G), |X| ≥ k and |[Formula: see text]| ≥ k. Moreover, G is called a super-γk graph if every γk-cut [X,[Formula: see text]] of G has the property that either |X| = k or |[Formula: see text]| = k. Let G be a graph of order at least 2k with k ≥ 2. In this paper, we prove that for any pair u, υ of nonadjacent vertices in G, if |N(u)∩N(υ)| ≥ k+1 when neither u nor υ lies on a triangle, or |N(u)∩N(υ)| ≥ 2k -1 when u or υ lies on a triangle, then G is γk-optimal. Moreover, if G is a triangle-free graph, and |N(u)∩υ(υ)| ≥ k +1 for all pairs u, υ of nonadjacent vertices in G, then G is either super-γk or isomorphic to Kk+1,k+1.

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