The Giant Component Threshold for Random Regular Graphs with Edge Faults

نویسنده

  • Andreas Goerdt
چکیده

Let G be a given graph (modelling a communication network) which we assume suuers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 ? p). In particular we are interested in ro-bustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree. Our result is: If the degree d is xed then p = 1=(d ? 1) is a threshold probability for the existence of a linear-sized component in a faulty version of almost all random regular graphs. We show: If each edge of an arbitrary graph G with maximum degree bounded above by d is present with probability p = =(d ? 1) where < 1 is xed then the faulted version of G has only components whose size is at most logarithmic in the number of nodes with high probability. If on the other hand G is a random regular graph with degree d and p = =(d ? 1) where > 1 then for almost all G the faulted version of G has a linear-sized component with high probability. Note that these results imply some kind of optimality of random regular graphs among the class of graphs with the same degree bound. The theme is: Use the known expansion properties of almost all random regular graphs to obtain strong robustness results. This has not been done systematically before.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Random Regular Graphs with Edge Faults: Expansion through Cores

Let G be a given graph (modelling a communication network) which we assume suuers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 ? p). In particular we are interested in ro-bustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree d. Her...

متن کامل

Vacant Sets and Vacant Nets: Component Structures Induced by a Random Walk

Given a discrete random walk on a finite graph G, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step t. Let Γ(t) be the subgraph of G induced by the vacant set of the walk at step t. Similarly, let Γ̂(t) be the subgraph of G induced by the edges of the vacant net. For random r-regular graphs Gr, it was previously est...

متن کامل

Sharp Threshold for Percolation on Expanders by Itai Benjamini,

We study the appearance of the giant component in random subgraphs of a given large finite graph G = (V , E) in which each edge is present independently with probability p. We show that if G is an expander with vertices of bounded degree, then for any c ∈ ]0, 1[, the property that the random sub-graph contains a giant component of size c|V | has a sharp threshold. 1. Introduction. Percolation t...

متن کامل

GIANT VACANT COMPONENT LEFT BY A RANDOM WALK IN A RANDOM d-REGULAR GRAPH

We study the trajectory of a simple random walk on a d-regular graph with d ≥ 3 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. We sho...

متن کامل

CERTAIN TYPES OF EDGE m-POLAR FUZZY GRAPHS

In this research paper, we present a novel frame work for handling $m$-polar information by combining the theory of $m-$polar fuzzy  sets with graphs. We introduce certain types of edge regular $m-$polar fuzzy graphs and edge irregular $m-$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular $m-$polar fuzzy graphs. W...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 1997