2,044 research outputs found
Expansion properties of a random regular graph after random vertex deletions
We investigate the following vertex percolation process. Starting with a
random regular graph of constant degree, delete each vertex independently with
probability p, where p=n^{-alpha} and alpha=alpha(n) is bounded away from 0. We
show that a.a.s. the resulting graph has a connected component of size n-o(n)
which is an expander, and all other components are trees of bounded size.
Sharper results are obtained with extra conditions on alpha. These results have
an application to the cost of repairing a certain peer-to-peer network after
random failures of nodes.Comment: 14 page
Robustness of the Rotor-Router Mechanism
International audienceThe rotor-router model, also called the Propp machine, was first considered as a deter-ministic alternative to the random walk. The edges adjacent to each node v (or equivalently, the exit ports at v) are arranged in a fixed cyclic order, which does not change during the exploration. Each node v maintains a port pointer π(v) which indicates the exit port to be adopted by an agent on the conclusion of the next visit to this node (the "next exit port"). The rotor-router mechanism guarantees that after each consecutive visit at the same node, the pointer at this node is moved to the next port in the cyclic order. It is known that, in an undirected graph G with m edges, the route adopted by an agent controlled by the rotor-router mechanism forms eventually an Euler tour based on arcs obtained via replacing each edge in G by two arcs with opposite direction. The process of ushering the agent to an Euler tour is referred to as the lock-in problem. In [Yanovski et al., Algorithmica 37(3), 165–186 (2003)], it was proved that, independently of the initial configuration of the rotor-router mechanism in G, the agent locks-in in time bounded by 2mD, where D is the diameter of G. In this paper we examine the dependence of the lock-in time on the initial configuration of the rotor-router mechanism. Our analysis is performed in the form of a game between a player P intending to lock-in the agent in an Euler tour as quickly as possible and its adversary A with the counter objective. We consider all cases of who decides the initial cyclic orders and the initial values π(v). We show, for example, that if A provides its own port numbering after the initial setup of pointers by P, the complexity of the lock-in problem is O(m·min{log m, D}). We also investigate the robustness of the rotor-router graph exploration in presence of faults in the pointers π(v) or dynamic changes in the graph. We show, for example, that after the exploration establishes an Eulerian cycle, if k edges are added to the graph, then a new Eulerian cycle is established within O(km) steps
Link deletion in directed complex networks
We present a systematic and detailed study of the robustness of directed
networks under random and targeted removal of links. We work with a set of
network models of random and scale free type, generated with specific features
of clustering and assortativity. Various strategies like random deletion of
links, or deletions based on betweenness centrality and degrees of source and
target nodes, are used to breakdown the networks. The robustness of the
networks to the sustained loss of links is studied in terms of the sizes of the
connected components and the inverse path lengths. The effects of clustering
and 2-node degree correlations, on the robustness to attack, are also explored.
We provide specific illustrations of our study on three real-world networks
constructed from protein-protein interactions and from transport data.Comment: 13 pages, 6 figures, submitted to Physica
Uncoverings on graphs and network reliability
We propose a network protocol similar to the -tree protocol of Itai and
Rodeh [{\em Inform.\ and Comput.}\ {\bf 79} (1988), 43--59]. To do this, we
define an {\em -uncovering-by-bases} for a connected graph to be a
collection of spanning trees for such that any -subset of
edges of is disjoint from at least one tree in , where is
some integer strictly less than the edge connectivity of . We construct
examples of these for some infinite families of graphs. Many of these infinite
families utilise factorisations or decompositions of graphs. In every case the
size of the uncovering-by-bases is no larger than the number of edges in the
graph and we conjecture that this may be true in general.Comment: 12 pages, 5 figure
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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