123 research outputs found
Revolutionaries and spies on random graphs
Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a
simplified model for network security. In the game we consider in this paper, a
team of revolutionaries tries to hold an unguarded meeting consisting of
revolutionaries. A team of spies wants to prevent this forever. For
given and , the minimum number of spies required to win on a graph
is the spy number . We present asymptotic results for the game
played on random graphs for a large range of , and
. The behaviour of the spy number is analyzed completely for dense
graphs (that is, graphs with average degree at least n^{1/2+\eps} for some
\eps > 0). For sparser graphs, some bounds are provided
On the limiting distribution of the metric dimension for random forests
The metric dimension of a graph G is the minimum size of a subset S of
vertices of G such that all other vertices are uniquely determined by their
distances to the vertices in S. In this paper we investigate the metric
dimension for two different models of random forests, in each case obtaining
normal limit distributions for this parameter.Comment: 22 pages, 5 figure
A bound for the diameter of random hyperbolic graphs
Random hyperbolic graphs were recently introduced by Krioukov et. al.
[KPKVB10] as a model for large networks. Gugelmann, Panagiotou, and Peter
[GPP12] then initiated the rigorous study of random hyperbolic graphs using the
following model: for , ,
, set and build the graph with
as follows: For each , generate i.i.d. polar coordinates
using the joint density function , with
chosen uniformly from and with density
for . Then,
join two vertices by an edge, if their hyperbolic distance is at most . We
prove that in the range a.a.s. for any two vertices
of the same component, their graph distance is , where
, thus answering a
question raised in [GPP12] concerning the diameter of such random graphs. As a
corollary from our proof we obtain that the second largest component has size
, thus answering a question of Bode, Fountoulakis and
M\"{u}ller [BFM13]. We also show that a.a.s. there exist isolated components
forming a path of length , thus yielding a lower bound on the
size of the second largest component.Comment: 5 figure
Cops and Invisible Robbers: the Cost of Drunkenness
We examine a version of the Cops and Robber (CR) game in which the robber is
invisible, i.e., the cops do not know his location until they capture him.
Apparently this game (CiR) has received little attention in the CR literature.
We examine two variants: in the first the robber is adversarial (he actively
tries to avoid capture); in the second he is drunk (he performs a random walk).
Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD),
which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being
the expected capture times in the adversarial and drunk CiR variants,
respectively. We show that these capture times are well defined, using game
theory for the adversarial case and partially observable Markov decision
processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD
for several special graph families such as -regular trees, give some bounds
for grids, and provide general upper and lower bounds for general classes of
graphs. We also give an infinite family of graphs showing that iCOD can be
arbitrarily close to any value in [2,infinty). Finally, we briefly examine one
more CiR variant, in which the robber is invisible and "infinitely fast"; we
argue that this variant is significantly different from the Graph Search game,
despite several similarities between the two games
The set chromatic number of random graphs
In this paper we study the set chromatic number of a random graph
for a wide range of . We show that the set chromatic number, as a
function of , forms an intriguing zigzag shape
Vertex-pursuit in random directed acyclic graphs
We examine a dynamic model for the disruption of information flow in
hierarchical social networks by considering the vertex-pursuit game Seepage
played in directed acyclic graphs (DAGs). In Seepage, agents attempt to block
the movement of an intruder who moves downward from the source node to a sink.
The minimum number of such agents required to block the intruder is called the
green number. We propose a generalized stochastic model for DAGs with given
expected total degree sequence. Seepage and the green number is analyzed in
stochastic DAGs in both the cases of a regular and power law degree sequence.
For each such sequence, we give asymptotic bounds (and in certain instances,
precise values) for the green number
On rigidity, orientability and cores of random graphs with sliders
Suppose that you add rigid bars between points in the plane, and suppose that
a constant fraction of the points moves freely in the whole plane; the
remaining fraction is constrained to move on fixed lines called sliders. When
does a giant rigid cluster emerge? Under a genericity condition, the answer
only depends on the graph formed by the points (vertices) and the bars (edges).
We find for the random graph the threshold value of
for the appearance of a linear-sized rigid component as a function of ,
generalizing results of Kasiviswanathan et al. We show that this appearance of
a giant component undergoes a continuous transition for and a
discontinuous transition for . In our proofs, we introduce a
generalized notion of orientability interpolating between 1- and
2-orientability, of cores interpolating between 2-core and 3-core, and of
extended cores interpolating between 2+1-core and 3+2-core; we find the precise
expressions for the respective thresholds and the sizes of the different cores
above the threshold. In particular, this proves a conjecture of Kasiviswanathan
et al. about the size of the 3+2-core. We also derive some structural
properties of rigidity with sliders (matroid and decomposition into components)
which can be of independent interest.Comment: 32 pages, 1 figur
- …