10 research outputs found
On the strengths of connectivity and robustness in general random intersection graphs
Random intersection graphs have received much attention for nearly two
decades, and currently have a wide range of applications ranging from key
predistribution in wireless sensor networks to modeling social networks. In
this paper, we investigate the strengths of connectivity and robustness in a
general random intersection graph model. Specifically, we establish sharp
asymptotic zero-one laws for -connectivity and -robustness, as well as
the asymptotically exact probability of -connectivity, for any positive
integer . The -connectivity property quantifies how resilient is the
connectivity of a graph against node or edge failures. On the other hand,
-robustness measures the effectiveness of local diffusion strategies (that
do not use global graph topology information) in spreading information over the
graph in the presence of misbehaving nodes. In addition to presenting the
results under the general random intersection graph model, we consider two
special cases of the general model, a binomial random intersection graph and a
uniform random intersection graph, which both have numerous applications as
well. For these two specialized graphs, our results on asymptotically exact
probabilities of -connectivity and asymptotic zero-one laws for
-robustness are also novel in the literature.Comment: This paper about random graphs appears in IEEE Conference on Decision
and Control (CDC) 2014, the premier conference in control theor
Engineering Emergence: A Survey on Control in the World of Complex Networks
Complex networks make an enticing research topic that has been increasingly attracting researchers from control systems and various other domains over the last two decades. The aim of this paper was to survey the interest in control related to complex networks research over time since 2000 and to identify recent trends that may generate new research directions. The survey was performed for Web of Science, Scopus, and IEEEXplore publications related to complex networks. Based on our findings, we raised several questions and highlighted ongoing interests in the control of complex networks.publishedVersio
On the Strength of Connectivity of Inhomogeneous Random K-out Graphs
Random graphs are an important tool for modelling and analyzing the
underlying properties of complex real-world networks. In this paper, we study a
class of random graphs known as the inhomogeneous random K-out graphs which
were recently introduced to analyze heterogeneous sensor networks secured by
the pairwise scheme. In this model, first, each of the nodes is classified
as type-1 (respectively, type-2) with probability (respectively,
independently from each other. Next, each type-1 (respectively,
type-2) node draws 1 arc towards a node (respectively, arcs towards
distinct nodes) selected uniformly at random, and then the orientation of the
arcs is ignored. From the literature on homogeneous K-out graphs wherein all
nodes select neighbors (i.e., ), it is known that when , the graph is -connected asymptotically almost surely (a.a.s.) as
gets large. In the inhomogeneous case (i.e., ), it was recently
established that achieving even 1-connectivity a.a.s. requires .
Here, we provide a comprehensive set of results to complement these existing
results. First, we establish a sharp zero-one law for -connectivity, showing
that for the network to be -connected a.a.s., we need to set for all .
Despite such large scaling of being required for -connectivity, we
show that the trivial condition of for all is sufficient to
ensure that inhomogeneous K-out graph has a connected component of size
whp
On Two Combinatorial Optimization Problems in Graphs: Grid Domination and Robustness
In this thesis, we study two problems in combinatorial optimization, the dominating set problem and the robustness problem. In the first half of the thesis, we focus on the dominating set problem in grid graphs and present a distributed algorithm for finding near optimal dominating sets on grids. The dominating set problem is a well-studied mathematical problem in which the goal is to find a minimum size subset of vertices of a graph such that all vertices that are not in that set have a neighbor inside that set. We first provide a simpler proof for an existing centralized algorithm that constructs dominating sets on grids so that the size of the provided dominating set is upper-bounded by the ceiling of (m+2)(n+2)/5 for m by n grids and its difference from the optimal domination number of the grid is upper-bounded by five. We then design a distributed grid domination algorithm to locate mobile agents on a grid such that they constitute a dominating set for it. The basis for this algorithm is the centralized grid domination algorithm. We also generalize the centralized and distributed algorithms for the k-distance dominating set problem, where all grid vertices are within distance k of the vertices in the dominating set.
In the second half of the thesis, we study the computational complexity of checking a graph property known as robustness. This property plays a key role in diffusion of information in networks. A graph G=(V,E) is r-robust if for all pairs of nonempty and disjoint subsets of its vertices A,B, at least one of the subsets has a vertex that has at least r neighbors outside its containing set. In the robustness problem, the goal is to find the largest value of r such that a graph G is r-robust. We show that this problem is coNP-complete. En route to showing this, we define some new problems, including the decision version of the robustness problem and its relaxed version in which B=V \ A. We show these two problems are coNP-hard by showing that their complement problems are NP-hard