51 research outputs found
Evaluation of effective resistances in pseudo-distance-regular resistor networks
In Refs.[1] and [2], calculation of effective resistances on distance-regular
networks was investigated, where in the first paper, the calculation was based
on the stratification of the network and Stieltjes function associated with the
network, whereas in the latter one a recursive formula for effective
resistances was given based on the Christoffel-Darboux identity. In this paper,
evaluation of effective resistances on more general networks called
pseudo-distance-regular networks [21] or QD type networks \cite{obata} is
investigated, where we use the stratification of these networks and show that
the effective resistances between a given node such as and all of the
nodes belonging to the same stratum with respect to
(, belonging to the -th stratum with respect
to the ) are the same. Then, based on the spectral techniques, an
analytical formula for effective resistances such that
(those nodes , of
the network such that the network is symmetric with respect to them) is given
in terms of the first and second orthogonal polynomials associated with the
network, where is the pseudo-inverse of the Laplacian of the network.
From the fact that in distance-regular networks,
is satisfied for all nodes
of the network, the effective resistances
for ( is diameter of the network which
is the same as the number of strata) are calculated directly, by using the
given formula.Comment: 30 pages, 7 figure
Bounds on Kemeny's constant of a graph and the Nordhaus-Gaddum problem
We study Nordhaus-Gaddum problems for Kemeny's constant of a
connected graph . We prove bounds on
and the product
for various families of graphs. In
particular, we show that if the maximum degree of a graph on vertices
is or , then
is at most
Disturbance Propagation in Interconnected Linear Dynamical Networks
We consider performance analysis of interconnected linear dynamical networks subject to external stochastic disturbances. For stable linear networks, we define scalar performance measures by considering weighted H2--norms of the underlying systems, which are defined from the disturbance input to a desired output. It is shown that the performance measure of a general stable linear network can be tightly bounded from above and below using some spectral functions of the state matrix of the network. This result is applied to a class of cyclic linear networks and shown that the performance measure of such networks scales quadratically with the network size. Next, we focus on first-- and second--order linear consensus networks and introduce the notion of Laplacian energy for such networks, which in fact measures the expected steady-state dispersion of the state of the entire network. We develop a graph-theoretic framework in order to relate graph characteristics to the Laplacian energy of the network and show that how the Laplacian energy scales asymptotically with the network size. We quantify several inherent fundamental limits on Laplacian energy in terms of graph diameter, node degrees, and the number of spanning trees, and several other graph specifications. Particularly we characterize several versions of fundamental tradeoffs between Laplacian energy and sparsity measures of a linear consensus network, showing that more sparse networks have higher levels of Laplacian energies. At the end, we show that several existing performance measures in real--world applications, such as total power loss in synchronous power networks and flock energy of a group of autonomous vehicles in a formation, are indeed special forms of Laplacian energies
GRASP/VND Optimization Algorithms for Hard Combinatorial Problems
Two hard combinatorial problems are addressed in this thesis. The first one is known as the ”Max CutClique”, a combinatorial problem introduced by P. Martins in 2012. Given a simple graph, the goal is to
find a clique C such that the number of links shared between C and its complement C
C is maximum.
In a first contribution, a GRASP/VND methodology is proposed to tackle the problem. In a second
one, the N P-Completeness of the problem is mathematically proved. Finally, a further generalization
with weighted links is formally presented with a mathematical programming formulation, and the
previous GRASP is adapted to the new problem.
The second problem under study is a celebrated optimization problem coming from network
reliability analysis. We assume a graph G with perfect nodes and imperfect links, that fail independently
with identical probability ρ ∈ [0,1]. The reliability RG(ρ), is the probability that the resulting subgraph
has some spanning tree. Given a number of nodes and links, p and q, the goal is to find the (p,q)-graph
that has the maximum reliability RG(ρ), uniformly in the compact set ρ ∈ [0,1]. In a first contribution,
we exploit properties shared by all uniformly most-reliable graphs such as maximum connectivity and
maximum Kirchhoff number, in order to build a novel GRASP/VND methodology. Our proposal finds
the globally optimum solution under small cases, and it returns novel candidates of uniformly
most-reliable graphs, such as Kantor-Mobius and Heawood graphs. We also offer a literature review, ¨
and a mathematical proof that the bipartite graph K4,4 is uniformly most-reliable.
Finally, an abstract mathematical model of Stochastic Binary Systems (SBS) is also studied. It is a
further generalization of network reliability models, where failures are modelled by a general logical
function. A geometrical approximation of a logical function is offered, as well as a novel method to find
reliability bounds for general SBS. This bounding method combines an algebraic duality, Markov
inequality and Hahn-Banach separation theorem between convex and compact sets
An extensive English language bibliography on graph theory and its applications
Bibliography on graph theory and its application
Analysis and Design of Robust and High-Performance Complex Dynamical Networks
In the first part of this dissertation, we develop some basic principles to investigate performance deterioration of dynamical networks subject to external disturbances. First, we propose a graph-theoretic methodology to relate structural specifications of the coupling graph of a linear consensus network to its performance measure. Moreover, for this class of linear consensus networks, we introduce new insights into the network centrality based not only on the network graph but also on a more structured model of network uncertainties. Then, for the class of generic linear networks, we show that the H_2-norm, as a performance measure, can be tightly bounded from below and above by some spectral functions of state and output matrices of the system. Finally, we study nonlinear autocatalytic networks and exploit their structural properties to characterize their existing hard limits and essential tradeoffs. In the second part, we consider problems of network synthesis for performance enhancement. First, we propose an axiomatic approach for the design and performance analysis of linear consensus networks by introducing a notion of systemic performance measure. We build upon this new notion and investigate a general form of combinatorial problem of growing a linear consensus network via minimizing a given systemic performance measure. Two efficient polynomial-time approximation algorithms are devised to tackle this network synthesis problem. Then, we investigate the optimal design problem of distributed system throttlers. A throttler is a mechanism that limits the flow rate of incoming metrics, e.g., byte per second, network bandwidth usage, capacity, traffic, etc. Finally, a framework is developed to produce a sparse approximation of a given large-scale network with guaranteed performance bounds using a nearly-linear time algorithm
Algorithms For Discovering Communities In Complex Networks
It has been observed that real-world random networks like the WWW, Internet, social networks, citation networks, etc., organize themselves into closely-knit groups that are locally dense and globally sparse. These closely-knit groups are termed communities. Nodes within a community are similar in some aspect. For example in a WWW network, communities might consist of web pages that share similar contents. Mining these communities facilitates better understanding of their evolution and topology, and is of great theoretical and commercial significance. Community related research has focused on two main problems: community discovery and community identification. Community discovery is the problem of extracting all the communities in a given network, whereas community identification is the problem of identifying the community, to which, a given set of nodes belong. We make a comparative study of various existing community-discovery algorithms. We then propose a new algorithm based on bibliographic metrics, which addresses the drawbacks in existing approaches. Bibliographic metrics are used to study similarities between publications in a citation network. Our algorithm classifies nodes in the network based on the similarity of their neighborhoods. One of the drawbacks of the current community-discovery algorithms is their computational complexity. These algorithms do not scale up to the enormous size of the real-world networks. We propose a hash-table-based technique that helps us compute the bibliometric similarity between nodes in O(m ?) time. Here m is the number of edges in the graph and ?, the largest degree. Next, we investigate different centrality metrics. Centrality metrics are used to portray the importance of a node in the network. We propose an algorithm that utilizes centrality metrics of the nodes to compute the importance of the edges in the network. Removal of the edges in ascending order of their importance breaks the network into components, each of which represent a community. We compare the performance of the algorithm on synthetic networks with a known community structure using several centrality metrics. Performance was measured as the percentage of nodes that were correctly classified. As an illustration, we model the ucf.edu domain as a web graph and analyze the changes in its properties like densification power law, edge density, degree distribution, diameter, etc., over a five-year period. Our results show super-linear growth in the number of edges with time. We observe (and explain) that despite the increase in average degree of the nodes, the edge density decreases with time
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