Generalizing Kronecker graphs in order to model searchable networks

This paper describes an extension to stochastic Kronecker graphs that provides the special structure required for searchability, by defining a “distance”-dependent Kronecker operator. We show how this extension of Kronecker graphs can generate several existing social network models, such as the Watts-Strogatz small-world model and Kleinberg’s latticebased model. We focus on a specific example of an expanding hypercube, reminiscent of recently proposed social network models based on a hidden hyperbolic metric space, and prove that a greedy forwarding algorithm can find very short paths of length O((log log n)^2) for graphs with n nodes

A Candour-based Trust and Reputation Management System for Mobile Ad Hoc Networks

The decentralized administrative controlled-nature of mobile ad hoc networks (MANETs) presents security vulnerabilities which can lead to attacks such as malicious modification of packets. To enhance security in MANETs, Trust and Reputation Management systems (TRM) have been developed to serve as measures in mitigating threats arising from unusual behaviours of nodes. In this paper we propose a candour-based trust and reputation system which measures and models reputation and trust propagation in MANETs. In the proposed model Dirichlet Probability Distribution is employed in modelling the individual reputation of nodes and the trust of each node is computed based on the node’s actual network performance and the quality of the recommendations it gives about other nodes. Cooperative nodes in our model will be rewarded for expanding their energy in forwarding packets for other nodes or for disseminating genuine recommenda-tions. Uncooperative nodes are isolated and denied the available network resources. We employed the Ruffle algorithm which will ensure that cooperative nodes are allowed to activate sleep mode when their service is not required in forwarding packets for its neighbouring trustworthy nodes. The proposed TRM system enshrines fairness in its mode of operation as well as creating an enabling environment free from bias. It will also ensure a connected and capacity preserving network of trustworthy node

Distance-Dependent Kronecker Graphs for Modeling Social Networks

This paper focuses on a generalization of stochastic Kronecker graphs, introducing a Kronecker-like operator and defining a family of generator matrices H dependent on distances between nodes in a specified graph embedding. We prove that any lattice-based network model with sufficiently small distance-dependent connection probability will have a Poisson degree distribution and provide a general framework to prove searchability for such a network. Using this framework, we focus on a specific example of an expanding hypercube and discuss the similarities and differences of such a model with recently proposed network models based on a hidden metric space. We also prove that a greedy forwarding algorithm can find very short paths of length O((log log n)^2) on the hypercube with n nodes, demonstrating that distance-dependent Kronecker graphs can generate searchable network models

The Laplacian Eigenvalues and Invariants of Graphs

In this paper, we investigate some relations between the invariants (including vertex and edge connectivity and forwarding indices) of a graph and its Laplacian eigenvalues. In addition, we present a sufficient condition for the existence of Hamiltonicity in a graph involving its Laplacian eigenvalues.Comment: 10 pages,Filomat, 201