Ore-degree threshold for the square of a Hamiltonian cycle

A classic theorem of Dirac from 1952 states that every graph with minimum degree at least n/2 contains a Hamiltonian cycle. In 1963, P\'osa conjectured that every graph with minimum degree at least 2n/3 contains the square of a Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac's theorem by proving that every graph with $deg(u) + deg(v) \geq n$ for every $uv \notin E(G)$ contains a Hamiltonian cycle. Recently, Ch\^au proved an Ore-type version of P\'osa's conjecture for graphs on $n\geq n_0$ vertices using the regularity--blow-up method; consequently the $n_0$ is very large (involving a tower function). Here we present another proof that avoids the use of the regularity lemma. Aside from the fact that our proof holds for much smaller $n_0$, we believe that our method of proof will be of independent interest.Comment: 24 pages, 1 figure. In addition to some fixed typos, this updated version contains a simplified "connecting lemma" in Section 3.

Permissioned Blockchain-Based Security for SDN in IoT Cloud Networks

The advancement in cloud networks has enabled connectivity of both traditional networked elements and new devices from all walks of life, thereby forming the Internet of Things (IoT). In an IoT setting, improving and scaling network components as well as reducing cost is essential to sustain exponential growth. In this domain, software-defined networking (SDN) is revolutionizing the network infrastructure with a new paradigm. SDN splits the control/routing logic from the data transfer/forwarding. This splitting causes many issues in SDN, such as vulnerabilities of DDoS attacks. Many solutions (including blockchain based) have been proposed to overcome these problems. In this work, we offer a blockchain-based solution that is provided in redundant SDN (load-balanced) to service millions of IoT devices. Blockchain is considered as tamper-proof and impossible to corrupt due to the replication of the ledger and consensus for verification and addition to the ledger. Therefore, it is a perfect fit for SDN in IoT Networks. Blockchain technology provides everyone with a working proof of decentralized trust. The experimental results show gain and efficiency with respect to the accuracy, update process, and bandwidth utilization.Comment: Accepted to International Conference on Advances in the Emerging Computing Technologies (AECT) 202

Affine-Invariant Outlier Detection and Data Visualization

A wealth of data is generated daily by social media websites that is an essential component of the Big Data Revolution. In many cases, the data is anonymized before being disseminated for research and analysis. This anonymization process distorts the data so that some essential characteristics are lost which may not be captured by methods that are not robust against such transformations. In this paper we propose novel algorithms, for two-dimensional data, for a recently discovered statistical data analysis measure, the Ray Shooting Depth (RSD) that provides an affineinvariant ranking of data points. In addition, we prove some complexity results and illustrate some of the desirable properties of RSD via comparisons with other similar notions. We develop an open-source data visualization tool based on RSD, and show its applications in distribution estimation, outlier detection, and 2D tolerance-region construction

Scalable Approximation Algorithm for Network Immunization

The problem of identifying important players in a given network is of pivotal importance for viral marketing, public health management, network security and various other fields of social network analysis. In this work we find the most important vertices in a graph G = (V;E) to immunize so as the chances of an epidemic outbreak is minimized. This problem is directly relevant to minimizing the impact of a contagion spread (e.g. flu virus, computer virus and rumor) in a graph (e.g. social network, computer network) with a limited budget (e.g. the number of available vaccines, antivirus software, filters). It is well known that this problem is computationally intractable (it is NP-hard). In this work we reformulate the problem as a budgeted combinational optimization problem and use techniques from spectral graph theory to design an efficient greedy algorithm to find a subset of vertices to be immunized. We show that our algorithm takes less time compared to the state of the art algorithm. Thus our algorithm is scalable to networks of much larger sizes than best known solutions proposed earlier. We also give analytical bounds on the quality of our algorithm. Furthermore, we evaluate the efficacy of our algorithm on a number of real world networks and demonstrate that the empirical performance of algorithm supplements the theoretical bounds we present, both in terms of approximation guarantees and computational efficiency