# American Institute of Mathematical Sciences

eISSN:
2577-8838

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## Mathematical Foundations of Computing

February 2018 , Volume 1 , Issue 1

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2018, 1(1): 1-10 doi: 10.3934/mfc.2018001 +[Abstract](1062) +[HTML](556) +[PDF](708.5KB)
Abstract:

Many approaches have been proposed to perform the analysis of network(graph) data correlated with Internet, social networks, communication networks, knowledge graph etc, but few of them have been applied in the coordinate system which is thought to be an efficient platform to processing graph data. In this survey, we provide a short yet structured analysis of network(graph) data research in the coordinate system. We first introduce the coordinate embedding and transforming of double-loop network with its symmetrical and regular structure. We then present two categories of approaches for Internet embedding in the coordinate system and the technology of graph embedding of social network. Finally, we draw our conclusions and discuss potential applications and future research direction.

2018, 1(1): 11-48 doi: 10.3934/mfc.2018002 +[Abstract](1089) +[HTML](655) +[PDF](653.98KB)
Abstract:

An individual's location history in the real world implies his or her interests and behaviors. This paper analyzes and understands the process of Collaborative Filtering (CF) approach, which mines an individual's preference from his/her geographic location histories and recommends locations based on the similarities between the user and others. We find that a CF-based recommendation process can be summarized as a sequence of multiplications between a transition matrix and visited-location matrix. The transition matrix is usually approximated by the user's interest matrix that reflect the similarity among users, regarding to their interest in visiting different locations. The visited-location matrix provides the history of visited locations of all users, which is currently available to the recommendation system. We find that recommendation results will converge if and only if the transition matrix remains unchanged; otherwise, the recommendations will be valid for only a certain period of time. Based on our analysis, a novel location-based accurate recommendation (LAR) method is proposed, which considers the semantic meaning and category information of locations, as well as the timeliness of recommending results, to make accurate recommendations. We evaluated the precision and recall rates of LAR, using a large-scale real-world data set collected from Brightkite. Evaluation results confirm that LAR offers more accurate recommendations, comparing to the state-of-art approaches.

2018, 1(1): 49-61 doi: 10.3934/mfc.2018003 +[Abstract](869) +[HTML](528) +[PDF](392.27KB)
Abstract:

The frequency assignment problem (FAP) is the assignment of frequencies to television and radio transmitters subject to restrictions imposed by the distance between transmitters. One of the graph theoretical models of FAP which is well elaborated is the concept of distance constrained labeling of graphs. Let \begin{document}$G = (V, E)$\end{document} be a graph. For two vertices \begin{document}$u$\end{document} and \begin{document}$v$\end{document} of \begin{document}$G$\end{document}, we denote \begin{document}$d(u, v)$\end{document} the distance between \begin{document}$u$\end{document} and \begin{document}$v$\end{document}. An \begin{document}$L(2, 1)$\end{document}-labeling for \begin{document}$G$\end{document} is a function \begin{document}$f: V → \{0, 1, ···\}$\end{document} such that \begin{document}$|f(u)-f(v)| ≥ 1$\end{document} if \begin{document}$d(u, v) = 2$\end{document} and \begin{document}$|f(u)-f(v)| ≥ 2$\end{document} if \begin{document}$d(u, v) = 1$\end{document}. The span of \begin{document}$f$\end{document} is the difference between the largest and the smallest number of \begin{document}$f(V)$\end{document}. The \begin{document}$λ$\end{document}-number for \begin{document}$G$\end{document}, denoted by \begin{document}$λ(G)$\end{document}, is the minimum span over all \begin{document}$L(2, 1)$\end{document}-labelings of \begin{document}$G$\end{document}. In this paper, we study the \begin{document}$λ$\end{document}-number of the Cartesian and strong product of two directed cycles. We show that for \begin{document}$m, n ≥ 4$\end{document} the \begin{document}$λ$\end{document}-number of \begin{document}$\overrightarrow{C_m} \Box \overrightarrow{C_n}$\end{document} is between 4 and 5. We also establish the \begin{document}$λ$\end{document}-number of \begin{document}$\overrightarrow{{{C}_{m}}}\boxtimes \overrightarrow{{{C}_{n}}}$\end{document} for \begin{document}$m ≤ 10$\end{document} and prove that the \begin{document}$λ$\end{document}-number of the strong product of cycles \begin{document}$\overrightarrow{{{C}_{m}}}\boxtimes \overrightarrow{{{C}_{n}}}$\end{document} is between 6 and 8 for \begin{document}$m, n ≥ 48$\end{document}.

2018, 1(1): 63-76 doi: 10.3934/mfc.2018004 +[Abstract](2748) +[HTML](2213) +[PDF](1032.13KB)
Abstract:

With the rapid development of Internet of Things (IoT) technologies, smart home systems are getting more and more popular in our daily life. Besides providing convenient functionality and tangible benefits, smart home systems also expose users to security risks. To enhance the functionality and the security, machine learning algorithms play an important role in a smart home ecosystem, e.g., ensuring biotechnology-based authentication and authorization, anomalous detection, etc. On the other side, attackers also treat learning algorithms as a tool, as well as a target, to exploit the security vulnerabilities in smart home systems. In this paper, we unify the system architectures suggested by the mainstream service providers, e.g., Samsung, Google, Apple, etc. Based on our proposed overall smart home system model, we investigate the application of learning algorithms in smart home IoT system security. Our study includes two angles. First, we discussed the functionality and security enhancing methods based on learning mechanisms; second, we described the security threats exposed by employing learning techniques. We also explored the potential solutions that may address the aforementioned security problems.

2018, 1(1): 77-100 doi: 10.3934/mfc.2018005 +[Abstract](1126) +[HTML](778) +[PDF](1438.97KB)
Abstract:

In this paper, we present a signcryption scheme called CP_ABSC based on Ciphertext-Policy Attribute Based Encryption (CP_ABE) [7] to secure the multicast communications in smart grids that require access control, data encryption, and authentication to ensure message integrity and confidentiality. CP_ABSC provides algorithms for key management, signcryption, and designcryption. It can be used to signcrypt a message based on the access rights specified by the message itself. A user can designcrypt a ciphertext if and only if it possesses the attributes required by the access structure of the data. Thus CP_ABSC effectively defines a multicast group based on the access rights of the data specified by the data itself, which differs significantly from the traditional Internet based multicast where the destination group is predetermined and must be known by the data source. CP_ABSC provides collusion attack resistance, message authentication, forgery prevention, and confidentiality. It can be easily applied in smart grids to secure the instructions/commands broadcast from a utility company to multiple smart meters (push-based multicast) and the data retrieved from a smart meter to multiple destinations (pull-based multicast). Compared to CP_ABE, CP_ABSC combines encryption with signature at a lower computational cost for signcryption and a slightly higher cost in designcryption for signature verification. We also consider the adoption of attribute-based signature (ABS), and conclude that CP_ABSC has a much lower computational cost than ABS.