A Unifying Model for Representing Time-Varying Graphs

Graph-based models form a fundamental aspect of data representation in Data Sciences and play a key role in modeling complex networked systems. In particular, recently there is an ever-increasing interest in modeling dynamic complex networks, i.e. networks in which the topological structure (nodes and edges) may vary over time. In this context, we propose a novel model for representing finite discrete Time-Varying Graphs (TVGs), which are typically used to model dynamic complex networked systems. We analyze the data structures built from our proposed model and demonstrate that, for most practical cases, the asymptotic memory complexity of our model is in the order of the cardinality of the set of edges. Further, we show that our proposal is an unifying model that can represent several previous (classes of) models for dynamic networks found in the recent literature, which in general are unable to represent each other. In contrast to previous models, our proposal is also able to intrinsically model cyclic (i.e. periodic) behavior in dynamic networks. These representation capabilities attest the expressive power of our proposed unifying model for TVGs. We thus believe our unifying model for TVGs is a step forward in the theoretical foundations for data analysis of complex networked systems.Comment: Also appears in the Proc. of the IEEE International Conference on Data Science and Advanced Analytics (IEEE DSAA'2015

Identification of the Emergent Leaders within a CSE Professional Development Program

The need for high quality, sustainable Computer Science Education (CSE) professional development (PD) at the grades K-12 level is essential to the success of the global CSE initiatives. This study investigates the use of Social Network Analysis (SNA) to identify emergent teacher leaders within a high quality CSE PD program. The CSE PD program was designed and implemented through collaboration between the computer science and teacher education units at a Midwestern metropolitan university in North America. A unique feature of this specific program is in the intentional development of a social network. This study discusses the importance of social networks, the development of social capital, and its impact on the sustainability of the goals of the CSE PD program. The role of emergent teacher leaders in the development of the social capital of the CSE PD cohort is investigated using SNA techniques. The cohort consisted of 16 in-service teachers in grades 6-12 representing seven districts and four distinct content areas. The instruments used involved a questionnaire and the results of a CSE PD program online course. The findings suggest a correlation between the emergent teacher leaders, the online course results, and the overall cohort social capital. Future uses of SNA within professional development programs are also discussed

MultiAspect Graphs: Algebraic representation and algorithms

We present the algebraic representation and basic algorithms for MultiAspect Graphs (MAGs). A MAG is a structure capable of representing multilayer and time-varying networks, as well as higher-order networks, while also having the property of being isomorphic to a directed graph. In particular, we show that, as a consequence of the properties associated with the MAG structure, a MAG can be represented in matrix form. Moreover, we also show that any possible MAG function (algorithm) can be obtained from this matrix-based representation. This is an important theoretical result since it paves the way for adapting well-known graph algorithms for application in MAGs. We present a set of basic MAG algorithms, constructed from well-known graph algorithms, such as degree computing, Breadth First Search (BFS), and Depth First Search (DFS). These algorithms adapted to the MAG context can be used as primitives for building other more sophisticated MAG algorithms. Therefore, such examples can be seen as guidelines on how to properly derive MAG algorithms from basic algorithms on directed graph. We also make available Python implementations of all the algorithms presented in this paper.Comment: 59 pages, 6 figure

Diffusion Dynamics in Interconnected Communities

In this dissertation, multi-community-based Susceptible-Infected-Recovered (SIR) and Susceptible-Infected-Susceptible (SIS) models of infection/innovation diffusion are introduced for heterogeneous social networks in which agents are viewed as belonging to one of a finite number of communities. Agents are assumed to have well-mixed interactions within and between communities. The communities are connected through a backbone graph which defines an overall network structure for the models. The models are used to determine conditions for outbreak of an initial infection. The role of the strengths of the connections between communities in the development of an outbreak as well as long-term behavior of the diffusion is also studied. Percolation theory is brought to bear on these questions as an independent approach separate from the main dynamic multi-community modeling approach of the dissertation. Results obtained using both approaches are compared and found to be in agreement in the limit of infinitely large populations in all communities. Based on the proposed models, three classes of marketing problems are formulated and studied: referral marketing, seeding marketing and dynamic marketing. It is found that referral marketing can be optimized relatively easily because the associated optimization problem can be formulated as a convex optimization. Also, both seeding marketing and dynamic marketing are shown to enjoy a useful property, namely ``continuous monotone submodularity." Based on this property, a greedy heuristic is proposed which yields solutions with approximation ratio no less than 1-1/e. Also, dynamic marketing for SIS models is reformulated into an equivalent convex optimization to obtain an optimal solution. Both cost minimization and trade-off of cost and profit are analyzed. Next, the proposed modeling framework is applied to study competition of multiple companies in marketing of similar products. Marketing of two classes of such products are considered, namely marketing of durable consumer goods (DCG) and fast-moving consumer goods (FMCG). It is shown that an epsilon-equilibrium exists in the DCG marketing game and a pure Nash equilibrium exists in the FMCG marketing game. The Price of Anarchy (PoA) in both marketing games is found to be bounded by 2. Also, it is shown that any two Nash equilibria for the FMCG marketing game agree almost everywhere, and a distributed algorithm converging to the Nash equilibrium is designed for the FMCG marketing game. Finally, a preliminary investigation is carried out to explore possible concepts of network centrality for diffusions. In a diffusion process, the centrality of a node should reflect the influence that the node has on the network over time. Among the preliminary observations in this work, it is found that when an infection does not break out, diffusion centrality is closely related to Katz centrality; when an infection does break out, diffusion centrality is closely related to eigenvector centrality