Ralucca Gera

On the Dominator Colorings in Bipartite Graphs

A graph has a dominator coloring if it has a proper coloring in which each vertex of the graph dominates every vertex of some color class. The dominator chromatic number _Xd(G) is the minimum number of color classes in a dominator coloring of a graph G. In this paper we study the dominator chromatic number for the hypercube, Q_n = Q_n- times K₂ (with Q₁ cong P₂, n ges 2), and more generally for bipartite graphs. We then conclude it with open questions for further research

The Marginal Benefit of Monitor Placement on Networks

Inferring the structure of an unknown network is a difficult problem of interest to researchers, academics , and industrialists . We develop a novel algorithm to infer nodes and edges in an unknown network. Our algorithm utilizes monitors that detect incident edges and adjacent nodes with their labels and degrees. The algorithm infers the network through a preferential random walk with a probabilistic restart at a previously discovered but unmonitored node, or a random teleportation to an unexplored node. Our algorithm outperforms random walk inference and random placement of monitors inference in edge discovery in all test cases. Our algorithm outperforms both methodologies in node inference in synthetic test networks; on real networks it outperforms them in the beginning of the inference. Finally, a website was created where these algorithms can be tested live on preloaded networks or custom networks as desired by the user. The visualization also displays the network as it is being inferred, and provides other statistics about the real and inferred networks.

On the hardness of recognizing triangular line graphs

Given a graph G, its triangular line graph is the graph T (G) with vertex set consisting of the edges of G and adjacencies between edges that are incident in G as well as being within a common triangle. Graphs with a representation as the triangular line graph of some graphG are triangular line graphs, which have been studied under many names including anti-Gallai graphs, 2-in-3 graphs, and link graphs. While closely related to line graphs, triangular line graphs have been dicult to understand and characterize. Van Bang Le asked if recognizing triangular line graphs has an ecient algorithm or is computationally complex. We answer this question by proving that the complexity of recognizing triangular line graphs is NP-complete via a reduction from 3-SAT.

Seeing Red: Locating People of Interest in Networks

The focus of the current research is to identify people of interest in social networks. We are especially interested in studying dark networks , which represent illegal or covert activity. In such networks, people are unlikely to disclose accurate information when queried. We present RedLearn, an algorithm for sampling dark networks with the goal of identifying as many nodes of interest as possible. We consider two realistic lying scenarios, which describe how individuals in a dark network may attempt to conceal their connections. We test and present our results on several real-world multilayered networks, and show that RedLearn achieves up to a 340% improvement over the next best strategy.

DOMINATION IN FUNCTIGRAPHS

Let G1 and G2 be disjoint copies of a graph G, and let f : V (G1) → V (G2) be a function. Then a functigraph C(G,f) = (V,E) has the vertex set V = V (G1) ∪ V (G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | u ∈ V (G1),v ∈ V (G2),v = f(u)}. A functigraph is a generalization of a permutation graph (also known as a generalized prism) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let γ(G) denote the domination number of G. It is readily seen that γ(G) ≤ γ(C(G,f)) ≤ 2γ(G). We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.

On stratification and domination in graphs

A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class), where the vertices in one class are colored red and those in the other class are colored blue. Let F be a 2-stratified graph rooted at some blue vertex v. An F -coloring of a graph is a red-blue coloring of the vertices of G in which every blue vertex v belongs to a copy of F rooted at v. The F domination number γF (G) is the minimum number of red vertices in an F -coloring of G. In this paper, we study F -domination, where F is a 2-stratified red-blue-blue path of order 3 rooted at a blue end-vertex. We present characterizations of connected graphs of order n with F domination number n or 1 and establish several realization results on F -domination number and other domination parameters.

Three is The Answer: Combining Relationships to Analyze Multilayered Terrorist Networks

In this paper we introduce a methodology to create multilayered terrorist networks, taking into account that the main challenges of the data behind the networks are incompleteness, fuzzy boundaries, and dynamic behavior. To account for these dark networks’ characteristics, we use knowledge sharing communities in determining the methodology to create 3-layered networks from each of our datasets. We analyze the resulting layers of three terrorist datasets and present explanations of why three layers should be used for these models. We also use the information of just one layer, to identify the Bali 2005 attack community.

Community evolution in multiplex layer aggregation

This research studies community detection in multiplex dark networks. Our method seeks to intelligently select appropriate layers for aggregation to approximate communities in the whole network, while reducing the impact of over-modeling the network. Community evolution is explored as layers of different types of information are added to the partial picture of the network. We determine the set of dominant layers needed to produce similar community partitions to the established ground truth aggregate network. The identification of dominant layers enhances the selection of which layers to choose for aggregation purposes. This reduces redundancy and noise, and increases the optimization of the available data to produce the desired network partitions. We use normalized mutual index (NMI), purity, density, and modularity for methodology evaluation and comparison metrics.

Graph Structure Similarity using Spectral Graph Theory

In understanding an unknown network we search for metrics to determine how close an inferred network that is being analyzed, is to the truth. We develop a metric to test for similarity between an inferred network and the true network. Our method uses the eigenvalues of the adjacency matrix and of the Laplacian at each step of the network discovery to decide on the comparison to the ground truth. We consider synthetic networks and real terrorist networks for our analysis.

Identifying network structure similarity using spectral graph theory

Most real networks are too large or they are not available for real time analysis. Therefore, in practice, decisions are made based on partial information about the ground truth network. It is of great interest to have metrics to determine if an inferred network (the partial information network) is similar to the ground truth. In this paper we develop a test for similarity between the inferred and the true network. Our research utilizes a network visualization tool, which systematically discovers a network, producing a sequence of snapshots of the network. We introduce and test our metric on the consecutive snapshots of a network, and against the ground truth.To test the scalability of our metric we use a random matrix theory approach while discovering Erdös-Rényi graphs. This scaling analysis allows us to make predictions about the performance of the discovery process.