Bounds for identifying codes in terms of degree parameters

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its neighbourhood within the identifying code. If \M(G) denotes the minimum size of an identifying code of a graph $G$, it was conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there exists a constant $c$ such that if a connected graph $G$ with $n$ vertices and maximum degree $d$ admits an identifying code, then \M(G)\leq n-\tfrac{n}{d}+c. We use probabilistic tools to show that for any $d\geq 3$, \M(G)\leq n-\tfrac{n}{\Theta(d)} holds for a large class of graphs containing, among others, all regular graphs and all graphs of bounded clique number. This settles the conjecture (up to constants) for these classes of graphs. In the general case, we prove \M(G)\leq n-\tfrac{n}{\Theta(d^{3})}. In a second part, we prove that in any graph $G$ of minimum degree $\delta$ and girth at least 5, \M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n. Using the former result, we give sharp estimates for the size of the minimum identifying code of random $d$-regular graphs, which is about $\tfrac{\log d}{d}n$

Topics in social network analysis and network science

This chapter introduces statistical methods used in the analysis of social networks and in the rapidly evolving parallel-field of network science. Although several instances of social network analysis in health services research have appeared recently, the majority involve only the most basic methods and thus scratch the surface of what might be accomplished. Cutting-edge methods using relevant examples and illustrations in health services research are provided

Synthetic Generation of Social Network Data With Endorsements

In many simulation studies involving networks there is the need to rely on a sample network to perform the simulation experiments. In many cases, real network data is not available due to privacy concerns. In that case we can recourse to synthetic data sets with similar properties to the real data. In this paper we discuss the problem of generating synthetic data sets for a certain kind of online social network, for simulation purposes. Some popular online social networks, such as LinkedIn and ResearchGate, allow user endorsements for specific skills. For each particular skill, the endorsements give rise to a directed subgraph of the corresponding network, where the nodes correspond to network members or users, and the arcs represent endorsement relations. Modelling these endorsement digraphs can be done by formulating an optimization problem, which is amenable to different heuristics. Our construction method consists of two stages: The first one simulates the growth of the network, and the second one solves the aforementioned optimization problem to construct the endorsements.Comment: 5 figures, 2 algorithms, Journal of Simulation 201