Random Graph-based Multiple Instance Learning for Structured IoT Smart City Applications

Because of the complex activities involved in IoT networks of a smart city, an important question arises: What are the core activities of the networks as a whole and its basic information flow structure? Identifying and discovering core activities and information flow is a crucial step that can facilitate the analysis. This is the question we are addressing—that is, to identify the core services as a common core substructure despite the probabilistic nature and the diversity of its activities. If this common substructure can be discovered, a systemic analysis and planning can then be performed and key policies related to the community can be developed. Here, a local IoT network can be represented as an attributed graph. From an ensemble of attributed graphs, identifying the common subgraph pattern is then critical in understanding the complexity. We introduce this as the common random subgraph (CRSG) modeling problem, aiming at identifying a subgraph pattern that is the structural “core” that conveys the probabilistically distributed graph characteristics. Given an ensemble of network samples represented as attributed graphs, the method generates a CRSG model that encompasses both structural and statistical characteristics from the related samples while excluding unrelated networks. In generating a CRSG model, our method using a multiple instance learning algorithm transforms an attributed graph (composed of structural elements as edges and their two endpoints) into a “bag” of instances in a vector space. Common structural components across positively labeled graphs are then identified as the common instance patterns among instances across different bags. The structure of the CRSG arises through the combining of common patterns. The probability distribution of the CRSG can then be estimated based on the connections and distributions from the common elements. Experimental results demonstrate that CRSG models are highly expressive in describing typical network characteristics.

Large complete minors in random subgraphs

Let G be a graph of minimum degree at least k and let G p be the random subgraph of G obtained by keeping each edge independently with probability p. We are interested in the size of the largest complete minor that G p contains when p = (1 + ε)/k with ε > 0. We show that with high probability G p contains a complete minor of order $\tilde{\Omega}(\sqrt{k})$ , where the ~ hides a polylogarithmic factor. Furthermore, in the case where the order of G is also bounded above by a constant multiple of k, we show that this polylogarithmic term can be removed, giving a tight bound.

The $Q_2$-Free Process in the Hypercube

The generation of a random triangle-saturated graph via the triangle-free process has been studied extensively. In this short note our aim is to introduce an analogous process in the hypercube. Specifically, we consider the $Q_2$-free process in $Q_d$ and the random subgraph of $Q_d$ it generates. Our main result is that with high probability the graph resulting from this process has at least $cd^{2/3} 2^d$ edges. We also discuss a heuristic argument based on the differential equations method which suggests a stronger conjecture, and discuss the issues with making this rigorous. We conclude with some open questions related to this process.