The shortest distance in random multi-type intersection graphs

Using an associated branching process as the basis of our approximation, we show that typical inter-point distances in a multitype random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph.Comment: 32 page

k-connectivity of Random Graphs and Random Geometric Graphs in Node Fault Model

k-connectivity of random graphs is a fundamental property indicating reliability of multi-hop wireless sensor networks (WSN). WSNs comprising of sensor nodes with limited power resources are modeled by random graphs with unreliable nodes, which is known as the node fault model. In this paper, we investigate k-connectivity of random graphs in the node fault model by evaluating the network breakdown probability, i.e., the disconnectivity probability of random graphs after stochastic node removals. Using the notion of a strongly typical set, we obtain universal asymptotic upper and lower bounds of the network breakdown probability. The bounds are applicable both to random graphs and to random geometric graphs. We then consider three representative random graph ensembles: the Erdos-Renyi random graph as the simplest case, the random intersection graph for WSNs with random key predistribution schemes, and the random geometric graph as a model of WSNs generated by random sensor node deployment. The bounds unveil the existence of the phase transition of the network breakdown probability for those ensembles.Comment: 6 page

You Only Transfer What You Share: Intersection-Induced Graph Transfer Learning for Link Prediction

Link prediction is central to many real-world applications, but its performance may be hampered when the graph of interest is sparse. To alleviate issues caused by sparsity, we investigate a previously overlooked phenomenon: in many cases, a densely connected, complementary graph can be found for the original graph. The denser graph may share nodes with the original graph, which offers a natural bridge for transferring selective, meaningful knowledge. We identify this setting as Graph Intersection-induced Transfer Learning (GITL), which is motivated by practical applications in e-commerce or academic co-authorship predictions. We develop a framework to effectively leverage the structural prior in this setting. We first create an intersection subgraph using the shared nodes between the two graphs, then transfer knowledge from the source-enriched intersection subgraph to the full target graph. In the second step, we consider two approaches: a modified label propagation, and a multi-layer perceptron (MLP) model in a teacher-student regime. Experimental results on proprietary e-commerce datasets and open-source citation graphs show that the proposed workflow outperforms existing transfer learning baselines that do not explicitly utilize the intersection structure.Comment: Accepted in TMLR (https://openreview.net/forum?id=Nn71AdKyYH

Data Analysis with Intersection Graphs

AbstractThis paper presents a new framework for multivariate data analysis, based on graph theory, using intersection graphs [1]. We have named this approach DAIG – Data Analysis with Intersection Graphs. This new framework represents data vectors as paths on a graph, which has a number of advantages over the classical table representation of data. To do so, each node represents an atom of information, i.e. a pair of a variable and a value, associated with the set of observations for which that pair occurs. An edge exists between a pair of nodes whenever the intersection of their respective sets is not empty. We show that this representation of data as an intersection graph allows an easy and intuitive geometric interpretation of data observations, groups of observations, and results of multivariate data analysis techniques such as biplots, principal components, cluster analysis, or multidimensional scaling. These will appear as paths on the graph, relating variables, values and observations. This approach allows for a compact and memory efficient representation of data that contains many missing values or multi-valued attributes. The basic principles and advantages of this approach are presented with an example of its application to a simple toy problem. The main features of this methodology are illustrated with the aid software specifically developed for this purpose

On Generalizations of Pairwise Compatibility Graphs

A graph $G$ is a PCG if there exists an edge-weighted tree such that each leaf of the tree is a vertex of the graph, and there is an edge $\{ x, y \}$ in $G$ if and only if the weight of the path in the tree connecting $x$ and $y$ lies within a given interval. PCGs have different applications in phylogenetics and have been lately generalized to multi-interval-PCGs. In this paper we define two new generalizations of the PCG class, namely k-OR-PCGs and k-AND-PCGs, that are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. The problems we consider can be also described in terms of the \emph{covering number} and the \emph{intersection dimension} of a graph with respect to the PCG class. In this paper we investigate how the classes of PCG, multi-interval-PCG, OR-PCG and AND-PCG are related to each other and to other graph classes known in the literature. In particular, we provide upper bounds on the minimum $k$ for which an arbitrary graph $G$ belongs to k-interval-PCG, k-OR-PCG and k-AND-PCG classes. Furthermore, for particular graph classes, we improve these general bounds. Moreover, we show that, for every integer $k$, there exists a bipartite graph that is not in the k-interval-PCG class, proving that there is no finite $k$ for which the k-interval-PCG class contains all the graphs. Finally, we use a Ramsey theory argument to show that for any $k$, there exist graphs that are not in k-AND-PCG, and graphs that are not in k-OR-PCG

Approximation Algorithms for Union and Intersection Covering Problems

In a classical covering problem, we are given a set of requests that we need to satisfy (fully or partially), by buying a subset of items at minimum cost. For example, in the k-MST problem we want to find the cheapest tree spanning at least k nodes of an edge-weighted graph. Here nodes and edges represent requests and items, respectively. In this paper, we initiate the study of a new family of multi-layer covering problems. Each such problem consists of a collection of h distinct instances of a standard covering problem (layers), with the constraint that all layers share the same set of requests. We identify two main subfamilies of these problems: - in a union multi-layer problem, a request is satisfied if it is satisfied in at least one layer; - in an intersection multi-layer problem, a request is satisfied if it is satisfied in all layers. To see some natural applications, consider both generalizations of k-MST. Union k-MST can model a problem where we are asked to connect a set of users to at least one of two communication networks, e.g., a wireless and a wired network. On the other hand, intersection k-MST can formalize the problem of connecting a subset of users to both electricity and water. We present a number of hardness and approximation results for union and intersection versions of several standard optimization problems: MST, Steiner tree, set cover, facility location, TSP, and their partial covering variants