Weighted degrees and heavy cycles in weighted graphs

AbstractA weighted graph is a graph provided with an edge-weighting function w from the edge set to nonnegative real numbers. Bondy and Fan [Annals of Discrete Math. 41 (1989), 53–69] began the study on the existence of heavy cycles in weighted graphs. Though several results with Dirac-type degree condition can be generalized to an Ore-type one in unweighted graphs, it is shown in Bondy et al. [Discuss. Math. Graph Theory 22 (2002), 7–15] that Bondy and Fan’s theorem, which uses Dirac-type condition, cannot be generalized analogously by using Ore-type condition.In this paper we investigate the property peculiar to weighted graphs, and prove a theorem on the existence of heavy cycles in weighted graphs under an Ore-type condition, which generalizes Bondy and Fan’s theorem. Moreover, we show the existence of heavy cycles passing through some specified vertices

A weighted configuration model and inhomogeneous epidemics

A random graph model with prescribed degree distribution and degree dependent edge weights is introduced. Each vertex is independently equipped with a random number of half-edges and each half-edge is assigned an integer valued weight according to a distribution that is allowed to depend on the degree of its vertex. Half-edges with the same weight are then paired randomly to create edges. An expression for the threshold for the appearance of a giant component in the resulting graph is derived using results on multi-type branching processes. The same technique also gives an expression for the basic reproduction number for an epidemic on the graph where the probability that a certain edge is used for transmission is a function of the edge weight. It is demonstrated that, if vertices with large degree tend to have large (small) weights on their edges and if the transmission probability increases with the edge weight, then it is easier (harder) for the epidemic to take off compared to a randomized epidemic with the same degree and weight distribution. A recipe for calculating the probability of a large outbreak in the epidemic and the size of such an outbreak is also given. Finally, the model is fitted to three empirical weighted networks of importance for the spread of contagious diseases and it is shown that $R_0$ can be substantially over- or underestimated if the correlation between degree and weight is not taken into account

An Even Faster and More Unifying Algorithm for Comparing Trees via Unbalanced Bipartite Matchings

A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with all their nodes labeled by the same symbol) and evolutionary trees (i.e., leaf-labeled trees with distinct symbols for distinct leaves). This paper presents an algorithm for comparing trees that are labeled in an arbitrary manner. In addition to this generality, this algorithm is faster than the previous algorithms. Another contribution of this paper is on maximum weight bipartite matchings. We show how to speed up the best known matching algorithms when the input graphs are node-unbalanced or weight-unbalanced. Based on these enhancements, we obtain an efficient algorithm for a new matching problem called the hierarchical bipartite matching problem, which is at the core of our maximum agreement subtree algorithm.Comment: To appear in Journal of Algorithm

The role of bipartite structure in R&D collaboration networks

A number of real-world networks are, in fact, one-mode projections of bipartite networks comprised of two types of nodes. For institutions engaging in collaboration for technological innovation, the underlying network is bipartite with institutions (agents) linked to the patents they have filed (artifacts), while the projection is the co-patenting network. Projected network topology is highly affected by the underlying bipartite structure, hence a lack of understanding of the bipartite network has consequences for the information that might be drawn from the one-mode co-patenting network. Here, we create an empirical bipartite network using data from 2.7 million patents. We project this network onto the agents (institutions) and look at properties of both the bipartite and projected networks that may play a role in knowledge sharing and collaboration. We compare these empirical properties to those of synthetic bipartite networks and their projections in order to understand the processes that might operate in the network formation. A good understanding of the topology is critical for investigating the potential flow of technological knowledge. We show how degree distributions and small cycles affect the topology of the one-mode projected network - specifically degree and clustering distributions, and assortativity. We propose new network-based metrics to quantify how collaborative agents are in the co-patenting network. We find that several large corporations that are the most collaborative agents in the network, however such organisations tend to have a low diversity of collaborators. In contrast, the most prolific institutions tend to collaborate relatively little but with a diverse set of collaborators. This indicates that they concentrate the knowledge of their core technical research, while seeking specific complementary knowledge via collaboration with smaller companies.Comment: 23 pages, 12 figures, 2 table