Personalized PageRank with Node-dependent Restart

Personalized PageRank is an algorithm to classify the improtance of web pages on a user-dependent basis. We introduce two generalizations of Personalized PageRank with node-dependent restart. The first generalization is based on the proportion of visits to nodes before the restart, whereas the second generalization is based on the probability of visited node just before the restart. In the original case of constant restart probability, the two measures coincide. We discuss interesting particular cases of restart probabilities and restart distributions. We show that the both generalizations of Personalized PageRank have an elegant expression connecting the so-called direct and reverse Personalized PageRanks that yield a symmetry property of these Personalized PageRanks

Generalized Erdos Numbers for network analysis

In this paper we consider the concept of `closeness' between nodes in a weighted network that can be defined topologically even in the absence of a metric. The Generalized Erd\H{o}s Numbers (GENs) satisfy a number of desirable properties as a measure of topological closeness when nodes share a finite resource between nodes as they are real-valued and non-local, and can be used to create an asymmetric matrix of connectivities. We show that they can be used to define a personalized measure of the importance of nodes in a network with a natural interpretation that leads to a new global measure of centrality and is highly correlated with Page Rank. The relative asymmetry of the GENs (due to their non-metric definition) is linked also to the asymmetry in the mean first passage time between nodes in a random walk, and we use a linearized form of the GENs to develop a continuum model for `closeness' in spatial networks. As an example of their practicality, we deploy them to characterize the structure of static networks and show how it relates to dynamics on networks in such situations as the spread of an epidemic

Network Capacity Bound for Personalized PageRank in Multimodal Networks

In a former paper the concept of Bipartite PageRank was introduced and a theorem on the limit of authority flowing between nodes for personalized PageRank has been generalized. In this paper we want to extend those results to multimodal networks. In particular we introduce a hypergraph type that may be used for describing multimodal network where a hyperlink connects nodes from each of the modalities. We introduce a generalisation of PageRank for such graphs and define the respective random walk model that can be used for computations. we finally state and prove theorems on the limit of outflow of authority for cases where individual modalities have identical and distinct damping factors.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1702.0373

The Minimum Wiener Connector

The Wiener index of a graph is the sum of all pairwise shortest-path distances between its vertices. In this paper we study the novel problem of finding a minimum Wiener connector: given a connected graph $G=(V,E)$ and a set $Q\subseteq V$ of query vertices, find a subgraph of $G$ that connects all query vertices and has minimum Wiener index. We show that The Minimum Wiener Connector admits a polynomial-time (albeit impractical) exact algorithm for the special case where the number of query vertices is bounded. We show that in general the problem is NP-hard, and has no PTAS unless $\mathbf{P} = \mathbf{NP}$. Our main contribution is a constant-factor approximation algorithm running in time $\widetilde{O}(|Q||E|)$. A thorough experimentation on a large variety of real-world graphs confirms that our method returns smaller and denser solutions than other methods, and does so by adding to the query set $Q$ a small number of important vertices (i.e., vertices with high centrality).Comment: Published in Proceedings of the 2015 ACM SIGMOD International Conference on Management of Dat