Axiomatic Characterization of PageRank

This paper examines the fundamental problem of identifying the most important nodes in a network. We use an axiomatic approach to this problem. Specifically, we propose six simple properties and prove that PageRank is the only centrality measure that satisfies all of them. Our work gives new conceptual and theoretical foundations of PageRank that can be used to determine suitability of this centrality measure in specific applications

Multitask Learning on Graph Neural Networks: Learning Multiple Graph Centrality Measures with a Unified Network

The application of deep learning to symbolic domains remains an active research endeavour. Graph neural networks (GNN), consisting of trained neural modules which can be arranged in different topologies at run time, are sound alternatives to tackle relational problems which lend themselves to graph representations. In this paper, we show that GNNs are capable of multitask learning, which can be naturally enforced by training the model to refine a single set of multidimensional embeddings $\in \mathbb{R}^d$ and decode them into multiple outputs by connecting MLPs at the end of the pipeline. We demonstrate the multitask learning capability of the model in the relevant relational problem of estimating network centrality measures, focusing primarily on producing rankings based on these measures, i.e. is vertex $v_1$ more central than vertex $v_2$ given centrality $c$?. We then show that a GNN can be trained to develop a \emph{lingua franca} of vertex embeddings from which all relevant information about any of the trained centrality measures can be decoded. The proposed model achieves $89\%$ accuracy on a test dataset of random instances with up to 128 vertices and is shown to generalise to larger problem sizes. The model is also shown to obtain reasonable accuracy on a dataset of real world instances with up to 4k vertices, vastly surpassing the sizes of the largest instances with which the model was trained ($n=128$). Finally, we believe that our contributions attest to the potential of GNNs in symbolic domains in general and in relational learning in particular.Comment: Published at ICANN2019. 10 pages, 3 Figure

How to choose the most appropriate centrality measure?

We propose a new method to select the most appropriate network centrality measure based on the user's opinion on how such a measure should work on a set of simple graphs. The method consists in: (1) forming a set $\cal F$ of candidate measures; (2) generating a sequence of sufficiently simple graphs that distinguish all measures in $\cal F$ on some pairs of nodes; (3) compiling a survey with questions on comparing the centrality of test nodes; (4) completing this survey, which provides a centrality measure consistent with all user responses. The developed algorithms make it possible to implement this approach for any finite set $\cal F$ of measures. This paper presents its realization for a set of 40 centrality measures. The proposed method called culling can be used for rapid analysis or combined with a normative approach by compiling a survey on the subset of measures that satisfy certain normative conditions (axioms). In the present study, the latter was done for the subsets determined by the Self-consistency or Bridge axioms.Comment: 26 pages, 1 table, 1 algorithm, 8 figure

Selection of Centrality Measures Using Self-Consistency and Bridge Axioms

We consider several families of network centrality measures induced by graph kernels, which include some well-known measures and many new ones. The Self-consistency and Bridge axioms, which appeared earlier in the literature, are closely related to certain kernels and one of the families. We obtain a necessary and sufficient condition for Self-consistency, a sufficient condition for the Bridge axiom, indicate specific measures that satisfy these axioms, and show that under some additional conditions they are incompatible. PageRank centrality applied to undirected networks violates most conditions under study and has a property that according to some authors is ``hard to imagine'' for a centrality measure. We explain this phenomenon. Adopting the Self-consistency or Bridge axiom leads to a drastic reduction in survey time in the culling method designed to select the most appropriate centrality measures.Comment: 23 pages, 5 figures. A reworked versio

SAKE: Estimating Katz Centrality Based on Sampling for Large-Scale Social Networks

Katz centrality is a fundamental concept to measure the influence of a vertex in a social network. However, existing approaches to calculating Katz centrality in a large-scale network are unpractical and computationally expensive. In this article, we propose a novel method to estimate Katz centrality based on graph sampling techniques, which object to achieve comparable estimation accuracy of the state-of-the-arts with much lower computational complexity. Specifically, we develop a Horvitz–Thompson estimate for Katz centrality by using a multi-round sampling approach and deriving an unbiased mean value estimator. We further propose SAKE, a Sampling-based Algorithm for fast Katz centrality Estimation. We prove that the estimator calculated by SAKE is probabilistically guaranteed to be within an additive error from the exact value. Extensive evaluation experiments based on four real-world networks show that the proposed algorithm can estimate Katz centralities for partial vertices with low sampling rate, low computation time, and it works well in identifying high influence vertices in social networks

Rank monotonicity in centrality measures

A measure of centrality is rank monotone if after adding an arc x -> y, all nodes with a score smaller than (or equal to) y have still a score smaller than (or equal to) y. If, in particular, all nodes with a score smaller than or equal to y get a score smaller than y (i.e., all ties with y are broken in favor of y), the measure is called strictly rank monotone. We prove that harmonic centrality is strictly rank monotone, whereas closeness is just rank monotone on strongly connected graphs, and that some other measures, including betweenness, are not rank monotone at all (sometimes not even on strongly connected graphs). Among spectral measures, damped scores such as Katz's index and PageRank are strictly rank monotone on all graphs, whereas the dominant eigenvector is strictly monotone on strongly connected graphs only