In this work we introduce and study a nonlocal version of the PageRank. In our approach, the random walker explores the graph using longer excursions than just moving between neighboring nodes. As a result, the corresponding ranking of the nodes, which takes into account a \textit{long-range interaction} between them, does not exhibit concentration phenomena typical of spectral rankings which take into account just local interactions. We show that the predictive value of the rankings obtained using our proposals is considerably improved on different real world problems.
While PageRank has been extensively used to rank sport tournament participants (teams or individuals), its superiority over simpler ranking methods has been never clearly demonstrated. We use sports results from 18 major leagues to calibrate a state-of-art model for synthetic sports results. Model data are then used to assess the ranking performance of PageRank in a controlled setting. We find that PageRank outperforms the benchmark ranking by the number of wins only when a small fraction of all games have been played. Increased randomness in the data, such as intrinsic randomness of outcomes …
PageRank is arguably the most popular ranking algorithm which is being applied in real systems ranging from information to biological and infrastructure networks. Despite its outstanding popularity and broad use in different areas of science, the relation between the algorithm's efficacy and properties of the network on which it acts has not yet been fully understood. We study here PageRank's performance on a network model supported by real data, and show that realistic temporal effects make PageRank fail in individuating the most valuable nodes for a broad range of model parameters. Results o…
This paper describes our solution for WSDM Cup 2016. Ranking the query independent importance of scholarly articles is a critical and challenging task, due to the heterogeneity and dynamism of entities involved. Our approach is called Ensemble enabled Weighted PageRank (EWPR). To do this, we first propose Time-Weighted PageRank that extends PageRank by introducing a time decaying factor. We then develop an ensemble method to assemble the authorities of the heterogeneous entities involved in scholarly articles. We finally propose to use external data sources to further improve the ranking accur…
We study the time evolution of ranking and spectral properties of the Google matrix of English Wikipedia hyperlink network during years 2003 - 2011. The statistical properties of ranking of Wikipedia articles via PageRank and CheiRank probabilities, as well as the matrix spectrum, are shown to be stabilized for 2007 - 2011. A special emphasis is done on ranking of Wikipedia personalities and universities. We show that PageRank selection is dominated by politicians while 2DRank, which combines PageRank and CheiRank, gives more accent on personalities of arts. The Wikipedia PageRank of universit…
Ranking on large-scale graphs plays a fundamental role in many high-impact application domains, ranging from information retrieval, recommender systems, sports team management, biology to neuroscience and many more. PageRank, together with many of its random walk based variants, has become one of the most well-known and widely used algorithms, due to its mathematical elegance and the superior performance across a variety of application domains. Important as it might be, state-of-the-art lacks an intuitive way to explain the ranking results by PageRank (or its variants), e.g., why it thinks the…
The basic indicators of a researcher's productivity and impact are still the number of publications and their citation counts. These metrics are clear, straightforward, and easy to obtain. When a ranking of scholars is needed, for instance in grant, award, or promotion procedures, their use is the fastest and cheapest way of prioritizing some scientists over others. However, due to their nature, there is a danger of oversimplifying scientific achievements. Therefore, many other indicators have been proposed including the usage of the PageRank algorithm known for the ranking of webpages and its…
PageRank has been widely used to measure the authority or the influence of a user in social networks. However, conventional PageRank only makes use of edge-based relations, which represent first-order relations between two connected nodes. It ignores higher-order relations that may exist between nodes. In this paper, we propose a novel framework, motif-based PageRank (MPR), to incorporate higher-order relations into the conventional PageRank computation. Motifs are subgraphs consisting of a small number of nodes. We use motifs to capture higher-order relations between nodes in a network and in…
The Football World Cup as world's favorite sporting event is a source of both entertainment and overwhelming amount of data about the games played. In this paper we analyse the available data on football world championships since 1930 until today. Our goal is to rank the national teams based on all matches during the championships. For this purpose, we apply the PageRank with restarts algorithm to a graph built from the games played during the tournaments. Several statistics such as matches won and goals scored are combined in different metrics that assign weights to the links in the graph. Fi…
In the light of the need to achieve a ranking which is understood by all tennis supporters, the ATP ranking is exposed to constant complaints from players and at the same time exposes new players to be benefited with a good tournament to be able to start progressing in their careers. Moreover, the ATP ranking is not powerful enough to predict with certainty who will be the winner of a match if we are based solely on the positions. In order to combat these problems, the idea of creating a new ranking that can indicate what are the real chances of victory of a player before the start of a new to…
Connections among entities are everywhere. From social media interactions to web page hyperlinks, networks are frequently used to represent such complex systems. Node ranking is a fundamental task that provides the strategy to identify central entities according to multiple criteria. Popular node ranking metrics include degree, closeness or betweenness centralities, as well as HITS authority or PageRank. In this work, we propose a novel node ranking metric, where we combine PageRank and the idea of node fatigue, in order to model a random explorer who wants to optimize coverage - it gets fatig…
Methods for ranking the importance of nodes in a network have a rich history in machine learning and across domains that analyze structured data. Recent work has evaluated these methods though the seed set expansion problem: given a subset $S$ of nodes from a community of interest in an underlying graph, can we reliably identify the rest of the community? We start from the observation that the most widely used techniques for this problem, personalized PageRank and heat kernel methods, operate in the space of landing probabilities of a random walk rooted at the seed set, ranking nodes according…
In network science, there is often the need to sort the graph nodes. While the sorting strategy may be different, in general sorting is performed by exploiting the network structure. In particular, the metric PageRank has been used in the past decade in different ways to produce a ranking based on how many neighbors point to a specific node. PageRank is simple, easy to compute and effective in many applications, however it comes with a price: as PageRank is an application of the random walker, the arc weights need to be normalized. This normalization, while necessary, introduces a series of un…
The objective assessment of the prestige of an academic institution is a difficult and hotly debated task. In the last few years, different types of University Rankings have been proposed to quantify the excellence of different research institutions in the world. Albeit met with criticism in some cases, the relevance of university rankings is being increasingly acknowledged: indeed, rankings are having a major impact on the design of research policies, both at the institutional and governmental level. Yet, the debate on what rankings are {\em exactly} measuring is enduring. Here, we address th…
Many complex systems can be described as multiplex networks in which the same nodes can interact with one another in different layers, thus forming a set of interacting and co-evolving networks. Examples of such multiplex systems are social networks where people are involved in different types of relationships and interact through various forms of communication media. The ranking of nodes in multiplex networks is one of the most pressing and challenging tasks that research on complex networks is currently facing. When pairs of nodes can be connected through multiple links and in multiple layer…
Identifying the importance of nodes of complex networks is of interest to the research of Social Networks, Biological Networks etc.. Current researchers have proposed several measures or algorithms, such as betweenness, PageRank and HITS etc., to identify the node importance. However, these measures are based on different aspects of properties of nodes, and often conflict with the others. A reasonable, fair standard is needed for evaluating and comparing these algorithms. This paper develops a framework as the standard for ranking the importance of nodes. Four intuitive rules are suggested to …
The importance of a node in a directed graph can be measured by its PageRank. The PageRank of a node is used in a number of application contexts - including ranking websites - and can be interpreted as the average portion of time spent at the node by an infinite random walk. We consider the problem of maximizing the PageRank of a node by selecting some of the edges from a set of edges that are under our control. By applying results from Markov decision theory, we show that an optimal solution to this problem can be found in polynomial time. Our core solution results in a linear programming for…
We study the problem of score and rank monotonicity for spectral ranking methods, such as eigenvector centrality and PageRank, in the case of undirected networks. Score monotonicity means that adding an edge increases the score at both ends of the edge. Rank monotonicity means that adding an edge improves the relative position of both ends of the edge with respect to the remaining nodes. It is known that common spectral rankings are both score and rank monotone on directed, strongly connected graphs. We show that, surprisingly, the situation is very different for undirected graphs, and in part…
Random teleportation is a necessary evil for ranking and clustering directed networks based on random walks. Teleportation enables ergodic solutions, but the solutions must necessarily depend on the exact implementation and parametrization of the teleportation. For example, in the commonly used PageRank algorithm, the teleportation rate must trade off a heavily biased solution with a uniform solution. Here we show that teleportation to links rather than nodes enables a much smoother trade-off and effectively more robust results. We also show that, by not recording the teleportation steps of th…
This paper describes the first results obtained by implementing a novel approach to rank vertices in a heterogeneous graph, based on the PageRank family of algorithms and applied here to the bipartite graph of papers and authors as a first evaluation of its relevance on real data samples. With this approach to evaluate research activities, the ranking of a paper/author depends on that of the papers/authors citing it/him or her. We compare the results against existing ranking methods (including methods which simply apply PageRank to the graph of papers or the graph of authors) through the analy…
In this paper, we consider the problem of diversity in ranking of the nodes in a graph. The task is to pick the top-k nodes in the graph which are both 'central' and 'diverse'. Many graph-based models of NLP like text summarization, opinion summarization involve the concept of diversity in generating the summaries. We develop a novel method which works in an iterative fashion based on random walks to achieve diversity. Specifically, we use negative reinforcement as a main tool to introduce diversity in the Personalized PageRank framework. Experiments on two benchmark datasets show that our alg…
Some social networks, such as LinkedIn and ResearchGate, allow user endorsements for specific skills. In this way, for each skill we get a directed graph where the nodes correspond to users' profiles and the arcs represent endorsement relations. From the number and quality of the endorsements received, an authority score can be assigned to each profile. In this paper we propose an authority score computation method that takes into account the relations existing among different skills. Our method is based on enriching the information contained in the digraph of endorsements corresponding to a s…
Measuring the influence of users in social networks is key for numerous applications. A recently proposed influence metric, coined as $ψ$-score, allows to go beyond traditional centrality metrics, which only assess structural graph importance, by further incorporating the rich information provided by the posting and re-posting activity of users. The $ψ$-score is shown in fact to generalize PageRank for non-homogeneous node activity. Despite its significance, it scales poorly to large datasets; for a network of $N$ users, it requires to solve $N$ linear systems of equations of size $N$. To addr…
Recent research on finding important intermediate nodes in a network suspected to contain criminal activity is highly dependent on network centrality values. Betweenness centrality, for example, is widely used to rank the nodes that act as brokers in the shortest paths connecting all source and all the end nodes in a network. However both the shortest path node betweenness and the linearly scaled betweenness can only show rankings for all the nodes in a network. In this paper we explore the mathematical concept of pair-dependency on intermediate nodes, adapting the concept to criminal relation…
To rank nodes in quasi-hierarchical networks of social nature, it is necessary to carry out a detailed analysis of the network and evaluate the results obtained according to all the given criteria and identify the most influential nodes. Existing ranking algorithms in the overwhelming majority estimate such networks in general, which does not allow to clearly determine the influence of nodes among themselves. In the course of the study, an analysis of the results of known algorithms for ranking the nodes of HITS, PageRank and compares the obtained data with the expert evaluation of the network…
We present Wikipedia Ranking of World Universities (WRWU) based on analysis of networks of 24 Wikipedia editions collected in May 2017. With PageRank and CheiRank algorithms we determine ranking of universities averaged over cultural views of these editions. The comparison with the Shanghai ranking gives overlap of 60% for top 100 universities showing that WRWU gives more significance to their historical development. We show that the new reduced Google matrix algorithm allows to determine interactions between leading universities on a scale of ten centuries. This approach also determines the i…
Node influence metrics have been applied to many applications, including ranking web pages on internet, or locations on spatial networks. PageRank is a popular and effective algorithm for estimating node influence. However, conventional PageRank method considers neither the heterogeneity of network structures nor additional network information, causing a major impedance to performance improvement and an underestimation of non-hub nodes' importance. As these problems are only partially studied, existing solutions are still not satisfying. This paper addresses the problems by presenting a genera…
Nanyang Technological University（南洋理工大学）；National University of Singapore（新加坡国立大学）；Hamad Bin Khalifa University（哈马德·本·哈利法大学）
Given an input graph G and a node v in G, homogeneous network embedding (HNE) maps the graph structure in the vicinity of v to a compact, fixed-dimensional feature vector. This paper focuses on HNE for massive graphs, e.g., with billions of edges. On this scale, most existing approaches fail, as they incur either prohibitively high costs, or severely compromised result utility. Our proposed solution, called Node-Reweighted PageRank (NRP), is based on a classic idea of deriving embedding vectors from pairwise personalized PageRank (PPR) values. Our contributions are twofold: first, we design a …
Citations between scientific papers and related bibliometric indices, such as the $h$-index for authors and the impact factor for journals, are being increasingly used - often in controversial ways - as quantitative tools for research evaluation. Yet, a fundamental research question remains still open: to which extent do quantitative metrics capture the significance of scientific works? We analyze the network of citations among the $449,935$ papers published by the American Physical Society (APS) journals between 1893 and 2009, and focus on the comparison of metrics built on the citation count…
Nodes can be ranked according to their relative importance within the network. Ranking algorithms based on random walks are particularly useful because they connect topological and diffusive properties of the network. Previous methods based on random walks, as for example the PageRank, have focused on static structures. However, several realistic networks are indeed dynamic, meaning that their structure changes in time. In this paper, we propose a centrality measure for temporal networks based on random walks which we call TempoRank. While in a static network, the stationary density of the ran…
Wikipedia is a huge global repository of human knowledge, that can be leveraged to investigate interwinements between cultures. With this aim, we apply methods of Markov chains and Google matrix, for the analysis of the hyperlink networks of 24 Wikipedia language editions, and rank all their articles by PageRank, 2DRank and CheiRank algorithms. Using automatic extraction of people names, we obtain the top 100 historical figures, for each edition and for each algorithm. We investigate their spatial, temporal, and gender distributions in dependence of their cultural origins. Our study demonstrat…
