The PageRank (PR) algorithm is the base of Google search engine. In this paper, we study the PageRank sequence for undirected graphs of order six by computing their PR vectors. In continuing, we provide an ordering of graphs by variance of PR vector whose variation is proportional with variance of degree sequence. Finally, we introduce a relation between domination number and PR-variance of given graphs.

One of the criteria for search engines to determine the popularity of pages is an analysis of links in the web graph, and various methods have already been presented in this regard. The PageRank algorithm is the oldest web page ranking methods based on web graph and is still used as one of the important factors of web pages on Google. Since the invention of this method, several bugs have been published and solutions have been proposed to correct them. The most important problem that is most noticed is pages without an out link or so-called suspended pages. In web graph analysis, we noticed another problem that occurs on some pages at the out degree of one, and the problem is that under conditions, the linked page score is more than the home page. This problem can generate unrealistic scores for pages, and the link chain can invalidate the web graph. In this paper, this problem has been investigated under the title "One-Two Gap", and a solution has been proposed to it. Experimental results show that fixing of the One-Two gap problem using the proposed solution. Test standard benchmark dataset, TREC2003, is applied to evaluate the proposed method. The experimental results show that our proposed method outperforms PageRank method theoretically and experimentally in the term of precision, accuracy, and sensitivity with such criteria as PD, P@n, NDCG@n, MAP, and Recall.

Influence maximization is one of the most important topics in the social network analysis field. As all the social networks can be considered signed, explicitly or implicitly, assessing influence maximization in these networks is inevitable. Due to the NP-hard nature of this problem, the category of node-ranking-based solutions is of concern, where, the PageRank algorithm is outstanding. Original PageRank is merely defined based on the trust relationships and it is not applicable in signed social networks. Upon an agreement on the scheme of trust propagation, where trust propagates step by step in the social network, the two main schemes of distrust propagation are: a) distrust propagates step by step throughout the social network, and b) distrust propagates up to one step of the neighborhood. Despite the claims made by related researches that scheme (b) is the dominant behavior compared to (a),available PageRank algorithms are updated to incorporate scheme (a). In this study, a new PageRank-based method, which adopts scheme (b) to model the distrust-based influence propagation in signed social networks, is proposed. Accordingly, the importance of each node is computed considering that every node propagates the received influence from its trusted neighbors to other nodes, while it blocks the received influence from its untrusted neighbors. Assessments run on the three real datasets reveal the superiority of this proposed method over other existing PageRank algorithms in maximizing influence in signed social networks. The outperformance is between 22% to 46% considering all experimental settings in comparison with the most effective benchmark method.

Due to the huge amount of data published on the Web, the Web search process has become more difficult, and it is sometimes hard to get the expected results, especially when the users are less certain about their information needs. Several efforts have been proposed to support exploratory search on the web by using query expansion, faceted search, or supplementary information extracted from external knowledge resources. However, these solutions are not well explored for the general web search in an open-domain setting. In addition, they mostly focus on supporting search in content expressed in English and Latin based languages. In this research, we propose a fully automated approach that aims to support exploratory search over the Arabic web content. It exploits the Arabic version of Wikipedia to extract complementary information that supports visual representation and deeper exploration of the search engine's results. Key Wikipedia entities are extracted from the text snippets produced by the search engine in response to the user's query. Entities are then filtered and ranked by using a novel ranking algorithm that extends the conventional PageRank algorithm. Finally, a graph is built and presented to the user to visually represent highly ranked topics and their relationships. The proposed approach was realized by developing ArabXplore, a system that integrates with the web browser to support the web search process by executing our approach in query time. It was assessed over a dataset of 100 Arabic search queries covering different domains, and results were assessed and rated by human subjects. The underlying ranking algorithm was also compared with the conventional PageRank.

The random walk technique, which has a reputation for excellent performance, is one method for complex networks sampling. However, reducing the input data size is still a considerable topic to increase the efficiency and speed of this algorithm. The two approaches discussed in this paper, the no-retracing and the seed node selection algorithms, inspired the development of random walk technique. The Google PageRank method is integrated with these different approaches. Input data size is decreased while critical nodes are preserved. A real database was used for this sampling. Significant sample characteristics were also covered, including average clustering coefficient, sampling effectiveness, degree distribution, and average degree. The no-retracing method, for example, performs better. The efficiency increases even further when the no-retracing technique is combined with the Google PageRank. When choosing between public transportation and aircraft, for example, these algorithms might be used since time is crucial. Additionally, these algorithms are more energy-efficient methods that were looked at.