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Using heat conduction mechanism on a social network we develop a systematic method to predict missing values as recommendations. This method can treat very large matrices that are typical of internet communities. In particular, with an innovative, exact formulation that accommodates arbitrary boundary condition, our method is easy to use in real applications. The performance is assessed by comparing with traditional recommendation methods using real data.

A Figure 1. Schematic representation of the preferential attachment process: The probability that a tie appears between node A and another node is a function of the number of ties of the other node. In this example, probabilities of new ties (in green) are indicated by line width. In this paper, we measure this relationship between the degree and the tie creation probability, modeling it as a power with an exponent whose values we explain by the processes underlying the network. ABSTRACTWe perform an empirical study of the preferential attachment phenomenon in temporal networks and show that on the Web, networks follow a nonlinear preferential attachment model in which the exponent depends on the type of network considered. The classical preferential attachment model for networks by Barabási and Albert (1999) assumes a linear relationship between the number of neighbors of a node in a network and the probability of attachment. Although this assumption is widely made in Web Science and related fields, the underlying linearity is rarely measured. To fill this gap, this paper performs an empirical longitudinal (timebased) study on forty-seven diverse Web network datasets from seven network categories and including directed, undirected and bipartite networks. We show that contrary to the usual assumption, preferential attachment is nonlinear in the networks under consideration. Furthermore, we observe that the deviation from linearity is dependent on the type of network, giving sublinear attachment in certain types of networks, and superlinear attachment in others. Thus, we introduce the preferential attachment exponent β as a novel numerical network measure that can be used to discriminate different types of networks. We propose explanations for the behavior of that network measure, based on the mechanisms that underly the growth of the network in question.

We develop a simple statistical method to find affinity relations in a large opinion network which is represented by a very sparse matrix. These relations allow us to predict missing matrix elements. We test our method on the Eachmovie data of thousands of movies and viewers. We found that significant prediction precision can be achieved and it is rather stable. There is an intrinsic limit to further improve the prediction precision by collecting more data, implying perfect prediction can never obtain via statistical means.

This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter α. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes N , time t and the model parameter α. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerical simulations of quantum walks on circulant graphs show complex interference phenomena and how complete suppression of transport is achieved near α = π 2. By proving two mirror symmetries around α = 0 and π 2 we show that these quantum walks have a period of π in α. We show that undirected edges lose their effect on the quantum walk at α = π 2 and present non-bipartite graphs that exhibit suppression of transport. Finally, we analytically compute the Hamiltonians of quantum walks on the directed ring graph.

The rich get richer principle, manifested by the Preferential attachment (PA) mechanism, is widely considered one of the major factors in the growth of real-world networks. PA stipulates that popular nodes are bound to be more attractive than less popular nodes; for example, highly cited papers are more likely to garner further citations. However, it overlooks the transient nature of popularity, which is often governed by trends. Here, we show that in a wide range of real-world networks the recent popularity of a node, i.e., the extent by which it accumulated links recently, significantly influences its attractiveness and ability to accumulate further links. We proceed to model this observation with a natural extension to PA, named Trending Preferential Attachment (TPA), in which edges become less influential as they age. TPA quantitatively parametrizes a fundamental network property, namely the network’s tendency to trends. Through TPA, we find that real-world networks tend to be moderately to highly trendy. Networks are characterized by different susceptibilities to trends, which determine their structure to a large extent. Trendy networks display complex structural traits, such as modular community structure and degree-assortativity, occurring regularly in real-world networks. In summary, this work addresses an inherent trait of complex networks, which greatly affects their growth and structure, and develops a unified model to address its interaction with preferential attachment.

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