Degree distributions of bipartite networks and their projections

Abstract

Bipartite (two-mode) networks are important in the analysis of social and economic systems as they explicitly show conceptual links between different types of entities. However, applications of such networks often work with a projected (one-mode) version of the original bipartite network. The topology of the projected network, and the dynamics that take place on it, are highly dependent on the degree distributions of the two different node types from the original bipartite structure. To date, the interaction between the degree distributions of bipartite networks and their one-mode projections is well understood for only a few cases, or for networks that satisfy a restrictive set of assumptions. Here we show a broader analysis in order to fill the gap left by previous studies. We use the formalism of generating functions to prove that the degree distributions of both node types in the original bipartite network affect the degree distribution in the projected version. To support our analysis, we simulate several types of synthetic bipartite networks using a configuration model where node degrees are assigned from specific probability distributions, ranging from peaked to heavy-tailed distributions. Our findings show that when projecting a bipartite network onto a particular set of nodes, the degree distribution for the resulting one-mode network follows the distribution of the nodes being projected on to, but only so long as the degree distribution for the opposite set of nodes does not have a heavier tail. Furthermore, we show that bipartite degree distributions are not the only feature driving topology formation of projected networks, in contrast to what is commonly described in the literature.

Modularity and projection of bipartite networks

“…al. [29] show that when the bottom nodes have a heavier-tailed degree distribution, µ 2 > µ 1 , the degree distribution of the projected graph follows the degree distribution of the bottom node set.…”

“…We generate synthetic bipartite networks with known degree distributions following [29]. For each of the N top nodes and M bottom nodes we assign a degree value from the chosen distributions to give us top and bottom degree sequences d j and q i .…”

“…In the case where the average degrees in each node set are different we will end up with unbalanced graphs, i.e. N = M , see [29] for details. To build in a target community structure we construct C bipartite networks, with the same top and bottom distributions and starting N and M .…”

Stability Analysis of Company Co-Mention Network and Market Graph Over Time Using Graph Similarity Measures

“…Thus, economic and financial connections between companies may be reflected by the correlation of the log returns of company assets [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. • Some researchers believed that a connection between companies arises, if both companies have common members of the board of directors [33][34][35][36][37]. • Several papers dealt with company co-mention networks in which connection between companies is reflected by the fact of mentioning the both companies in one news item [38][39][40][41].…”

Application of Methods of Statistical Analysis of Network Structures to Study the Dynamics of Market Graphs and Graphs of Joint References

“…Некоторые исследователи использовали и иные методы косвенного оценивания взаимосвязей, основанные на географической близости, отраслевой принадлежности, сведений о составах советов директоров и пр. [34][35][36][37][38][39][40][41][42]. Мы, кроме данных о результатах биржевых торгов, использовали материалы, предоставляемые провайдерами новостной аналитики, так как деятельность крупных корпораций находит свое отражение в огромном количестве публикаций, таких как отчеты, пресс-релизы самих компаний, решений арбитражных судов, публикаций рейтинговых и аналитических агентств, публикаций в специализированных изданиях и т. д. Про-Приложение методов статистического анализа сетевых структур для исследования динамики рыночных графов и графов совместных упоминаний вайдеры новостной аналитики, крупнейшими из которых являются Thompson Reuters и Raven Pack, собирают, обрабатывают и представляют пользователям в оцифрованном виде доступ к подобной информации [43,44].…”