Bipartite networks are currently regarded as providing a major insight into URLanization of  real-world systems, unveiling the mechanisms shaping the interactions occurring between distinct groups of nodes. One of the major problems encountered when dealing with bipartite networks is obtaining a (monopartite) projection over the layer of interest which preserves as much as possible the information encoded into the original bipartite structure  . In the present paper we propose an algorithm to obtain statistically-validated monopartite projections of bipartite networks, which implements a simple rule: in order for any two nodes to be linked, a significantly-large number of neighbors must be shared . Naturally, assessing the statistical significance of nodes similarity requires the definition of a proper statistical benchmark: here we consider two recently-proposed null modelsfor bipartite networks, opportunely defined through the exponential random graph formalism . Our algorithm outputs a matrix of link-specific p-values, from which a validated projection can be straightforwardly obtained, upon running a multiple hypothesis test and retaining only the statistically significant links. Finally,  to test our method we analyze a social network (i.e. the MovieLens dataset, a bipartite network of users and rated movies)and an economic network (i. e. the countries-products World Trade Web representation): while, in the first case, projecting MovieLens on the films layer allows clusters of movies belonging to similar genres to be detected, in the second case, projecting the World Trade Web on the countries layer reveals a modular structure of similarly-industrialized clusters of nations .
Bipartite networks are currently regarded as providing a major insight into URLanization of many real-world systems, unveiling the mechanisms driving the interactions which occur between distinct groups of nodes. One of the most important issues encountered when modeling bipartite networks is devising a way to obtain a (monopartite) projection on the layer of interest , which preserves the information encoded into the original bipartite structure as much as possible . In the present paper we propose an algorithm to obtain statistically-validated projections of bipartite networks, which implements a simple rule: in order for any two nodes to be linked, the number of shared neighbors must be statistically significant . Naturally, assessing the statistical significance of nodes similarity requires the definition of a proper statistical benchmark: here we consider a set of four null models, defined within the Exponential Random Graph framework . Our algorithm outputs a matrix of link-specific p-values, from which a validated projection can be straightforwardly obtained, upon running a multiple hypothesis test and retaining only the statistically-significant links. Finally, in order to test our method , we analyze an economic network (i.e. the countries-products World Trade Web representation) and a social network (i.e. the MovieLens dataset, collecting the users' ratings of a list of movies). In both cases non-trivial communities are detected. In the first case, while projecting the World Trade Web on the countries layer reveals modules of similarly-industrialized nations, projecting it on the products layer allows communities characterized by an increasing level of complexity to be detected; in the second case, projecting MovieLens on the films layer allows clusters of movies whose affinity cannot be fully accounted for by genre similarity to be individuated .
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many
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shaping the interactions occurring
driving the interactions which occur
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major problems encountered when dealing with
most important issues encountered when modeling
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obtaining a
devising a way to obtain a
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over
on
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which preserves as much as possible
, which preserves
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as much as possible
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monopartite
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a significantly-large number of
the number of shared
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shared
statistically significant
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two recently-proposed null modelsfor bipartite networks, opportunely defined through the exponential random graph formalism
a set of four null models, defined within the Exponential Random Graph framework
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statistically significant
statistically-significant
A
in order
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we analyze
, we analyze an economic network (i.e. the countries-products World Trade Web representation) and
R
a bipartite network of users and rated movies)and an economic network (i. e. the countries-products
collecting the users' ratings of a list of movies). In both cases non-trivial communities are detected. In the first case, while projecting the
R
representation): while, in the first case, projecting MovieLens on the films layer allows clusters of movies belonging to similar genres to be detected,
on the countries layer reveals modules of similarly-industrialized nations, projecting it on the products layer allows communities characterized by an increasing level of complexity to be detected;
R
the World Trade Web on the countries layer reveals a modular structure of similarly-industrialized clusters of nations
MovieLens on the films layer allows clusters of movies whose affinity cannot be fully accounted for by genre similarity to be individuated