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 MovieLensdataset , 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.
Bipartite networks are currently regarded as providing a major insight into URLanization of many real-world systems, unveiling the mechanisms driving the interactions occurring 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 as much as possible the information encoded into the original bipartite structure . In the present paper we propose an algorithm to obtain statistically-validated projections of bipartite networks, according to which any two nodes sharing a statistically-significant number of neighbors are linked. Since assessing the statistical significance of nodes similarity requires 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 is straightforwardly obtainable , upon running a multiple hypothesis testing procedure . Finally, we test our method on an economic network (i.e. the countries-products World Trade Web representation) and a social network (i.e. MovieLens , collecting the users' ratings of a list of movies). In both cases non-trivial communities are detected : 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.
R
which occur
occurring
A
as much as possible
D
as much as possible
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which implements a simple rule: in order for
according to which
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to be linked, the number of shared neighbors must be statistically significant. Naturally,
sharing a statistically-significant number of neighbors are linked. Since
D
the definition of
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:
,
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Exponential Random Graph
exponential random graph
R
can be straightforwardly obtained
is straightforwardly obtainable
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test and retaining only the statistically-significant links
testing procedure
R
in order to
we
R
, we analyze
on
R
the MovieLensdataset
MovieLens
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. In the first case,
: