Such colored matrix can be considered as a factorization of the complete

bipartite graph [K.sub.n,m].

Based on the theory of

bipartite graph [18], we construct the Sample/ Grain-Size bipartite weighted network model which can objectively reflect the association relationships between sediment samples and grain sizes.

For a complete

bipartite graph G = (V, E, w) with n vertices, we have

User-location

bipartite graph is established via the check-in data.

For example, consider a complete

bipartite graph [K.sub.r,s] with partite sets U and V.

In the proposed methodology, users are able to provide constraints on the target mode, specifying the multiple connecting relationships in each

bipartite graph. Our goal is to improve the quality of community structure by multiview learning in all modes of nodes and linking.

A

bipartite graph is called semiregular if each vertex in the same part of a bipartition has the same degree.

Therefore, when the topology of the system is a

bipartite graph, the matrix D + A associated with the system can be rewritten as [mathematical expression not reproducible].

However, it can be observed that, given the constraints on user association, the optimization problem is equivalent to an optimal matching problem in the

bipartite graph theory, which can then be solved based on the classical algorithms such as K-M algorithm [24].

(iv) For any

bipartite graph [K.sub.r,r+1], r, even, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For instance, the complete

bipartite graph [K.sub.m,n] and wheel graph [W.sub.n] are 2-distance balanced, but not distance balanced.

Such a matrix can efficiently be represented by a

bipartite graph which consists of bit and check nodes corresponding to columns and rows in H.