We discuss three principal missing data mechanisms: network boundary specification (non-inclusion of actors or affiliations), survey non-response, and censoring by vertex degree (fixed choice design), examining their impact on the scientific collaboration network from the Los Alamos E-print Archive as well as random bipartite graphs.
Hence, our negative set was constructed by shuffling the links between the proteins in the positive set while keeping the degree fixed, i.e. the number of connections for each protein.
Some SigShrink graphs are plotted in Figure 2 for different values of the attenuation degree (fixed threshold ). Figure 2 Shapes of SigShrink functions for different values of the attenuation degree : for the continuous (blue) curve, for the dotted (red) curve, and for the dashed (magenta) curve.
Particularly, one may use it to establish a prior bound of solutions for nonlinear elliptic equations by the blow-up method, then various methods, such as topological degree, fix point theorems etc., can be used to obtain the existence of solutions for such problems; see for instance [1] and [2].
By advanced techniques, based on the Lefschetz-type fixed point theorems and sophisticated degree arguments (fixed point index techniques), we are able to treat not only the sole existence problems, but also to guarantee certain sorts of a weak stability called nonejectivity and essentiality.
The existence and multiplicity of positive solutions for (1.1) with (lambdaequiv1) have been extensively studied by many authors using topological degree theory, fixed point theorems, lower and upper solution methods, and critical point theory (see, for example, [2 17] and the references therein).
The method to show our main results is the topological degree and fixed point theory of nonlinear operator on lattice.
Many classical methods and tools, such as coincidence degree theory, fixed point theory and the method of upper and lower solutions, have been used to study them; see [11 15] and references therein.
By the topological degree and fixed point theory of superlinear operator on lattice (the definition of lattice will be given in Section 2), we obtain the existence of nontrivial solutions for the non-positive BVP (P) and the existence of positive solutions for the semipositive BVP (P).
Although number fields are fundamental objects in algebraic number theory, some basic facts are unknown, such as how many number fields there are for a fixed degree and fixed discriminant.
If the spline degree k and the Lagrange polynomial degree are fixed to 2 and 3 respectively, and the number of finite elements is taken as ntime−1, as the finite elements are situated between the measurement time points, the dimensions of this problem depend on the total number of internal knots ng, the number of time points ntime and the characteristics of the network.
