geometrylinear algebra
Is it possible to compare two vectors with different dimension?(e.g. the degree sequence of graphs with different graph size.)
I know it can be done by using some statistic measures, but will it make sense if I project the one with higher dimension to the lower dimensional space?
Yes, you get ellipses and spheres. There are also 3D equivalents to a parabola and hyperbola. There are actually two types of 4D conic surfaces: one is a single connected surface, and the other has two separate surfaces. Here are some animations I just made on the topic:
3D View of Conic Sections from $x^2+y^2=z^2$
2D View
4D Conic Sections of $x^2+y^2=z^2+w^2$
4D Conic Sections of $x^2+y^2+z^2=w^2$
The easiest way to do this is to extend $X = (x_1, x_2)$ to $X' = (X_1, x_2, 0)$ (and otherwise just always extend vectors with as many zeroes as you need). The resulting space is infinite dimensional, and it is in some sense the "smallest" infinite-dimensional vector space construction there is. Polynomials, for instance, work this way if you ignore multiplication.
Best Answer
For the problem at hand, one can define the degree sequence of a finite graph $G$ as the integer valued sequence $D(G)=(D_k(G))_{k\geqslant0}$ where, for every nonnegative integer $k$, $D_k(G)$ is the number of vertices in $G$ with degree $k$. Of course, $D_k(G)=0$ for every large enough $k$, but $N(G)$ lives in the same space for every finite graph $G$ hence, for every finite graphs $G$ and $H$, one can compare the sequences $D(G)$ and $D(H)$ by the method of one's choice.
Another possibility is to consider the sequence $N(G)=(N_k(G))_{k\geqslant0}$ defined by $N_k(G)=D_k(G)/|D(G)|$ for every nonnegative $k$, where $|D(G)|=\sum\limits_{k\ge0}D_k(G)$ is the total number of vertices of $G$. Thus $N(G)$ is a probability measure with finite support on the set $\{0,1,2,\ldots\}$ and one can likewise compare $N(G)$ and $N(H)$ by the method of one's choice.