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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?

(Sorry that I don't know what tag should I tag so I just simply pick one)

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1 Answer 1

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.

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In essence, you're projecting the one with lower dimension to the higher dimensional space, the opposite of what OP considered doing! –  Gerry Myerson Nov 1 '11 at 12:02
    
Maybe. And I do not know what it is the OP describes as project(ing) the one with higher dimension to the lower dimensional space. –  Did Nov 1 '11 at 13:03
    
Indeed, that clause is open to many interpretations. –  Gerry Myerson Nov 1 '11 at 21:43

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