## Metrics for sparse graphs

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Bollobas, B.

Riordan, O.

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Recently, Bollob\'as, Janson and Riordan introduced a very general family of
random graph models, producing inhomogeneous random graphs with $\Theta(n)$
edges. Roughly speaking, there is one model for each {\em kernel}, i.e., each
symmetric measurable function from $[0,1]^2$ to the non-negative reals,
although the details are much more complicated. A different connection between
kernels and random graphs arises in the recent work of Borgs, Chayes, Lov\'asz,
S\'os, Szegedy and Vesztergombi. They introduced several natural metrics on
dense graphs (graphs with $n$ vertices and $\Theta(n^2)$ edges), showed that
these metrics are equivalent, and gave a description of the completion of the
space of all graphs with respect to any of these metrics in terms of {\em
graphons}, which are essentially bounded kernels. One of the most appealing
aspects of this work is the message that sequences of inhomogeneous
quasi-random graphs are in a sense completely general: any sequence of dense
graphs contains such a subsequence.
Our aim here is to briefly survey these results, and then to investigate to
what extent they can be generalized to graphs with $o(n^2)$ edges. Although
many of the definitions extend in a simple way, the connections between the
various metrics, and between the metrics and random graph models, turn out to
be much more complicated than in the dense case. We shall prove many partial
results, and state even more conjectures and open problems, whose resolution
would greatly enhance the currently rather unsatisfactory theory of metrics on
sparse graphs. This paper deals mainly with graphs with $o(n^2)$ but
$\omega(n)$ edges: a companion paper [arXiv:0812.2656] will discuss the (more
problematic still) case of {\em extremely sparse} graphs, with O(n) edges.

Comment: 83 pages, 1 figure. References updated and one corrected

Comment: 83 pages, 1 figure. References updated and one corrected

##### Keywords

Mathematics - Combinatorics, Mathematics - Probability, 05C99, 05C80