Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Differential Calculus on Graphon Space   Recently, the theory of dense graph limits has received attention from
multiple disciplines including graph theory, computer science, statistical
physics, probability, statistics, and group theory. In this paper we initiate
the study of the general structure of differentiable graphon parameters $F$. We
derive consistency conditions among the higher G\^ateaux derivatives of $F$
when restricted to the subspace of edge weighted graphs $\mathcal{W}_{\bf p}$.
Surprisingly, these constraints are rigid enough to imply that the multilinear
functionals $\Lambda: \mathcal{W}_{\bf p}^n \to \mathbb{R}$ satisfying the
constraints are determined by a finite set of constants indexed by isomorphism
classes of multigraphs with $n$ edges and no isolated vertices. Using this
structure theory, we explain the central role that homomorphism densities play
in the analysis of graphons, by way of a new combinatorial interpretation of
their derivatives. In particular, homomorphism densities serve as the monomials
in a polynomial algebra that can be used to approximate differential graphon
parameters as Taylor polynomials. These ideas are summarized by our main
theorem, which asserts that homomorphism densities $t(H,-)$ where $H$ has at
most $N$ edges form a basis for the space of smooth graphon parameters whose
$(N+1)$st derivatives vanish. As a consequence of this theory, we also extend
and derive new proofs of linear independence of multigraph homomorphism
densities, and characterize homomorphism densities. In addition, we develop a
theory of series expansions, including Taylor's theorem for graph parameters
and a uniqueness principle for series. We use this theory to analyze questions
raised by Lov\'asz, including studying infinite quantum algebras and the
connection between right- and left-homomorphism densities.

23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci