Polynomial graph invariants from homomorphism numbers

We give a method of generating strongly polynomial sequences of graphs, i.e., sequences $(H_{\mathbf{k}})$ indexed by a multivariate parameter $\mathbf{k}=(k_1,\ldots, k_h)$ such that, for each fixed graph $G$, there is a multivariate polynomial $p(G;x_1,\ldots, x_h)$ such that the number of homomorphisms from $G$ to $H_{\mathbf{k}}$ is given by the evaluation $p(G;k_1,\ldots, k_h)$. A classical example is the sequence $(K_k)$ of complete graphs, for which ${\rm hom}(G,K_k)=P(G;k)$ is the evaluation of the chromatic polynomial at $k$. Our construction produces a large family of graph polynomials that includes the Tutte polynomial, the Averbouch-Godlin-Makowsky polynomial and the Tittmann-Averbouch-Makowsky polynomial. We also introduce a new graph parameter, the {\em branching core size} of a simple graph, related to how many involutive automorphisms with fixed points it has. We prove that a countable family of graphs of bounded branching core size (which in particular implies bounded tree-depth) is always contained in a finite union of strongly polynomial sequences.Comment: 40 pages, 12 figure

Sampling random graph homomorphisms and applications to network data analysis

A graph homomorphism is a map between two graphs that preserves adjacency relations. We consider the problem of sampling a random graph homomorphism from a graph $F$ into a large network $\mathcal{G}$. We propose two complementary MCMC algorithms for sampling a random graph homomorphisms and establish bounds on their mixing times and concentration of their time averages. Based on our sampling algorithms, we propose a novel framework for network data analysis that circumvents some of the drawbacks in methods based on independent and neigborhood sampling. Various time averages of the MCMC trajectory give us various computable observables, including well-known ones such as homomorphism density and average clustering coefficient and their generalizations. Furthermore, we show that these network observables are stable with respect to a suitably renormalized cut distance between networks. We provide various examples and simulations demonstrating our framework through synthetic networks. We also apply our framework for network clustering and classification problems using the Facebook100 dataset and Word Adjacency Networks of a set of classic novels.Comment: 51 pages, 33 figures, 2 table

From the Ising and Potts models to the general graph homomorphism polynomial

In this note we study some of the properties of the generating polynomial for homomorphisms from a graph to at complete weighted graph on $q$ vertices. We discuss how this polynomial relates to a long list of other well known graph polynomials and the partition functions for different spin models, many of which are specialisations of the homomorphism polynomial. We also identify the smallest graphs which are not determined by their homomorphism polynomials for $q=2$ and $q=3$ and compare this with the corresponding minimal examples for the $U$-polynomial, which generalizes the well known Tutte-polynomal.Comment: V2. Extended versio

Definable decompositions for graphs of bounded linear cliquewidth

We prove that for every positive integer $k$, there exists an $\text{MSO}_1$-transduction that given a graph of linear cliquewidth at most $k$ outputs, nondeterministically, some cliquewidth decomposition of the graph of width bounded by a function of $k$. A direct corollary of this result is the equivalence of the notions of $\text{CMSO}_1$-definability and recognizability on graphs of bounded linear cliquewidth.Comment: 39 pages, 5 figures. The conference version of the manuscript appeared in the proceedings of LICS 201

Regular and First Order List Functions

We define two classes of functions, called regular (respectively, first-order) list functions, which manipulate objects such as lists, lists of lists, pairs of lists, lists of pairs of lists, etc. The definition is in the style of regular expressions: the functions are constructed by starting with some basic functions (e.g. projections from pairs, or head and tail operations on lists) and putting them together using four combinators (most importantly, composition of functions). Our main results are that first-order list functions are exactly the same as first-order transductions, under a suitable encoding of the inputs; and the regular list functions are exactly the same as MSO-transductions

A formalisation of deep metamodelling

The final publication is available at Springer via http://dx.doi.org/10.1007/s00165-014-0307-xMetamodelling is one of the pillars of model-driven engineering, used for language engineering and domain modelling. Even though metamodelling is traditionally based on a two-metalevel approach, several researchers have pointed out limitations of this solution and proposed an alternative deep (also called multi-level) approach to obtain simpler system specifications. However, this approach currently lacks a formalisation that can be used to explain fundamental concepts such as deep characterisation, double linguistic/ontological typing and linguistic extension. This paper provides such a formalisation based on the Diagram Predicate Framework, and discusses its practical realisation in the metaDepth tool.This work was partially funded by the SpanishMinistry of Economy and Competitiveness (project “Go Lite” TIN2011- 24139)

Metric uniformization of morphisms of Berkovich curves

We show that the metric structure of morphisms $f\colon Y\to X$ between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton $\Gamma=(\Gamma_Y,\Gamma_X)$ of $f$, the sets $N_{f,\ge n}$ of points of $Y$ of multiplicity at least $n$ in the fiber are radial around $\Gamma_Y$ with the radius changing piecewise monomially along $\Gamma_Y$. In this case, for any interval $l=[z,y]\subset Y$ connecting a rigid point $z$ to the skeleton, the restriction $f|_l$ gives rise to a $profile$ piecewise monomial function $\varphi_y\colon [0,1]\to[0,1]$ that depends only on the type 2 point $y\in\Gamma_Y$. In particular, the metric structure of $f$ is determined by $\Gamma$ and the family of the profile functions $\{\varphi_y\}$ with $y\in\Gamma_Y^{(2)}$. We prove that this family is piecewise monomial in $y$ and naturally extends to the whole $Y^{\mathrm{hyp}}$. In addition, we extend the theory of higher ramification groups to arbitrary real-valued fields and show that $\varphi_y$ coincides with the Herbrand's function of $\mathcal{H}(y)/\mathcal{H}(f(y))$. This gives a curious geometric interpretation of the Herbrand's function, which applies also to non-normal and even inseparable extensions.Comment: second version, 28 page