Clustering Phase Transitions and Hysteresis: Pitfalls in Constructing Network Ensembles

Ensembles of networks are used as null models in many applications. However, simple null models often show much less clustering than their real-world counterparts. In this paper, we study a model where clustering is enhanced by means of a fugacity term as in the Strauss (or "triangle") model, but where the degree sequence is strictly preserved -- thus maintaining the quenched heterogeneity of nodes found in the original degree sequence. Similar models had been proposed previously in [R. Milo et al., Science 298, 824 (2002)]. We find that our model exhibits phase transitions as the fugacity is changed. For regular graphs (identical degrees for all nodes) with degree k > 2 we find a single first order transition. For all non-regular networks that we studied (including Erdos - Renyi and scale-free networks) we find multiple jumps resembling first order transitions, together with strong hysteresis. The latter transitions are driven by the sudden emergence of "cluster cores": groups of highly interconnected nodes with higher than average degrees. To study these cluster cores visually, we introduce q-clique adjacency plots. We find that these cluster cores constitute distinct communities which emerge spontaneously from the triangle generating process. Finally, we point out that cluster cores produce pitfalls when using the present (and similar) models as null models for strongly clustered networks, due to the very strong hysteresis which effectively leads to broken ergodicity on realistic time scales.Comment: 13 pages, 11 figure

Recursion Schemes and Logical Reflection

International audienceLet R be a class of generators of node-labelled infinite trees, and L be a logical language for describing correctness properties of these trees. Given r in R and phi in L, we say that r_phi is a phi-reflection of r just if (i) r and r_phi generate the same underlying tree, and (ii) suppose a node u of the tree t(r) generated by r has label f, then the label of the node u of t(r_phi) is f* if u in t(r) satisfies phi; it is f otherwise. Thus if t(r) is the computation tree of a program r, we may regard r_phi as a transform of r that can internally observe its behaviour against a specification phi. We say that R is (constructively) reflective w.r.t. L just if there is an algorithm that transforms a given pair (r,phi) to r_phi. In this paper, we prove that higher-order recursion schemes are reflective w.r.t. both modal mu-calculus and monadic second order (MSO) logic. To obtain this result, we give the first characterisation of the winning regions of parity games over the transition graphs of collapsible pushdown automata (CPDA): they are regular sets defined by a new class of automata. (Order-n recursion schemes are equi-expressive with order-n CPDA for generating trees.) As a corollary, we show that these schemes are closed under the operation of MSO-interpretation followed by tree unfolding a la Caucal

Recursion Schemes and Logical Reflection

International audienceLet R be a class of generators of node-labelled infinite trees, and L be a logical language for describing correctness properties of these trees. Given r in R and phi in L, we say that r_phi is a phi-reflection of r just if (i) r and r_phi generate the same underlying tree, and (ii) suppose a node u of the tree t(r) generated by r has label f, then the label of the node u of t(r_phi) is f* if u in t(r) satisfies phi; it is f otherwise. Thus if t(r) is the computation tree of a program r, we may regard r_phi as a transform of r that can internally observe its behaviour against a specification phi. We say that R is (constructively) reflective w.r.t. L just if there is an algorithm that transforms a given pair (r,phi) to r_phi. In this paper, we prove that higher-order recursion schemes are reflective w.r.t. both modal mu-calculus and monadic second order (MSO) logic. To obtain this result, we give the first characterisation of the winning regions of parity games over the transition graphs of collapsible pushdown automata (CPDA): they are regular sets defined by a new class of automata. (Order-n recursion schemes are equi-expressive with order-n CPDA for generating trees.) As a corollary, we show that these schemes are closed under the operation of MSO-interpretation followed by tree unfolding a la Caucal

Automatic enumeration of regular objects

We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer Sequence

Enumeration of labelled 4-regular planar graphs

We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps

On the expected number of perfect matchings in cubic planar graphs

A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.Comment: 19 pages, 4 figure