相关论文: Breaking of ensemble equivalence for perturbed Erd…
In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales…
Breaking of ensemble equivalence between the microcanonical ensemble and the canonical ensemble may occur for random graphs whose size tends to infinity, and is signaled by a non-zero specific relative entropy of the two ensembles. In [3]…
For random systems subject to a constraint, the microcanonical ensemble requires the constraint to be met by every realisation ("hard constraint"), while the canonical ensemble requires the constraint to be met only on average ("soft…
Breaking of equivalence between the microcanonical ensemble and the canonical ensemble, describing a large system subject to hard and soft constraints, respectively, was recently shown to occur in large random graphs. Hard constraints must…
Two ensembles are often used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when…
In this paper we consider the Erd\H{o}s-R\'enyi random graph in the sparse regime in the limit as the number of vertices $n$ tends to infinity. We are interested in what this graph looks like when it contains many triangles, in two…
For a random graph subject to a topological constraint, the microcanonical ensemble requires the constraint to be met by every realisation of the graph (`hard constraint'), while the canonical ensemble requires the constraint to be met only…
The entropy of random graph ensembles has gained widespread attention in the field of graph theory and network science. We consider microcanonical ensembles of simple graphs with prescribed degree sequences. We demonstrate that the…
We consider an inhomogeneous Erd\H{o}s-R\'enyi random graph ensemble with exponentially decaying random disconnection probabilities determined by an i.i.d. field of variables with heavy tails and infinite mean associated to the vertices of…
The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of…
We study the graph alignment problem over two independent Erd\H{o}s-R\'enyi graphs on $n$ vertices, with edge density $p$ falling into two regimes separated by the critical window around $p_c=\sqrt{\log n/n}$. Our result reveals an…
When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations.…
We study the intersection of a random geometric graph with an Erd\H{o}s-R\'enyi graph. Specifically, we generate the random geometric graph $G(n, r)$ by choosing $n$ points uniformly at random from $D=[0, 1]^2$ and joining any two points…
An $n$-vertex graph is called $C$-Ramsey if it has no clique or independent set of size $C\log_2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge-statistics in Ramsey graphs, in particular obtaining very…
Networks (graphs) permeate scientific fields such as biology, social science, economics, etc. Empirical studies have shown that real-world networks are often heterogeneous, that is, the degrees of nodes do not concentrate on a number.…
A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large…
We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together…
What does an Erdos-Renyi graph look like when a rare event happens? This paper answers this question when p is fixed and n tends to infinity by establishing a large deviation principle under an appropriate topology. The formulation and…
It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the…
This paper examines a model involving two dynamic Erd\H{o}s-R\'enyi random graphs that evolve in parallel, with edges in each graph alternating between being present and absent according to specified on- and off-time distributions. A key…