arXiv subjects

Yasmin Tousinejad

Publications and source records attributed to Yasmin Tousinejad.

4 recordsLinked to original sources

arXiv2026

Exponential random graph models with soft clique constraints

Let $r\geq3$ be fixed, and let $\mathbf{G}_n$ be the set of all simple graphs with vertex set $[n]=\{1,\ldots,n\}$. We consider an exponential random graph model which gives higher probability to $G \in \mathbf{G}_n$ than to $H \in \mathbf{G}_n$ if $G$ has fewer $r$-cliques than $H$. But all graphs in $\mathbf{G}_n$ have positive probability. The degree to which graphs with fewer $r$-cliques are given higher probability is determined by a positive weight $w$. We prove that, asymptotically almost surely as $n \to \infty$, a random graph from $\mathbf{G}_n$ has a vertex partition into $r-1$ parts of roughly equal size, the density of edges between the parts is close to $1/2$, and for every $\varepsilon > 0$ the density of edges within any part is less than $\varepsilon$. The asymptotic structural properties are independent of the weight $w$ as long as it is positive. We also extend the result to the context of several clique sizes, each one with its own weight.

math.CO↗

arXiv2026

Random coloured digraphs defined by a Markov logic network

A Markov Logic Network (MLN) is a probabilistic relational model used in Statistical Relational Artificial Intelligence for defining a probability distribution on the set of possible worlds with domain $D$ for an arbitrary finite domain $D$. An MLN consists of soft constraints with associated weights which are nonnegative real numbers. In this study we consider a language speaking about a property $P(x)$ and a relation $R(x, y)$. We consider an MLN for which every Boolean combination of $P(x)$ and $R(x, y)$ is a soft constraint (with associated weight). Let $n$ denote the size (cardinality) of the domain. We show that, for every choice of weights, if the weights are scaled by $1/n$ then, for every first-order sentence $φ$, the probability that $φ$ holds tends to either 0 or 1 as $n \to \infty$; that is, a 0-1 law for first-order logic holds. Morover, the limit probability does {\em not} depend on the weights. If we instead use the standard semantics of MLNs, in the case of which the weights are {\em not} scaled, then the limit behaviour is more complicated and {\em depends} on the weights. With unscaled weights we get 7 qualitatively different cases which depend on the weights. In some cases we have a 0-1 law for first-order logic, in some cases not, but we may still have a convergence law. The influence of the weights on the asymptotic probability of a first-order sentence may be in the form of a sudden ``phase transition'' from one of the 7 cases to another. The presence of a convergence law has positive implications for inference on large domains.

math.LO↗

arXiv2026

The stochastic block model has the overlap graph property for modularity

The overlap gap property (OGP) is a statement about the geometry of near-optimal solutions. Exhibiting OGP implies failure of a class of local algorithms; and has been observed to coincide with conjectured algorithmic limits in problems with statistical computational gap. We consider the Stochastic Block Model (SBM), where the graph has a planted partition with $k$ equal-size blocks which form the `communities', and where, for parameters $p>q$, vertices within the same community connect with probability $p$, while vertices in different communities connect with probability $q$, independently across pairs of vertices. Modularity--based clustering algorithms have become ubiquitous in applications. This article studies theoretical limits of local algorithms based on the modularity score on the SBM. We establish that modularity exhibits OGP on the SBM. This rules out a class of local algorithms based on modularity for recovery in the SBM, and shows slow mixing time for a related Markov Chain. Theoretically this is one of the few instances where OGP has been established for a `planted' model, as most such analyses to date consider the `null' model. As part of our analysis, we extend a result by Bickel and Chen 2009, who established that with high probability, the modularity optimal partition of SBM is $o(n)$ local moves away from the planted partition, where $n$ is the graph size. We show that, with high probability, any partition with modularity score sufficiently near the optimal value is close to the planted partition.

math.PR↗

arXiv2025

Random expansions of trees with bounded height

We consider a sequence $\mathbf{T} = (\mathcal{T}_n : n \in \mathbb{N}^+)$ of trees $\mathcal{T}_n$ where, for some $Δ\in \mathbb{N}^+$ every $\mathcal{T}_n$ has height at most $Δ$ and as $n \to \infty$ the minimal number of children of a nonleaf tends to infinity. We can view every tree as a (first-order) $τ$-structure where $τ$ is a signature with one binary relation symbol. For a fixed (arbitrary) finite and relational signature $σ\supseteq τ$ we consider the set $\mathbf{W}_n$ of expansions of $\mathcal{T}_n$ to $σ$ and a probability distribution $\mathbb{P}_n$ on $\mathbf{W}_n$ which is determined by a (parametrized/lifted) Probabilistic Graphical Model (PGM) $\mathbb{G}$ which can use the information given by $\mathcal{T}_n$. The kind of PGM that we consider uses formulas of a many-valued logic that we call $PLA^*$ with truth values in the unit interval $[0, 1]$. We also use $PLA^*$ to express queries, or events, on $\mathbf{W}_n$. With this setup we prove that, under some assumptions on $\mathbf{T}$, $\mathbb{G}$, and a (possibly quite complex) formula $φ(x_1, \ldots, x_k)$ of $PLA^*$, as $n \to \infty$, if $a_1, \ldots, a_k$ are vertices of the tree $\mathcal{T}_n$ then the value of $φ(a_1, \ldots, a_k)$ will, with high probability, be almost the same as the value of $ψ(a_1, \ldots, a_k)$, where $ψ(x_1, \ldots, x_k)$ is a ``simple'' formula the value of which can always be computed quickly (without reference to $n$), and $ψ$ itself can be found by using only the information that defines $\mathbf{T}$, $\mathbb{G}$ and $φ$. A corollary of this, subject to the same conditions, is a probabilistic convergence law for $PLA^*$-formulas.

cs.LO↗