Exponential random graph models with soft clique constraints

• URL: http://arxiv.org/abs/2608.30869v1
• Date: Mon, 31 Aug 2026 14:31:57 GMT
• Title: Exponential random graph models with soft clique constraints
• Abstract summary: We consider an exponential random graph model which gives higher probability to $G in mathbfG_n than to $H in mathbfG_n$ if $G$ has fewer $r$-cliques than $H$.<n>The degree to which graphs with fewer $r$-cliques are given higher probability is determined by a positive weight $w$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• On the Capacity Region of Individual Key Rates in Vector Linear Secure Aggregation [55.126702858312456]
  We show that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature.<n>Our results uncover the novel fact that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature.
  arXiv  Detail & Related papers  (2026-01-06T18:34:07Z)
• A Mathematical Theory of Top-$k$ Sparse Attention via Total Variation Distance [7.014801584517052]
  We develop a unified framework certified Top-$k$ attention truncation that quantifies error at both the distribution and output levels.<n>We show that the total-variation distance coincides with the discarded softmax tail mass and satisfies $mathrmTV(P,hat P)=1-e-mathrmTV(P,hat P)=1-e-mathrmTV(P,hat P)$.
  arXiv  Detail & Related papers  (2025-12-08T15:36:41Z)
• Approximating the operator norm of local Hamiltonians via few quantum states [53.16156504455106]
  Consider a Hermitian operator $A$ acting on a complex Hilbert space of $2n$.<n>We show that when $A$ has small degree in the Pauli expansion, or in other words, $A$ is a local $n$-qubit Hamiltonian.<n>We show that whenever $A$ is $d$-local, textiti.e., $deg(A)le d$, we have the following discretization-type inequality.
  arXiv  Detail & Related papers  (2025-09-15T14:26:11Z)
• Sharp Gap-Dependent Variance-Aware Regret Bounds for Tabular MDPs [54.28273395444243]
  We show that the Monotonic Value Omega (MVP) algorithm achieves a variance-aware gap-dependent regret bound of $$tildeOleft(left(sum_Delta_h(s,a)>0 fracH2 log K land mathttVar_maxtextc$.
  arXiv  Detail & Related papers  (2025-06-06T20:33:57Z)
• Improved convergence rate of kNN graph Laplacians [11.93971616098517]
  General class of $k$NN graph where the graph affinity is $W_ij = epsilon-d/2 . We prove the point-wise convergence of the $k$NN graph Laplacian to the limiting manifold operator.
  arXiv  Detail & Related papers  (2024-10-30T17:01:00Z)
• Efficient Continual Finite-Sum Minimization [52.5238287567572]
  We propose a key twist into the finite-sum minimization, dubbed as continual finite-sum minimization. Our approach significantly improves upon the $mathcalO(n/epsilon)$ FOs that $mathrmStochasticGradientDescent$ requires. We also prove that there is no natural first-order method with $mathcalOleft(n/epsilonalpharight)$ complexity gradient for $alpha 1/4$, establishing that the first-order complexity of our method is nearly tight.
  arXiv  Detail & Related papers  (2024-06-07T08:26:31Z)
• A note on estimating the dimension from a random geometric graph [2.3020018305241337]
  We study the problem of estimating the dimension $d$ of the underlying space when we have access to the adjacency matrix of the graph. We also show that, without any condition on the density, a consistent estimator of $d$ exists when $n r_nd to infty$ and $r_n = o(1)$.
  arXiv  Detail & Related papers  (2023-11-21T23:46:44Z)
• Detection of Dense Subhypergraphs by Low-Degree Polynomials [72.4451045270967]
  Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem. We consider detecting the presence of a planted $Gr(ngamma, n-alpha)$ subhypergraph in a $Gr(n, n-beta) hypergraph. Our results are already new in the graph case $r=2$, as we consider the subtle log-density regime where hardness based on average-case reductions is not known.
  arXiv  Detail & Related papers  (2023-04-17T10:38:08Z)
• Planted Bipartite Graph Detection [13.95780443241133]
  We consider the task of detecting a hidden bipartite subgraph in a given random graph. Under the null hypothesis, the graph is a realization of an ErdHosR'enyi random graph over $n$ with edge density $q$. Under the alternative, there exists a planted $k_mathsfR times k_mathsfL$ bipartite subgraph with edge density $p>q$.
  arXiv  Detail & Related papers  (2023-02-07T18:18:17Z)
• Random Graph Matching in Geometric Models: the Case of Complete Graphs [21.689343447798677]
  This paper studies the problem of matching two complete graphs with edge weights correlated through latent geometries. We derive an approximate maximum likelihood estimator, which provably achieves, with high probability, perfect recovery of $pi*$. As a side discovery, we show that the celebrated spectral algorithm of [Ume88] emerges as a further approximation to the maximum likelihood in the geometric model.
  arXiv  Detail & Related papers  (2022-02-22T04:14:45Z)
• Fast Graph Sampling for Short Video Summarization using Gershgorin Disc Alignment [52.577757919003844]
  We study the problem of efficiently summarizing a short video into several paragraphs, leveraging recent progress in fast graph sampling. Experimental results show that our algorithm achieves comparable video summarization as state-of-the-art methods, at a substantially reduced complexity.
  arXiv  Detail & Related papers  (2021-10-21T18:43:00Z)
• Exact Matching of Random Graphs with Constant Correlation [2.578242050187029]
  This paper deals with the problem of graph matching or network alignment for ErdHos--R'enyi graphs. It can be viewed as a noisy average-case version of the graph isomorphism problem.
  arXiv  Detail & Related papers  (2021-10-11T05:07:50Z)
• Near-Optimal SQ Lower Bounds for Agnostically Learning Halfspaces and ReLUs under Gaussian Marginals [49.60752558064027]
  We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. Our lower bounds provide strong evidence that current upper bounds for these tasks are essentially best possible.
  arXiv  Detail & Related papers  (2020-06-29T17:10:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.