The principle of least action for random graphs

• URL: http://arxiv.org/abs/2507.01468v1
• Date: Wed, 02 Jul 2025 08:29:28 GMT
• Title: The principle of least action for random graphs
• Authors: Ioannis Kleftogiannis, Ilias Amanatidis,
• Abstract summary: We study the statistical properties of the physical action $S$ for random graphs.<n>We calculate the Lagrangian of the degree field by using a lattice quantum field theory approach.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the statistical properties of the physical action $S$ for random graphs, by treating the number of neighbors at each vertex of the graph (degree), as a scalar field. For each configuration (run) of the graph we calculate the Lagrangian of the degree field by using a lattice quantum field theory(LQFT) approach. Then the corresponding action is calculated by integrating the Lagrangian over all the vertices of the graph. We implement an evolution mechanism for the graph by removing one edge per a fundamental quantum of time, resulting in different evolution paths based on the run that is chosen at each evolution step. We calculate the action along each of these evolution paths, which allows us to calculate the probability distribution of $S$. We find that the distribution approaches the normal(Gaussian) form as the graph becomes denser, by adding more edges between its vertices. The maximum of the probability distribution of the action corresponds to graph configurations whose spacing between the values of $S$ becomes zero $\Delta S=0$, corresponding to the least-action (Hamilton) principle, which gives the path that the physical system follows classically. In addition, we calculate the fluctuations(variance) of the degree field showing that the graph configurations corresponding to the maximum probability of $S$, which follow the Hamilton's principle, have a balanced structure between regular and irregular graphs.

Related papers

• Scalable Generative Modeling of Weighted Graphs [1.0923877073891446]
  We develop an autoregressive model BiGG-E that learns a joint distribution over weighted graphs while still exploiting sparsity to generate a weighted graph with $n$ nodes and $m$ edges in $O((n + m)log n)$ time.<n> Simulation studies and experiments on a variety of benchmark datasets demonstrate that BiGG-E best captures distributions over weighted graphs while remaining scalable and computationally efficient.
  arXiv  Detail & Related papers  (2025-07-30T21:28:28Z)
• Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions [84.01980526069075]
  We study the generalization capabilities of geometric graph neural networks (GNNs)<n>We prove a generalization gap between the optimal empirical risk and the optimal statistical risk of this GNN.<n>We verify this theoretical result with experiments on multiple real-world datasets.
  arXiv  Detail & Related papers  (2024-09-08T18:55:57Z)
• Fine-grained Graph Rationalization [51.293401030058085]
  We propose fine-grained graph rationalization (FIG) for graph machine learning.<n>Our idea is driven by the self-attention mechanism, which provides rich interactions between input nodes.<n>Our experiments involve 7 real-world datasets, and the proposed FIG shows significant performance advantages compared to 13 baseline methods.
  arXiv  Detail & Related papers  (2023-12-13T02:56:26Z)
• Graph Neural Networks with a Distribution of Parametrized Graphs [27.40566674759208]
  We introduce latent variables to parameterize and generate multiple graphs. We obtain the maximum likelihood estimate of the network parameters in an Expectation-Maximization framework.
  arXiv  Detail & Related papers  (2023-10-25T06:38:24Z)
• Analysis and Approximate Inference of Large Random Kronecker Graphs [4.417282202068703]
  We show that the adjacency of a large random Kronecker graph can be decomposed. We propose a denoise-and-solve'' approach to infer the key graph parameters.
  arXiv  Detail & Related papers  (2023-06-14T13:09:38Z)
• Graph Fourier MMD for Signals on Graphs [67.68356461123219]
  We propose a novel distance between distributions and signals on graphs. GFMMD is defined via an optimal witness function that is both smooth on the graph and maximizes difference in expectation. We showcase it on graph benchmark datasets as well as on single cell RNA-sequencing data analysis.
  arXiv  Detail & Related papers  (2023-06-05T00:01:17Z)
• Graph Condensation via Receptive Field Distribution Matching [61.71711656856704]
  This paper focuses on creating a small graph to represent the original graph, so that GNNs trained on the size-reduced graph can make accurate predictions. We view the original graph as a distribution of receptive fields and aim to synthesize a small graph whose receptive fields share a similar distribution.
  arXiv  Detail & Related papers  (2022-06-28T02:10:05Z)
• Non-separable Spatio-temporal Graph Kernels via SPDEs [69.4678086015418]
  A lack of justified graph kernels for principled-temporal modelling has held back their use in graph problems.<n>We leverage a link between partial differential equations (SPDEs) and onsepa-temporal graphs, introduce a framework for deriving graph kernels via SPDEs.<n>We show that by providing novel tools for GP modelling on graphs, we outperform pre-existing graph kernels in real-world applications.
  arXiv  Detail & Related papers  (2021-11-16T14:53:19Z)
• Settling the Sharp Reconstruction Thresholds of Random Graph Matching [19.54246087326288]
  We study the problem of recovering the hidden correspondence between two edge-correlated random graphs. For dense graphs with $p=n-o(1)$, we prove that there exists a sharp threshold. For sparse ErdHos-R'enyi graphs with $p=n-Theta(1)$, we show that the all-or-nothing phenomenon no longer holds.
  arXiv  Detail & Related papers  (2021-01-29T21:49:50Z)
• Random Geometric Graphs on Euclidean Balls [2.28438857884398]
  We consider a latent space model for random graphs where a node $i$ is associated to a random latent point $X_i$ on the Euclidean unit ball. For certain link functions, the model considered here generates graphs with degree distribution that have tails with a power-law-type distribution.
  arXiv  Detail & Related papers  (2020-10-26T17:21:57Z)
• Asymptotic entropy of the Gibbs state of complex networks [68.8204255655161]
  The Gibbs state is obtained from the Laplacian, normalized Laplacian or adjacency matrices associated with a graph. We calculated the entropy of the Gibbs state for a few classes of graphs and studied their behavior with changing graph order and temperature. Our results show that the behavior of Gibbs entropy as a function of the temperature differs for a choice of real networks when compared to the random ErdHos-R'enyi graphs.
  arXiv  Detail & Related papers  (2020-03-18T18:01:28Z)
• Block-Approximated Exponential Random Graphs [77.4792558024487]
  An important challenge in the field of exponential random graphs (ERGs) is the fitting of non-trivial ERGs on large graphs. We propose an approximative framework to such non-trivial ERGs that result in dyadic independence (i.e., edge independent) distributions. Our methods are scalable to sparse graphs consisting of millions of nodes.
  arXiv  Detail & Related papers  (2020-02-14T11:42:16Z)

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.