[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81479-en":3,"doc-seo-81479-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81479,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Size-varying Reversible Causal Graph Dynamics","Reversible nearest-neighbours causal graph dynamics are studied under the specific challenge of allowing the network size to change. Although reversibility and bijectivity suggest that deleting nodes may threaten invertibility, bijective graph transformations can be non-vertex-preserving. Earlier work ruled out such size variation negatively for multiple reasons. This paper establishes reversible local node creation and destruction in three relaxed settings, proves their equivalence for robustness, and supports the model with links to reversible computing and discrete quantum-gravity formalisms.","arXiv : 1805 . 10330v4 [ cs .DM] 10 Jul 2026  \nSize-varying reversible causal graph dynamics  \nPablo Arrighia , Am´elia Durbecb,, Aur´elien Emmanuelc  \na Universit´e Paris-Saclay, Inria, CNRS, LMF, Gif-sur-Yvette, 91190, France b Universit´e Paris-Saclay, CEA, List, Palaiseau, 91120, France c Universit´e d’Orl´eans, LIFO EA 4022, Orl´eans, 45067, France  \nAbstract  \nConsider a network that evolves according to a reversible, nearest neighbours dynamics. Is the dynamics allowed to vary the size of the network? On the one hand it seems that, being the principal carriers of information, nodes cannot be destroyed without jeopardising bijectivity. On the other hand, there are plenty of bijective functions from the set of graphs to the set of graphs that are non-vertex-preserving. The question has been settled negatively—for three different reasons. Yet, in this paper we do obtain reversible local node creation/destruction—in three relaxed settings, whose equivalence we prove for robustness. We motivate our work both by theoretical computer science considerations (reversible computing, cellular automata extensions) and theoretical physics concerns (basic formalisms towards discrete quantum gravity) .  \nKeywords: Reversible Causal Graph Dynamics, Reversible Cellular  \nAutomata, Network Growth Dynamics, Reversibility, Invertible, One-to-one  \n1. Introduction  \nCellular Automata (CA) consist in a Zn grid of identical cells, each of which may take a state in Σ . Thus the configurations are in ΣZn . The next state of a cell is given by applying a fixed local rule f to the cell and its neighbours, synchronously and homogeneously across space. CA thus have a number of physics-like symmetries: shift-invariance (the dynamics acts everywhere and everywhen the same) and causality (information has abounded speed of propagation) . They constitute one of the most established models of computation that accounts for Euclidean space: they are widely used to model spatially-dependent computational problems (self-replicating  \nPreprint submitted to Elsevier July 13, 2026  \nmachines, synchronization...), and multi-agents phenomena (traffic jams, demographics...) . But their origin lies in Physics, where they are constantly used to model waves or particles ([e.g. as](e.g. as) numerical schemes for Partial Differential Equations) .  \nSince both quantum and classical mechanics are reversible, it was natural to endow CA with this other, physics-like symmetry. The study of Reversible CA (RCA) was further motivated by the promise of lower energy consumption in reversible computation. RCA have turned out to have an elegant mathematical theory, which relies on a topological characterization in order to prove for instance that the inverse of a CA is a CA [23]—which clearly is non-trivial due to [24] . Another fundamental property of RCA is that they can be expressed as a finite-depth circuits of local reversible permutations or ‘blocks’[25, 26, 17] .  \nCausal Graph Dynamics (CGD) [1 , 3 , 9 , 30 , 29] are a twofold extension of CA. First, the underlying grid is extended to arbitrary bounded-degree graphs. Informally, this means that each vertex of a graph G may take a state among a set Σ, so that configurations are in ΣV (G), whereas edges dictate the locality of the evolution: the next state of a vertex v depends only upon the subgraph Gru induced by the vertices lying at graph distance at most r of u. Second, the graph itself is allowed to evolve over time. Informally, this means that configurations are in the union of ΣV (G) for every possible bounded-degree graph G, i.e. SG ΣV (G) . This leads to a model where the local rule f is applied synchronously and homogeneously on every possible sub-disk of the input graph, thereby producing small patches of the output graphs, whose union constitutes the output graph. Figure 1 illustrates the concept. CGD were motivated by the countless situations featuring nearest-neighbours interactions with time-varying neighbourh","cbCaimwHsRjir5Tr","https://ap.wps.com/l/cbCaimwHsRjir5Tr","pdf",663512,2,1,59,"English","en",105,"# Introduction\n## Cellular Automata and Reversibility\n## Causal Graph Dynamics\n## Reversible Causal Graph Dynamics and Motivation","[{\"question\":\"Why does allowing node deletion or creation threaten reversibility in reversible causal graph dynamics?\",\"answer\":\"Reversibility relies on bijectivity, and removing nodes can jeopardize the one-to-one correspondence required for an inverse global evolution.\"},{\"question\":\"What does the paper achieve regarding size-varying reversible dynamics?\",\"answer\":\"It constructs reversible local node creation/destruction in three relaxed settings and proves the equivalence of these settings for robustness.\"},{\"question\":\"What are the motivations behind this work?\",\"answer\":\"The study is motivated both by theoretical computer science topics such as reversible computing and cellular automata extensions, and by theoretical physics interests like discrete quantum-gravity-related 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does allowing node deletion or creation threaten reversibility in reversible causal graph dynamics?","Question",{"text":75,"@type":76},"Reversibility relies on bijectivity, and removing nodes can jeopardize the one-to-one correspondence required for an inverse global evolution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the paper achieve regarding size-varying reversible dynamics?",{"text":80,"@type":76},"It constructs reversible local node creation/destruction in three relaxed settings and proves the equivalence of these settings for robustness.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the motivations behind this work?",{"text":84,"@type":76},"The study is motivated both by theoretical computer science topics such as reversible computing and cellular automata extensions, and by theoretical physics interests like discrete quantum-gravity-related 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