[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85917-en":3,"doc-seo-85917-105":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},85917,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Pulse Graphs: Prime-Activated Boolean Dynamics on Directed Graphs","Pulse Graphs are synchronous Boolean dynamics on finite loopless directed graphs where a vertex becomes active at the next time step exactly when the count of its active in-neighbors is prime. The largest realizable attractor period L(n) on n vertices satisfies L(1..5) = 1,1,1,3,9 and the exponential growth is bounded by 2^n−3−1 ≤ L(n) ≤ 2^n−1 for n ≥ 5. Complete digraphs allow an exact update formula, attractor classification, orbit-time bounds, and attractor counting, with additional mean-field analysis for sparse random graphs.","arXiv :2607 . 10453v1 [math .CO] 11 Jul 2026  \nPulse Graphs: Prime-Activated Boolean Dynamics on Directed  \nGraphs  \nPakin Methawisal  \nMahidol University International College, Mahidol University [pakin. met@student. mahidol. edu](pakin. met@student. mahidol. edu)  \nJuly 12, 2026  \nAbstract  \nWe study synchronous Boolean dynamics on finite loopless directed graphs in which a vertex is active at the next time step exactly when its number of active in-neighbors is prime. We call these systems Pulse Graphs. Let L (n) denote the largest attractor period realizable on n vertices. Exhaustive enumeration gives  \nL(1),..., L(5) = 1, 1 , 1 , 3 , 9.  \nOur main result determines the exponential order of the maximum period:  \n2n−3 − 1 ≤ L (n) ≤ 2n − 1 (n ≥ 5) .  \nThe lower bound is obtained by implementing a maximal-length affine feedback register using prime-count logic gates. For n ≥ 6, the construction is loopless, has maximum in-degree five, and uses only O (n) edges.  \nFor complete directed graphs, we derive an exact update formula, classify all attractors as fixed points or complement two-cycles, prove that every orbit reaches its eventual attractor within three updates, and count the attractors explicitly. We also derive the activation probability under independent random inputs. For sparse random directed graphs, the associated prime-Poisson mean-field map undergoes a nondegenerate fold at  \nc∗ ≈ 3.824963, ρ∗ ≈ 0.368241 ,  \nwith local bistability immediately above the threshold.  \n1 Introduction  \nBoolean networks are finite-state dynamical systems in which interacting components take values in {0, 1} and update according to prescribed local rules. They have been studied as abstract models of regulatory, computational, and network dynamics since the early work of Kauffman [Kauffman, 1969] . Their behavior depends on the directed graph, the local update functions, and the update schedule.  \nIn this paper, we choose the prime numbers as the activating input counts. This gives a simple arithmetic rule that is nonmonotone: for example, a vertex receiving three active inputs becomes active, whereas one receiving four active inputs becomes inactive. More generally, increasing the number of active inputs can switch the output between active and inactive. Prime activation therefore provides a natural setting for studying how an elementary arithmetic condition can generate nontrivial global dynamics on a directed graph.  \nThe rule is totalistic because the next state of a vertex depends only on the number of its active in-neighbors, rather than on their identities. Totalistic rules on general graphs have been studied as extensions of cellular-automaton dynamics [Marr and H¨utt, 2009, Goles et al., 2021] .  \nPulse Graphs differ from traditional monotone threshold systems. For symmetric threshold networks updated synchronously, periodic attractors have period at most two [Goles and Olivos, 1980] . Kuhlman et al. obtained a similar restriction for synchronous bi-threshold systems, while showing that asynchronous bi-threshold systems may have longer periodic orbits [Kuhlmanet al., 2011] . The prime-activation rule is nonmonotone and does not satisfy the assumptions of those results. This leads to the extremal question that motivates the present paper: how large can an attractor period be on n vertices?  \nThe central question of this paper is how much dynamical complexity can be generated by this single arithmetic update rule, and how strongly that complexity depends on the underlying graph. We approach this question from two complementary directions. On general directed graphs, prime activation can implement Boolean operations and feedback mechanisms that produce very long periodic orbits. On highly symmetric graphs like complete graphs, however, the same rule can collapse to much simpler dynamics. This contrast shows that the behavior of Pulse Graphs is governed not only by the nonmonotonicity of prime activation, but also by the underlying ","cbCaif5GiyVa21Ma","https://ap.wps.com/l/cbCaif5GiyVa21Ma","pdf",330688,4,1,19,"English","en",105,"# Abstract\n# Introduction\n## Definition and setup\n## Fixed-point facts","[{\"question\":\"What defines a Pulse Graph and its update rule?\",\"answer\":\"A Pulse Graph is a finite loopless directed graph with Boolean states on vertices. At each synchronous update, a vertex becomes active at the next time step iff the number of its active in-neighbors is a prime number.\"},{\"question\":\"How large can the attractor period be on n vertices?\",\"answer\":\"Let L(n) be the largest attractor period realizable on n vertices. Exhaustive enumeration gives L(1..5)=1,1,1,3,9, and for n≥5 the paper proves 2^{n−3}−1 ≤ L(n) ≤ 2^{n−1}−1.\"},{\"question\":\"What additional results are obtained for complete directed graphs?\",\"answer\":\"For complete directed graphs, the work derives an exact update formula, classifies all attractors as fixed points or complement two-cycles, shows every orbit reaches its eventual attractor within three updates, and counts attractors explicitly.\"}]",1784207162,48,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"pulse-graphs-prime-activated-boolean-dynamics-on-directed-graphs","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/pulse-graphs-prime-activated-boolean-dynamics-on-directed-graphs/85917/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What defines a Pulse Graph and its update rule?","Question",{"text":75,"@type":76},"A Pulse Graph is a finite loopless directed graph with Boolean states on vertices. At each synchronous update, a vertex becomes active at the next time step iff the number of its active in-neighbors is a prime number.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How large can the attractor period be on n vertices?",{"text":80,"@type":76},"Let L(n) be the largest attractor period realizable on n vertices. Exhaustive enumeration gives L(1..5)=1,1,1,3,9, and for n≥5 the paper proves 2^{n−3}−1 ≤ L(n) ≤ 2^{n−1}−1.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional results are obtained for complete directed graphs?",{"text":84,"@type":76},"For complete directed graphs, the work derives an exact update formula, classifies all attractors as fixed points or complement two-cycles, shows every orbit reaches its eventual attractor within three updates, and counts attractors explicitly.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},"General","general"]