Branching Network Model of Observable Influence and Latent Source

Branching Network Model of Observable Influence and Latent Source | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Branching Network Model of Observable Influence and Latent Source Yonten Gyatso Lama This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8669960/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We present a branching network model that distinguishes between observable influence and a latent original source. Influence is divided equally across generations, producing exponential dilution, while the original source remains invariant and latent. As branching depth increases, observable traces of the origin diminish and are cognitively forgotten, yet they remain structurally implicit. This separation between structural persistence and perceptual visibility provides a conceptual framework for understanding how small-world networks maintain connectivity to their origins without explicit awareness. Figures Figure 1 Figure 2 Introduction Understanding how influence spreads through networks requires distinguishing between observable effects and hidden structural relationships. In many real-world systems, the observable impact of an initial source diminishes as it propagates through successive generations, even though the source continues to shape the network’s architecture. To formalize this process, we propose a branching network model in which influence is divided equally among descendants, producing exponential dilution of observable effects while preserving a latent connection to the original source. This distinction between observable influence and latent structural persistence provides a framework for understanding how networks retain long-range connectivity and memory traces without requiring direct awareness of their origins. Previous studies of network dynamics have primarily focused on observable propagation patterns, often overlooking hidden structural connections that preserve information across generations. By explicitly separating observable influence from latent persistence, our model explains how networks can maintain memory of their origins even when individual traces appear cognitively forgotten. In the following sections, we describe the branching network model, analyze the dynamics of influence dilution, and discuss implications for connectivity, memory, and small-world phenomena in real-world networks. Methods We construct a branching network in which each node represents an agent influenced by a single parent, and influence is equally divided among all offspring nodes at each generation. Let b denote the branching factor and d the network depth. Influence from the original source is represented by a unit value of 1, which is divided equally among all offspring at each branching event, producing exponential decay of observable influence with increasing depth. The original source node remains invariant and latent, serving as a structural anchor for the network. Let I_g represent the observable influence at generation g. Influence is divided equally among b offspring at each step, giving the recursive relation: I_0 = 1, I_g = I_{g-1} / b, g > = 1 which simplifies to I_g = 1 / b^g. Figure 1 illustrates a branching network with b = 2 over d = 4 generations. The original source node is shown at the top (latent and unobserved), while observable influence is represented at each descendant node. Node color intensity corresponds to the magnitude of I_g, demonstrating exponential dilution across generations. Despite the apparent weakening of influence, all nodes maintain a structural connection to the original source. This model captures the distinction between cognitive perception and structural persistence. As influence propagates and divides among descendants, observable traces diminish exponentially, mirroring how human cognition forgets distant origins. Nevertheless, the latent source ensures the network retains full connectivity, illustrating how structural memory persists independently of perceptual awareness. Results We simulated branching networks with branching factors b = 2, 3, 4 and depths d = 5, 6, 7. Observable influence I_g decreased exponentially with generation number g, consistent with I_g = 1 / b^g. Even at moderate depths, influence on descendant nodes became negligible, while the latent source maintained full structural connectivity to all nodes. These results confirm that the network’s architecture preserves links to the origin despite the apparent disappearance of influence in observable measures. These findings have direct implications for real-world networks, where connections to original sources are often imperceptible yet structurally preserved. In small-world networks, short path lengths and high clustering allow influence to propagate widely while individual traces dilute beyond conscious perception. Our model provides a conceptual explanation for how networks retain memory of their origins and ensure robust connectivity, even when observable effects appear fragmented or forgotten. Discussion Our branching network model highlights a fundamental distinction between observable influence and latent structural persistence. By formalizing how influence dilutes across generations while the original source remains structurally intact, we provide a framework for understanding cognitive forgetting in networks and the preservation of long-range connectivity. This perspective applies to a variety of real-world systems, including social networks, information diffusion, and neural circuits, where origins may be imperceptible yet still exert influence. The model assumes equal division of influence among offspring and a fixed branching factor, which may not hold in all empirical networks. Future work could incorporate heterogeneous branching, stochastic influence, or feedback loops, enabling more realistic simulations. Overall, the separation between perceptual visibility and structural persistence offers a novel lens for studying how networks maintain connections to their origins, providing both theoretical insight and practical relevance for network analysis and cognitive science. Conclusion We introduced a branching network model distinguishing observable influence from latent structural persistence. While influence dilutes exponentially across generations, the original source remains structurally connected to all nodes. This framework explains how networks retain memory of their origins without explicit awareness, offering insights into cognitive forgetting, small-world connectivity, and information propagation in complex systems. By highlighting the interplay between perceptual visibility and structural persistence, this work provides a foundation for future studies on the dynamics of influence and memory in both natural and artificial networks. Declarations Author Contribution Yonten Gyatso Lama conceived the study, developed the branching network model, performed all simulations and analyses, created all figures, and wrote the manuscript. Data Availability Author Contribution: Yonten Gyatso Lama conceived the study, developed the branching network model, performed all simulations and analyses, created all figures, and wrote the manuscript.Competing Interests: The author declares no competing financial or non-financial interests.Third-Party Material: The author confirms that all material in this manuscript and supplementary files is original and created by the author. No third-party material is included.Research Data: Yes, the study generated original simulation data, which is included as supplementary material and available upon request. References Watts DJ, Strogatz SH (1998) Collective dynamics of 'small-world' networks. Nature 393(6684):440–442 Newman MEJ (2010) Networks: An Introduction. Oxford University Press Barabási A-L, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512 Bollobás B (2001) Random Graphs. Cambridge University Press Kempe D, Kleinberg J, Tardos É (2003) Maximizing the spread of influence through a social network. Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 137–146 Cowan N, Saults JS (2013) Working memory capacity, attention, and long-term memory. Curr Dir Psychol Sci 22(2):133–138 Latora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87(19):198701 Tables Table 1: Observable Influence Across Generations Generation (g) b = 2 b = 3 b = 4 0 1.0 1.0 1.0 1 0.5 0.333 0.25 2 0.25 0.111 0.0625 3 0.125 0.037 0.0156 4 0.0625 0.012 0.0039 5 0.0313 0.004 0.001 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8669960","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":580269546,"identity":"5202065d-b9a6-4c47-b2a3-ae8e019be3ed","order_by":0,"name":"Yonten Gyatso Lama","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/0lEQVRIiWNgGAWjYPACCyBmP3DgQwWQZmZuIKScEahCAkjzJB6ccQakhZFoLQzGh3nbYAJ4gMHx3uMPPu6QkNdtP5BwgHdebTR/O1DLj4ptuLWcOZfYOPOMhOG2M4kHDkhuO5474zBjA2PPmdu4tdzIMWzmbZNg3HYgIeGA4bZjuQ1ALcyMbQS0/G2TsN92/oHBgcQ5x3LnE6WFsU0icduNBIMDBxtqcjcQ0iJ55ozhzN42ieRtN94kHGw4diB3I1DLQXx+4TveY/DhZ5uN7bbz6Yc//6mpy513/vDBBz8qcGtROIDKPwwmD2CoQwLyDaj8OnyKR8EoGAWjYIQCAOefaq73toNNAAAAAElFTkSuQmCC","orcid":"","institution":"Independent researcher","correspondingAuthor":true,"prefix":"","firstName":"Yonten","middleName":"Gyatso","lastName":"Lama","suffix":""}],"badges":[],"createdAt":"2026-01-22 12:55:26","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8669960/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8669960/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":101297826,"identity":"4e7c46bc-de42-4be2-a0d5-81e05d2c04a9","added_by":"auto","created_at":"2026-01-28 09:28:56","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":431435,"visible":true,"origin":"","legend":"\u003cp\u003eBranching network with branching factor b = 2 and depth d = 4. Node color intensity corresponds to observable influence I_g = 1/b^g, illustrating exponential dilution across generations. The original source remains latent while structurally connected to all nodes.\u003c/p\u003e","description":"","filename":"88F5BD297C7B4885B7DB41B835F53545.png","url":"https://assets-eu.researchsquare.com/files/rs-8669960/v1/4ca6f18f3ee89cfa3cb09af1.png"},{"id":101250480,"identity":"0fa2c0bc-fbe8-4c35-9c3d-266b6670d50e","added_by":"auto","created_at":"2026-01-27 17:30:22","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":76350,"visible":true,"origin":"","legend":"\u003cp\u003eExponential decay of observable influence I_g across generations g for branching factors b = 2, 3, and 4 (logarithmic scale). Higher branching factors produce more rapid dilution of observable influence.\u003c/p\u003e","description":"","filename":"B34FFD5B77154A93BDE7F616776FC9D5.png","url":"https://assets-eu.researchsquare.com/files/rs-8669960/v1/bf22d02703568e93009e5896.png"},{"id":101301766,"identity":"17c160aa-b1f6-4907-aec5-2747d6a91ce4","added_by":"auto","created_at":"2026-01-28 09:52:28","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":663038,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8669960/v1/389f5b01-7208-4d7d-85f9-829c6c370a01.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Branching Network Model of Observable Influence and Latent Source","fulltext":[{"header":"Introduction","content":"\u003cp\u003eUnderstanding how influence spreads through networks requires distinguishing between observable effects and hidden structural relationships. In many real-world systems, the observable impact of an initial source diminishes as it propagates through successive generations, even though the source continues to shape the network\u0026rsquo;s architecture.\u003c/p\u003e \u003cp\u003eTo formalize this process, we propose a branching network model in which influence is divided equally among descendants, producing exponential dilution of observable effects while preserving a latent connection to the original source. This distinction between observable influence and latent structural persistence provides a framework for understanding how networks retain long-range connectivity and memory traces without requiring direct awareness of their origins.\u003c/p\u003e \u003cp\u003ePrevious studies of network dynamics have primarily focused on observable propagation patterns, often overlooking hidden structural connections that preserve information across generations. By explicitly separating observable influence from latent persistence, our model explains how networks can maintain memory of their origins even when individual traces appear cognitively forgotten.\u003c/p\u003e \u003cp\u003eIn the following sections, we describe the branching network model, analyze the dynamics of influence dilution, and discuss implications for connectivity, memory, and small-world phenomena in real-world networks.\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003eWe construct a branching network in which each node represents an agent influenced by a single parent, and influence is equally divided among all offspring nodes at each generation. Let b denote the branching factor and d the network depth. Influence from the original source is represented by a unit value of 1, which is divided equally among all offspring at each branching event, producing exponential decay of observable influence with increasing depth. The original source node remains invariant and latent, serving as a structural anchor for the network.\u003c/p\u003e \u003cp\u003eLet I_g represent the observable influence at generation g. Influence is divided equally among b offspring at each step, giving the recursive relation:\u003c/p\u003e \u003cp\u003eI_0\u0026thinsp;=\u0026thinsp;1, I_g\u0026thinsp;=\u0026thinsp;I_{g-1} / b, g\u0026thinsp;\u0026gt;\u0026thinsp;=\u0026thinsp;1\u003c/p\u003e \u003cp\u003ewhich simplifies to\u003c/p\u003e \u003cp\u003eI_g\u0026thinsp;=\u0026thinsp;1 / b^g.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e illustrates a branching network with b\u0026thinsp;=\u0026thinsp;2 over d\u0026thinsp;=\u0026thinsp;4 generations. The original source node is shown at the top (latent and unobserved), while observable influence is represented at each descendant node. Node color intensity corresponds to the magnitude of I_g, demonstrating exponential dilution across generations. Despite the apparent weakening of influence, all nodes maintain a structural connection to the original source.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThis model captures the distinction between cognitive perception and structural persistence. As influence propagates and divides among descendants, observable traces diminish exponentially, mirroring how human cognition forgets distant origins. Nevertheless, the latent source ensures the network retains full connectivity, illustrating how structural memory persists independently of perceptual awareness.\u003c/p\u003e"},{"header":"Results","content":" \u003cp\u003eWe simulated branching networks with branching factors b\u0026thinsp;=\u0026thinsp;2, 3, 4 and depths d\u0026thinsp;=\u0026thinsp;5, 6, 7. Observable influence I_g decreased exponentially with generation number g, consistent with I_g\u0026thinsp;=\u0026thinsp;1 / b^g. Even at moderate depths, influence on descendant nodes became negligible, while the latent source maintained full structural connectivity to all nodes. These results confirm that the network\u0026rsquo;s architecture preserves links to the origin despite the apparent disappearance of influence in observable measures.\u003c/p\u003e \u003cp\u003eThese findings have direct implications for real-world networks, where connections to original sources are often imperceptible yet structurally preserved. In small-world networks, short path lengths and high clustering allow influence to propagate widely while individual traces dilute beyond conscious perception. Our model provides a conceptual explanation for how networks retain memory of their origins and ensure robust connectivity, even when observable effects appear fragmented or forgotten.\u003c/p\u003e"},{"header":"Discussion","content":" \u003cp\u003eOur branching network model highlights a fundamental distinction between observable influence and latent structural persistence. By formalizing how influence dilutes across generations while the original source remains structurally intact, we provide a framework for understanding cognitive forgetting in networks and the preservation of long-range connectivity.\u003c/p\u003e \u003cp\u003eThis perspective applies to a variety of real-world systems, including social networks, information diffusion, and neural circuits, where origins may be imperceptible yet still exert influence.\u003c/p\u003e \u003cp\u003eThe model assumes equal division of influence among offspring and a fixed branching factor, which may not hold in all empirical networks. Future work could incorporate heterogeneous branching, stochastic influence, or feedback loops, enabling more realistic simulations.\u003c/p\u003e \u003cp\u003eOverall, the separation between perceptual visibility and structural persistence offers a novel lens for studying how networks maintain connections to their origins, providing both theoretical insight and practical relevance for network analysis and cognitive science.\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eWe introduced a branching network model distinguishing observable influence from latent structural persistence. While influence dilutes exponentially across generations, the original source remains structurally connected to all nodes. This framework explains how networks retain memory of their origins without explicit awareness, offering insights into cognitive forgetting, small-world connectivity, and information propagation in complex systems. By highlighting the interplay between perceptual visibility and structural persistence, this work provides a foundation for future studies on the dynamics of influence and memory in both natural and artificial networks.\u003c/p\u003e "},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eYonten Gyatso Lama conceived the study, developed the branching network model, performed all simulations and analyses, created all figures, and wrote the manuscript.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eAuthor Contribution: Yonten Gyatso Lama conceived the study, developed the branching network model, performed all simulations and analyses, created all figures, and wrote the manuscript.Competing Interests: The author declares no competing financial or non-financial interests.Third-Party Material: The author confirms that all material in this manuscript and supplementary files is original and created by the author. No third-party material is included.Research Data: Yes, the study generated original simulation data, which is included as supplementary material and available upon request.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eWatts DJ, Strogatz SH (1998) Collective dynamics of 'small-world' networks. Nature 393(6684):440\u0026ndash;442\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eNewman MEJ (2010) Networks: An Introduction. Oxford University Press\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBarab\u0026aacute;si A-L, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509\u0026ndash;512\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBollob\u0026aacute;s B (2001) Random Graphs. Cambridge University Press\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKempe D, Kleinberg J, Tardos \u0026Eacute; (2003) Maximizing the spread of influence through a social network. Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 137\u0026ndash;146\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eCowan N, Saults JS (2013) Working memory capacity, attention, and long-term memory. Curr Dir Psychol Sci 22(2):133\u0026ndash;138\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLatora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87(19):198701\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"},{"header":"Tables","content":"\u003cp\u003eTable 1: Observable Influence Across Generations\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003eGeneration (g)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003eb = 2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003eb = 3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003eb = 4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e1.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e1.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e1.0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.333\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.25\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.111\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.0625\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.125\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.037\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.0156\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.0625\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.012\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.0039\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.0313\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.004\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 25%;\"\u003e\n \u003cp\u003e0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8669960/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8669960/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe present a branching network model that distinguishes between observable influence and a latent original source. Influence is divided equally across generations, producing exponential dilution, while the original source remains invariant and latent. As branching depth increases, observable traces of the origin diminish and are cognitively forgotten, yet they remain structurally implicit. This separation between structural persistence and perceptual visibility provides a conceptual framework for understanding how small-world networks maintain connectivity to their origins without explicit awareness.\u003c/p\u003e","manuscriptTitle":"Branching Network Model of Observable Influence and Latent Source","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-27 17:30:17","doi":"10.21203/rs.3.rs-8669960/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"31b53542-b7c1-433b-abe7-3e06a320d1d0","owner":[],"postedDate":"January 27th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-28T09:25:06+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-27 17:30:17","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8669960","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8669960","identity":"rs-8669960","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}