Evolution of Networks: A New Kind of Science | Online by Stephen Wolfram [Page 508]

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<a href="/nks/p1--an-outline-of-basic-ideas/"> <span>1</span> <span>An Outline of Basic Ideas</span> </a> </li> <li class="section section-2"> <a href="/nks/p7--relations-to-other-areas/"> <span>2</span> <span>Relations to Other Areas</span> </a> </li> <li class="section section-3"> <a href="/nks/p12--some-past-initiatives/"> <span>3</span> <span>Some Past Initiatives</span> </a> </li> <li class="section section-4"> <a href="/nks/p17--the-personal-story-of-the-science-in-this-book/"> <span>4</span> <span>The Personal Story of the Science in This Book</span> </a> </li> </ul> </li> <li class="chapter chapter-2"> <a href="/nks/chap-2--the-crucial-experiment/"> <span>2</span> <span>The Crucial Experiment</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p23--how-do-simple-programs-behave/"> <span>1</span> <span>How Do Simple Programs Behave?</span> </a> </li> <li class="section section-2"> <a href="/nks/p39--the-need-for-a-new-intuition/"> <span>2</span> <span>The Need for a New Intuition</span> </a> </li> <li class="section section-3"> <a href="/nks/p42--why-these-discoveries-were-not-made-before/"> <span>3</span> <span>Why These Discoveries Were Not Made Before</span> </a> </li> </ul> </li> <li class="chapter chapter-3"> <a href="/nks/chap-3--the-world-of-simple-programs/"> <span>3</span> <span>The World of Simple Programs</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p51--the-search-for-general-features/"> <span>1</span> <span>The Search for General Features</span> </a> </li> <li class="section section-2"> <a href="/nks/p53--more-cellular-automata/"> <span>2</span> <span>More Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p71--mobile-automata/"> <span>3</span> <span>Mobile Automata</span> </a> </li> <li class="section section-4"> <a href="/nks/p78--turing-machines/"> <span>4</span> <span>Turing Machines</span> </a> </li> <li class="section section-5"> <a href="/nks/p82--substitution-systems/"> <span>5</span> <span>Substitution Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p88--sequential-substitution-systems/"> <span>6</span> <span>Sequential Substitution Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p93--tag-systems/"> <span>7</span> <span>Tag Systems</span> </a> </li> <li class="section section-8"> <a href="/nks/p95--cyclic-tag-systems/"> <span>8</span> <span>Cyclic Tag Systems</span> </a> </li> <li class="section section-9"> <a href="/nks/p97--register-machines/"> <span>9</span> <span>Register Machines</span> </a> </li> <li class="section section-10"> <a href="/nks/p102--symbolic-systems/"> <span>10</span> <span>Symbolic Systems</span> </a> </li> <li class="section section-11"> <a href="/nks/p105--some-conclusions/"> <span>11</span> <span>Some Conclusions</span> </a> </li> <li class="section section-12 selected"> <a href="/nks/p108--how-the-discoveries-in-this-chapter-were-made/"> <span>12</span> <span>How the Discoveries in This Chapter Were Made</span> </a> </li> </ul> </li> <li class="chapter chapter-4"> <a href="/nks/chap-4--systems-based-on-numbers/"> <span>4</span> <span>Systems Based on Numbers</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p115--the-notion-of-numbers/"> <span>1</span> <span>The Notion of Numbers</span> </a> </li> <li class="section section-2"> <a href="/nks/p117--elementary-arithmetic/"> <span>2</span> <span>Elementary Arithmetic</span> </a> </li> <li class="section section-3"> <a href="/nks/p128--recursive-sequences/"> <span>3</span> <span>Recursive Sequences</span> </a> </li> <li class="section section-4"> <a href="/nks/p132--the-sequence-of-primes/"> <span>4</span> <span>The Sequence of Primes</span> </a> </li> <li class="section section-5"> <a href="/nks/p136--mathematical-constants/"> <span>5</span> <span>Mathematical Constants</span> </a> </li> <li class="section section-6"> <a href="/nks/p145--mathematical-functions/"> <span>6</span> <span>Mathematical Functions</span> </a> </li> <li class="section section-7"> <a href="/nks/p149--iterated-maps-and-the-chaos-phenomenon/"> <span>7</span> <span>Iterated Maps and the Chaos Phenomenon</span> </a> </li> <li class="section section-8"> <a href="/nks/p155--continuous-cellular-automata/"> <span>8</span> <span>Continuous Cellular Automata</span> </a> </li> <li class="section section-9"> <a href="/nks/p161--partial-differential-equations/"> <span>9</span> <span>Partial Differential Equations</span> </a> </li> <li class="section section-10"> <a href="/nks/p167--continuous-versus-discrete-systems/"> <span>10</span> <span>Continuous Versus Discrete Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-5"> <a href="/nks/chap-5--two-dimensions-and-beyond/"> <span>5</span> <span>Two Dimensions and Beyond</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p169--introduction/"> <span>1</span> <span>Introduction</span> </a> </li> <li class="section section-2"> <a href="/nks/p170--cellular-automata/"> <span>2</span> <span>Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p184--turing-machines/"> <span>3</span> <span>Turing Machines</span> </a> </li> <li class="section section-4"> <a href="/nks/p187--substitution-systems-and-fractals/"> <span>4</span> <span>Substitution Systems and Fractals</span> </a> </li> <li class="section section-5"> <a href="/nks/p193--network-systems/"> <span>5</span> <span>Network Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p204--multiway-systems/"> <span>6</span> <span>Multiway Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p210--systems-based-on-constraints/"> <span>7</span> <span>Systems Based on Constraints</span> </a> </li> </ul> </li> <li class="chapter chapter-6"> <a href="/nks/chap-6--starting-from-randomness/"> <span>6</span> <span>Starting from Randomness</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p223--the-emergence-of-order/"> <span>1</span> <span>The Emergence of Order</span> </a> </li> <li class="section section-2"> <a href="/nks/p231--four-classes-of-behavior/"> <span>2</span> <span>Four Classes of Behavior</span> </a> </li> <li class="section section-3"> <a href="/nks/p250--sensitivity-to-initial-conditions/"> <span>3</span> <span>Sensitivity to Initial Conditions</span> </a> </li> <li class="section section-4"> <a href="/nks/p255--systems-of-limited-size-and-class-2-behavior/"> <span>4</span> <span>Systems of Limited Size and Class 2 Behavior</span> </a> </li> <li class="section section-5"> <a href="/nks/p261--randomness-in-class-3-systems/"> <span>5</span> <span>Randomness in Class 3 Systems</span> </a> </li> <li class="section section-6"> <a href="/nks/p266--special-initial-conditions/"> <span>6</span> <span>Special Initial Conditions</span> </a> </li> <li class="section section-7"> <a href="/nks/p275--the-notion-of-attractors/"> <span>7</span> <span>The Notion of Attractors</span> </a> </li> <li class="section section-8"> <a href="/nks/p281--structures-in-class-4-systems/"> <span>8</span> <span>Structures in Class 4 Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-7"> <a href="/nks/chap-7--mechanisms-in-programs-and-nature/"> <span>7</span> <span>Mechanisms in Programs and Nature</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p297--universality-of-behavior/"> <span>1</span> <span>Universality of Behavior</span> </a> </li> <li class="section section-2"> <a href="/nks/p299--three-mechanisms-for-randomness/"> <span>2</span> <span>Three Mechanisms for Randomness</span> </a> </li> <li class="section section-3"> <a href="/nks/p301--randomness-from-the-environment/"> <span>3</span> <span>Randomness from the Environment</span> </a> </li> <li class="section section-4"> <a href="/nks/p304--chaos-theory-and-randomness-from-initial-conditions/"> <span>4</span> <span>Chaos Theory and Randomness from Initial Conditions</span> </a> </li> <li class="section section-5"> <a href="/nks/p315--the-intrinsic-generation-of-randomness/"> <span>5</span> <span>The Intrinsic Generation of Randomness</span> </a> </li> <li class="section section-6"> <a href="/nks/p327--the-phenomenon-of-continuity/"> <span>6</span> <span>The Phenomenon of Continuity</span> </a> </li> <li class="section section-7"> <a href="/nks/p337--origins-of-discreteness/"> <span>7</span> <span>Origins of Discreteness</span> </a> </li> <li class="section section-8"> <a href="/nks/p342--the-problem-of-satisfying-constraints/"> <span>8</span> <span>The Problem of Satisfying Constraints</span> </a> </li> <li class="section section-9"> <a href="/nks/p351--origins-of-simple-behavior/"> <span>9</span> <span>Origins of Simple Behavior</span> </a> </li> </ul> </li> <li class="chapter chapter-8"> <a href="/nks/chap-8--implications-for-everyday-systems/"> <span>8</span> <span>Implications for Everyday Systems</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p363--issues-of-modelling/"> <span>1</span> <span>Issues of Modelling</span> </a> </li> <li class="section section-2"> <a href="/nks/p369--the-growth-of-crystals/"> <span>2</span> <span>The Growth of Crystals</span> </a> </li> <li class="section section-3"> <a href="/nks/p374--the-breaking-of-materials/"> <span>3</span> <span>The Breaking of Materials</span> </a> </li> <li class="section section-4"> <a href="/nks/p376--fluid-flow/"> <span>4</span> <span>Fluid Flow</span> </a> </li> <li class="section section-5"> <a href="/nks/p383--fundamental-issues-in-biology/"> <span>5</span> <span>Fundamental Issues in Biology</span> </a> </li> <li class="section section-6"> <a href="/nks/p400--growth-of-plants-and-animals/"> <span>6</span> <span>Growth of Plants and Animals</span> </a> </li> <li class="section section-7"> <a href="/nks/p422--biological-pigmentation-patterns/"> <span>7</span> <span>Biological Pigmentation Patterns</span> </a> </li> <li class="section section-8"> <a href="/nks/p429--financial-systems/"> <span>8</span> <span>Financial Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-9 selected"> <a href="/nks/chap-9--fundamental-physics/"> <span>9</span> <span>Fundamental Physics</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p433--the-problems-of-physics/"> <span>1</span> <span>The Problems of Physics</span> </a> </li> <li class="section section-2"> <a href="/nks/p435--the-notion-of-reversibility/"> <span>2</span> <span>The Notion of Reversibility</span> </a> </li> <li class="section section-3"> <a href="/nks/p441--irreversibility-and-the-second-law-of-thermodynamics/"> <span>3</span> <span>Irreversibility and the Second Law of Thermodynamics</span> </a> </li> <li class="section section-4"> <a href="/nks/p458--conserved-quantities-and-continuum-phenomena/"> <span>4</span> <span>Conserved Quantities and Continuum Phenomena</span> </a> </li> <li class="section section-5"> <a href="/nks/p465--ultimate-models-for-the-universe/"> <span>5</span> <span>Ultimate Models for the Universe</span> </a> </li> <li class="section section-6"> <a href="/nks/p472--the-nature-of-space/"> <span>6</span> <span>The Nature of Space</span> </a> </li> <li class="section section-7"> <a href="/nks/p475--space-as-a-network/"> <span>7</span> <span>Space as a Network</span> </a> </li> <li class="section section-8"> <a href="/nks/p481--the-relationship-of-space-and-time/"> <span>8</span> <span>The Relationship of Space and Time</span> </a> </li> <li class="section section-9"> <a href="/nks/p486--time-and-causal-networks/"> <span>9</span> <span>Time and Causal Networks</span> </a> </li> <li class="section section-10"> <a href="/nks/p497--the-sequencing-of-events-in-the-universe/"> <span>10</span> <span>The Sequencing of Events in the Universe</span> </a> </li> <li class="section section-11"> <a href="/nks/p504--uniqueness-and-branching-in-time/"> <span>11</span> <span>Uniqueness and Branching in Time</span> </a> </li> <li class="section section-12 selected"> <a href="/nks/p508--evolution-of-networks/"> <span>12</span> <span>Evolution of Networks</span> </a> </li> <li class="section section-13"> <a href="/nks/p516--space-time-and-relativity/"> <span>13</span> <span>Space, Time and Relativity</span> </a> </li> <li class="section section-14"> <a href="/nks/p525--elementary-particles/"> <span>14</span> <span>Elementary Particles</span> </a> </li> <li class="section section-15"> <a href="/nks/p530--the-phenomenon-of-gravity/"> <span>15</span> <span>The Phenomenon of Gravity</span> </a> </li> <li class="section section-16"> <a href="/nks/p537--quantum-phenomena/"> <span>16</span> <span>Quantum Phenomena</span> </a> </li> </ul> </li> <li class="chapter chapter-10"> <a href="/nks/chap-10--processes-of-perception-and-analysis/"> <span>10</span> <span>Processes of Perception and Analysis</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p547--introduction/"> <span>1</span> <span>Introduction</span> </a> </li> <li class="section section-2"> <a href="/nks/p548--what-perception-and-analysis-do/"> <span>2</span> <span>What Perception and Analysis Do</span> </a> </li> <li class="section section-3"> <a href="/nks/p552--defining-the-notion-of-randomness/"> <span>3</span> <span>Defining the Notion of Randomness</span> </a> </li> <li class="section section-4"> <a href="/nks/p557--defining-complexity/"> <span>4</span> <span>Defining Complexity</span> </a> </li> <li class="section section-5"> <a href="/nks/p560--data-compression/"> <span>5</span> <span>Data Compression</span> </a> </li> <li class="section section-6"> <a href="/nks/p572--irreversible-data-compression/"> <span>6</span> <span>Irreversible Data Compression</span> </a> </li> <li class="section section-7"> <a href="/nks/p577--visual-perception/"> <span>7</span> <span>Visual Perception</span> </a> </li> <li class="section section-8"> <a href="/nks/p585--auditory-perception/"> <span>8</span> <span>Auditory Perception</span> </a> </li> <li class="section section-9"> <a href="/nks/p588--statistical-analysis/"> <span>9</span> <span>Statistical Analysis</span> </a> </li> <li class="section section-10"> <a href="/nks/p598--cryptography-and-cryptanalysis/"> <span>10</span> <span>Cryptography and Cryptanalysis</span> </a> </li> <li class="section section-11"> <a href="/nks/p606--traditional-mathematics-and-mathematical-formulas/"> <span>11</span> <span>Traditional Mathematics and Mathematical Formulas</span> </a> </li> <li class="section section-12 selected"> <a href="/nks/p620--human-thinking/"> <span>12</span> <span>Human Thinking</span> </a> </li> <li class="section section-13"> <a href="/nks/p632--higher-forms-of-perception-and-analysis/"> <span>13</span> <span>Higher Forms of Perception and Analysis</span> </a> </li> </ul> </li> <li class="chapter chapter-11"> <a href="/nks/chap-11--the-notion-of-computation/"> <span>11</span> <span>The Notion of Computation</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p637--computation-as-a-framework/"> <span>1</span> <span>Computation as a Framework</span> </a> </li> <li class="section section-2"> <a href="/nks/p638--computations-in-cellular-automata/"> <span>2</span> <span>Computations in Cellular Automata</span> </a> </li> <li class="section section-3"> <a href="/nks/p642--the-phenomenon-of-universality/"> <span>3</span> <span>The Phenomenon of Universality</span> </a> </li> <li class="section section-4"> <a href="/nks/p644--a-universal-cellular-automaton/"> <span>4</span> <span>A Universal Cellular Automaton</span> </a> </li> <li class="section section-5"> <a href="/nks/p656--emulating-other-systems-with-cellular-automata/"> <span>5</span> <span>Emulating Other Systems with Cellular Automata</span> </a> </li> <li class="section section-6"> <a href="/nks/p664--emulating-cellular-automata-with-other-systems/"> <span>6</span> <span>Emulating Cellular Automata with Other Systems</span> </a> </li> <li class="section section-7"> <a href="/nks/p674--implications-of-universality/"> <span>7</span> <span>Implications of Universality</span> </a> </li> <li class="section section-8"> <a href="/nks/p675--the-rule-110-cellular-automaton/"> <span>8</span> <span>The Rule 110 Cellular Automaton</span> </a> </li> <li class="section section-9"> <a href="/nks/p690--the-significance-of-universality-in-rule-110/"> <span>9</span> <span>The Significance of Universality in Rule 110</span> </a> </li> <li class="section section-10"> <a href="/nks/p691--class-4-behavior-and-universality/"> <span>10</span> <span>Class 4 Behavior and Universality</span> </a> </li> <li class="section section-11"> <a href="/nks/p694--the-threshold-of-universality-in-cellular-automata/"> <span>11</span> <span>The Threshold of Universality in Cellular Automata</span> </a> </li> <li class="section section-12 selected"> <a href="/nks/p706--universality-in-turing-machines-and-other-systems/"> <span>12</span> <span>Universality in Turing Machines and Other Systems</span> </a> </li> </ul> </li> <li class="chapter chapter-12"> <a href="/nks/chap-12--the-principle-of-computational-equivalence/"> <span>12</span> <span>The Principle of Computational Equivalence</span> </a> <ul class="sections"> <li class="section section-1"> <a href="/nks/p715--basic-framework/"> <span>1</span> <span>Basic Framework</span> </a> </li> <li class="section section-2"> <a href="/nks/p716--outline-of-the-principle/"> <span>2</span> <span>Outline of the Principle</span> </a> </li> <li class="section section-3"> <a href="/nks/p719--the-content-of-the-principle/"> <span>3</span> <span>The Content of the Principle</span> </a> </li> <li class="section section-4"> <a href="/nks/p726--the-validity-of-the-principle/"> <span>4</span> <span>The Validity of the Principle</span> </a> </li> <li class="section section-5"> <a href="/nks/p735--explaining-the-phenomenon-of-complexity/"> <span>5</span> <span>Explaining the Phenomenon of Complexity</span> </a> </li> <li class="section section-6"> <a href="/nks/p737--computational-irreducibility/"> <span>6</span> <span>Computational Irreducibility</span> </a> </li> <li class="section section-7"> <a href="/nks/p750--the-phenomenon-of-free-will/"> <span>7</span> <span>The Phenomenon of Free Will</span> </a> </li> <li class="section section-8"> <a href="/nks/p753--undecidability-and-intractability/"> <span>8</span> <span>Undecidability and Intractability</span> </a> </li> <li class="section section-9"> <a href="/nks/p772--implications-for-mathematics-and-its-foundations/"> <span>9</span> <span>Implications for Mathematics and Its Foundations</span> </a> </li> <li class="section section-10"> <a href="/nks/p822--intelligence-in-the-universe/"> <span>10</span> <span>Intelligence in the Universe</span> </a> </li> <li class="section section-11"> <a href="/nks/p840--implications-for-technology/"> <span>11</span> <span>Implications for Technology</span> </a> </li> <li class="section section-12 selected"> <a href="/nks/p844--historical-perspectives/"> <span>12</span> <span>Historical Perspectives</span> </a> </li> </ul> </li> <li class="chapter notes"> <a href="/nks/notes-for-chapters/"> <span></span> <span>Notes</span> </a> </li> </ul> <div class="search"> </div> </div> </div> </nav> </div> </div> </div> <div id="search"> <div class="text-align-c__600"> <div class="inline-grid cols-3 cols-1__600"> <form id="search-index-form" action="/nks/index/"><div><label for="search-index-input">Look up in index</label><input type="search" id="search-index-input" name="search-index-input"></div></form> <form id="search-all-form" action="/nks/search/"><div><label for="search-all-input">Text search</label><input type="search" id="search-all-input" name="search-all-input"></div></form> <form id="search-jump-form"><div><label for="search-jump-input">Jump to page</label><input type="search" id="search-jump-input" name="search-jump-input"></div></form> </div> </div> </div> </header>  <main id="main"> <div class="wide-wrapper"> <div id="previous"><a href="/nks/p507--uniqueness-and-branching-in-time/"><span>‹</span></a></div> <div id="next"><a href="/nks/p509--evolution-of-networks/"><span>›</span></a></div> <div id="content" class="ribbon-top"> <div class="ribbon-wrapper top"> <div class="ribbon"> <a href="https://www.wolframphysics.org/" target="_blank" class="ribbon-link"> <div class="ribbon-content mini-m"> <span class="icon"><img src="/nks/img/future-icon.svg" height="23" width="23"></span> <span class="title">Further development: <span class="text">Wolfram Physics Project »</span></span> </div> </a> </div></div> <div class="side-tabs"> <ul> <li class="page-img"><a href="/nks/p508--evolution-of-networks/">Page Image</a></li> <li class="web-view"><a href="/nks/p508--evolution-of-networks--webview/">eBook View</a></li> <li class="notes"><a href="#related-notes">Notes</a></li> <li class="resources hide"><a href="#resources">Resources</a></li> </ul> </div> <div class="xlarge"> <img src="/nks/pages/p508--evolution-of-networks/image-xlarge.png" alt="" width="1000" height="1205" class="fill"> </div> <div class="large"> <img src="/nks/pages/p508--evolution-of-networks/image-large.png" alt="" width="850" height="1025" class="fill"> </div> <div class="medium"> <img src="/nks/pages/p508--evolution-of-networks/image-medium.png" alt="" width="750" height="904" class="fill"> </div> <div class="small"> <img src="/nks/pages/p508--evolution-of-networks/image-small.png" alt="" width="650" height="784" class="fill"> </div><div class="text-content wide-padding"><p class="textnoindent">regardless of the path one chooses, the overall form of causal network will be essentially the same. And what this means is that on a sufficiently large scale, the universe will appear to have a unique history, even though at the level of individual events there will be considerable arbitrariness.</p><p class="text">If there is not enough convergence in the multiway system it will still be possible to get stuck with different types of strings that never lead to each other. And if this happens, then it means that the history of the universe can in effect follow many truly separate branches. But whenever there is significant randomness produced by the evolution of the multiway system, this does not typically appear to occur.</p><p class="text">So this suggests that in fact it is at some level not too difficult for multiway systems to reproduce our everyday perception that more or less definite things happen in the universe. But while this means that it might be possible for there to be arbitrariness in the causal network for the universe, it still tends to be my suspicion that there is not—and that in fact the particular rules followed by the universe do in the end have the property that they always yield the same causal network.</p><h2 class="text-section-heading">Evolution of Networks</h2><p class="text">Earlier in <a href="/nks/chap-9--fundamental-physics/">this chapter</a>, I suggested that at the lowest level space might consist of a giant network of nodes. But how might such a network evolve?</p><p class="text">The most straightforward possibility is that it could work much like the substitution systems that we have discussed in the past few sections [<a href="/nks/chap-9--fundamental-physics/#sect-9-9--time-and-causal-networks">9</a>, <a href="/nks/chap-9--fundamental-physics/#sect-9-10--the-sequencing-of-events-in-the-universe">10</a>, <a href="/nks/chap-9--fundamental-physics/#sect-9-11--uniqueness-and-branching-in-time">11</a>]—and that at each step some piece or pieces of the network could be replaced by others according to some fixed rule.</p><p class="text">The pictures at the top of the <a href="/nks/p509--evolution-of-networks--webview/">facing page</a> show two very simple examples. Starting with a network whose connections are like the edges of a tetrahedron, both the rules shown work by replacing each node at each step by a certain fixed cluster of nodes.</p><p class="text">This setup is very much similar to the neighbor-independent substitution systems that we discussed on pages <a href="/nks/p83--substitution-systems--webview/">83</a> and <a href="/nks/p187--substitution-systems-and-fractals--webview/">187</a>. And just as in these systems, it is possible for intricate structures to be produced, but the structures always turn out to have a highly regular nested form.</p></div> <div class="ribbon-wrapper stripe"> <div class="ribbon"> <a href="https://www.wolframphysics.org/" target="_blank" class="ribbon-link"> <div class="ribbon-content mini-m"> <span class="icon"><img src="/nks/img/future-icon.svg" height="23" width="23"></span> <span class="title">Wolfram Physics Project <span class="text">— An Approach to the Fundamental Theory of Physics & More</span></span> </div> </a> </div> </div> <footer id="footer" class="wide-padding"> <section id="related-notes" class="width-full max-width-900 center"> <h2>Notes In the Book Related To This Page:</h2> <div> <div class="thumbnails cf"> <a href="/nks/notes-9-12--neighbor-independent-network-substitution-rules/"><img src="/nks/img/thumbnails/notes-9-12--neighbor-independent-network-substitution-rules--textonly.png" alt="Neighbor-independent [network substitution] rules" title="Neighbor-independent [network substitution] rules" width="76" height="62"></a> <a href="/nks/notes-9-12--implementation-of-network-substitution-rules/"><img src="/nks/img/thumbnails/notes-9-12--implementation-of-network-substitution-rules--textonly.png" alt="Implementation [of network substitution rules]" title="Implementation [of network substitution rules]" width="76" height="62"></a> <a href="/nks/notes-9-12--number-of-possible-network-replacements/"><img src="/nks/img/thumbnails/notes-9-12--number-of-possible-network-replacements--textonly.png" alt="Number of [possible network] replacements" title="Number of [possible network] replacements" width="76" height="62"></a> <a href="/nks/notes-9-12--cellular-structures-in-nature/"><img src="/nks/img/thumbnails/notes-9-12--cellular-structures-in-nature--textonly.png" alt="Cellular structures [in nature]" title="Cellular structures [in nature]" width="76" height="62"></a> <a href="/nks/notes-9-12--graph-grammars/"><img src="/nks/img/thumbnails/notes-9-12--graph-grammars--textonly.png" alt="Graph grammars" title="Graph grammars" width="76" height="62"></a> <a href="/nks/notes-9-5--initial-conditions-for-the-universe/"><img src="/nks/img/thumbnails/notes-9-5--initial-conditions-for-the-universe--textonly.png" alt="Initial conditions [for the universe]" title="Initial conditions [for the universe]" width="76" height="62"></a> <a href="/nks/notes-9-7--maintaining-simple-rules-for-networks/"><img src="/nks/img/thumbnails/notes-9-7--maintaining-simple-rules-for-networks--textonly.png" alt="Maintaining simple rules [for networks]" title="Maintaining simple rules [for networks]" width="76" height="62"></a> <a href="/nks/notes-9-10--distributed-computing/"><img src="/nks/img/thumbnails/notes-9-10--distributed-computing--textonly.png" alt="Distributed computing" title="Distributed computing" width="76" height="62"></a> <a href="/nks/notes-9-11--parallel-universes/"><img src="/nks/img/thumbnails/notes-9-11--parallel-universes--textonly.png" alt="Parallel universes" title="Parallel universes" width="76" height="62"></a> </div> <ul class="cols-2 cols-1__600 heirs-width-full__600"> <li><span><a href="/nks/notes-9-12--neighbor-independent-network-substitution-rules/"><span><img src="/nks/img/thumbnails/notes-9-12--neighbor-independent-network-substitution-rules--textonly.png" alt="" width="76" height="62"></span><span>Neighbor-independent [network substitution] rules</span></a></span></li> <li><span><a href="/nks/notes-9-12--implementation-of-network-substitution-rules/"><span><img src="/nks/img/thumbnails/notes-9-12--implementation-of-network-substitution-rules--textonly.png" alt="" width="76" height="62"></span><span>Implementation [of network substitution rules]</span></a></span></li> <li><span><a href="/nks/notes-9-12--number-of-possible-network-replacements/"><span><img src="/nks/img/thumbnails/notes-9-12--number-of-possible-network-replacements--textonly.png" alt="" width="76" height="62"></span><span>Number of [possible network] replacements</span></a></span></li> <li><span><a href="/nks/notes-9-12--cellular-structures-in-nature/"><span><img src="/nks/img/thumbnails/notes-9-12--cellular-structures-in-nature--textonly.png" alt="" width="76" height="62"></span><span>Cellular structures [in nature]</span></a></span></li> <li><span><a href="/nks/notes-9-12--graph-grammars/"><span><img src="/nks/img/thumbnails/notes-9-12--graph-grammars--textonly.png" alt="" width="76" height="62"></span><span>Graph grammars</span></a></span></li> <li><span><a href="/nks/notes-9-5--initial-conditions-for-the-universe/"><span><img src="/nks/img/thumbnails/notes-9-5--initial-conditions-for-the-universe--textonly.png" alt="" width="76" height="62"></span><span>Initial conditions [for the universe]</span></a></span></li> <li><span><a href="/nks/notes-9-7--maintaining-simple-rules-for-networks/"><span><img src="/nks/img/thumbnails/notes-9-7--maintaining-simple-rules-for-networks--textonly.png" alt="" width="76" height="62"></span><span>Maintaining simple rules [for networks]</span></a></span></li> <li><span><a href="/nks/notes-9-10--distributed-computing/"><span><img src="/nks/img/thumbnails/notes-9-10--distributed-computing--textonly.png" alt="" width="76" height="62"></span><span>Distributed computing</span></a></span></li> <li><span><a href="/nks/notes-9-11--parallel-universes/"><span><img src="/nks/img/thumbnails/notes-9-11--parallel-universes--textonly.png" alt="" width="76" height="62"></span><span>Parallel universes</span></a></span></li> <li><span><a href="/nks/notes-9-11--many-worlds-models/"><span><img src="/nks/img/thumbnails/notes-9-11--many-worlds-models--textonly.png" alt="" width="76" height="62"></span><span>Many-worlds models</span></a></span></li> <li><span><a href="/nks/notes-9-11--conditions-for-convergence-in-string-rewriting/"><span><img src="/nks/img/thumbnails/notes-9-11--conditions-for-convergence-in-string-rewriting--textonly.png" alt="" width="76" height="62"></span><span>Conditions for convergence [in string rewriting]</span></a></span></li> <li><span><a href="/nks/notes-9-11--confluence-in-string-rewriting/"><span><img src="/nks/img/thumbnails/notes-9-11--confluence-in-string-rewriting--textonly.png" alt="" width="76" height="62"></span><span>Confluence [in string rewriting]</span></a></span></li> <li><span><a href="/nks/notes-9-11--completion-in-multiway-systems/"><span><img src="/nks/img/thumbnails/notes-9-11--completion-in-multiway-systems--textonly.png" alt="" width="76" height="62"></span><span>Completion [in multiway systems]</span></a></span></li> <li><span><a href="/nks/notes-9-15--manifold-properties-and-undecidability/"><span><img src="/nks/img/thumbnails/notes-9-15--manifold-properties-and-undecidability--textonly.png" alt="" width="76" height="62"></span><span>Manifold [properties and] undecidability</span></a></span></li> <li><span><a href="/nks/notes-3-10--network-analogs-of-symbolic-systems/"><span><img src="/nks/img/thumbnails/notes-3-10--network-analogs-of-symbolic-systems--textonly.png" alt="" width="76" height="62"></span><span>Network analogs [of symbolic systems]</span></a></span></li> </ul> </div> </section> <section id="resources"> <section id="exportable-images" class="width-full max-width-900 center hide"> <hr class="even"> <h2>Exportable Images for This Page:</h2> <div> <ul class="grid width-full"> </ul> </div> </section> <section id="related-resources" class="width-full max-width-900 hide"> <hr class="even"> <h2>External Resources Related to this Page:</h2> <div> <ul class="grid cols-2 width-full heirs-width-1-2 cols-1__600 heirs-width-full__600"> </ul> </div> </section> </section>  <footer id="inline-footer"> <div class="heirs-display-ib heirs-margin-bt-1-2__600 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Wolfram uses his approach to tackle a remarkable array of fundamental problems in science, from the origins of apparent randomness in physical systems, to the development of complexity in biology, the ultimate scope and limitations of mathematics, the possibility of a truly fundamental theory of physics, the interplay between free will and determinism, and the character of intelligence in the universe.", "author": { "@type": "Person", "name": "Stephen Wolfram" }, "publisher": { "@type": "Corporation", "name": "Wolfram Media, Inc.", "url": "http://www.wolfram-media.com/" }, "isbn": "1-57955-008-8", "datePublished": "2002", "numberOfPages": "1197", "bookFormat": "print, online, iPad", "genre": "Science, Cellular Automata, Computation, Biology, Physics", "audience": "Scientists, Researchers, Science Educators, Biologists, Physicists, Mathematicians, Programmers", "discussionUrl": "http://community.wolfram.com/content?curTag=wolfram%20science", "inLanguage": "en" } </script>   </footer> </div>  <footer id="citation"> <p>From Stephen Wolfram: <em>A New Kind of Science</em> [<a href="/nks/citation/">citation</a>]   <a href="https://creativecommons.org/licenses/by-sa/4.0/" target="_blank"> <svg id="icon-cc-by-sa" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 110 32"> <g transform="translate(76 0)"> <path d="M15.7,0.9c4.2,0,7.7,1.5,10.7,4.4c2.9,2.9,4.4,6.4,4.4,10.6c0,4.2-1.4,7.7-4.3,10.6c-3,3-6.6,4.4-10.7,4.4 c-4.1,0-7.6-1.5-10.6-4.4C2.2,23.5,0.7,20,0.7,15.9c0-4.1,1.5-7.6,4.4-10.6C8.1,2.3,11.6,0.9,15.7,0.9z M15.8,3.6 c-3.4,0-6.3,1.2-8.7,3.6c-2.4,2.5-3.7,5.4-3.7,8.7c0,3.3,1.2,6.2,3.6,8.6c2.4,2.4,5.3,3.6,8.7,3.6c3.3,0,6.2-1.2,8.7-3.7 c2.4-2.3,3.5-5.2,3.5-8.6c0-3.4-1.2-6.3-3.6-8.7C22.1,4.8,19.2,3.6,15.8,3.6z M9.1,13.8c0.3-1.8,1-3.3,2.2-4.3 c1.2-1,2.7-1.5,4.4-1.5c2.4,0,4.2,0.8,5.6,2.3s2.1,3.5,2.1,5.8c0,2.3-0.7,4.2-2.2,5.7c-1.4,1.5-3.3,2.3-5.6,2.3 c-1.7,0-3.2-0.5-4.4-1.5c-1.2-1-2-2.5-2.2-4.3h3.8c0.1,1.8,1.2,2.7,3.3,2.7c1.1,0,1.9-0.5,2.5-1.4c0.6-0.9,1-2.1,1-3.6 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Evolution of Networks: A New Kind of Science | Online by Stephen Wolfram [Page 508]