The phase transitions of diameters in random axis-parallel hyperrectangle intersection graphs

Congsong Zhang, Yong Gao, James Nastos

Abstract: We study the behaviours of diameters in two models of random axis-parallel hyperrectangle intersection graphs: G(n,r,l)<math><mrow is="true"><mi is="true">G</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">r</mi><mo is="true">,</mo><mi is="true">l</mi><mo is="true">)</mo></mrow></mrow></math> and Gu(n,r,l)<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mi is="true">u</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">r</mi><mo is="true">,</mo><mi is="true">l</mi><mo is="true">)</mo></mrow></mrow></math>. These two models use axis-parallel l<math><mi is="true">l</mi></math>-dimensional hyperrectangles to represent vertices, and vertices are adjacent if and only if their corresponding axis-parallel hyperrectangles intersect. In the model G(n,r,l)<math><mrow is="true"><mi is="true">G</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">r</mi><mo is="true">,</mo><mi is="true">l</mi><mo is="true">)</mo></mrow></mrow></math>, we distribute n<math><mi is="true">n</mi></math> points within [0,1]l<math><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">[</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo is="true">]</mo></mrow></mrow><mrow is="true"><mi is="true">l</mi></mrow></msup></math> uniformly and independently, and each point is the centre of an axis-parallel l<math><mi is="true">l</mi></math>-dimensional hypercube with edge length r<math><mi is="true">r</mi></math>. The model Gu(n,r,l)<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mi is="true">u</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">r</mi><mo is="true">,</mo><mi is="true">l</mi><mo is="true">)</mo></mrow></mrow></math>, distributing the centres of n<math><mi is="true">n</mi></math> axis-parallel l<math><mi is="true">l</mi></math>-dimensional hyperrectangles within [0,1]l<math><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">[</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo is="true">]</mo></mrow></mrow><mrow is="true"><mi is="true">l</mi></mrow></msup></math> exactly as the model G(n,r,l)<math><mrow is="true"><mi is="true">G</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">r</mi><mo is="true">,</mo><mi is="true">l</mi><mo is="true">)</mo></mrow></mrow></math>, assigns a length from a uniform distribution over [0,r]<math><mrow is="true"><mo is="true">[</mo><mn is="true">0</mn><mo is="true">,</mo><mi is="true">r</mi><mo is="true">]</mo></mrow></math> to each edge of the n<math><mi is="true">n</mi></math> axis-parallel l<math><mi is="true">l</mi></math>-dimensional hyperrectangles.We prove that in the model G(n,r,l)<math><mrow is="true"><mi is="true">G</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">r</mi><mo is="true">,</mo><mi is="true">l</mi><mo is="true">)</mo></mrow></mrow></math>, there is a phase transition for the event that the diameter is at most d(n)<math><mrow is="true"><mi is="true">d</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">)</mo></mrow></mrow></math> occurring at r=d(n)−1<math><mrow is="true"><mi is="true">r</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">d</mi><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">)</mo></mrow></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">1</mn></mrow></msup></mrow></math> if n⋅d(n)−l(n)≥nϵ,<math><mrow is="true"><mi is="true">n</mi><mi is="true">⋅</mi><mi is="true">d</mi><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">)</mo></mrow></mrow><mrow is="true"><mo is="true">−</mo><mi is="true">l</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">)</mo></mrow></mrow></msup><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">ϵ</mi></mrow></msup><mo is="true">,</mo></mrow></math> where 0<ϵ<1<math><mrow is="true"><mn is="true">0</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">ϵ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mn is="true">1</mn></mrow></math> is an arbitrary small constant, and l=l(n)=o(1)⋅(lnn)⋅(lnlnn)−1.<math><mrow is="true"><mi is="true">l</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">l</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">o</mi><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">)</mo></mrow><mi is="true">⋅</mi><mrow is="true"><mo is="true">(</mo><mo class="qopname" is="true">ln</mo><mi is="true">n</mi><mo is="true">)</mo></mrow><mi is="true">⋅</mi><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">(</mo><mo class="qopname" is="true">ln</mo><mo class="qopname" is="true">ln</mo><mi is="true">n</mi><mo is="true">)</mo></mrow></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">1</mn></mrow></msup><mo is="true">.</mo></mrow></math> In the model Gu(n,r,l)<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">G</mi></mrow><mrow is="true"><mi is="true">u</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">r</mi><mo is="true">,</mo><mi is="true">l</mi><mo is="true">)</mo></mrow></mrow></math>, this phase transition occurs at r=(d(n)−1)−1<math><mrow is="true"><mi is="true">r</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msup is="true"><mrow is="true"><mrow is="true"><mo class="bigl" fence="true" is="true">(</mo><mrow is="true"><mi is="true">d</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">)</mo></mrow><mo is="true">−</mo><mn is="true">1</mn></mrow><mo class="bigr" fence="true" is="true">)</mo></mrow></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">1</mn></mrow></msup></mrow></math>.