Opinion forming in Erdős-Rényi random graph and expanders

Ahad N. Zehmakan

Abstract: Assume for a graph G=V,E<math><mrow is="true"><mi is="true">G</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mfenced close=")" open="(" is="true"><mrow is="true"><mi is="true">V</mi><mo is="true">,</mo><mi is="true">E</mi></mrow></mfenced></mrow></math> and an initial configuration, where each node is blue or red, in each discrete-time round all nodes simultaneously update their color to the most frequent color in their neighborhood and a node keeps its color in case of a tie. We study the behavior of this basic process, which is called the majority model, on the Erdős–Rényi random graph Gn,p<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">G</mi></mrow><mrow is="true"><mi is="true">n</mi><mo is="true">,</mo><mi is="true">p</mi></mrow></msub></math> and regular expanders. First we consider the behavior of the majority model on Gn,p<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">G</mi></mrow><mrow is="true"><mi is="true">n</mi><mo is="true">,</mo><mi is="true">p</mi></mrow></msub></math> with an initial random configuration, where each node is blue independently with probability pb<math><msub is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">b</mi></mrow></msub></math> and red otherwise. It is shown that in this setting the process goes through a phase transition at the connectivity threshold, namely logn∕n<math><mrow is="true"><mo class="qopname" is="true">log</mo><mi is="true">n</mi><mo is="true">∕</mo><mi is="true">n</mi></mrow></math>. Furthermore, we say a graph G<math><mi is="true">G</mi></math> is λ<math><mi is="true">λ</mi></math>-expander if the second-largest absolute eigenvalue of its adjacency matrix is λ<math><mi is="true">λ</mi></math>. We prove that for a Δ<math><mi is="true">Δ</mi></math>-regular λ<math><mi is="true">λ</mi></math>-expander graph if λ∕Δ<math><mrow is="true"><mi is="true">λ</mi><mo is="true">∕</mo><mi is="true">Δ</mi></mrow></math> is sufficiently small, then the majority model by starting from 1∕2−δn<math><mrow is="true"><mfenced close=")" open="(" is="true"><mrow is="true"><mn is="true">1</mn><mo is="true">∕</mo><mn is="true">2</mn><mo is="true">−</mo><mi is="true">δ</mi></mrow></mfenced><mi is="true">n</mi></mrow></math> blue nodes (for an arbitrarily small constant δ>0<math><mrow is="true"><mi is="true">δ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></mrow></math>) results in fully red configuration in sub-logarithmically many rounds. Roughly speaking, this means the majority model is an “efficient” and “fast” density classifier on regular expanders. As a by-product of our results, we show that regular Ramanujan graphs are asymptotically optimally immune, that is for an n<math><mi is="true">n</mi></math>-node Δ<math><mi is="true">Δ</mi></math>-regular Ramanujan graph if the initial number of blue nodes is s≤βn<math><mrow is="true"><mi is="true">s</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≤</mo><mi is="true">β</mi><mi is="true">n</mi></mrow></math>, the number of blue nodes in the next round is at most cs∕Δ<math><mrow is="true"><mi is="true">c</mi><mi is="true">s</mi><mo is="true">∕</mo><mi is="true">Δ</mi></mrow></math> for some constants c,β>0<math><mrow is="true"><mi is="true">c</mi><mo is="true">,</mo><mi is="true">β</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></mrow></math>.