Effect of a static phase transition on searching dynamics

TL;DR

This paper investigates how a static phase transition at a critical point in a Euclidean network influences the efficiency of various search strategies, analyzing path lengths and success rates.

Contribution

It introduces and analyzes three search strategies on a preferential attachment network with a known static phase transition, highlighting their sensitivity to this transition.

Findings

Search strategies are significantly affected by the static phase transition.

Distributions of search path lengths are marginally affected by the transition.

Success rates vary with the static critical point.

Abstract

We consider a one dimensional Euclidean network which is grown using a preferential attachment. Here the $j$th incoming node gets attached to the $i$th existing node with the probability $\Pi_i \propto k_i {{l}}_{ij}^\alpha$, where ${l}_{ij}$ is the Euclidean distance between them and $k_i$ the degree of the $i$th node. This network is known to have a static phase transition point at $\alpha_c \simeq 0.5$. On this network, we employ three different searching strategies based on degrees or distances or both, where the possibility of termination of search chains is allowed. A detailed analysis shows that these strategies are significantly affected by the presence of the static critical point. The distributions of the search path lengths and the success rates are also estimated and compared for the different strategies. These distributions appear to be marginally affected by the static phase transition.