Degree product rule tempers explosive percolation in the absence of global information

Abstract

We introduce a guided network growth model, which we call the degree product rule process, that uses solely local information when adding new edges. For small numbers of candidate edges our process gives rise to a second-order phase transition, but becomes first-order in the limit of global choice. We provide the set of critical exponents required to characterize the nature of this percolation transition. Such a process permits interventions which can delay the onset of percolation while tempering the explosiveness caused by cluster product rule processes.Introduction.-Network based approaches continue to see growing applications in a wide array of fields, from epidemiology [1, 2] to finance [3,4], neuroscience [5,6], and machine learning [7]. As we increasingly rely on networks, understanding how they form out of complex conditions becomes all the more consequential [8][9][10][11][12]. Many the networks we entrust to support our modernized society-transportation, financial, social, etc.-are formed with some amount of agency, meaning that potential new members have control over how they connect and interact with the network. This agency can lead to markedly different behavior compared to the classical case of purely random network growth [13]. In particular, networks subject to competitive edge addition break time-reversal symmetry, as there is no well-defined method for running the process in reverse that achieves a statistically identical growth curve [14]. Furthermore, edge competition can be used as a means of control over cluster growth and connectivity within a growing network. Depending on the desired outcome (delayed connectivity for contagion spreading, increased connectivity for communication networks, etc.), intervening on growing networks can help produce more specialized and responsive networks.

Tuning the tricritical points of percolation transitions in random networks

“…In 2009, a seemingly discontinuous percolation transition, called explosive percolation [14], is found in an edge competitive percolation process where two candidate edges are considered and only the edge connecting two clusters with smaller product of their sizes is added. Inspired by this work, to achieve explosive percolation phenomena, various competitive percolation models [15][16][17][18][19][20][21][22][23][24][25][26] are proposed, as well as some weighted rules [27][28][29][30][31][32][33][34][35][36] choosing occupied edge according to a certain probability are introduced. Later, Riordan and Warnke [37] mathematically showed that any rule with fixed number of random vertices leads to a continuous phase transition in the thermodynamical limit, and that discontinuous phase transition indeed can occur if the number of the competitive edges grows with the system size.…”