Reaction-diffusion processes on interconnected scale-free networks

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Reaction-diffusion processes on interconnected scale-free networks

Antonios Garas

Phys. Rev. E 92, 020801(R) – Published 10 August 2015

Abstract

We study the two-particle annihilation reaction $A + B \to \emptyset$ on interconnected scale-free networks, using different interconnecting strategies. We explore how the mixing of particles and the process evolution are influenced by the number of interconnecting links, by their functional properties, and by the interconnectivity strategies in use. We show that the reaction rates on this system are faster than what was observed in other topologies, due to the better particle mixing that suppresses the segregation effect, in line with previous studies performed on single scale-free networks.

Received 25 July 2014
Revised 9 July 2015

DOI:https://doi.org/10.1103/PhysRevE.92.020801

Authors & Affiliations

Antonios Garas^*

Chair of Systems Design, ETH Zurich, Weinbergstrasse 56/58, 8092 Zurich, Switzerland

^*agaras@ethz.ch

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Vol. 92, Iss. 2 — August 2015

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Figure 1
(a) Illustration of the three different interconnectivity strategies between two scale-free networks. The CC strategy connects hub nodes from one network to hub nodes from the other, the CP strategy connects hub nodes from one network to peripheral nodes in the other, and the PP strategy connects peripheral nodes from one network to peripheral nodes in the other. (b)–(d) Reaction progress $1 / ρ - 1 / ρ_{0}$ as a function of time for the three different interconnecting strategies of $n$ interconnections, applied on interconnected SFNs with $γ = 2.5, 3.0, 3.5$ and $q = 0.2$ . Open symbols show results for the well-mixed case, while dashed lines show results for the polarized case. The results are averaged over 100 realizations and the standard errors are smaller than the size of the symbols. The solid lines show the best fit at the asymptotic limit.
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Figure 2
Reaction progress $1 / ρ - 1 / ρ_{0}$ as a function of time for the well-mixed case with(a) and (i) $n$ interconnections and (b) and (j) $t$ interconnections and for the polarized case with (e) and (m) $n$ interconnections and (f) and (n) $t$ interconnections. Also shown is the ratio $Q_{A B}$ over time for the well-mixed case with (c) and (k) $n$ interconnections and (d) and (l) $t$ interconnections and for the polarized case with (g) and (o) $n$ interconnections and (h) and (p) $t$ interconnections. All cases are for two coupled SFNs with $γ = 3.0$ . The results are averaged over 100 realizations and the standard errors are smaller than the size of the symbols. Solid lines in (a), (b), (e), (f), (i), (j), (m), and (n) show the $1 / ρ \sim t$ behavior and the dashed line at $Q_{A B} = 1$ in (c), (d), (g), (h), (k), (l), (o), and (p) corresponds to perfect mixing. The left column shows that the fraction of interconnecting links is $q = 0.2$ arranged according to the CC (circles), CP (triangles), and PP (squares) strategies. The right column shows that the fraction of interconnecting links is $q = 0.1$ (circles), $q = 0.2$ (triangles), and $q = 0.5$ (squares), arranged using a CC strategy.
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Figure 3
Reaction progress $1 / ρ - 1 / ρ_{0}$ as a function of time for the well-mixed case with (a) $n$ interconnections and (b) $t$ interconnections and for the polarized case with (e) $n$ interconnections and (f) $t$ interconnections. Ratio $Q_{A B}$ over time for the well-mixed case with (c) $n$ interconnections and (d) $t$ interconnections and for the polarized case with (g) $n$ interconnections and (h) $t$ interconnections. All cases are for two coupled SFNs with $γ = 3.0$ , with a fixed number $L = 100$ (symbols) or $L = 1000$ (lines) of interconnecting links. The results are averaged over 100 realizations and the standard errors are smaller than the size of the symbols.
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