TY - CHAP

T1 - Distributed Averaging in Opinion Dynamics

AU - Berenbrink, Petra

AU - Cooper, Colin

AU - Gava, Cristina

AU - Kohan Marzagão, David

AU - Mallmann-Trenn, Frederik

AU - Radzik, Tomasz

AU - Rivera, Nicolás

N1 - Funding Information:
P. Berenbrink was in part supported by the Deutsche Forschungs-gemeinschaft (DFG) - Project number 491453517 and number DFG-FOR 2975. F. Mallmann-Trenn was in part supported by the EPSRC grant EP/W005573/1. N. Rivera was supported by ANID FONDE-CYT 3210805 and ANID SIA 85220033.
Funding Information:
P. Berenbrink was in part supported by the Deutsche Forschungsgemeinschaft (DFG) - Project number 491453517 and number DFGFOR 2975. F. Mallmann-Trenn was in part supported by the EPSRC grant EP/W005573/1. N. Rivera was supported by ANID FONDECYT 3210805 and ANID SIA 85220033.
Publisher Copyright:
© 2023 ACM.

PY - 2023/6/16

Y1 - 2023/6/16

N2 - We consider two simple asynchronous opinion dynamics on arbitrary graphs where every node u of the graph has an initial value ζu(0). In the first process, which we call the NodeModel, at each time step t ≥ 0, a random node u and a random sample of k of its neighbours v1, v2, ⋯ , vk are selected. Then, u updates its current value ζu(t) to [EQUATION], where α ĝ (0, 1) and k ≥ 1 are parameters of the process. In the second process, called the EdgeModel, at each step a random pair of adjacent nodes (u, v) is selected, and then node u updates its value equivalently to the NodeModel with k = 1 and v as the selected neighbour.For both processes, the values of all nodes converge to the same value F, which is a random variable depending on the random choices made in each step. For the NodeModel and regular graphs, and for the EdgeModel and arbitrary graphs, the expectation of F is the average of the initial values [EQUATION]. For the NodeModel and non-regular graphs, the expectation of F is the degree-weighted average of the initial values.Our results are two-fold. We consider the concentration of F and show tight bounds on the variance of F for regular graphs. We show that when the initial values do not depend on the number of nodes, then the variance is negligible, and hence the nodes are able to estimate the initial average of the node values. Interestingly, this variance does not depend on the graph structure. For the proof we introduce a duality between our processes and a process of two correlated random walks. We also analyse the convergence time for both models and for arbitrary graphs, showing bounds on the time Tϵ required to make all node values 'ϵ-close' to each other. Our bounds are asymptotically tight under some assumptions on the distribution of the initial values.

AB - We consider two simple asynchronous opinion dynamics on arbitrary graphs where every node u of the graph has an initial value ζu(0). In the first process, which we call the NodeModel, at each time step t ≥ 0, a random node u and a random sample of k of its neighbours v1, v2, ⋯ , vk are selected. Then, u updates its current value ζu(t) to [EQUATION], where α ĝ (0, 1) and k ≥ 1 are parameters of the process. In the second process, called the EdgeModel, at each step a random pair of adjacent nodes (u, v) is selected, and then node u updates its value equivalently to the NodeModel with k = 1 and v as the selected neighbour.For both processes, the values of all nodes converge to the same value F, which is a random variable depending on the random choices made in each step. For the NodeModel and regular graphs, and for the EdgeModel and arbitrary graphs, the expectation of F is the average of the initial values [EQUATION]. For the NodeModel and non-regular graphs, the expectation of F is the degree-weighted average of the initial values.Our results are two-fold. We consider the concentration of F and show tight bounds on the variance of F for regular graphs. We show that when the initial values do not depend on the number of nodes, then the variance is negligible, and hence the nodes are able to estimate the initial average of the node values. Interestingly, this variance does not depend on the graph structure. For the proof we introduce a duality between our processes and a process of two correlated random walks. We also analyse the convergence time for both models and for arbitrary graphs, showing bounds on the time Tϵ required to make all node values 'ϵ-close' to each other. Our bounds are asymptotically tight under some assumptions on the distribution of the initial values.

UR - http://www.scopus.com/inward/record.url?scp=85164022291&partnerID=8YFLogxK

U2 - 10.1145/3583668.3594593

DO - 10.1145/3583668.3594593

M3 - Conference paper

T3 - Proceedings of the Annual ACM Symposium on Principles of Distributed Computing

SP - 211

EP - 221

BT - Proceedings of the 2023 ACM Symposium on Principles of Distributed Computing

PB - ACM

ER -