Effect of Network Geometry and Interference on Consensus in Wireless Networks

Part of the Springer Optimization and Its Applications book series (SOIA, volume 40)


We study the convergence of the average consensus algorithm in wireless networks in the presence of interference. It is well known that convergence of the consensus algorithm improves with network connectivity. However, from a networking standpoint, highly connected wireless networks may have lower throughput because of increased interference. This raises an interesting question: what is the effect of increased network connectivity on the convergence of the consensus algorithm, given that this connectivity comes at the cost of lower network throughput? We address this issue for two types of networks: regular lattices with periodic boundary conditions, and a hierarchical network where a backbone of nodes arranged as a regular lattice supports a collection of randomly placed nodes. We characterize the properties of an optimal Time Division Multiple Access (TDMA) protocol that maximizes the speed of convergence on these networks, and provide analytical upper and lower bounds for the achievable convergence rate. Our results show that in an interference-limited scenario the fastest converging interconnection topology for the consensus algorithm crucially depends on the geometry of node placement. In particular, we prove that asymptotically in the number of nodes, forming long-range interconnections improves the convergence rate in one-dimensional tori, while it has the opposite effect in higher dimensions.


Wireless Network Time Slot Hierarchical Network Consensus Algorithm Node Placement 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blondel, V., Hendrickx, J., Olshevsky, A., Tsitsiklis, J.: Convergence in multiagent coordination, consensus and flocking. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 2996–3000 (2005) Google Scholar
  2. 2.
    Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Randomized gossip algorithms. IEEE Trans. Inf. Theory 52(6), 2508–2530 (2005) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Fang, L., Antsaklis, P.: On communication requirements for multi-agents consensus seeking. In: Proceedings of Workshop NESC05: University of Notre Dame. Lecture Notes in Control and Information Sciences (LNCIS), vol. 331, pp. 53–68. Springer, Berlin (2006) Google Scholar
  5. 5.
    Olfati Saber, R., Murray, R.M.: Consensus problems in networks of agents with switcing topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ren, W., Beard, R.W., McLain, T.W.: Coordination variables and consensus building in multiple vehicle systems. In: Proceedings of the Block Island Workshop on Cooperative Control. Lecture Notes in Control and Information Sciences, vol. 309, pp. 171–188. Springer, Berlin (2004) Google Scholar
  7. 7.
    Xiao, L., Boyd, S.: Fast linear iterations for distributed averaging. In: Proceedings of 42th IEEE Conference on Decision and Control, pp. 4997–5002 (2003) Google Scholar
  8. 8.
    Xiao, L., Boyd, S., Lall, S.: A scheme for robust distributed sensor fusion based on average consensus. In: Proceedings of International Conference on Information Processing in Sensor Networks, pp. 63–70 (2005) Google Scholar
  9. 9.
    Nedich, A., Olshevsky, A., Ozdaglar, A., Tsitsiklis, J.N.: On distributed averaging algorithms and quantization effects. LIDS Technical Report 2778, MIT, Lab. for Information and Decision Systems Google Scholar
  10. 10.
    Hovareshti, P., Gupta, V., Bars, J.S.: Average consensus over small world networks: a probabilistic framework. In: Proceedings of the IEEE Conference on Decision and Control (CDC’ 08), pp. 375–380 (December 2008). Google Scholar
  11. 11.
    Huang, M., Manton, J.H.: Stochastic double array analysis and convergence of consensus algorithms with noisy measurements. In: Proc. American Control Conference, New York, pp. 705–710, July 2007 Google Scholar
  12. 12.
    Huang, M., Manton, J.H.: Stochastic Lyapunov analysis for consensus algorithms with noisy measurements. In: Proc. American Control Conference, New York, pp. 1419–1424, July 2007 Google Scholar
  13. 13.
    Nedich, A., Ozdaglar, A.: Convergence rate for consensus with delays. LIDS Technical Report 2774, MIT, Lab. for Information and Decision Systems Google Scholar
  14. 14.
    Desai, M.P., Rao, V.B.: A new eigenvalue bound for reversible Markov chains with applications to the temperature-asymptotics of simulated annealing. Proc. IEEE Int. Symp. Circuits Syst. 2, 1211–1214 (1990) CrossRefGoogle Scholar
  15. 15.
    Seneta, E.: Nonnegative Matrices and Markov Chains, 2nd edn. Springer, Berlin (1981) Google Scholar
  16. 16.
    Gupta, P., Kumar, P.R.: The capacity of wireless networks. IEEE Trans. Inf. Theory 46(2), 388–404 (2000) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Liu, X., Haenggi, M.: Throughput analysis of fading sensor networks with regular and random topologies. EURASIP J. Wirel. Commun. Netw. 4, 554–564 (2005). Special Issue on Wireless Sensor Networks CrossRefGoogle Scholar
  18. 18.
    Xie, M., Haenggi, M.: Delay performance of different MAC schemes for multihop wireless networks. In: IEEE Global Communications Conference (GLOBECOM’05), St. Louis, MO, November 2005 Google Scholar
  19. 19.
    Vanka, S., Gupta, V., Haenggi, M.: Power-delay analysis of consensus algorithms on wireless networks with interference. Int. J. Syst. Control Commun. 2(1), 256–274 (2010) CrossRefGoogle Scholar
  20. 20.
    Penrose, M.: Random Geometric Graphs. Oxford University Press, London (2003) MATHCrossRefGoogle Scholar
  21. 21.
    Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Mixing times for random walks on geometric random graphs. In: SIAM Workshop on Analytic Algorithmics & Combinatorics (ANALCO), Vancouver, January 2005 Google Scholar
  22. 22.
    Carli, R., Fagnani, F., Speranzon, A., Zampieri, S.: Communication constraints in the average consensus problem. Automatica 44(3), 671–684 (2008) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Electrical EngineeringUniversity of Notre DameNotre DameUSA

Personalised recommendations