Maximizing Algebraic Connectivity in the Space of Graphs With a Fixed Number of Vertices and Edges

The second smallest eigenvalue of the Laplacian matrix, also known as the algebraic connectivity, characterizes the performance of some dynamic processes on networks, such as consensus in multiagent networks, synchronization of coupled oscillators, random walks on graphs, and so on. In a multiagent network, for example, the larger the algebraic connectivity of the graph representing interactions between agents is, the faster the convergence speed of a representative consensus algorithm is. This paper tackles the problem of finding graphs that maximize or locally maximize the algebraic connectivity in the space of graphs with a fixed number of vertices and edges. It is shown that some well-known classes of graphs such as star graphs, cycle graphs, complete bipartite graphs, and circulant graphs are algebraic connectivity maximizers or local maximizers under certain conditions.

[1]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

[2]  Stephen P. Boyd,et al.  A scheme for robust distributed sensor fusion based on average consensus , 2005, IPSN 2005. Fourth International Symposium on Information Processing in Sensor Networks, 2005..

[3]  Sandro Zampieri,et al.  Average consensus with packet drop communication , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[4]  Marc Moonen,et al.  Seeing the Bigger Picture: How Nodes Can Learn Their Place Within a Complex Ad Hoc Network Topology , 2013, IEEE Signal Processing Magazine.

[5]  Caterina M. Scoglio,et al.  Optimizing algebraic connectivity by edge rewiring , 2013, Appl. Math. Comput..

[6]  Richard M. Murray,et al.  INFORMATION FLOW AND COOPERATIVE CONTROL OF VEHICLE FORMATIONS , 2002 .

[7]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[8]  Reza Olfati-Saber,et al.  Flocking for multi-agent dynamic systems: algorithms and theory , 2006, IEEE Transactions on Automatic Control.

[9]  Lin Huang,et al.  Consensus of Multiagent Systems and Synchronization of Complex Networks: A Unified Viewpoint , 2016, IEEE Transactions on Circuits and Systems I: Regular Papers.

[10]  Daizhan Cheng,et al.  Leader-following consensus of multi-agent systems under fixed and switching topologies , 2010, Syst. Control. Lett..

[11]  Albert Y. Zomaya,et al.  On the Characterization of the Structural Robustness of Data Center Networks , 2013, IEEE Transactions on Cloud Computing.

[12]  Damon Mosk-Aoyama,et al.  Maximum algebraic connectivity augmentation is NP-hard , 2008, Operations Research Letters.

[13]  T. D. Morley,et al.  Eigenvalues of the Laplacian of a graph , 1985 .

[14]  Christoph Maas,et al.  Transportation in graphs and the admittance spectrum , 1987, Discret. Appl. Math..

[15]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[16]  Randal W. Beard,et al.  Consensus seeking in multiagent systems under dynamically changing interaction topologies , 2005, IEEE Transactions on Automatic Control.

[17]  Piet Van Mieghem,et al.  Graph Spectra for Complex Networks , 2010 .

[18]  Francesca Cuomo,et al.  Keeping the connectivity and saving the energy in the internet , 2011, 2011 IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS).

[19]  Norikazu Takahashi.  On Topology of Networked Multi-Agent Systems for Fast Consensus , 2011 .

[20]  Stephen P. Boyd,et al.  Fast linear iterations for distributed averaging , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[21]  Dengfeng Sun,et al.  Algebraic connectivity maximization of an air transportation network: The flight routes' addition/deletion problem , 2012 .

[22]  Theodore Kolokolnikov,et al.  Maximizing algebraic connectivity for certain families of graphs , 2014, ArXiv.

[23]  Z. Duan,et al.  Network synchronizability analysis: a graph-theoretic approach. , 2008, Chaos.

[24]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[25]  Mehran Mesbahi,et al.  Agreement over random networks , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[26]  Shaun M. Fallat,et al.  Extremizing algebraic connectivity subject to graph theoretic constraints , 1998 .

[27]  Alexandre M. Bayen,et al.  Optimal network topology design in multi-agent systems for efficient average consensus , 2010, 49th IEEE Conference on Decision and Control (CDC).

[28]  Josef Leydold,et al.  Graphs of given order and size and minimum algebraic connectivity , 2012 .

[29]  Piet Van Mieghem,et al.  Algebraic connectivity optimization via link addition , 2008, BIONETICS.

[30]  Robert E. Kooij,et al.  Graphs with given diameter maximizing the algebraic connectivity , 2010 .

[31]  Yongsun Kim.  Bisection algorithm of increasing algebraic connectivity by adding an edge , 2009 .

[32]  Jonathan H. Manton,et al.  Coordination and Consensus of Networked Agents with Noisy Measurements: Stochastic Algorithms and Asymptotic Behavior , 2009, SIAM J. Control. Optim..

[33]  Stephen P. Boyd,et al.  Growing Well-connected Graphs , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[34]  N. Abreu.  Old and new results on algebraic connectivity of graphs , 2007 .

[35]  P. Hansen,et al.  Variable Neighborhood Search for Extremal Graphs. XI. Bounds on Algebraic Connectivity , 2004 .

[36]  M. A. Muñoz,et al.  Optimal network topologies: expanders, cages, Ramanujan graphs, entangled networks and all that , 2006, cond-mat/0605565.

[37]  Pierre Hansen,et al.  Variable Neighborhood Search for Extremal Graphs: IV: Chemical Trees with Extremal Connectivity Index , 1998, Comput. Chem..

[38]  James P. G. Sterbenz,et al.  Cost-efficient algebraic connectivity optimisation of backbone networks , 2014, Opt. Switch. Netw..

[39]  Ran Dai,et al.  Optimal topology design for dynamic networks , 2011, IEEE Conference on Decision and Control and European Control Conference.

[40]  Jean-Charles Delvenne,et al.  Optimal strategies in the average consensus problem , 2007, 2007 46th IEEE Conference on Decision and Control.

[41]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[42]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

[43]  Francesca Cuomo,et al.  Network pruning for energy saving in the Internet , 2012, Comput. Networks.

[44]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[45]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[46]  Dengfeng Sun,et al.  Algebraic Connectivity Maximization for Air Transportation Networks , 2014, IEEE Transactions on Intelligent Transportation Systems.

[47]  Mehran Mesbahi,et al.  On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian , 2006, IEEE Transactions on Automatic Control.

[48]  Stephen P. Boyd,et al.  Upper bounds on algebraic connectivity via convex optimization , 2006 .

[49]  Richard M. Murray,et al.  Consensus Protocols for Undirected Networks of Dynamic Agents with Communication Time-Delays , 2003 .