A graph theoretic approach to ultrafast information distribution: Borel Cayley graph resizing algorithm

A graph theoretic approach is proposed to formulate communication graphs that enable ultrafast information distribution. In our earlier work, we reported that Borel Cayley graph (BCG) is a promising candidate as a logical topology for fast information distribution. However, the practical applications of BCG have been challenging because of its inflexible sizes. In this paper, we propose a simple but effective graph resizing algorithm that removes nodes from an oversized BCG to achieve a desired network size. The proposed resizing algorithm consists of two parts: a random pruning algorithm that identifies nodes to be removed uniformly at random; and a novel cut-through rewiring (CTR) algorithm that rewires the remaining nodes. The proposed resizing algorithm preserves the superior properties of the original BCGs, including a small diameter, a short average path length, a large algebraic connectivity, and ultrafast information distribution performance. Furthermore, analytical formulae were derived to compute the graph disconnection probability of the BCGs after resizing. Analytical results showed that the resized graphs are almost surely connected even after 80~90% size reduction, depending on the original BCG size.

[1]  Beom Jun Kim,et al.  Factors that predict better synchronizability on complex networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  R. Olfati-Saber,et al.  Consensus Filters for Sensor Networks and Distributed Sensor Fusion , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[3]  Tian He,et al.  Differentiated surveillance for sensor networks , 2003, SenSys '03.

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

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

[6]  Shashi Shekhar,et al.  Environmental Sensor Networks , 2008, ACM SIGSPATIAL International Workshop on Advances in Geographic Information Systems.

[7]  Christian Bettstetter,et al.  On the Connectivity of Ad Hoc Networks , 2004, Comput. J..

[8]  K. Wendy Tang,et al.  Representations and routing for Cayley graphs [computer networks] , 1991, IEEE Trans. Commun..

[9]  Kumar N. Sivarajan,et al.  Lightwave networks based on de Bruijn graphs , 1994, TNET.

[10]  Richard M. Murray,et al.  Information flow and cooperative control of vehicle formations , 2004, IEEE Transactions on Automatic Control.

[11]  Adilson E Motter,et al.  Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? , 2003, Physical review letters.

[12]  Mauricio Barahona,et al.  Synchronization in small-world systems. , 2002, Physical review letters.

[13]  A. Uhlig S. Jamakovic,et al.  On the relationships between topological metrics in real-world networks , 2007 .

[14]  K. Wendy Tang,et al.  Dense and Symmetric Graph Formulation and Generation for Wireless Information Networks , 2009, 2009 International Conference on Wireless Networks and Information Systems.

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

[16]  K. Wendy Tang,et al.  Representations of Borel Cayley Graphs , 1993, SIAM J. Discret. Math..

[17]  George W. Polites,et al.  An introduction to the theory of groups , 1968 .

[18]  B. Bollobás.  The evolution of random graphs , 1984 .

[19]  K. Wendy Tang,et al.  Diagonal and Toroidal Mesh Networks , 1994, IEEE Trans. Computers.

[20]  Sheldon M. Ross,et al.  Introduction to probability models , 1975 .

[21]  K. Wendy Tang,et al.  Vertex-transitivity and routing for Cayley graphs in GCR representations , 1992, SAC '92.

[22]  Paolo Santi.  Topology control in wireless ad hoc and sensor networks , 2005 .

[23]  Michael William Newman,et al.  The Laplacian spectrum of graphs , 2001 .

[24]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[25]  Deborah Estrin,et al.  Impact of network density on data aggregation in wireless sensor networks , 2002, Proceedings 22nd International Conference on Distributed Computing Systems.

[26]  Sheldon M. Ross,et al.  Introduction to Probability Models, Eighth Edition , 1972 .

[27]  Mark Newman,et al.  Detecting community structure in networks , 2004 .

[28]  Dmitri Loguinov,et al.  Graph-theoretic analysis of structured peer-to-peer systems: routing distances and fault resilience , 2003, IEEE/ACM Transactions on Networking.

[29]  David Lazer,et al.  The Network Structure of Exploration and Exploitation , 2007 .

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

[31]  Wolfgang Nejdl,et al.  HyperCuP - Hypercubes, Ontologies, and Efficient Search on Peer-to-Peer Networks , 2002, AP2PC.

[32]  Sonia Martínez,et al.  Robust rendezvous for mobile autonomous agents via proximity graphs in arbitrary dimensions , 2006, IEEE Transactions on Automatic Control.

[33]  R. Olfati-Saber.  Ultrafast consensus in small-world networks , 2005, Proceedings of the 2005, American Control Conference, 2005..

[34]  K. Wendy Tang,et al.  Pseudo-Random Graphs for Fast Consensus Protocol , 2009, PDPTA.

[35]  P. Varaiya,et al.  Sensor Networks for Monitoring Traffic , 2004 .

[36]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[37]  Sergio D. Servetto,et al.  Asymptotically optimal time synchronization in dense sensor networks , 2003, WSNA '03.

[38]  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..

[39]  Jie Lin,et al.  The multi-agent rendezvous problem , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[40]  Francesc Comellas,et al.  Synchronizability of complex networks , 2007 .

[41]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[42]  Daniel A. Spielman,et al.  Fault diagnosis in a small constant number of parallel testing rounds , 1993, SPAA '93.

[43]  JOSEP DÍAZ,et al.  A survey of graph layout problems , 2002, CSUR.

[44]  K. Wendy Tang,et al.  Degree Constrained Topology Control for Very Dense Wireless Sensor Networks , 2010, 2010 IEEE Global Telecommunications Conference GLOBECOM 2010.

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

[46]  K. Wendy Tang,et al.  Representations and Routing for Cayley Graphs , 2009 .

[47]  Abhijit Sengupta,et al.  On a Lightwave Network Topology Using Kauts Digraphs , 1999, IEEE Trans. Computers.

[48]  R. Olfati-Saber,et al.  Algebraic Connectivity Ratio of Ramanujan Graphs , 2007, 2007 American Control Conference.

[49]  Charles Delorme,et al.  Strategies for Interconnection Networks: Some Methods from Graph Theory , 1986, J. Parallel Distributed Comput..

[50]  Cem Ersoy,et al.  MAC protocols for wireless sensor networks: a survey , 2006, IEEE Communications Magazine.

[51]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[52]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[53]  Masafumi Yamashita,et al.  Distributed memoryless point convergence algorithm for mobile robots with limited visibility , 1999, IEEE Trans. Robotics Autom..