Technical Report CSTN-083 Detecting and Labelling Wireless Community Network Structures from Eigen-spectra K.A. Hawick Institute of Information and Mathematical Sciences Massey University – Albany, North Shore 102-904, Auckland, New Zealand Email: k.a.hawick@massey.ac.nz Tel: +64 9 414 0800 Fax: +64 9 441 8181 March 2010 Abstract Wireless and ad hoc networks often have local regions of highly intra-connected modules or communities of nodes against a backdrop of sparser longer-range inter- community connectivities. We report on empirical observations and a prototype algorithm for correctly counting and labelling the number of dense community structures or modules within a fully connected network. We describe the eigen-spectral method applied to the Laplacian characteristic matrix. We illustrate the ef- fect of different network characteristics on the eigen- spectrum and on properties of the eigenvectors corre- sponding to the N c smallest eigenvalues in networks with N c such dense community modules. Keywords: wireless networks; ad hoc networks; community structure; modules; eigen-spectra; smallest eigen-values. 1 Introduction Wireless networks are often highly complex systems [1,2] that are difficult to analyse for optimal design. Im- portant considerations for the design of mobile, wireless networks [3] include: maintaining connectivity when nodes move; deploying resources only where required [4]; and managing power consumption [5]. Many ad hoc and mobile wireless device deployments give rise to highly structured connectivity patterns based upon localised regions or groups of units that are highly intra-connected amongst nodes of the same group, but where the groups themselves may be only sparsely inter- connected. Such networks are fully connected in the sense that there are path-ways from any node to any other node, but have variations in connectivity strength Figure 1: Spatial deployment pattern of connected de- vices on Massey’s Albany campus, divided into 15 mod- ular communities. such that nodes form tightly connected communities or modules with sparser “trunk routes” connecting com- munities. This sort of community or modular structure within a fully-connected graph [6] is common in a variety of ap- plication contexts including: the Internet and World- Wide-Web; social and battle deployment situations; and also in gene-sequenced and neuronal biological net- works. Mobile and wireless networks are particularly interesting however as we can easily visualise a spatial pattern [7] amongst the nodes of the network that is separate from the actual links or edges connecting nodes together. The actual bandwidth or number of packet hops can therefore be embedded in a spatial coordinate system but is completely separate from it in terms of its connectivity properties. Figure 1 shows a possible spa- 1