Unravelling small world networks

TY - JOUR

T1 - Unravelling small world networks

AU - Higham, D.J.

PY - 2003/9

Y1 - 2003/9

N2 - New classes of random graphs have recently been shown to exhibit the small world phenomenon - they are clustered like regular lattices and yet have small average pathlengths like traditional random graphs. Small world behaviour has been observed in a number of real life networks, and hence these random graphs represent a useful modelling tool. In particular, Grindrod [Phys. Rev. E 66 (2002) 066702-1] has proposed a class of range dependent random graphs for modelling proteome networks in bioinformatics. A property of these graphs is that, when suitably ordered, most edges in the graph are short-range, in the sense that they connect near-neighbours, and relatively few are long-range. Grindrod also looked at an inverse problem - given a graph that is known to be an instance of a range dependent random graph, but with vertices in arbitrary order, can we reorder the vertices so that the short-range/long-range connectivity structure is apparent? When the graph is viewed in terms of its adjacency matrix, this becomes a problem in sparse matrix theory: find a symmetric row/column reordering that places most nonzeros close to the diagonal. Algorithms of this general nature have been proposed for other purposes, most notably for reordering to reduce fill-in and for clustering large data sets. Here, we investigate their use in the small world reordering problem. Our numerical results suggest that a spectral reordering algorithm is extremely promising, and we give some theoretical justification for this observation via the maximum likelihood principle.

AB - New classes of random graphs have recently been shown to exhibit the small world phenomenon - they are clustered like regular lattices and yet have small average pathlengths like traditional random graphs. Small world behaviour has been observed in a number of real life networks, and hence these random graphs represent a useful modelling tool. In particular, Grindrod [Phys. Rev. E 66 (2002) 066702-1] has proposed a class of range dependent random graphs for modelling proteome networks in bioinformatics. A property of these graphs is that, when suitably ordered, most edges in the graph are short-range, in the sense that they connect near-neighbours, and relatively few are long-range. Grindrod also looked at an inverse problem - given a graph that is known to be an instance of a range dependent random graph, but with vertices in arbitrary order, can we reorder the vertices so that the short-range/long-range connectivity structure is apparent? When the graph is viewed in terms of its adjacency matrix, this becomes a problem in sparse matrix theory: find a symmetric row/column reordering that places most nonzeros close to the diagonal. Algorithms of this general nature have been proposed for other purposes, most notably for reordering to reduce fill-in and for clustering large data sets. Here, we investigate their use in the small world reordering problem. Our numerical results suggest that a spectral reordering algorithm is extremely promising, and we give some theoretical justification for this observation via the maximum likelihood principle.

KW - adjacency matrix

KW - bandwidth

KW - bioinformatics

KW - Cuthill-McKee

KW - proteome networks

KW - small world phenomenon

KW - sparse matrix

U2 - 10.1016/S0377-0427(03)00471-0

DO - 10.1016/S0377-0427(03)00471-0

M3 - Article

VL - 158

SP - 61

EP - 74

JO - Journal of Computational and Applied Mathematics

T2 - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

IS - 1

ER -