## Abstract

Current analyses of complex biological networks focus either on their global statistical connectivity properties (e.g. topological path lengths and nodes connectivity ranks) or the statistics of specific local connectivity circuits (motifs). Here we present a different approach - Functional Topology, to enable identification of hidden topological and geometrical fingerprints of biological computing networks that afford their functioning - the form-function fingerprints. To do so we represent the network structure in terms of three matrices: 1. Topological connectivity matrix - each row (i) is the shortest topological path lengths of node (i) with all other nodes; 2. Topological correlation matrix - the element (i,j) s the correlation between the topological connectivity of nodes (i) and (j); and 3. Weighted graph matrix - in this case the links represent the conductance between nodes that can be simply one over the geometrical length, the synaptic strengths in case of neural networks or other quantity that represents the strengths of the connections. Various methods (e.g. clustering algorithms, random matrix theory, eigenvalues spectrum etc.), can be used to analyze these matrices, here we use the newly developed functional holography approach which is based on clustering of the matrices following their collective normalization. We illustrate the approach by analyzing networks of different topological and geometrical properties: 1. Artificial networks, including - random, regular 4-fold and 5-fold lattice and a tree-like structure; 2. Cultured neural networks: A single network and a network composed of three linked sub-networks; and 3. Model neural network composed of two overlapping sub-networks. Using these special networks, we demonstrate the method so ability to reveal functional topology features of the networks.

Original language | English |
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Pages (from-to) | 339-361 |

Number of pages | 23 |

Journal | Natural Computing |

Volume | 4 |

Issue number | 4 |

DOIs | |

State | Published - 1 Sep 2005 |

## Keywords

- Connectivity networks
- Functional topology
- Graph theory
- Information processing
- Scale free networks
- Small word networks
- Topological correlations