# Extraction of Components from a Social Network-A Graph Theoretic Approach.

December 2002

## Date of Award

Winter 12-12-2003

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Computer and Statistical Services Centre (CSSC)

## Abstract (Summary of the Work)

Social Network is a graph model that represents the relationship among the members of a community. For example, let us consider a village that has a number of households. If the members of a house commu- nicate with another house, a directional edge is provided from the first to the second. If both the houses approach each other, then the edge is bidirectional. This way, the inter-relationship among the households of a village gives rise to a digraph. So a collection of villages would be represented by a collection of such digraphs. The social scientists are interested to extract certain properties of these digraphs in order to compare the social structures of the villages under study. The prop- erties are extracted either from the adjacency matrix of the digraph representing a community or by a suitably designed graph algorithm. The properties may be enumerated as :1. A community may have a few members who have no connection with the other members of the community. These nodes can be easily identified in the adjancency matrix. The corresponding row and col- umn for any such ISOLATOR(an absolutely isolated member of a community) would have only zero entries signifying that both the inde- gree and outdegree for such a node are equal to zero. If a community has high percentage of ISOLATORS, it has possibly a new settlement. 2. Since all the members of a community may not communicate with one another, it is possible that a number of disjoint subgraphs may be present in the social network. These disjoint subgraphs are the Weakly Connected Components (WCC) The nodes of a WCC are somwhow connected but all the nodes may not be reachable from all others. Larger the size of the WCCS compared to the size of the actual community, higher is the cohesiveness of the community it is representing. An ideal society, would have a single WCC covering all the nodes of the community. 3. Each WCC within the network is considered as a group of mem- bers within the community who are socially close to each other. Each WCC is a digraph by itself. If in a WCC, each node is reachable from each other node, it is a "Strongly Connected Component (SCC)". Many algorithms are available for finding SCCS in a graph [1,2,3,4,5]. However, in a social network, an SCC shows properties usually not present in other graphs. Since in a social network bidirectional links may be present between any two nodes, it is possible that an SCC or a WCC may have nested cycles. The standard cycle detection algorithms usually fail to identify these nested cycles.This disscrtation first identifics the ISOLATORS and then augments the adjancency matrix removing the rows and columns belonging to the ISOLATORS.The augmented matrix is then used to find the WCCS present in the network. The dissertation then offers an algorithm to find the different cycles (including the nested cycles) present in a social network.Two households in a community may contact each other for more than one purpose. It may be for economic or social or for any other purpose. In such a case two nodes may be connected by more than one

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843085

ISI-DISS-2002-95