Abstract
Social networks often has the graph structure of giant strongly connected component (GSCC) and its upstream and downstream portions (IN and OUT), known as a bow-tie structure since a pioneering study on the World Wide Web (WWW). GSCC, on the other hand, has community structure, namely tightly knitted clusters, reflecting how the networks developed in time. By using our visualization of enhanced multidimensional scaling (MDS) and force-directed graph drawing for large and directed graphs, we discovered that a bow-tie in the WWW usually has clusters, which are locally-located mini bow-ties that are loosely connected to each other, resulting in a formation of GSCC as a whole. To quantify the mutual connectivity among such local bow-tie, we define a quantity to measure how a local bow-tie connects to others in comparison with random graphs. We found that there are striking difference between the WWW and other social and artificial networks including a million firms’ nationwide supply chain network in Japan and thousands of symbols’ dependency in the programming language of Emacs LISP, in which a global bow-tie exits. Presumably the difference comes from a self-similar structure and development of the WWW speculated by others.
Highlights
Two decades studies of complex networks have revealed some aspects of complex systems
Broder and colleagues found that the giant weakly connected component (GWCC) consists of the equal size of the giant strongly connected component (GSCC or SCC) (27.7%), the IN set (21.3%), the OUT set (21.2%), tendrils (21.5%), and tubes
Donato et al (2005, 2008) investigated three national (Italy, Indochina, and UK) webs and suggested that these tree national webs almost consist of the GSCC and the OUT set, and investigated the global web and suggested that the global web resembles the bow-tie structure (Donato et al 2005; 2008)
Summary
Two decades studies of complex networks have revealed some aspects of complex systems. Donato et al (2005, 2008) investigated three national (Italy, Indochina, and UK) webs (collected by the “Language Observatory Project" and the “Institute Informatica e Telematica) and suggested that these tree national webs almost consist of the GSCC and the OUT set, and investigated the global web (collected by the WebBase project at Stanford in 2001) and suggested that the global web resembles the bow-tie structure (Donato et al 2005; 2008) They investigated the detailed structure of the IN set and the OUT set and found that IN and OUT sets are fragmented into a large number of small and shallow “petals" (weakly connected components; WCCs) hanging from the GSCC. “Analysis and results” section is the main part of this paper, and we show the self-similar property, i.e., the bow-tie structures of each community in the web.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
