12,906 research outputs found
Graphs, Friends and Acquaintances
As is well known, a graph is a mathematical object modeling the
existence of a certain relation between pairs of elements of a given set.
Therefore, it is not surprising that many of the first results concerning
graphs made reference to relationships between people or groups of
people. In this article, we comment on four results of this kind, which
are related to various general theories on graphs and their applications:
the Handshake lemma (related to graph colorings and Boolean
algebra), a lemma on known and unknown people at a cocktail party
(to Ramsey theory), a theorem on friends in common (to distanceregularity
and coding theory), and Hall’s Marriage theorem (to the
theory of networks). These four areas of graph theory, often with
problems which are easy to state but difficult to solve, are extensively
developed and currently give rise to much research work. As examples
of representative problems and results of these areas, which are
discussed in this paper, we may cite the following: the Four Colors
Theorem (4CTC), the Ramsey numbers, problems of the existence of
distance-regular graphs and completely regular codes, and finally the
study of topological proprieties of interconnection networks.Preprin
Extraction and Analysis of Facebook Friendship Relations
Online Social Networks (OSNs) are a unique Web and social phenomenon, affecting tastes and behaviors of their users and helping them to maintain/create friendships. It is interesting to analyze the growth and evolution of Online Social Networks both from the point of view of marketing and other of new services and from a scientific viewpoint, since their structure and evolution may share similarities with real-life social networks. In social sciences, several techniques for analyzing (online) social networks have been developed, to evaluate quantitative properties (e.g., defining metrics and measures of structural characteristics of the networks) or qualitative aspects (e.g., studying the attachment model for the network evolution, the binary trust relationships, and the link prediction problem).\ud
However, OSN analysis poses novel challenges both to Computer and Social scientists. We present our long-term research effort in analyzing Facebook, the largest and arguably most successful OSN today: it gathers more than 500 million users. Access to data about Facebook users and their friendship relations, is restricted; thus, we acquired the necessary information directly from the front-end of the Web site, in order to reconstruct a sub-graph representing anonymous interconnections among a significant subset of users. We describe our ad-hoc, privacy-compliant crawler for Facebook data extraction. To minimize bias, we adopt two different graph mining techniques: breadth-first search (BFS) and rejection sampling. To analyze the structural properties of samples consisting of millions of nodes, we developed a specific tool for analyzing quantitative and qualitative properties of social networks, adopting and improving existing Social Network Analysis (SNA) techniques and algorithms
Classes of behavior of small-world networks
Small-world networks are the focus of recent interest because they appear to
circumvent many of the limitations of either random networks or regular
lattices as frameworks for the study of interaction networks of complex
systems. Here, we report an empirical study of the statistical properties of a
variety of diverse real-world networks. We present evidence of the occurrence
of three classes of small-world networks: (a) scale-free networks,
characterized by a vertex connectivity distribution that decays as a power law;
(b) broad-scale networks, characterized by a connectivity distribution that has
a power-law regime followed by a sharp cut-off; (c) single-scale networks,
characterized by a connectivity distribution with a fast decaying tail.
Moreover, we note for the classes of broad-scale and single-scale networks that
there are constraints limiting the addition of new links. Our results suggest
that the nature of such constraints may be the controlling factor for the
emergence of different classes of networks
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