the very small world of the well-connected. (19 june 2008 ) lada adamic school of information...
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The Very Small World of the Well-connected.
(19 june 2008 )
Lada AdamicSchool of InformationUniversity of Michigan
Ann Arbor, MI [email protected]
Anna C. GilbertDepartment of Mathematics
University of MichiganAnn Arbor, MI 48109-1043
Xiaolin ShiDepartment of EECS
University of MichiganAnn Arbor, MI 48109-2121
Matthew BonnerDepartment of EECS
University of MichiganAnn Arbor, MI 48109-2121
School of Information. University of Michigan Ann Arbor, MI 48109-1107
PRELIMINARIES Importance measures Network datasets proprieties description
IMPORTANT VERTICES Network properties and important vertices Original vs. subgraph properties
Summary
Introduction
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Importance measuresLet the graph G (V,E ) have |V | = n
vertices
1 Degree D (vi ): Is the number of edges incident to vi. Degree reflects a local property of the
vertices in the graph.
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Importance measuresLet the graph G (V,E ) have |V | = n
vertices 1 Degree D (vi ).
2 Betweenness B (vi ) : a measure of how many pairs of vertices go
through vi in order to connect through shortest paths in G:
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Importance measuresLet the graph G (V,E ) have |V | = n vertices 1 Degree D (vi ). 2 Betweenness B (vi ).
3 Closeness C (vi ): a measure of the distances from all other
vertices in G to vertex vi closeness means that vertices that are in the
“middle” of the network are important.
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Importance measuresLet the graph G (V,E ) have |V | = n vertices 1 Degree D (vi ). 2 Betweenness B (vi ). 3 Closeness C (vi ). 4 PageRank :
a variant of the Eigenvector centrality measure and assigns greater importance to vertices that are themselves neighbors of important vertices
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Network datasets proprieties description
Data sets is a representative of web.
Data sets as an online social network data.
Data sets will be interested in examining the properties of important vertices and their graph synopsis.
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Network datasets proprieties description
prototypical random graph
1 Erdos-Renyi random graph : each pair of vertices having an equal
probability p of being joined by an edge. |V | = 10000 ; p = 0.001 ; d = p × |V | =
10.
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Network datasets proprieties description
prototypical random graph 1 Erdos-Renyi random graph. 2 Budyzoo dataset :
Considered as the first real-world network producing an undirected graph from AOL
Instant Messenger (AIM) Users >> Nodes Contact list >> edges
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Network datasets proprieties description prototypical random graph
1 Erdos-Renyi random graph. 2 Budyzoo dataset. 3 TREC (Text REtrieval Conference).
Considered as the second real-world graph is a network of blog connections It is a crawl of 100,649 RSS and Atom feeds collected
The TREC dataset contains Hyperlinks, comments, trackbacks, etc.
removed feeds and feeds without a homepage or permalinks are. over 300 Technorati tags. which are in fact automatically
generated are not true indicators of social linking.
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Network datasets proprieties description
prototypical random graph 1 Erdos-Renyi random graph. 2 Budyzoo dataset. 3 TREC (Text REtrieval Conference). 4 Web graph dataset
259,794 websites 50 million pages Collected in 1998
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Network datasets proprieties description prototypical random graph
1 Erdos-Renyi random graph. 2 Budyzoo dataset. 3 TREC (Text REtrieval Conference). 4 Web graph dataset
PRELIMINARIES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
recent blog datasets
the decade old website-level data
set
== Similarity ==applicable to larger, ore current webcrawls
Network properties and important vertices 1 Degree distributions.
The degree distributions of online networks
IMPORTANT VERTICES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Type : social networks
due to the limitation of The data sampling
Network properties and important vertices
1 Degree distributions. 2 Correlation of importance values of
different measures. relationships of importance measures in
different networks. Analysis of correlation
Higher : degree, betweenness and PageRank Lower : closeness.
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
IMPORTANT VERTICES
Network properties and important vertices
1 Degree distributions. 2 Correlation of importance values of different measures.
3 Assortativity. The concept of assortativity or
assortative mixing is defined as the preference of the vertices in a network to have edges with others that are similar.
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
IMPORTANT VERTICES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
•Assortativity :
Important vertices in their subgraphs.
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
ConnectivityThe Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
DensityThe Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Network properties and important vertices
1 Degree distributions. 2 Correlation of importance values of different
measures. Assortativity.
3 Important vertices in their subgraphs. Connectivity Density
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
IMPORTANT VERTICES
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
IMPORTANT VERTICES
Original vs. subgraph properties 1 Density 2 distance. 3 Relative importance.
DensityThe Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
distanceThe Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Relative importanceThe Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Original vs. subgraph properties 1 Density.
2 distance.
3 Relative importance.
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
IMPORTANT VERTICES
two overall observations about the four networks: Different importance measures yield subgraphs of varying
density and topology However, in spite of these differences, “important
vertices” in the online networks have some properties that agree with each other
Thus, we know that in the real online networks, in contrast to random graph model
the important vertices tend to preserve information about the relationships among important vertices
we can use the subgraphs to study the properties of important
vertices in the original graphs.
The Very Small World of the Well-connected.
School of Information. University of Michigan Ann Arbor, MI 48109-1107
Summary and conclusion