Analysis and Modeling of Social Networks Foudalis Ilias

Introduction Online social networks have become a ubiquitous part of everyday life Opportunity to study social interactions in a large-scale worldwide environment Why model such networks?  Understand their evolution and formation  Improve current systems and build better applications  Advance the state of the art in closely related fields (such as diffusion of information)

Social and Information Networks Social Networks  Mainly undirected graphs  Connect people  Nodes with more similar degrees (limited capacity of social ties) Information Networks  Tend to be directed graphs  Connect web pages or other units of information  Few nodes with extremely large number of incoming links

Statistical characteristics of social networks Exhibit small diameter and small average path length  Also known as the “small world phenomenon” Clustering coefficients tend to be larger Distribution of nodes tend to exhibit fat tails High degree nodes tend to be connected with other high degree nodes Neighbors of a high degree node are less likely to be connected with each other

Related work Internet  Wats and Strogatz (1998), simple model that exhibits small world characteristics  Barabasi and Albert (1999), preferential attachment models, power law distributions  Kumar et al. (2000), link copying model, power law distributions  Klemm, Eguiluz (2002), preferential attachment with fertile nodes, small world properties Social Networks  Jackson and Rogers (2006), random meetings and local search  Kumar et al. (2006), preferential attachment, different types of nodes

Our algorithm, General Description People by default are part of certain groups A person will have a high chance to connect to people in the same group People also make connections to people they meet at random  To capture this effect we introduce random walks  In a random walk a person will have a higher chance to connect with social or famous persons  As time passes “older” persons will do less random walks

Our algorithm, Group Formation First Pass Clique Formation

Our algorithm, Group Formation Second Pass Clique Formation

Our Algorithm, Group Formation Clique generation (Imaginary graph)  For FIRST_PASS times While the total number of nodes in cliques are less than N  Get m nodes and put them in a clique  m will be chosen according to a power law distribution with exponent γ  Let M be the number of cliques generated from the first pass  For M times Get m nodes and put them in a clique m will be chosen according to a power law distribution with exponent γ

Our Algorithm, Graph Generation Connection to groups  At each time step t a node will enter the graph The node will try to connect to all nodes with id < t with probability:

Our Algorithm, Graph Generation Random walks  All nodes with id ≤ t will try RW_TIMES to start a random walk with probability 1/(t-id+1)  During the random walk node i will try to connect with node j with probability social i *quality j  At each step the probability to stop will be (1 – 1/DEPTH)

Metrics 1/3 Degree distribution  Description of the relative frequencies of nodes that have different degrees Diameter and average path length  Diameter is the largest distance between any two pairs of nodes in the network Distance is defined as the length of the shortest path between two nodes  Average path length is the average over all the shortest paths Betweenness Centrality  Gives information on how important a node is in terms of connecting other nodes  Computed as:  Where P i (k,j) denotes the number of shortest paths from k and j that i lies on

Metrics 2/3 Clustering  Indicates whether two neighbors of the same node are also connected with each other  Clustering coefficient for each node i is: Assortativity coefficient  In real networks the degrees in the endpoints of any edge tend not to be independent  This feature can be captured by computing the assortativity coefficient:  Where m is the average degree of the graph

Metrics 3/3  Neighbor degree distribution Average degree of the nearest neighbors of a vertex with degree k : Where P(k’|k) is the conditional probability that a node with degree k will be connected to a node with degree k’ Positive assortativity is translated as an increasing k nn (k) function

Data Description Facebook data from 4 large U.S. universities Number of nodes is small compared to the real Facebook graph Nodes represent a closed society Much better way to analyze a social network Large sample presents disadvantages  Difficult to analyze  How good is the sampling?

Results and Comparisons 1/5 Average degree does not depend on the size of network

Results and Comparisons 1/5 Average degree does not depend on the size of network All networks present positive assortativity  High degree nodes tend to connect with other high degree nodes

Results and Comparisons 1/5 Average degree does not depend on the size of network All networks present positive assortativity  High degree nodes tend to connect with other high degree nodes High clustering coefficients Average degree does not depend on the size of network All networks present positive assortativity  High degree nodes tend to connect with other high degree nodes High clustering coefficients

Results and Comparisons 1/5 Average degree does not depend on the size of network All networks present positive assortativity  High degree nodes tend to connect with other high degree nodes High clustering coefficients Small diameter and average path length

Results and Comparisons 2/5 Increasing k nn (k) functions As expected due to positive assortativity Nodes with high degree tend to be connected to each other

Results and Comparisons 3/5 Small betweenness values Almost independent of node degree No central authorities Information flows are distributed

Results and Comparisons 4/5 No clear power law phenomena On the log scale we see fat tails as expected

Results and Comparisons 5/5 Overall clustering is a simple summary characteristic Clear clustering pattern emerges High node degrees have small clustering Neighbors of high degree nodes less likely to be connected to each other

Current Work Analysis of information networks  Very large datasets from LiveJournal, YouTube, Flickr  As expected, different structure  Clear power law distributions  Introduction of a new metric: How close is pagerank with in-degree?

Future Work Make our model mathematically tractable Graph evolution over time  Densification laws  Shrinking diameters Community detection and formation New focus on coevolutionary models

Thank you! aiw.cs.aueb.gr/projects.html