ecs289m Spring, 2008 Social Network Models S. Felix Wu Computer Science Department University of California, Davis

SOURCE: Brandes, Raab and Wagner (2001) Organization Chart

Activities of Actual Advice Seeking Who is the most power? Can you determine that for OSN?

Real Social Organization

OECD Trade Flows SOURCE: Lothar Krempel

9-11 Hijackers Network SOURCE: Valdis Krebs

03/14/2008Davis Social Links7 The Web ???

Social Network Analysis “Structural relationships” as explanations: Network Formation Influence and collective actions

03/14/2008Davis Social Links9 Social Network Analysis 1. Degree Centrality: The number of direct connections a node has. What really matters is where those connections lead to and how they connect the otherwise unconnected. 2. Betweenness Centrality: A node with high betweenness has great influence over what flows in the network indicating important links and single point of failure. 3. Closeness Centrality: The measure of closeness of a node which are close to everyone else. The pattern of the direct and indirect ties allows the nodes any other node in the network more quickly than anyone else. They have the shortest paths to all others. 4. Eigenvector Centrality: It assigns relative scores to all nodes in the network based on the principle that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.

03/14/2008Davis Social Links10 Random Graphs G(n, p): n nodes and each edge with prob p

03/14/2008Davis Social Links11 Random Graphs G(n, p): n nodes and each edge with prob p When p < 1/n, disconnected components When p is sufficiently large, 1 giant component How about diameter? –The maximum distance (in hops) between any two nodes.

03/14/2008Davis Social Links12 Random Graph (Erdos/Renyi) Probabilistically, each node has (N-1)p direct neighbors ~ Z Z D = N (D is the diameter) D = logN / logZ In two hops, each node will have Z 2 neighbors in (equal) probability?

03/14/2008Davis Social Links13 Small World Model Low Diameter –Logarithmic or poly-logarithmic to N “High” Cluster Coefficient –cluster coefficient: the portion of X’s neighbors directly connecting to one of X’s other neighbors

03/14/2008Davis Social Links14 Cluster Coefficient Mesh network: C cluster = 1 Lattice Network (with degree K): C cluster = 0 –E.g., a linear line How about C cluster for Random Graph?

03/14/2008Davis Social Links15 Re-wiring (Watts/Strogatz) Trade off between D and C cluster ! Structured/Clustered

03/14/2008Davis Social Links16 A Cycle plus a Random Matching A dual combinatorial problem: –For given integers n and k, find a graph on n vertices with maximum degree k. –For givens integers k and D, find a graph, with bounded degree k and diameter at most D, having as many vertices as possible. How?

03/14/2008Davis Social Links17 A Cycle plus a Random Matching Cycle & Random “disjointed” match Bollobas/Chung: (logN) < D(G) < (logN + loglogN)

03/14/2008Davis Social Links18 Degree Centrality Degree distribution and the expected number of neighbors –Random graph (Poisson Distribution) Lower-law Tail for real world networks –P(k) ~ k -r –Scale-free: invariant to the size of N

03/14/2008Davis Social Links19 Exponential Distribution

03/14/2008Davis Social Links20 Power Law (function or dist.) f(x) = ax k + o(x k ) f(cx) = ?

03/14/2008Davis Social Links21 Zipf’s law Discrete Power-Law Ranking in the frequency table –{“the” (7%), “of” (3.5%), “and”, …} f(k;s,N) = k -s /(sum[n=1-N] n -s )

03/14/2008Davis Social Links22 Re-wiring (Watts/Strogatz) Trade off between D and C cluster ! Structured/Clustered

03/14/2008Davis Social Links23 Two Issues about Low Diameters Why should there exist short chains of acquaintances linking together arbitrary pairs of strangers? Why should arbitrary pairs of strangers be able to find the short chains of acquaintances that link them together?

03/14/2008Davis Social Links24 Kleinberg’s Basic setting

03/14/2008Davis Social Links25 p, q, r p: lattice distance between one node and all its local neighbors q: number of long range contacts r: inverse probability [d(u,v)] -r –What is the intuition about r? –What about r = 0

03/14/2008Davis Social Links26 Kleinberg’s results A decentralized routing problem –For nodes s,t with known lattice coordinates, find a short path from s to t. –At any step, can only use local information, –Kleinberg suggests a simple greedy algorithm and analyzes it:

03/14/2008Davis Social Links27 Local Information Local contacts Coordinate for the target The locations and long-range contacts of all nodes that have come in contact with the message.

03/14/2008Davis Social Links28 Results If r = 0, expected delivery time is at least a 0 n 2/3. –Lower bound If r = 2, p = q = 1, a 2 (log n) 2 –Martel/Nguyen’s newer results 0 <= r < 2 ~ a r n (2-r)/3 r > 2 ~ a r n (r-2)(r-1)

03/14/2008Davis Social Links29 Skip Lists The basic idea: Keep a doubly-linked list of elements –Min, max, successor, predecessor: O(1) time –Delete is O(1) time, Insert is O(1)+Search time During insert, add each level-i element to level i+1 with probability p (e.g., p = 1/2 or p = 1/4) level level 2 level 3

03/14/2008Davis Social Links30 Skip Graphs Based on “skip list”: –A randomized balanced tree structure organized as a tower of increasingly sparse linked lists –All nodes join the link list of level 0 –For other levels, each node joins with a fixed probability p –Each node has 2/(1-p) pointers –Average search time: O(log(n/((1-p)*log1/p)))

03/14/2008Davis Social Links31 Skip Graph: Skip List is not suitable for P2P environment –No redundancy, Hotspot problem –Vulnerable to failure and contention Skip Graph: Extension of Skip List –Level 0 link list builds a Chord ring –Multiple (max 2 i ) lists for level i (i = 1, … logn) –Each node participate in all levels, but different lists –Membership vector m(x): decide which list to join –Every node sees its own skip list

03/14/2008Davis Social Links32 Degree Optimal P2P Routing Different routing schemes –Viceroy [MNR02] : emulates the butterfly network Constant degree O(log n) hops for routing –Constructions emulating De-Bruijn graphs Can achieve any degree/number of hops tradeoff –In particular degree O(log n) and O(log n/ log log n) hops Routing is not greedy –Recent construction [AM] fixes that. Even if target and source are close in label space message might be routed away No (natural) prefix search –Random keys are necessary.

03/14/2008Davis Social Links33 Skip – Graphs [AS02],[HDJ+03] Each node (resource) has a name. Nodes are arranged on a line sorted by name. Each node chooses a random string of bits. An edge is established if two nodes share a prefix which is not shared by the nodes between them. Allows prefix search a b cf e d

03/14/2008Davis Social Links34 Routing in Skip – Graphs Greedy Routing – use longest edge possible. Path length is  (log n) w.h.p. The NoN algorithm optimizes over two hops Theorem: Using the NoN algorithm, the expected path length of any lookup is.

03/14/2008Davis Social Links35 Kleinberg’s Lattice Model Graph embedded in a metric space (e.g., 2D lattice) “Search efficiently” using only Local information + long range contact(s) –~ inverse probability [d(u,v)] -r –r = 2, a special case

03/14/2008Davis Social Links36 Some Extensions Hierarchical Network Models Group Structure Models Constant Number of Out-Links “Small World Phenomena and the Dynamics of Information” by J. Kleinberg, NIPS, 2001

03/14/2008Davis Social Links37 Generation & Search There is a data structure behind and among all the social peers –Lattice, Tree, Group/Community The link probability depends on this “social data structure” –And, using it to generate the social network Searching may use “direct contacts” plus the knowledge about the social data structure

03/14/2008Davis Social Links38 Hierarchical Network Models Representation –a complete b-ary tree, T –All social nodes are “leaves” Distance and Link Probability – = the height of the least common ancestor of v and w in T – probability proportional – normalization in probability – out-degree in graph

03/14/2008Davis Social Links39 the Critical Value

03/14/2008Davis Social Links40 Interpretation (1) /Science/Computer_Science/Algorithms /Arts/Music/Opera /Science/Computer_Science/Machine_Learning

03/14/2008Davis Social Links41 Interpretation (2) Target: “stock Boston, MA” Next hop: Cambridge, MA” New York City, NY”

03/14/2008Davis Social Links42 Results Otherwise, no polylogarithmic search

03/14/2008Davis Social Links43 How to Search in HNM??

03/14/2008Davis Social Links44 Useful Neighbor Is “v” useful to reach “t”? vt

03/14/2008Davis Social Links45 Useful Neighbor Is “v” useful to reach “t”? v u t

03/14/2008Davis Social Links46 Useful Neighbor Is “v” useful to reach “t”? v u tw

03/14/2008Davis Social Links47 Useful Neighbor Is “v” useful to reach “t”? v u tw

03/14/2008Davis Social Links48 Useful Neighbor Recursively Is “v” useful to reach “t”? v u wt

03/14/2008Davis Social Links49 Search Find one “useful” neighbor in G as the next step What happens if NO useful neighbor? Expected steps to reach “t”.

03/14/2008Davis Social Links50 Probability to have 1 U.N. One leave All out-links

03/14/2008Davis Social Links51 HNM High probability to be useful How about “constant links”?

03/14/2008Davis Social Links52 Group Structures R is a group; R’ is a strict smaller subgroup R 1, R 2,R 3,… all contain v, then q(v,w) : minimum size of a group containing both v and w

03/14/2008Davis Social Links53 How to Search in Group Structure??

03/14/2008Davis Social Links54 Idea (v, t) R is the minimum-sized group containing both v and t. With property (1) Then: How to define “usefulness” of v?

03/14/2008Davis Social Links55 Usefulness of v (v, t) R is the minimum-sized group containing both v and t. With property (1) Then:

03/14/2008Davis Social Links56 Probability to have 1 U.N. One leave All out-links

03/14/2008Davis Social Links57 Probability to have 1 U.N.

03/14/2008Davis Social Links58 Results Otherwise, no polylogarithmic search

03/14/2008Davis Social Links59 Fixed Number of Out-Links Relax “t” to “a cluster of t” vtCl t x v w r: Resolution

03/14/2008Davis Social Links60 Question #1 Why can’t we just treat “Cluster” as “Super Node” and we go home (by applying the HNM results)? Cl t x v w

03/14/2008Davis Social Links61 Not necessarily Cl t x v w p q

03/14/2008Davis Social Links62 Probability

03/14/2008Davis Social Links63 Question #2 For any out-link of v, what is the probability that the end point of the out- link is in the same cluster of v ?

03/14/2008Davis Social Links64 Answer

03/14/2008Davis Social Links65 Results If the resolution is polylogarithmic, the the search is polylogarithmic if alpha = 1.

03/14/2008Davis Social Links66 A “Similar” Process v u wt Coloring the Links

03/14/2008Davis Social Links67 Reading “Small World Phenomena and the Dynamics of Information” by J. Kleinberg, NIPS, 2001