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دانشجو: مجتبی بلبلی ، مجتبی صادقی ، احمد رحمانی استاد درس: دکتر شیخ اسماعیلی درس : شبکه های اجتماعی بهار91 دانشگاه آزاد اسلامی واحد علوم و تحقیقات کردستان SRB of Islamic Azad University, Kurdistan

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Six degrees of separation refers to the idea that everyone is on average approximately six steps away, by way of introduction, from any other person on Earth, so that a chain of, "a friend of a friend" statements can be made, on average, to connect any two people in six steps or fewer. The small world : take any two people in a social network, and they will be connected by a short chain of acquaintances. SRB of Islamic Azad University, Kurdistan2

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3 Human subject Target person Deliver a letter Friend Pass along a letter Know target ? The surprising outcome was that a reasonably large fraction of the letters reached the target and moreover, they did so in very few hops.

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Social network : co-authorships of computer science papers. Two people in this networks are “friends” if they are co- authors. Space of interests: computer science topics. we go from person to person by moving to the friend of the current person that has more interests in common with the target. SRB of Islamic Azad University, Kurdistan4

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We give a new evolutionary model that captures the small world phenomenon together with other important properties of real-world social networks. The main aspect of the model is its built in “interest space”. An important issue with Milgram's small world hypothesis is the difficulty of its verification. The advent of the internet has made it possible : ◦ To perform large-scale replicas of the experiment. ◦ to genuine “in silico” experiments, where there is no human participation. SRB of Islamic Azad University, Kurdistan5

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A snapshot of the social networking site Live Journal was downloaded to obtain a social network of roughly 15 million individuals. The experiment was simulated by picking source and target at random, and by moving toward the target according to geographical proximity SRB of Islamic Azad University, Kurdistan6

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the current node X we move to the neighbor of X that is closest to the target. (Compatible with small world hypothesis.) The main problem is how to make them realistic. Our experiments strongly reinforce two significant pieces of work in the sociology literature: ◦ The importance of weak ties ◦ The significance of the social status of the target node in Milgram's experiment. SRB of Islamic Azad University, Kurdistan7

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An affiliation network is a bipartite graph with people on one side and interests on the other. The affiliation network comes with an associated friendship graph in which two people are friends if they share an interest. SRB of Islamic Azad University, Kurdistan8

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The first graph is a bipartite graph, denoted as B(P, I), that represents the affiliation network, with a set P of people on one side and a set of interests I on the other. An edge (p, i) represents the fact that p is interested in i. The second graph is a friendship network, denoted as G(P,E), representing friendship relations within the same set P of people. People can be friends : ◦ If they share an interest or because of preferential attachment SRB of Islamic Azad University, Kurdistan9

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10 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2

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SRB of Islamic Azad University, Kurdistan11 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4

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SRB of Islamic Azad University, Kurdistan12 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4

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SRB of Islamic Azad University, Kurdistan13 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4

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SRB of Islamic Azad University, Kurdistan14 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4

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SRB of Islamic Azad University, Kurdistan15 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4 P3P3 P1P1 P2P2 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 P4P4

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Three different properties of social networks: ◦ Small world phenomena ◦ Evolutionary properties ◦ Navigability of the interest space Our model exhibit all of them simultaneously. They assume that every person knows the distance between its neighbors and the target, while we only assume that every person knows how similar interests are. The number of nodes in the graph increase with time SRB of Islamic Azad University, Kurdistan16

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Effective Diameter Core and hubs SRB of Islamic Azad University, Kurdistan 17

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Links in a social network are of two types, local and long range, also called weak, ties. Weak ties form bridges between different communities and, in particular, they are the crucial ingredient that makes small worlds possible. Weak ties are crucial to bound the effective diameter of the friendship graph by a constant. SRB of Islamic Azad University, Kurdistan 18

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Theorem4. For every q < 1, there is a constant q such that the q-effective diameter of G(P,E)is Bounded From above above by q. Lemma: Let C be the core of the affliation network B(P,I). There exists a constant d, independent of n, Such that, with high probability, the distance in the affliation network between any pair of nodes in the core is at most d. SRB of Islamic Azad University, Kurdistan 19

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Any two hubs are at constant distance in G(P,E) and B(P,I), with high probability. SRB of Islamic Azad University, Kurdistan 20

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Two more aspects make the following analysis Interesting in our opinion: 1.This is the first study of the performance of Local routing algorithm with an evolving model. 2.ours is the first model that can explain Milgram's experiment if we assume some constant attrition. SRB of Islamic Azad University, Kurdistan 21

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In this Figure we shown an example of affliation Network(A) with the induced friendship network(B) and hierarchy of interests(C). A B C SRB of Islamic Azad University, Kurdistan 22 P3P3 P1P1 P2P2 P4P4 P3P3 P1P1 P2P2 I3I3 I1I1 I2I2 P4P4 I4I4 I2I2 I1I1 I3I3 I4I4

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Distance between interests Local Routing algorithm Lemma: In every step of the local routing algorithm, either the interest distance between the message holder and the destination is reduced or the message is delivered to the target. SRB of Islamic Azad University, Kurdistan 23

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Let : Theorem8 Expanded interests setting Theorem9 Theorem10 SRB of Islamic Azad University, Kurdistan 24

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Mathematical model of social networks, building on the affiliation network model. A source vertex s and a target vertex t. Identify the interests of s and t in the underlying affiliation network and identify the neighbor of s whose interests are closer to that of t. SRB of Islamic Azad University, Kurdistan25

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Modestly-sized social network Consists of authors as nodes and edges defined by co-authorship of one or more articles. Randomly selected about 575 pairs of source- target SRB of Islamic Azad University, Kurdistan26

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SRB of Islamic Azad University, Kurdistan27 Probability that two randomly selected nodes belong to the largest connected component is roughly 64%. Shortest path between nodes in this component is roughly 6.3.

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▶ Two authors are friends if they coauthor a paper. ▶ Define the interest space with set of bigrams and unigrams. ▶ Unigrams and Bigrams??? ▶Consider "Small world experiments for everyone“ ▶Unigrams IS `small',`world', `experiments', and `everyone‘ ▶Bigrams IS `small world', `world experiments' SRB of Islamic Azad University, Kurdistan28

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We define s(i, a) to be the strength of interest i for author a. If we are currently at node x, we move to the neighbor y of x whose interest profile is closest to the target t. Measure of proximity of y to t is computed according to the formula. SRB of Islamic Azad University, Kurdistan29

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Another of the algorithm that the proximity measure strictly increase in each step of the routing this version is called Local Monotone, and the version without this restriction is called Local. SRB of Islamic Azad University, Kurdistan30

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We not only evaluate neighbors of x, but also evaluate neighbors of neighbors of x. in real social networks, one not only has knowledge about their friends, one often has partial knowledge about friends-of-friends SRB of Islamic Azad University, Kurdistan31

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Figure 3, 4 report the percentage of successful chains for the eight variations of the decentralized routing algorithm are studied. The success percentage' in Figure 3, 4 is SRB of Islamic Azad University, Kurdistan32 percentage of source-target pairs successfully routed 0.64

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Figure 3 SRB of Islamic Azad University, Kurdistan33

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Figure 4 SRB of Islamic Azad University, Kurdistan34

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The results are presented in four groups, each corresponding to one value of a parameter called 1. Navigation based on interests is an extremely powerful paradigm. the success of the basic algorithm Local in achieving 21%. 2. Adding one of two natural cues to local routing raising the success rate to about 57%, and reducing the path length from about 24 to about 12. SRB of Islamic Azad University, Kurdistan35

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3. Adding both interest expansion and lookahead results in 80% successful routing, with extremely short paths. 4. Insisting on monotonically better proximity to the target's interests typically reduces success rate, but significantly improves the length of the path constructed, for each of the four variations of the algorithm. SRB of Islamic Azad University, Kurdistan36

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5. Picking the target from a distribution that is restricted to targets of certain minimum degree dramatically improves the success rate and path length for decentralized routing algorithms. 6. These findings reinforce the strength of weak ties, and importance of core nodes for decentralized routing. SRB of Islamic Azad University, Kurdistan37

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با تشکر از توجه شما SRB of Islamic Azad University, Kurdistan38

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