Lecture 1 September 4, 2013

Transcription

1 CS 84r: Incentives an Information in Networks Fall 013 Prof. Yaron Singer Lecture 1 September 4, 013 Scribe: Bo Waggoner 1 Overview In this course we will try to evelop a mathematical unerstaning for the way in which information can be transmitte through social networks. For this purpose we will introuce mathematical moels of social networks, communication, an human behavior, an then evelop algorithms an incentive-base mechanisms that have provable guarantees. The premise of the course can summarize as follows. Human interactions follow mathematical patterns that can be leverage by algorithms an mechanisms for acquiring, isseminating, an learning information. In the course we will broaly focus on three main topics: acquisition, issemination, an learning of information. In the first part of information acquisition we will be intereste in the question of how to esign algorithms that fin information using a social network. In the secon part, we will be intereste in how to esign algorithms that can trigger large information cascaes. In the last part, we will stuy mathematical moels of learning that attempt to capture the way in which people upate their beliefs base on their knowlege of beliefs of others in the network. Information Acquisition Through Social Networks In the first part of the course, we will be intereste in unerstaning how one can acquire information using a social network: given a piece of information hel by an iniviual, how can we esign algorithms that fin this information efficiently? In 1967 while a grauate stuent at Harvar, Stanley Milgram performe an experiment in an attempt to verify the theory of small worl. The theory of small worl argues that every two people are separate by a short chain of acquaintances. To verify this theory, Milgram selecte a set of people at ranom in the Miwest an aske them to forwar an envelope to a specific person they i not know in Massachusetts. After some trial an error, Milgram launche an experiment where a goo fraction of the envelopes reache their estination, in a mean number of steps of six, which coine the term six-egrees of separation. What kin of graphs allow for these short paths to exist? What kin of graphs allow for people to be able to sen information to strangers through acquaintances so efficiently? We will attempt to answer these questions in the next few lectures. 1

2 3 Graphs as Moels of Small Worl Networks Throughout this course, we will moel social networks as graphs. A graph G consists of a set of noes (or vertices) V an a set of eges E. If u, v V an (u, v) E, then they are connecte by an ege. For an example of a graph, the Facebook network can be rawn as a graph where people are the noes, an two people are connecte by an ege if they are friens. We will sometime introuce weights on the graph, either on eges or noes. We will be intereste in escribing moels that are somewhat escriptive of social networks. In particular, we woul like to unerstan the topological properties that explain phenomena like the one observe in Milgram s experiment. As a first attempt for a social network moel, we can consier the clique: a graph in which every noe is connecte to every other noe in the graph. This moel fails to capture our intuitive unerstaning of a social network, as the egree of every noe is very large. We therefore wish to explore moels in which every noe has constant egree. As a secon attempt, we can consier trees with constant egree. This moel however fails to capture another intuitive feature of social networks an it is that some of our friens are likely to be friens themselves. In their seminal paper, Watts an Strogatz [?] formalize this using the following efinition: Definition 1 (Clustering coefficient). Given a graph G = (V, E), the clustering coefficient of a vertex v, enote C(v) is the fraction, over all pairs of neighbors of v, of those pairs who are neighbors of each other. Formally, {(u, w) E : u, w N (v)} C(v) = ). Intuitively, if you were to pick two of your friens uniformly at ranom, then the probability that they are also friens is your clustering coefficient. We will efine the clustering coefficient of a graph to be the average clustering coefficient of its noes. For real-worl social networks, we expect reasonably high clustering coefficients; ieally constant (that is, inepenent of the graph size). Note that trees an cycles have a clustering coefficient of zero, an so they also are not escriptive of the unerlying structure of social networks. ( (v) 4 Constructing a Small Worl moel We will now construct a moel for small worl networks which has two main builing blocks: ring lattices an -regular ranom graphs. 4.1 Ring lattices A ring lattice is a graph which is obtaine by taking a cycle an connecting each vertex to its neighbors two hops away, giving a 4-regular graph; We can generalize the efinition to other even constants greater than 4 (connecting each vertex to its neighbors three hops away, giving a 6-regular graph, an so on).

3 Figure 1: A 4-regular ring lattice on 1 vertices. Figure : A 6-regular ring lattice on 1 vertices. In the problem set we will show that the clustering coefficient of a -regular ring lattice is 3 ( ) 4 ( 1). For a 4-regular lattice it is easy to see that the clustering coefficient is 1/. The iameter of a -regular lattice, however, is Θ ( ) n. 1 So the ring lattice has a high clustering coefficient, which nicely captures that aspect of social networks. For the purposes of explaining short istances the ring lattice fails since it has a large iameter. 4. Expansion an ranom graphs We like to formalize a graph property that will guarantee small iameter (short istances). Definition. The expansion of a graph G = (V, E) is the minimum, over all cuts we can make (iviing the graph in two pieces), of the number of eges crossing the cut ivie by the number of vertices in the smaller half of the cut. Formally, it is = min S V,1 S n where e(s) is the set of eges leaving the set of noes S. e(s) S Theorem 3. Suppose the graph G is -regular, for some 3 an has expansion. Then, the iameter of G is O ( log n). Proof. We will show that any two vertices s an t are a istance at most O ( log n) apart. Let S j be the set of vertices reachable from s in at most j steps. (We can think of S j as being forme by a breath-first-search that starts at s.) Suppose that S j n. Because G is an -expaner, there are at least S j eges leaving S j. Consier all the vertices outsie of S j that these eges touch (as they are the ones who will be ae to S j to get S j+1 ). 1 The iameter of a graph is the longest istance (number of hops) between any two vertices. Big-Θ (theta) notation is use, meaning on the orer of : There are constants c 1 < c so that the iameter of -regular ring n lattices on n vertices is between c 1 an c n. See also big-o notation. 3

4 Each such vertex uses up at most of the eges (because G is -regular), so we will a at least S j new vertices to S j to get S j+1. That is, ( S j ) S j. Because S 0 = {s}, this gives that S j ( 1 + ) j. Now pick j = log n. Then we have S (1 log n + ) log n. Use the fact that ( ) k k for k 1 to get that S log n n. Therefore, the size of S j reaches at least n before this point, i.e. before j = log n. Now, by the exact same reasoning, if we start at t an consier the sets T j, we fin that the size of T j reaches at least n before j = log n. But then T log n an S log n must have some vertex in common (since both are larger than n ). That means there is a path from s to t of length at most log n, because we can go from s to this common vertex in at most log n steps, an then similarly to get from this vertex to t. So the above theorem implies that for 3, a -regular graph with constant expansion gives us the small worl property that we want: It has small (logarithmic) iameter. So how o create -regular graphs with goo expansion? Using ranomness, we can construct expaners quite easily: For 3, it turns out that -ranom graphs have expansion that is a function of only, not of the number of vertices (we ll skip the proof). 3 While -regular ranom graphs have small iameter, they have low clustering coefficients (about n ). So we have two types of graphs with ifferent weaknesses. A moel for small worl networks. To get the best of both worls we can simply combine the ring-lattice with a -regular ranom graph, for some constant 3. By that we mean that we ll take a -regular ranom graph an a the ring-lattice eges to the noes of the ranom graph. The result will be a graph with a constant egree that has both short istances an high clustering coefficient. The Watts-Strogatz moel. The above construction is a simple variant of a moel suggeste by Watts an Strogatz [?]. In the Watts-Strogatz moel we take the ring lattice an rewire every ege to a ranom noe in the graph with some probability p. For p = 0 we have a ring-lattice, Note: This assumes that k = is bigger than 1. But if it is less, then we can fix the argument by taking j = c log n for some large c in orer to bring ( ) 1 + j back up to log n. The constant c only epens on, not n, so it gets eaten up by the big-o, so it oesn t affect the conclusion. 3 A -regular ranom graph is a graph where all noes have egree an every two noes have the same likelihoo to be connecte. That is, fix some number of vertices n, consier all graphs on n vertices that are -regular, an pick one of these uniformly at ranom. Accoring to the fact we state, with very high probability, it will have constant expansion. 4

5 an p = 1 a regular ranom graph. Watts an Strogatz showe that there is a region of p s.t. the moel has both short istances an a high clustering coefficient. In summary: iameter clustering coefficient ring lattice ba (too big) goo (constant) -regular ranom goo (logarithmic) ba (too small) Watts-Strogatz goo goo 5