Web Graph Characteristics Kira Radinsky All of the following slides are courtesy of Ronny Lempel (Yahoo!)

2 The Web as a Graph Pages as graph nodes, hyperlinks as edges. – Sometimes sites are taken as the nodes Some natural questions: 1.Distribution of the number of in-links to a page. 2.Distribution of the number of out-links from a page. 3.Distribution of the number of pages in a site. 4.Connectivity: is it possible to reach most pages from most pages? 5.Is there a theoretical model that fits the graph?

3 Mathematical Background: Power-Law Distributions A non-negative random variable X is said to have a Power-Law distribution if, for some constants c>0 and α >0: Prob[X>x] ~ x - α, or equivalently f(x) ~ x -( α+1) Taking logs from both sides, we have: log Prob[X>x] = - α log(x) + c Power Law distributions have “ heavy/long tails ”, i.e. the probability mass of events whose value is far from the expectancy or median of the distribution is significant – Unlike Normal or Geometric/Exponential distributions, where the probability mass of the tail decreases exponentially, in Power Law distributions the mass of the tail decreases by the constant power of α – Another point of view: in an Exponential distribution, f(x)/f(x+k) is constant, whereas in a Power-Law distribution, f(x)/f(kx) is constant. – The “ average ” quantity in a Power-Law distribution is not “ typical ” Examples of Power-Law distributions are Pareto and Zipf distributions (see next slides)

4 Mathematical Background: The Pareto Distribution A continuous, positive random variable X in the range [L,  ] is said to be distributed Pareto(L,k) if its probability density function is: f(X=x;k;L) = k L k / x k+1 This implies that Prob(X>x) = (L/x) k – Has finite expectancy of Lk/(k-1) only for k>1 – Has finite variance only for k>2 Named after the Italian economist Vilfredo Pareto ( ), who modeled with it the distribution of wealth in society – Most people have little income; 20% of society holds 80% of the wealth

5 Mathematical Background: Zipf ’ s Law A random variable X follows Zipf ’ s Law (is “ Zipfian ” ) with parameter α when the j ’ th most popular value of X occurs with probability that is proportional to j - α – Essentially the distribution is over the discrete ranks Whenever α >1, X may take an infinite number of values (i.e. have infinitely many different value popularities) Named after the American Linguist George Kingsley Zipf ( ), who observed it on the frequencies of words in the English language – On a large corpus of English text, the 135 most frequently occurring words accounted for half of the text

6 Mathematical Background:An Observed Zipfian Sample Implies a Power-Law The following analysis is due to Lada Adamic: Assume that N units of wealth (coins) are distributed to M individuals – There are N observations of a random variable Y that can take on the discrete values 1,2, …,M Y k =j (k=1, … N, j=1..M) means that person j got coin k – Denote by X 1 [X m ] the number of coins of the richest[poorest] individual at the end of the process For simplicity, assume that N>>M and the X j ’ s are all distinct Assume that a perfect Zipfian behavior is observed, i.e. X r /N ~ r -b for all r=1, … M – This trivially implies X r ~ r -b

7 Mathematical Background:An Observed Zipfian Sample Implies a Power-Law (cont.) Recap: we distributed N coins to M individuals, and denoted by X 1 [X m ] the number of coins of the richest[poorest] individual at the end of the process By assuming Zipfian wealth: X r ~ r -b, or X r =cr -b Let Z be the random variable of a person ’ s wealth, i.e. the number of coins a person gets by this process Observation: if the r ’ th richest person got X r coins, then exactly r people out of M got X r coins or more Pr[Z  X r ]=Pr[Z  cr -b ]=r/M Define y= cr -b, and so r=(y/c) -(1/b), and so Pr[Z  y]= y -(1/b) c (1/b) /M Hence Pr[Z  y] ~ y -(1/b), and Z obeys a Power-Law

8 Distribution of Inlinks * Image taken from “ Graph Structure in the Web ”, Broder et al., WWW ’ A plot of the number of nodes having each value of in-degree Both axes are in log-scale Denoting the size of the sample crawl by N (over 200M here), we have: Log (N*Prob[node has in-degree x])  -a*log(x)+c Log (Prob[node has in-degree x])  -a*log(x)+c ’ Which indicates the Power-Law Prob[node has in-degree x] ~ x -a Note that the number of nodes with small in-degree is over-estimated while the number of nodes with very high in-degree is under-estimated

9 More Power-Laws on the Web We ’ ve seen that the in-degree of pages exhibits a Power-Law. Furthermore: Out-degree (somewhat surprising) Degrees of the inter-host graph Number of pages in Web sites Number of visits to Web sites/pages PageRank scores – With an exponent very close to that of the in-degree distribution – Curiously, degrees in the telephone call graph have the same 2.1 exponent Frequencies of words (as observed by Zipf) Popularities of queries submitted to search engines (will be discussed later in the course)

10 The Web as a Graph Connectivity: is it possible to reach most pages from most pages? The Web is a bow-tie! The Web graph is also scale-free, fractal: many slices and subgraphs exhibit similar properties. Image taken from “ Graph Structure in the Web ”, Broder et al., WWW ’ 2000.

11 Self-Similarity on the Web Dill et al., ACM TOIT 2002 Created large Thematically Unified Clusters (TUCs) Pages containing a certain keyword Pages of large Web sites/Intranets Pages containing a geographical reference in the Western US The host graph In general, the TUCs display very similar graph properties, e.g. In/out degree distributions Bow-tie structure (relative sizes of the components) Also discovered that the SCC of the different TUCs are strongly connected, i.e. it is possible to browse between the TUCs