Important problems of graph theory and the Internet
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1 Important problems of graph theory and the Internet Andrei Raigorodskii Lomonosov Moscow State University, Moscow Institute of Physics and Technology, Yandex Division of Theoretical and Applied Research, Moscow, Russia GraphHPC-2014, 04 March 2014 Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
2 The main objects Real-world web-graph =(Î ), whereî Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
3 The main objects Real-world web-graph =(Î ), whereî set of web-pages, Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
4 The main objects Real-world web-graph =(Î ), whereî set of web-pages, set of web-sites, Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
5 The main objects Real-world web-graph =(Î ), whereî set of web-pages, set of web-sites, set of web-hosts, Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
6 The main objects Real-world web-graph =(Î ), whereî set of web-pages, set of web-sites, set of web-hosts, and the set of all hyperlinks between the vertices (nodes). Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
7 The main objects Real-world web-graph =(Î ), whereî set of web-pages, set of web-sites, set of web-hosts, and the set of all hyperlinks between the vertices (nodes). Sometimes multiple edges are identified. Sometimes multiple edges and even loops are allowed. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
8 The main objects Real-world web-graph =(Î ), whereî set of web-pages, set of web-sites, set of web-hosts, and the set of all hyperlinks between the vertices (nodes). Sometimes multiple edges are identified. Sometimes multiple edges and even loops are allowed. How can we use this graph for practical purposes? Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
9 The main objects Real-world web-graph =(Î ), whereî set of web-pages, set of web-sites, set of web-hosts, and the set of all hyperlinks between the vertices (nodes). Sometimes multiple edges are identified. Sometimes multiple edges and even loops are allowed. How can we use this graph for practical purposes? Use its structure to find features improving learning to rank; Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
10 The main objects Real-world web-graph =(Î ), whereî set of web-pages, set of web-sites, set of web-hosts, and the set of all hyperlinks between the vertices (nodes). Sometimes multiple edges are identified. Sometimes multiple edges and even loops are allowed. How can we use this graph for practical purposes? Use its structure to find features improving learning to rank; Adjust algorithms including, say, those for crawling; Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
11 The main objects Real-world web-graph =(Î ), whereî set of web-pages, set of web-sites, set of web-hosts, and the set of all hyperlinks between the vertices (nodes). Sometimes multiple edges are identified. Sometimes multiple edges and even loops are allowed. How can we use this graph for practical purposes? Use its structure to find features improving learning to rank; Adjust algorithms including, say, those for crawling; Find unexpected structures such as news, spam, etc. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
12 Some important properties/features, which must be fastly checked on real data Barabási Albert, Watts Strogatz, Newman, and many others in 90s 00s. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
13 Some important properties/features, which must be fastly checked on real data Barabási Albert, Watts Strogatz, Newman, and many others in 90s 00s. Web-graphs are sparse, i.e., their numbers of edges (links) are proportional to their numbers of vertices. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
14 Some important properties/features, which must be fastly checked on real data Barabási Albert, Watts Strogatz, Newman, and many others in 90s 00s. Web-graphs are sparse, i.e., their numbers of edges (links) are proportional to their numbers of vertices. Web-graphs have a unique giant connected component. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
15 Some important properties/features, which must be fastly checked on real data Barabási Albert, Watts Strogatz, Newman, and many others in 90s 00s. Web-graphs are sparse, i.e., their numbers of edges (links) are proportional to their numbers of vertices. Web-graphs have a unique giant connected component. Every two vertices in the giant component are connected by a path of short length (5 6, depending on what we mean by web-graph): diam 6 (the rule of 6 handshakes). Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
16 Some important properties/features, which must be fastly checked on real data Barabási Albert, Watts Strogatz, Newman, and many others in 90s 00s. Web-graphs are sparse, i.e., their numbers of edges (links) are proportional to their numbers of vertices. Web-graphs have a unique giant connected component. Every two vertices in the giant component are connected by a path of short length (5 6, depending on what we mean by web-graph): diam 6 (the rule of 6 handshakes). Web-graphs are robust when random vertices are destroyed (a giant component survives). Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
17 Some important properties/features, which must be fastly checked on real data Barabási Albert, Watts Strogatz, Newman, and many others in 90s 00s. Web-graphs are sparse, i.e., their numbers of edges (links) are proportional to their numbers of vertices. Web-graphs have a unique giant connected component. Every two vertices in the giant component are connected by a path of short length (5 6, depending on what we mean by web-graph): diam 6 (the rule of 6 handshakes). Web-graphs are robust when random vertices are destroyed (a giant component survives). Web-graphs are vulnerable to attacks onto hubs (many small components appear after a threshold is surpassed). Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
18 Some important properties/features, which must be fastly checked on real data Barabási Albert, Watts Strogatz, Newman, and many others in 90s 00s. Web-graphs are sparse, i.e., their numbers of edges (links) are proportional to their numbers of vertices. Web-graphs have a unique giant connected component. Every two vertices in the giant component are connected by a path of short length (5 6, depending on what we mean by web-graph): diam 6 (the rule of 6 handshakes). Web-graphs are robust when random Ò ÓÒ Ø vertices are destroyed (a giant component survives). Web-graphs are vulnerable to attacks onto hubs (many small components appear after a threshold is surpassed). The degree distribution is close to a power-law: {Ú Î: degú= } where (2 3) depends on what we mean by web-graph. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
19 More of important properties/features: global clustering Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
20 More of important properties/features: global clustering 2ÒÚ Let =(Î ) be neighbours ofúin. SetÒÚ= ÆÚ, i.e.,òú= degú. IfÒÚ 2, clustering coefficient of the vertexúis Ú= {{Ü Ý} :Ü Ý ÆÚ} a simple graph, Î =Ò. LetÚ Î. Denote byæúthe set of then the Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
21 More of important properties/features: global clustering 2ÒÚ Let =(Î ) be neighbours ofúin. SetÒÚ= ÆÚ, i.e.,òú= degú. IfÒÚ 2, clustering coefficient of the vertexúis Ú= {{Ü Ý} :Ü Ý ÆÚ} Ú Î 2ÒÚ Ú Ú Î 2ÒÚ Clustering coefficient 1 The global clustering coefficient of is Ì( ) = a simple graph, Î =Ò. LetÚ Î. Denote byæúthe set of then the Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
22 More on global clustering Let (À ) be the number of copies of a graphàin a graph. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
23 More on global clustering Ú Î 2ÒÚ Ú (È2 ) Let (À ) be Clearly 3 (Ã3 ) Ì( ) = Ú Î 2ÒÚ= whereã3 is a triangle andè2 is a 2-path. the number of copies of a graphàin a graph. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
24 More on global clustering Ú Î 2ÒÚ Ú (È2 ) Let (À ) be Clearly 3 (Ã3 ) Ì( ) = Ú Î 2ÒÚ= whereã3 is a triangle andè2 is a 2-path. the number of copies of a graphàin a graph. Thus, roughly speaking,ì( ) is the probability that two neighbours of a vertex of are themselves joined by an edge. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
25 More on global clustering Ú Î 2ÒÚ Ú (È2 ) Let (À ) be Clearly 3 (Ã3 ) Ì( ) = Ú Î 2ÒÚ= whereã3 is a triangle andè2 is a 2-path. the number of copies of a graphàin a graph. Thus, roughly speaking,ì( ) is the probability that two neighbours of a vertex of are themselves joined by an edge. Clearly the last formula forì( ) may be used for any graph, not only a simple one. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
26 More of important properties/features: local clustering Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
27 2ÒÚ More of important properties/features: local clustering {{Ü Ý} :Ü Ý ÆÚ} Ú= Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
28 More of important properties/features: local clustering 1Ò Ú Î Ú 2ÒÚ {{Ü Ý} :Ü Ý ÆÚ} Ú= Clustering coefficient 2 The local clustering coefficient of is ( ) = Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
29 More of important properties/features: local clustering 1Ò Ú Î Ú 2ÒÚ {{Ü Ý} :Ü Ý ÆÚ} Ú= Clustering coefficient 2 The local clustering coefficient of is ( ) = The valuesì( ) and ( ) are rather different. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
30 More of important properties/features: local clustering 1Ò Ú Î Ú 2ÒÚ {{Ü Ý} :Ü Ý ÆÚ} Ú= Clustering coefficient 2 The local clustering coefficient of is ( ) = The valuesì( ) and ( ) are rather different. As a rule,ì( ) ( ). Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
31 More of important properties/features: local clustering 1Ò Ú Î Ú 2ÒÚ {{Ü Ý} :Ü Ý ÆÚ} Ú= Clustering coefficient 2 The local clustering coefficient of is ( ) = The valuesì( ) and ( ) are rather different. As a rule,ì( ) ( ). Both values are constant in real-world graphs (high clustering). Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
32 More of important properties/features: local clustering 1Ò Ú Î Ú 2ÒÚ {{Ü Ý} :Ü Ý ÆÚ} Ú= Clustering coefficient 2 The local clustering coefficient of is ( ) = The valuesì( ) and ( ) are rather different. As a rule,ì( ) ( ). Both values are constant in real-world graphs (high clustering). ( ) is much easier to calculate. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
33 More of important properties/features: local clustering 1Ò Ú Î Ú 2ÒÚ {{Ü Ý} :Ü Ý ÆÚ} Ú= Clustering coefficient 2 The local clustering coefficient of is ( ) = The valuesì( ) and ( ) are rather different. As a rule,ì( ) ( ). Both values are constant in real-world graphs (high clustering). ( ) is much easier to calculate. ( ) cannot be naturally defined for graphs with multiple edges and loops. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
34 More of important properties/features: local clustering 1Ò Ú Î Ú 2ÒÚ {{Ü Ý} :Ü Ý ÆÚ} Ú= Clustering coefficient 2 The local clustering coefficient of is ( ) = The valuesì( ) and ( ) are rather different. As a rule,ì( ) ( ). Both values are constant in real-world graphs (high clustering). ( ) is much easier to calculate. ( ) cannot be naturally defined for graphs with multiple edges and loops. ( ) is more complicated for theoretical study (in random graphs). Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
35 Degree correlations Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
36 Degree correlations Assortativity For a simple graph, let ÒÒ( ) = 1 { : deg = } : deg = : { } deg Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
37 Degree correlations Assortativity For a simple graph, let ÒÒ( ) = As a rule, ÒÒ( ) Æ. 1 { : deg = } : deg = : { } deg Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
38 Degree correlations Assortativity For a simple graph, let ÒÒ( ) = 1 { : deg = } : deg = As a rule, ÒÒ( ) Æ. For web-graphs,æ 0 (called dissassortative network). : { } deg Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
39 Degree correlations Assortativity For a simple graph, let ÒÒ( ) = 1 { : deg = } : deg = As a rule, ÒÒ( ) Æ. For web-graphs,æ 0 (called dissassortative network). For social networks,æ 0 (called assortative networks). : { } deg Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
40 Page Ranks Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
41 Page Ranks Classical PageRank Let Ò= (ÎÒ Ò) be a web-graph. Denote byèê(ú) a PageRank of a vertexú. DefineÈÊ( ), where {1 ÎÒ }, as the solution of the following system of linear equations: outdeg + ÎÒ =1 ÎÒ = ÈÊ( ) DÈÊ( ) ÈÊ( ) + 1 ÎÒ where (0 1) is a constant and D is the set of vertices having zero outdegrees. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
42 Page Ranks Classical PageRank Let Ò= (ÎÒ Ò) be a web-graph. Denote byèê(ú) a PageRank of a vertexú. DefineÈÊ( ), where {1 ÎÒ }, as the solution of the following system of linear equations: outdeg + ÎÒ =1 ÎÒ = ÈÊ( ) DÈÊ( ) ÈÊ( ) + 1 ÎÒ where (0 1) is a constant and D is the set of vertices having zero outdegrees. Very complicated calculations and many algorithms for approximation. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
43 Page Ranks Classical PageRank Let Ò= (ÎÒ Ò) be a web-graph. Denote byèê(ú) a PageRank of a vertexú. DefineÈÊ( ), where {1 ÎÒ }, as the solution of the following system of linear equations: outdeg + ÎÒ =1 ÎÒ = ÈÊ( ) DÈÊ( ) ÈÊ( ) + 1 ÎÒ where (0 1) is a constant and D is the set of vertices having zero outdegrees. Very complicated calculations and many algorithms for approximation. However, now Classical PR is not a strong feature! Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
44 Page Ranks Classical PageRank Let Ò= (ÎÒ Ò) be a web-graph. Denote byèê(ú) a PageRank of a vertexú. DefineÈÊ( ), where {1 ÎÒ }, as the solution of the following system of linear equations: outdeg + ÎÒ =1 ÎÒ = ÈÊ( ) DÈÊ( ) ÈÊ( ) + 1 ÎÒ where (0 1) is a constant and D is the set of vertices having zero outdegrees. Very complicated calculations and many algorithms for approximation. However, now Classical PR is not a strong feature! So many even more sophisticated algorithms. Andrei Raigorodskii (MSU) Important problems of graph theory and the Internet GraphHPC-2014, 04 March / 8
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