Random Graph Models of large networks Alan Frieze with special thanks to Abraham Flaxman

Random graph models of large networks - Alan Frieze, CMU Graphs A graph G is defined by a vertex set V and an edge set E. It may look like this:

Random graph models of large networks - Alan Frieze, CMU “Real” Graphs Large graphs are all around us in the world today, for example, the World Wide Web. The vertex set V consists of all web pages, and the edge set E consists of all hyperlinks. |V | ¼ 109, average 7 links per page.

Random graph models of large networks - Alan Frieze, CMU More Examples Internet Metabolic Networks Social Networks Neural Networks Peer to Peer Networks Tunnels of an Ant Colony

Random graph models of large networks - Alan Frieze, CMU Modeling Graphs We assume that they arise via some random process. Why not model them as random graphs: Erdos and Renyi model : Vertex set {1,2,…,n}, Each of the graphs with edges equally likely.

Random graph models of large networks - Alan Frieze, CMU Problem Suppose For small k the number of vertices degree k is whp In many real world cases the number of vertices of degree k for some A, (close to 3 in WWW graph) e.g. Faloutsos,Faloutsos and Faloutsos.

Random graph models of large networks - Alan Frieze, CMU Fixed Degree Sequence Models: Choose a degree sequence d1,…,dn and then choose a graph uniformly from graphs with this degree sequence.

Random graph models of large networks - Alan Frieze, CMU Molloy and Reed Let <0 implies all components of G are small whp. >0 implies that G contains a giant component (size (n)) whp.

Random graph models of large networks - Alan Frieze, CMU Explanation: represent a vertex of degree k by a set of size k. Randomly pair up points of sets. Vertex of Degree 3 Expected increase in “free dots” is idi(di-2)/idi

Random graph models of large networks - Alan Frieze, CMU Aiello, Chung, Lu Given , they considered random graphs where the number of vertices y of degree x satisfies For various ranges for ,.

Random graph models of large networks - Alan Frieze, CMU Flavour of results: existence of giant component whp: >0=3.478.. implies there is no giant component <0 implies there is a unique giant component. Bounds given on size of second largest component too.

Cooper and Frieze: Random Digraphs Random graph models of large networks - Alan Frieze, CMU Cooper and Frieze: Random Digraphs They consider random digraphs D with n vertices where the number of vertices with in-degree i and out-degree j is li,j . Let  n=i,jli,j be the number of arcs in D. Let d=i,jijli,j/( n).

Random graph models of large networks - Alan Frieze, CMU Flavour of results: size of largest strong component (size (n)) whp: d<1 implies there is no giant strong component. d>1 implies there is a giant strong component S.

Random graph models of large networks - Alan Frieze, CMU More on d>1: Let L+ be the set of vertices with giant “fan-out” and L- be the set of vertices with giant “fan-in”. Then whp S=L+Å L-

Papadimitriou and Mihail: Random graph models of large networks - Alan Frieze, CMU Papadimitriou and Mihail: Model: Fix a1,a2,…an and then add an edge between vertices i and j with probability aiaj/An where An= ai. Let 1¸ 2¸ … be the largest eigenvalues of the adjacency matrix of the random graph produced.

Random graph models of large networks - Alan Frieze, CMU Suppose ½<<1. Suppose that ai=a1i- for small i. Then for small i we have i» ai where i denotes the i’th largest degree, and

Dynamic models:Preferential Attachment Model (PAM) Random graph models of large networks - Alan Frieze, CMU Dynamic models:Preferential Attachment Model (PAM) We build the graph dynamically: 1. At time t (a) add vt (b) connect vt to u chosen randomly 2. Every m steps contract the most recently added m vertices into a single vertex.

Random graph models of large networks - Alan Frieze, CMU 1. (b) Connect vt to u chosen randomly Randomly how? “The rich get richer”

Preferential Attachment Model Random graph models of large networks - Alan Frieze, CMU Preferential Attachment Model History: Yule, 1925 - Zoology Simon, 1955 - Word frequencies, academic papers, cities, income, more zoology Barabasi and Albert, 1999 - WWW

Heuristic Analysis of Degrees Random graph models of large networks - Alan Frieze, CMU Heuristic Analysis of Degrees Let dv(t) denote the degree of v at time t. Then,

Random graph models of large networks - Alan Frieze, CMU Suppose that v is added at time s. Then we get Thus the number of vertices of degree exceeding k at time t is

Random graph models of large networks - Alan Frieze, CMU And the number of vertices of degree exactly k is

Random graph models of large networks - Alan Frieze, CMU A rigorous proof of the following is given in Bollobas, Riordan, Spencer, Tusnady: With probablity 1-o(1), as , the number of vertices of degree k is

Random graph models of large networks - Alan Frieze, CMU More Work B. Bollobas, O. Riordan: Diameter: Robustness: Suppose we delete the first ct vertices for some c<1.

Random graph models of large networks - Alan Frieze, CMU Researchers have developed similar, but more complex models which are mixtures of preferential and random attachment. These give arbitrary exponents for the power law.

Random graph models of large networks - Alan Frieze, CMU Copying Model Communities: A large dense bipartite sub-graph of the WWW indicates a “community”. Experiments indicate a larger number of these than you would get from say the simple model PAM. The next model does give many though: it is due to Kumar,Raghavan,Rajakopalan,Sivakumar and Upfal.

Random graph models of large networks - Alan Frieze, CMU As in PAM at each stage we add a new vertex vt and we give it m incident edges. Its construction rests on a parameter . Then a vertex u is chosen uniformly at random from Vt={v1,v2,…,vt-1} and then for i=1,2,…,m we 1. With probability  we create edge (vt,x) where x is chosen randomly from Vt. 2. With probability 1- we create edge (vt,y) where y was the i’th choice of u.

Random graph models of large networks - Alan Frieze, CMU 1. Whp the degree sequence has a power law with exponent . 2. Whp the number of copies of Ki,i, i· m is (te-i). This contrasts with the simple preferential model where the expected number is O(1).

Random graph models of large networks - Alan Frieze, CMU More on PAM Let Di = i’th max degree at time t: Fenner,Flaxman,Frieze: Fix k independent of t. Whp for any f(t) with f(t)!1 as t!1, and i<k t1/2/f(t) · i· t1/2f(t) and i+1· i - t1/2/f(t).

Random graph models of large networks - Alan Frieze, CMU Furthermore, if 1,…k are the k’th largest eigenvalues then whp

Random graph models of large networks - Alan Frieze, CMU Crawling on web graphs Cooper and Frieze considered the following scenario: We have the model PAM. Plus there is a spider S which does a random walk as the graph is growing. Let m(t) denote the expected number of vertices not visited at least once by the spider up to time t.

Random graph models of large networks - Alan Frieze, CMU Let then

Random graph models of large networks - Alan Frieze, CMU Heuristically Optimized Trade-Offs – Fabrikant, Koutsoupias and Papadimitriou This a random graph model of the growth of the internet that exhibits a power law in its degree sequence.

Random graph models of large networks - Alan Frieze, CMU The model builds a tree on n random points X1,X2,…,Xn in the unit square [0,1]2 n is a parameter of the model. Suppose that we have built a tree T on X1,X2,…,Xi-1 and we wish to connect Xi up a “close” point on T.

Random graph models of large networks - Alan Frieze, CMU We connect Xi to Xj where j minimises ndi,j+hj Here di,j is Euclidean distance and hj is the tree distance (in edge count) from Xj to the root X1.

Random graph models of large networks - Alan Frieze, CMU Results n<2-1/2 implies that T is a star with root X1 n=(n1/2) implies that the degree distribution of T is exponential. 4· n=o(n1/2) gives a power law for the degree distribution of T.

Random graph models of large networks - Alan Frieze, CMU