Upgrade Report - Department of Informatics - King's College London

6 GRAPH SAMPLING AND CRAWLING 16In addition to the above problem there is also the more general problem of when to stop the sampling.When do we know that we’ve sampled enough of the network? This question is very critical as it has manyside-effects. What if our method is appropriate but we are stopping it too soon? What if we sampled toomuch and distorted our perfect image? The answer to this question is still largely an open problem. Howeverin [27] some convergence tactics are discussed and analyzed. This offers a great tool for the crawler to beable to determine when it is best to halt the crawling.In general we can consider numerous other sampling goals which are sometimes case specific. For exampleone could consider the goal of measuring cover time of a graph, or of determining where a SRW of s stepsand starting from a vertex u is most likely to terminate at. These are some examples of possible interestingquestions that arise which always depends on the problem at hand. In general, sampling with SRWs is usedextensively in the context of property testing seen in Section 4.8, for example in the work of Czumaj etal [18] SRWs were used to determine whether or not a graph is an α-expander (defined in Section 4.3).6.1 Simple Random WalkThe most popular crawling method used in practice is the SRW. The reason for this is its simplicity andsurprising efficiency which made it a very attractive method to study and analyze.Let G = (V, E) be a connected graph. A SRW W u , u ∈ V on the undirected graph G = (V, E) is aMarkov chain X 0 = u, X 1 , . . . , X t , . . . on the vertices V associated to a particle that moves from vertex tovertex according to a transition rule. The probability of a transition from vertex i to vertex j is p(i, j) if{i, j} ∈ E, and 0 otherwise.Let d(v) = d(v, t) be the degree of vertex v ∈ G(t), and let N(v) denote the neighbours of v in thisgraph. A vertex u is a neighbor of v if there exists an edge e = (v, u).In the above general case the stopping criterion can be arbitrary. The most common use case is to stopwhen a sample of sufficient size and quality has been obtained, in other cases the method may stop whenall the vertices have been visited (covered).The SRW on graphs is in general the algorithm presented in Algorithm 1Algorithm 1 SRWv ← start vertexwhile not done doVisit(v)v ← random vertex from N(v)end whileThis simple method has certain properties:Transition matrix Given a graph G(N, M) the transition matrix P is a matrix which corresponds to theprobability that a certain transition will occur in a random walk. For example let P uv denote theelement at (u, v) in the transition matrix P uv = P r[“Given that we are at vertex u we move to vertexv”].{ 1P uv =d u, v ∈ N(u)0, otherwiseWhere d u is the degree of vertex u and N(u) denotes the neighborhood of u.Stationary Distribution Given a distribution π over a graph which is the proportion of time a randomwalk has spent over specific vertices, we determine the distribution at the next step of a SRW π ′ asπ ′ = P T π. The stationary distribution π s is the distribution with the property P T π s = π s .Mixing time The mixing time t m is the number of steps where the distribution π tm → π sIt is proven that in SRWs, as the number of steps t → ∞ the stationary distribution of a vertex u is:π u = d u2M(6.1)