Since the first investigations on web-graph compression, it has been clear that the ordering of the nodes of a web graph has a fundamental influence on the compression rate (usually expressed as the number of bits per link). The authors of the LINK database, for instance, investigated three different approaches: an extrinsic ordering (URL ordering) and two intrinsic orderings based on the rows of the adjacency matrix (lexicographic and Gray code); they concluded that URL ordering has many advantages in spite of a small penalty in compression. In this paper we approach this issue in a more systematic way, testing some known orderings and proposing some new ones. Our experiments are made in the WebGraph framework, and show that the compression technique and the structure of the graph can produce significantly different results. In particular, we show that for a transposed web graph, URL ordering is significantly less effective, and that some new mixed orderings combining host information and Gray/lexicographic orderings outperform all previous methods: in some large transposed graphs they yield the quite incredible compression rate of 1 bit per link. We experiment with these simple ideas on some nonweb social networks and obtain results that are extremely promising and are very close to those recently achieved using shingle orderings and backlink compression schemes.

We present a deterministic model for online social networks (OSNs) based on transitivity and local knowledge in social interactions. In the iterated local transitivity (ILT) model, at each time step and for every existing node $x$, a new node appears that joins to the closed neighbor set of $x$. The ILT model provably satisfies a number of both local and global properties that have been observed in OSNs and other real-world complex networks, such as a densification power law, decreasing average distance, and higher clustering than in random graphs with the same average degree. Experimental studies of social networks demonstrate poor expansion properties as a consequence of the existence of communities with low numbers of intercommunity edges. Bounds on the spectral gap for both the adjacency and normalized Laplacian matrices are proved for graphs arising from the ILT model indicating such bad expansion properties. The cop and domination numbers are shown to remain the same as those of the graph from the initial time step $G_0$, and the automorphism group of G0 is a subgroup of the automorphism group of graphs generated at all later time steps. A randomized version of the ILT model is presented that exhibits a tunable densification power-law exponent and maintains several properties of the deterministic model.

We give an improved local partitioning algorithm using heat kernel pagerank, a modified version of PageRank. For a subset $S$ with Cheeger ratio (or conductance) $h$, we show that at least a quarter of the vertices in $S$ can serve as seeds for heat kernel pagerank that lead to local cuts with Cheeger ratio at most $O(\sqrt{h})$, improving the previous bound by a factor of $\sqrt{\log s}$, where s denotes the volume of $S$.

We consider a random subgraph $G_p$ of a host graph $G$ formed by retaining each edge of $G$ with probability $p$. We address the question of determining the critical value $p$ (as a function of $G$) for which a giant component emerges. Suppose $G$ satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second-order average degree $\overline{d}$ to be $\overline{d} = \sum_v d^2_v/(\sum_v d_v)$, where $d_v$ denotes the degree of $v$. We prove that for any $\epsilon > 0$, if $p > (1 + \epsilon)/\overline{d}$, then asymptotically almost surely, the percolated subgraph $G_p$ has a giant component. In the other direction, if $p < (1 − \epsilon)/\overline{d}$, then almost surely, the percolated subgraph $G_p$ contains no giant component. An extended abstract of this paper appeared in the WAW 2009 proceedings. The main theorems are strengthened with much weaker assumptions.

The World Wide Web, the Internet, coupled biological and chemical systems, neural networks, and social interacting species are only a few examples of systems comprising a large number of highly interconnected dynamical units. These networks contain characteristic patterns, network motifs, that occur far more often than in randomized networks with the same degree sequence. Several algorithms have been suggested for counting or detecting the number of occurrences of network motifs as trees and bounded treewidth subgraphs of size $O(\log n)$, at most 7 for some motifs. In addition, local motif counting, counting the number of motifs in which a node participates, was recently suggested as a method of classifying nodes in the network. The premise is that the distribution of motifs in which a node participates is an indication of its function in the network. Therefore, local counting of network motifs provides a major challenge. However, no such practical algorithm exists other than local counting of triangles. We present several algorithms with time complexity $O(((3e)^k • n • |E| • \log \frac{1}{δ})/\epsilon^2)$ that approximate for every vertex the number of occurrences of the motif in which the vertex participates, for k-length cycles and $k$-length cycles with a chord, where $k = O(\log n)$, and algorithms with time complexity $O((n • |E| • \log \frac{1}{δ} )/\epsilon^2 + |E|^2 • \log n + |E| • n \log n)$ that approximate for every vertex the number of noninduced occurrences of the motif in which the vertex participates for all motifs of size four. In addition, we show algorithms that approximate the total number of occurrences of these network motifs when no efficient algorithm exists. Some of our algorithms use the “color-coding” technique.

A variety of lossless compression schemes has been proposed to reduce the storage requirements of web graphs. One successful approach is virtual-node compression, in which often-used patterns of links are replaced by links to virtual nodes, creating a compressed graph that succinctly represents the original. In this paper, we show that several important classes of web graph algorithms can be extended to run directly on virtual-node-compressed graphs, such that their running times depend on the size of the compressed graph rather than on that of the original. These include algorithms for link analysis, estimating the size of vertex neighborhoods, and a variety of algorithms based on matrix-vector products and random walks. Similar speedups have been obtained previously for classical graph algorithms such as shortest paths and maximum bipartite matching. We measure the performance of our modified algorithms on several publicly available web graph data sets, and demonstrate significant empirical speedups that nearly match the compression ratios.

We study personalized web-ranking algorithms based on the existence of document clusterings. Motivated by topic-sensitive page ranking, we develop and implement an efficient "local-cluster" algorithm by extending the web search algorithm of Achilioptas et al., Web Search via Hub Synthesis. We propose some formal criteria for evaluating such personalized ranking algorithms and provide some preliminary experiments in support of our analysis. Both theoretically and experimentally, our algorithm differs significantly from Topic-Sensitive PageRank.