In this paper, we address the problem of evaluating historical queries on graphs. To this end, we investigate the use of graph deltas, i.e., a log of time-annotated graph operations. Our storage model maintains the current graph snapshot and the delta. We reconstruct past snapshots by applying appropriate parts of the graph delta on the current snapshot. Query evaluation proceeds on the reconstructed snapshots but we also propose algorithms based mostly on deltas for efficiency. We introduce various techniques for improving performance, including materializing intermediate snapshots, partial reconstruction and indexing deltas.
There is also a large body of work that studies the evolution of real-world networks over time. In @cite_13 , it was shown that for a variety of real-world networks, graph density increases following a power-low and the graph diameter shrinks. Works on specific networks such as Flickr @cite_8 and Facebook @cite_9 also study network growth and their results can be exploited to enrich our model.
@cite_9 @cite_13 @cite_8
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How do real graphs evolve over timeq What are normal growth patterns in social, technological, and information networksq Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network or in a very small number of snapshots; these include heavy tails for in- and out-degree distributions, communities, small-world phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time. Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time with the number of edges growing superlinearly in the number of nodes. Second, the average distance between nodes often shrinks over time in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) or O(log(log n)). Existing graph generation models do not exhibit these types of behavior even at a qualitative level. We provide a new graph generator, based on a forest fire spreading process that has a simple, intuitive justification, requires very few parameters (like the flammability of nodes), and produces graphs exhibiting the full range of properties observed both in prior work and in the present study. We also notice that the forest fire model exhibits a sharp transition between sparse graphs and graphs that are densifying. Graphs with decreasing distance between the nodes are generated around this transition point. Last, we analyze the connection between the temporal evolution of the degree distribution and densification of a graph. We find that the two are fundamentally related. We also observe that real networks exhibit this type of relation between densification and the degree distribution. In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one non-trivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the cross-over from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Pade approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model. Appears in Phys. Rev. E 60, 7332-7342 (1999). Based on the empirical analysis of the dependency network in 18 Java projects, we develop a novel model of network growth which considersboth: an attachment mechanism and the addition of new nodes with a heterogeneous distribution of their initial degree, k0. Empirically we find that the cumulative degree distributions of initial degrees and of the final network, follow power-law behaviors: P(k0) ∝ k01−α, and P(k) ∝ k1−γ, respectively. For the total number of links as a function of the network size, we find empirically K(N) ∝ Nβ, where β is (at the beginning of thenetwork evolution) between 1.25 and 2, while converging to ∼ 1 for large N. This indicates a transition from a growth regime with increasing network density towards a sustainable regime, which revents a collapse because of ever increasing dependencies. Our theoretical framework is able to predict relations between the exponents α, β, γ, which also link issues of software engineering and developer activity. These relations are verified by means of computer simulations and empirical investigations. They indicate that the growth of real Open Source Software networks occurs on the edge between two regimes, which are either dominated by the initial degree distribution of added nodes, or by the preferential attachment mechanism. Hence, the heterogeneous degree distribution of newly added nodes, found empirically, is essential to describe the laws of sustainable growth in networks. We show that nodes of high degree tend to occur infrequently in random graphs but frequently in a wide variety of graphs associated with real world search problems. We then study some alternative models for randomly generating graphs which have been proposed to give more realistic topologies. For example, we show that Watts and Strogatz's small world model has a narrow distribution of node degree. On the other hand, Barabasi and Albert's power law model, gives graphs with both nodes of high degree and a small world topology. These graphs may therefore be useful for benchmarking. We then measure the impact of nodes of high degree and a small world topology on the cost of coloring graphs. The long tail in search costs observed with small world graphs disappears when these graphs are also constructed to contain nodes of high degree. We conjecture that this is a result of the small size of their "backbone", pairs of edges that are frozen to be the same color.