Power-law Graphs Research Articles

A Simple Yet Effective Balanced Edge Partition Model for Parallel Computing

Graph edge partition models have recently become an appealing alternative to graph vertex partition models for distributed computing due to both their flexibility in balancing loads and their performance in reducing communication cost. In this paper, we propose a simple yet effective graph edge partitioning algorithm. In practice, our algorithm provides good partition quality while maintaining low partition overhead. It also outperforms similar state-of-the-art edge partition approaches, especially for power-law graphs. In theory, previous work showed that an approximation guarantee of O(d max √(log n log k)) apply to the graphs with m=Ω(k 2 ) edges (n is the number of vertices, and k is the number of partitions). We further rigorously proved that this approximation guarantee hold for all graphs. We also demonstrate the applicability of the proposed edge partition algorithm in real parallel computing systems. We draw our example from GPU program locality enhancement and demonstrate that the graph edge partition model does not only apply to distributed computing with many computer nodes, but also to parallel computing in a single computer node with a many-core processor.

Predicting epidemic evolution on contact networks from partial observations

The massive employment of computational models in network epidemiology calls for the development of improved inference methods for epidemic forecast. For simple compartment models, such as the Susceptible-Infected-Recovered model, Belief Propagation was proved to be a reliable and efficient method to identify the origin of an observed epidemics. Here we show that the same method can be applied to predict the future evolution of an epidemic outbreak from partial observations at the early stage of the dynamics. The results obtained using Belief Propagation are compared with Monte Carlo direct sampling in the case of SIR model on random (regular and power-law) graphs for different observation methods and on an example of real-world contact network. Belief Propagation gives in general a better prediction that direct sampling, although the quality of the prediction depends on the quantity under study (e.g. marginals of individual states, epidemic size, extinction-time distribution) and on the actual number of observed nodes that are infected before the observation time.

Scalable subgraph enumeration in MapReduce: a cost-oriented approach

Subgraph enumeration, which aims to find all the subgraphs of a large data graph that are isomorphic to a given pattern graph, is a fundamental graph problem with a wide range of applications. However, existing sequential algorithms for subgraph enumeration fall short in handling large graphs due to the involvement of computationally intensive subgraph isomorphism operations. Thus, some recent researches focus on solving the problem using MapReduce. Nevertheless, exiting MapReduce approaches are not scalable to handle very large graphs since they either produce a huge number of partial results or consume a large amount of memory. Motivated by this, in this paper, we propose a new algorithm $$\mathsf {Twin}$$Twin$$\mathsf {Twig}$$Twig$$\mathsf {Join}$$Join based on a left-deep-join framework in MapReduce, in which the basic join unit is a $$\mathsf {Twin}$$Twin$$\mathsf {Twig}$$Twig (an edge or two incident edges of a node). We show that in the Erdos---Renyi random graph model, $$\mathsf {Twin}$$Twin$$\mathsf {Twig}$$Twig$$\mathsf {Join}$$Join is instance optimal in the left-deep-join framework under reasonable assumptions, and we devise an algorithm to compute the optimal join plan. We further discuss how our approach can be adapted to handle the power-law random graph model. Three optimization strategies are explored to improve our algorithm. Ultimately, by aggregating equivalent nodes into a compressed node, we construct the compressed graph, upon which the subgraph enumeration is further improved. We conduct extensive performance studies in several real graphs, one of which contains billions of edges. Our approach significantly outperforms existing solutions in all tests.

The Semilinear Heat Equation on Sparse Random Graphs

Using the theory of $L^p$-graphons (Borgs et al, 2014), we derive and rigorously justify the continuum limit for systems of differential equations on sparse random graphs. Specifically, we show that the solutions of the initial value problems for the discrete models can be approximated by those of an appropriate nonlocal diffusion equation. Our results apply to a range of spatially extended dynamical models of different physical, biological, social, and economic networks. Importantly, our assumptions cover network topologies featured in many important real-world networks. In particular, we derive the continuum limit for coupled dynamical systems on power law graphs. The latter is the main motivation for this work.

Push is Fast on Sparse Random Graphs

We consider the classical push broadcast process on a large class of sparse random multigraphs that includes random power law graphs and multigraphs. Our analysis shows that for every $\varepsilon>0$, whp $O(\log n)$ rounds are sufficient to inform all but an $\varepsilon$-fraction of the vertices. It is not hard to see that, e.g. for random power law graphs, the push process needs whp $n^{\Omega(1)}$ rounds to inform all vertices. Fountoulakis, Panagiotou and Sauerwald proved that for random graphs that have power law degree sequences with $\beta>3$, the push-pull protocol needs $\Omega(\log n)$ to inform all but $\varepsilon n$ vertices whp. Our result demonstrates that, for such random graphs, the pull mechanism does not (asymptotically) improve the running time. This is surprising as it is known that, on random power law graphs with $2<\beta<3$, push-pull is exponentially faster than pull.

On the approximation of Laplacian eigenvalues in graph disaggregation

Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient computation in parallel. In particular, we consider computations involving the graph Laplacian, which has significant applications, including diffusion mapping and graph partitioning, among others. We prove results regarding the spectral approximation of the Laplacian of the original graph by the Laplacian of the disaggregated graph. In addition, we construct an alternate disaggregation operator whose eigenvalues interlace those of the original Laplacian. Using this alternate operator, we construct a uniform preconditioner for the original graph Laplacian.

An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence

We extend the $L^{p}$ theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of $W$-random graphs.

Semi-External Memory Sparse Matrix Multiplication for Billion-Node Graphs

Sparse matrix multiplication is traditionally performed in memory and scales to large matrices using the distributed memory of multiple nodes. In contrast, we scale sparse matrix multiplication beyond memory capacity by implementing sparse matrix dense matrix multiplication (SpMM) in a semi-external memory (SEM) fashion; i.e., we keep the sparse matrix on commodity SSDs and dense matrices in memory. Our SEM-SpMM incorporates many in-memory optimizations for large power-law graphs. It outperforms the in-memory implementations of Trilinos and Intel MKL and scales to billion-node graphs, far beyond the limitations of memory. Furthermore, on a single large parallel machine, our SEM-SpMM operates as fast as the distributed implementations of Trilinos using five times as much processing power. We also run our implementation in memory (IM-SpMM) to quantify the overhead of keeping data on SSDs. SEM-SpMM achieves almost 100% performance of IM-SpMM on graphs when the dense matrix has more than four columns; it achieves at least 65% performance of IM-SpMM on all inputs. We apply our SpMM to three important data analysis tasks--PageRank, eigensolving, and non-negative matrix factorization--and show that our SEM implementations significantly advance the state of the art.

Minimum vertex cover in generalized random graphs with power law degree distribution

In this paper we study the approximability of the minimum vertex cover problem in power law graphs. In particular, we investigate the behavior of a standard 2-approximation algorithm together with a simple pre-processing step when the input is a random sample from a generalized random graph model with power law degree distribution. More precisely, if the probability of a vertex of degree i to be present in the graph is ci−β, where β>2 and c is a normalizing constant, the expected approximation ratio is 1+ζ(β)−1Liβ(e−ρ(β)), where ζ(β) is the Riemann Zeta function of β, Li(β) is the polylogarithmic special function of β and ρ(β)=Liβ−2(1e)ζ(β−1).

New analytical lower bounds on the clique number of a graph

This paper proposes three new analytical lower bounds on the clique number of a graph and compares these bounds with those previously established in the literature. Two proposed bounds are derived from the well-known Motzkin–Straus quadratic programming formulation for the maximum clique problem. Theoretical results on the comparison of various bounds are established. Computational experiments are performed on random graph models such as the Erdös-Rényi model for uniform graphs and the generalized random graph model for power-law graphs that simulate graphs with different densities and assortativity coefficients. Computational results suggest that the proposed new analytical bounds improve the existing ones on many graph instances.

Approximation algorithm for partial positive influence problem in social network

Influence problem is one of the central problems in the study of online social networks, the goal of which is to influence all nodes with the minimum number of seeds. However, in the real world, it might be too expensive to influence all nodes. In many cases, it is satisfactory to influence nodes only up to some percent p. In this paper, we study the minimum partial positive influence dominating set (MPPIDS) problem. In fact, we presented an approximation algorithm for a more general problem called minimum partial set multicover problem. As a consequence, the MPPIDS problem admits an approximation with performance ratio $$\gamma H(\Delta )$$źH(Δ), where $$H(\cdot )$$H(·) is the Harmonic number, $$\gamma =1/(1-(1-p)\eta ),\eta \approx \Delta ^2/\delta $$ź=1/(1-(1-p)ź),źźΔ2/ź, and $$\Delta ,\delta $$Δ,ź are the maximum degree and the minimum degree of the graph, respectively. For power-law graphs, we show that our algorithm has a constant performance ratio.

Signal Recovery on Graphs: Fundamental Limits of Sampling Strategies

This paper builds theoretical foundations for the recovery of a newly proposed class of smooth graph signals, approximately bandlimited graph signals, under three sampling strategies: uniform sampling, experimentally designed sampling, and active sampling. We then state minimax lower bounds on the maximum risk for the approximately bandlimited class under these three sampling strategies and show that active sampling cannot fundamentally outperform experimentally designed sampling. We propose a recovery strategy to compare uniform sampling with experimentally designed sampling. As the proposed recovery strategy lends itself well to statistical analysis, we derive the exact mean square error for each sampling strategy. To study convergence rates, we introduce two types of graphs and find that 1) the proposed recovery strategy achieves the optimal rates; and 2) the experimentally designed sampling fundamentally outperforms uniform sampling for Type-2 class of graphs. To validate our proposed recovery strategy, we test it on five specific graphs: a ring graph with $k$ nearest neighbors, an Erdős–Renyi graph, a random geometric graph, a small-world graph, and a power-law graph and find that experimental results match the proposed theory well. This paper also presents a comprehensive explanation for when and why sampling for semi-supervised learning with graphs works.

Randomized rumor spreading in poorly connected small-world networks

The Push-Pull protocol is a well-studied round-robin rumor spreading protocol defined as follows: initially a node knows a rumor and wants to spread it to all nodes in a network quickly. In each round, every informed node sends the rumor to a random neighbor, and every uninformed node contacts a random neighbor and gets the rumor from her if she knows it. We analyze the behavior of this protocol on random -trees, a class of power law graphs, which are small-world and have large clustering coefficients, built as follows: initially we have a -clique. In every step a new node is born, a random -clique of the current graph is chosen, and the new node is joined to all nodes of the -clique. When is fixed, we show that if initially a random node is aware of the rumor, then with probability after rounds the rumor propagates to nodes, where is the number of nodes and is any slowly growing function. Since these graphs have polynomially small conductance, vertex expansion and constant treewidth, these results demonstrate that Push-Pull can be efficient even on poorly connected networks. On the negative side, we prove that with probability the protocol needs at least rounds to inform all nodes. This exponential dichotomy between time required for informing almost all and all nodes is striking. Our main contribution is to present, for the first time, a natural class of random graphs in which such a phenomenon can be observed. Our technique for proving the upper bound successfully carries over to a closely related class of graphs, the random -Apollonian networks, for which we prove an upper bound of rounds for informing nodes with probability when is fixed. Here, © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Fast Generation of Sparse Random Kernel Graphs.

The development of kernel-based inhomogeneous random graphs has provided models that are flexible enough to capture many observed characteristics of real networks, and that are also mathematically tractable. We specify a class of inhomogeneous random graph models, called random kernel graphs, that produces sparse graphs with tunable graph properties, and we develop an efficient generation algorithm to sample random instances from this model. As real-world networks are usually large, it is essential that the run-time of generation algorithms scales better than quadratically in the number of vertices n. We show that for many practical kernels our algorithm runs in time at most 𝒪(n(logn)2). As a practical example we show how to generate samples of power-law degree distribution graphs with tunable assortativity.

On positive-influence target-domination

Consider a graph $$G=(V,E)$$ and a vertex subset $$A \subseteq V$$ . A vertex v is positive-influence dominated by A if either v is in A or at least half the number of neighbors of v belong to A. For a target vertex subset $$S \subseteq V$$ , a vertex subset A is a positive-influence target-dominating set for target set S if every vertex in S is positive-influence dominated by A. Given a graph G and a target vertex subset S, the positive-influence target-dominating set (PITD) problem is to find the minimum positive-influence dominating set for target S. In this paper, we show two results: (1) The PITD problem has a polynomial-time $$(1 + \log \lceil \frac{3}{2} \Delta \rceil )$$ -approximation in general graphs where $$\Delta $$ is the maximum vertex-degree of the input graph. (2) For target set S with $$|S|=\Omega (|V|)$$ , the PITD problem has a polynomial-time O(1)-approximation in power-law graphs.

Capacity of an associative memory model on random graph architectures

We analyze the storage capacity of the Hopfield models on classes of random graphs. While such a setup has been analyzed for the case that the underlying random graph model is an Erdös–Renyi graph, other architectures, including those investigated in the recent neuroscience literature, have not been studied yet. We develop a notion of storage capacity that highlights the influence of the graph topology and give results on the storage capacity for not too irregular random graph models. The class of models investigated includes the popular power law graphs for some parameter values.

Achieving Optimal Inter-Node Communication in Graph Partitioning Using Random Selection and Breadth-First Search

Processing large graph datasets represents an increasingly important area in computing research and applications. The size of many graph datasets has increased well beyond the processing capacity of a single computing node, thereby necessitating distributed approaches. As these datasets are processed over a distributed system of nodes, this leads to an inter-node communication cost problem that negatively affects system performance. Previously proposed algorithms implemented breadth-first search (BFS) for graph searching and focused on the execution, parallel performance and not the communication. In this paper a new methodology is proposed that combines BFS with random selection in order to partition large graph datasets and effectively minimize inter-node communication. The new method is discussed and applied to the single-source shortest path and PageRank algorithms using three graphs that are representative of real-world scenarios. Experimental results show that graph inter-node communication for canonical graphs representative of real-world data is improved up to 42 % in case of Powerlaw graph, up to 27 % in case of Random near K-regular graph (with low degree), and up to 7 % in case of Random near K-regular graph (with high degree).

Minimum edge blocker dominating set problem

This paper introduces and studies the minimum edge blocker dominating set problem (EBDP), which is formulated as follows. Given a vertex-weighted undirected graph and r > 0, remove a minimum number of edges so that the weight of any dominating set in the remaining graph is at least r. Dominating sets are used in a wide variety of graph-based applications such as the analysis of wireless and social networks. We show that the decision version of EBDP is NP-hard for any fixed r > 0. We present an analytical lower bound for the value of an optimal solution to EBDP and formulate this problem as a linear 0–1 program with a large number of constraints. We also study the convex hull of feasible solutions to EBDP and identify facet-inducing inequalities for this polytope. Furthermore, we develop the first exact algorithm for solving EBDP, which solves the proposed formulation by a branch-and-cut approach where nontrivial constraints are applied in a lazy fashion. Finally, we also provide the computational results obtained by using our approach on a test-bed of randomly generated instances and real-life power-law graphs.

Robustness of power-law networks: its assessment and optimization

Many practical complex networks, such as the Internet, WWW and social networks, are discovered to follow power-law distribution in their degree sequences, i.e., the number of nodes with degree $$i$$i in these networks is proportional to $$i^{-\beta }$$i-β for some exponential factor $$\beta > 0$$β>0. However, these networks also expose their vulnerabilities to a great number of threats such as adversarial attacks on the Internet, cyber-crimes on the WWW or malware propagations on social networks. Although power-law networks have been found robust under random attacks and vulnerable to intentional attacks via experimental observations, how to better understand their vulnerabilities from a theoretical point of view still remains open. In this paper, we study the vulnerability of power-law networks under random attacks and adversarial attacks using the in-depth probabilistic analysis on the theory of random power-law graph models. Our results indicate that power-law networks are able to tolerate random failures if their exponential factor $$\beta $$β is $$ 2.5$$β>2.5 the network robustness is unpredictable since it depends on the specific attacking strategy.

Synchronization of coupled chaotic maps

We prove a sufficient condition for synchronization for coupled one-dimensional maps and estimate the size of the window of parameters where synchronization takes place. It is shown that coupled systems on graphs with positive eigenvalues of the normalized graph Laplacian concentrated around 1 are more amenable for synchronization. In the light of this condition, we review spectral properties of Cayley, quasirandom, power-law graphs, and expanders and relate them to synchronization of the corresponding networks. The analysis of synchronization on these graphs is illustrated with numerical experiments. The results of this paper highlight the advantages of random connectivity for synchronization of coupled chaotic dynamical systems.