Efficient Laplacian spectral density computations for networks with arbitrary degree distributions

2021 ◽  
Vol 9 (3) ◽  
pp. 312-327
Author(s):  
Grover E. C. Guzman ◽  
Peter F. Stadler ◽  
André Fujita
Keyword(s):  
Spectral Density ◽  
Degree Distribution ◽  
Message Passing ◽  
Arbitrary Degree ◽  
Density Calculation ◽  
Diagonalization Method ◽  
Computationally Intensive ◽  
And Mathematics ◽  
Edge Based

AbstractThe network Laplacian spectral density calculation is critical in many fields, including physics, chemistry, statistics, and mathematics. It is highly computationally intensive, limiting the analysis to small networks. Therefore, we present two efficient alternatives: one based on the network’s edges and another on the degrees. The former gives the exact spectral density of locally tree-like networks but requires iterative edge-based message-passing equations. In contrast, the latter obtains an approximation of the spectral density using only the degree distribution. The computational complexities are 𝒪(|E|log(n)) and 𝒪(n), respectively, in contrast to 𝒪(n3) of the diagonalization method, where n is the number of vertices and |E| is the number of edges.

scholarly journals Anti-conformism in the Threshold Model of Collective Behavior

2019 ◽  
Vol 10 (2) ◽  
pp. 444-477 ◽  
Author(s):  
Michel Grabisch ◽  
Fen Li
Keyword(s):  
Uniform Distribution ◽  
Gaussian Distribution ◽  
Collective Behavior ◽  
Degree Distribution ◽  
Threshold Model ◽  
Opinion Dynamics ◽  
Time Step ◽  
Arbitrary Degree ◽  
Seminal Work ◽  
Cascade Effects

Abstract We provide a detailed study of the threshold model, where both conformist and anti-conformist agents coexist. Our study bears essentially on the convergence of the opinion dynamics in the society of agents, i.e., finding absorbing classes, cycles, etc. Also, we are interested in the existence of cascade effects, as this may constitute an undesirable phenomenon in collective behavior. We divide our study into two parts. In the first one, we basically study the threshold model supposing a fixed complete network, where every one is connected to every one, like in the seminal work of Granovetter. We study the case of a uniform distribution of the threshold, of a Gaussian distribution, and finally give a result for arbitrary distributions, supposing there is one type of anti-conformist. In a second part, we suppose that the neighborhood of an agent is random, drawn at each time step from a distribution. We distinguish the case where the degree (number of links) of an agent is fixed, and where there is an arbitrary degree distribution. We show the existence of cascades and that for most societies, the opinion converges to a chaotic situation.

scholarly journals Modularity and anti-modularity in networks with arbitrary degree distribution

2010 ◽  
Vol 5 (1) ◽  
pp. 32 ◽  
Author(s):  
Arend Hintze ◽  
Christoph Adami
Keyword(s):  
Degree Distribution ◽  
Arbitrary Degree

scholarly journals Arbitrary degree distribution networks with perturbations

AIP Advances ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 025301
Author(s):  
Xiaomin Wang ◽  
Fei Ma ◽  
Bing Yao
Keyword(s):  
Degree Distribution ◽  
Distribution Networks ◽  
Arbitrary Degree

scholarly journals A generalized configuration model with degree correlations and its percolation analysis

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Duan-Shin Lee ◽  
Cheng-Shang Chang ◽  
Miao Zhu ◽  
Hung-Chih Li
Keyword(s):  
Correlation Function ◽  
Closed Form ◽  
Computer Simulations ◽  
Degree Distribution ◽  
Configuration Model ◽  
Arbitrary Degree ◽  
Degree Correlation ◽  
Degree Correlations

AbstractIn this paper we present a generalization of the classical configuration model. Like the classical configuration model, the generalized configuration model allows users to specify an arbitrary degree distribution. In our generalized configuration model, we partition the stubs in the configuration model into b blocks of equal sizes and choose a permutation function h for these blocks. In each block, we randomly designate a number proportional to q of stubs as type 1 stubs, where q is a parameter in the range [0,1]. Other stubs are designated as type 2 stubs. To construct a network, randomly select an unconnected stub. Suppose that this stub is in block i. If it is a type 1 stub, connect this stub to a randomly selected unconnected type 1 stub in block h(i). If it is a type 2 stub, connect it to a randomly selected unconnected type 2 stub. We repeat this process until all stubs are connected. Under an assumption, we derive a closed form for the joint degree distribution of two random neighboring vertices in the constructed graph. Based on this joint degree distribution, we show that the Pearson degree correlation function is linear in q for any fixed b. By properly choosing h, we show that our construction algorithm can create assortative networks as well as disassortative networks. We present a percolation analysis of this model. We verify our results by extensive computer simulations.

GPU Computation and Platforms

2016 ◽  
pp. 136-174
Author(s):  
K. Bhargavi ◽  
Sathish Babu B.
Keyword(s):  
Message Passing ◽  
High Performance ◽  
Message Passing Interface ◽  
Gpu Computing ◽  
Graphics Processing Unit ◽  
General Purpose ◽  
Processing Unit ◽  
Computing Platforms ◽  
Computationally Intensive ◽  
Graphics Processing

The GPUs (Graphics Processing Unit) were mainly used to speed up computation intensive high performance computing applications. There are several tools and technologies available to perform general purpose computationally intensive application. This chapter primarily discusses about GPU parallelism, applications, probable challenges and also highlights some of the GPU computing platforms, which includes CUDA, OpenCL (Open Computing Language), OpenMPC (Open MP extended for CUDA), MPI (Message Passing Interface), OpenACC (Open Accelerator), DirectCompute, and C++ AMP (C++ Accelerated Massive Parallelism). Each of these platforms is discussed briefly along with their advantages and disadvantages.

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