scholarly journals Algorithms for generating large-scale clustered random graphs

2014 ◽  
Vol 2 (3) ◽  
pp. 403-415 ◽  
Author(s):  
CHENG WANG ◽  
OMAR LIZARDO ◽  
DAVID HACHEN
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Real World ◽  
Local Structure ◽  
Topological Structure ◽  
Large Scale ◽  
Random Processes ◽  
Clustering Coefficient ◽  
Computation Efficiency ◽  
Large Scale Networks

AbstractReal-world networks are often compared to random graphs to assess whether their topological structure could be a result of random processes. However, a simple random graph in large scale often lacks social structure beyond the dyadic level. As a result we need to generate clustered random graph to compare the local structure at higher network levels. In this paper a generalized version of Gleeson's algorithm G(VS, VT, ES, ET, S, T) is advanced to generate a clustered random graph in large-scale which persists the number of vertices |V|, the number of edges |E|, and the global clustering coefficient CΔ as in the real network and it works successfully for nine large-scale networks. Our new algorithm also has advantages in randomness evaluation and computation efficiency when compared with the existing algorithms.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

Characteristics of disrupted topological organization in white matter functional connectome in schizophrenia

2020 ◽  
pp. 1-11
Author(s):  
Yuchao Jiang ◽  
Dezhong Yao ◽  
Jingyu Zhou ◽  
Yue Tan ◽  
Huan Huang ◽  
...  
Keyword(s):  
White Matter ◽  
Large Scale ◽  
Negative Symptoms ◽  
Functional Organization ◽  
Clustering Coefficient ◽  
Theoretical Approaches ◽  
Topological Organization ◽  
Large Scale Networks ◽  
Nodal Level

Abstract Background Neuroimaging characteristics have demonstrated disrupted functional organization in schizophrenia (SZ), involving large-scale networks within grey matter (GM). However, previous studies have ignored the role of white matter (WM) in supporting brain function. Methods Using resting-state functional MRI and graph theoretical approaches, we investigated global topological disruptions of large-scale WM and GM networks in 93 SZ patients and 122 controls. Six global properties [clustering coefficient (Cp), shortest path length (Lp), local efficiency (Eloc), small-worldness (σ), hierarchy (β) and synchronization (S) and three nodal metrics [nodal degree (Knodal), nodal efficiency (Enodal) and nodal betweenness (Bnodal)] were utilized to quantify the topological organization in both WM and GM networks. Results At the network level, both WM and GM networks exhibited reductions in Eloc, Cp and S in SZ. The SZ group showed reduced σ and β only for the WM network. Furthermore, the Cp, Eloc and S of the WM network were negatively correlated with negative symptoms in SZ. At the nodal level, the SZ showed nodal disturbances in the corpus callosum, optic radiation, posterior corona radiata and tempo-occipital WM tracts. For GM, the SZ manifested increased nodal centralities in frontoparietal regions and decreased nodal centralities in temporal regions. Conclusions These findings provide the first evidence for abnormal global topological properties in SZ from the perspective of a substantial whole brain, including GM and WM. Nodal centralities enhance GM areas, along with a reduction in adjacent WM, suggest that WM functional alterations may be compensated for adjacent GM impairments in SZ.

scholarly journals A simple differential geometry for complex networks

2020 ◽  
pp. 1-28
Author(s):  
Emil Saucan ◽  
Areejit Samal ◽  
Jürgen Jost
Keyword(s):  
Differential Geometry ◽  
Complex Networks ◽  
Ricci Curvature ◽  
Real World ◽  
Large Scale ◽  
Metric Spaces ◽  
Geodesic Curvature ◽  
Higher Dimensional ◽  
Large Scale Networks ◽  
Bonnet Theorem

Abstract We introduce new definitions of sectional, Ricci, and scalar curvatures for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks and are more intuitive and easier to compute than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss–Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behavior, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely used network measures.

scholarly journals Analysis of E-Commerce Product Graphs

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Graph Analysis ◽  
Random Graph Model ◽  
Product Graphs ◽  
Clustering Coefficients

<p>Consumer behavior in retail stores gives rise to product graphs based on copurchasing</p><p>or co-viewing behavior. These product graphs can be analyzed using</p><p>the known methods of graph analysis. In this paper, we analyze the product graph</p><p>at Target Corporation based on the Erd˝os-Renyi random graph model. In particular,</p><p>we compute clustering coefficients of actual and random graphs, and we find that</p><p>the clustering coefficients of actual graphs are much higher than random graphs.</p><p>We conduct the analysis on the entire set of products and also on a per category</p><p>basis and find interesting results. We also compute the degree distribution and</p><p>we find that the degree distribution is a power law as expected from real world</p><p>networks, contrasting with the ER random graph.</p>

A novel method for identifying influential nodes in complex networks based on multiple attributes

2018 ◽  
Vol 32 (28) ◽  
pp. 1850307 ◽  
Author(s):  
Dong Liu ◽  
Hao Nie ◽  
Baowen Zhang
Keyword(s):  
Large Scale ◽  
Real Life ◽  
Entropy Method ◽  
Viral Marketing ◽  
Clustering Coefficient ◽  
Node Location ◽  
Influential Nodes ◽  
Multiple Attributes ◽  
Comparison Algorithms ◽  
Large Scale Networks

Identifying influential nodes is a crucial issue in epidemic spreading, controlling the propagation process of information and viral marketing. Thus, algorithms for exploring vital nodes have aroused more and more concern among researchers. Recently, scholars have proposed various types of algorithms based on different perspectives. However, each of these methods has their own strengths and weaknesses. In this work, we introduce a novel multiple attributes centrality for identifying significant nodes based on the node location and neighbor information attributes. We call our proposed method the MAC. Specifically, we utilize the information of the number of iterations per node to enhance the accuracy of the K-shell algorithm, so that the location attribute can be used to distinguish the important nodes more deeply. And the neighbor information attribute we selected can effectively avoid the overlapping problem of neighbor information propagation caused by large clustering coefficient of networks. Because these two indexes have different emphases, we use entropy method to assign them reasonable weights. In addition, MAC has low time complexity O(n), which makes the algorithm suitable for large-scale networks. In order to objectively assess its performance, we utilize the Susceptible-Infected-Recovered (SIR) model to verify the propagation capability of each node and compare the MAC method with several classic methods in six real-life datasets. Extensive experiments verify the superiority of our algorithm to other comparison algorithms.

scholarly journals Greed is Good for Deterministic Scale-Free Networks

Algorithmica ◽  
2020 ◽  
Vol 82 (11) ◽  
pp. 3338-3389
Author(s):  
Ankit Chauhan ◽  
Tobias Friedrich ◽  
Ralf Rothenberger
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
High Probability ◽  
Maximum Independent Set ◽  
Graph Models ◽  
Random Graph Models ◽  
Graph Classes

Abstract Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306–1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung–Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical $$\textsf {NP}$$ NP -hard optimization problems on PLB networks. It is known that on general graphs with maximum degree $$\Delta$$ Δ , a greedy algorithm, which chooses nodes in the order of their degree, only achieves a $$\Omega (\ln \Delta )$$ Ω ( ln Δ ) -approximation for Minimum Vertex Cover and Minimum Dominating Set, and a $$\Omega (\Delta )$$ Ω ( Δ ) -approximation for Maximum Independent Set. We prove that the PLB-U property with $$\beta >2$$ β > 2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are -hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless = . Furthermore, we prove that all three problems are in if the PLB-U property holds.

scholarly journals Local dependency in networks

2015 ◽  
Vol 25 (2) ◽  
pp. 281-293 ◽  
Author(s):  
Miloš Kudĕlka ◽  
Šárka Zehnalová ◽  
Zdenĕk Horák ◽  
Pavel Krömer ◽  
Václav Snášel
Keyword(s):  
Real World ◽  
Large Scale ◽  
Real World Data ◽  
High Quality ◽  
High Effectiveness ◽  
Local Dependency ◽  
Node Ranking ◽  
Large Scale Networks ◽  
The Relationship ◽  
Entire Network

Abstract Many real world data and processes have a network structure and can usefully be represented as graphs. Network analysis focuses on the relations among the nodes exploring the properties of each network. We introduce a method for measuring the strength of the relationship between two nodes of a network and for their ranking. This method is applicable to all kinds of networks, including directed and weighted networks. The approach extracts dependency relations among the network’s nodes from the structure in local surroundings of individual nodes. For the tasks we deal with in this article, the key technical parameter is locality. Since only the surroundings of the examined nodes are used in computations, there is no need to analyze the entire network. This allows the application of our approach in the area of large-scale networks. We present several experiments using small networks as well as large-scale artificial and real world networks. The results of the experiments show high effectiveness due to the locality of our approach and also high quality node ranking comparable to PageRank.

scholarly journals Analysis of E-Commerce Product Graphs

2020 ◽  
Author(s):  
Shalin Shah
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Real World ◽  
Degree Distribution ◽  
Graph Model ◽  
Graph Analysis ◽  
Random Graph Model ◽  
Product Graphs ◽  
Clustering Coefficients

<p>Consumer behavior in retail stores gives rise to product graphs based on copurchasing</p><p>or co-viewing behavior. These product graphs can be analyzed using</p><p>the known methods of graph analysis. In this paper, we analyze the product graph</p><p>at Target Corporation based on the Erd˝os-Renyi random graph model. In particular,</p><p>we compute clustering coefficients of actual and random graphs, and we find that</p><p>the clustering coefficients of actual graphs are much higher than random graphs.</p><p>We conduct the analysis on the entire set of products and also on a per category</p><p>basis and find interesting results. We also compute the degree distribution and</p><p>we find that the degree distribution is a power law as expected from real world</p><p>networks, contrasting with the ER random graph.</p>

scholarly journals Efficiently counting complex multilayer temporal motifs in large-scale networks

2019 ◽  
Vol 6 (1) ◽  
Author(s):  
Hanjo D. Boekhout ◽  
Walter A. Kosters ◽  
Frank W. Takes
Keyword(s):  
Communication Networks ◽  
Real World ◽  
Large Scale ◽  
Level Structure ◽  
Dynamic Networks ◽  
Computational Cost ◽  
Network Motifs ◽  
Interaction Patterns ◽  
Counting Algorithms ◽  
Large Scale Networks

Abstract This paper proposes novel algorithms for efficiently counting complex network motifs in dynamic networks that are changing over time. Network motifs are small characteristic configurations of a few nodes and edges, and have repeatedly been shown to provide insightful information for understanding the meso-level structure of a network. Here, we deal with counting more complex temporal motifs in large-scale networks that may consist of millions of nodes and edges. The first contribution is an efficient approach to count temporal motifs in multilayer networks and networks with partial timing, two prevalent aspects of many real-world complex networks. We analyze the complexity of these algorithms and empirically validate their performance on a number of real-world user communication networks extracted from online knowledge exchange platforms. Among other things, we find that the multilayer aspects provide significant insights in how complex user interaction patterns differ substantially between online platforms. The second contribution is an analysis of the viability of motif counting algorithms for motifs that are larger than the triad motifs studied in previous work. We provide a novel categorization of motifs of size four, and determine how and at what computational cost these motifs can still be counted efficiently. In doing so, we delineate the “computational frontier” of temporal motif counting algorithms.

scholarly journals Efficient network immunization under limited knowledge

2020 ◽  
Author(s):  
Yangyang Liu ◽  
Hillel Sanhedrai ◽  
GaoGao Dong ◽  
Louis M Shekhtman ◽  
Fan Wang ◽  
...  
Keyword(s):  
Real World ◽  
Large Scale ◽  
Complete Information ◽  
Analytical Framework ◽  
Epidemic Spreading ◽  
Immunization Strategy ◽  
Scale Free ◽  
Limited Knowledge ◽  
Small N ◽  
Large Scale Networks

Abstract Targeted immunization of centralized nodes in large-scale networks has attracted significant attention. However, in real-world scenarios, knowledge and observations of the network may be limited, thereby precluding a full assessment of the optimal nodes to immunize (or quarantine) in order to avoid epidemic spreading such as that of the current coronavirus disease (COVID-19) epidemic. Here, we study a novel immunization strategy where only n nodes are observed at a time and the most central among these n nodes is immunized. This process can globally immunize a network. We find that even for small n (≈10) there is significant improvement in the immunization (quarantine), which is very close to the levels of immunization with full knowledge. We develop an analytical framework for our method and determine the critical percolation threshold pc and the size of the giant component P∞ for networks with arbitrary degree distributions P(k). In the limit of n → ∞ we recover prior work on targeted immunization, whereas for n = 1 we recover the known case of random immunization. Between these two extremes, we observe that, as n increases, pc increases quickly towards its optimal value under targeted immunization with complete information. In particular, we find a new general scaling relationship between |pc(∞) − pc(n)| and n as |pc(∞) − pc(n)| ∼ n−1exp(−αn). For scale-free (SF) networks, where P(k) ∼ k−γ, 2 &lt; γ &lt; 3, we find that pc has a transition from zero to nonzero when n increases from n = 1 to O(log N) (where N is the size of the network). Thus, for SF networks, having knowledge of  ≈log N nodes and immunizing the most optimal among them can dramatically reduce epidemic spreading. We also demonstrate our limited knowledge immunization strategy on several real-world networks and confirm that in these real networks, pc increases significantly even for small n.

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