scholarly journals Network processes on clique-networks with high average degree: the limited effect of higher-order structure

2021 ◽  
Author(s):  
Clara Stegehuis ◽  
Thomas Peron
Keyword(s):  
Random Graph ◽  
Graph Model ◽  
Average Degree ◽  
Kuramoto Model ◽  
Order Structure ◽  
Bond Percolation ◽  
Random Graph Model ◽  
Analytical Approximations ◽  
Clustered Networks ◽  
The Kuramoto Model

Abstract In this paper, we investigate the effect of local structures on network processes. We investigate a random graph model that incorporates local clique structures, and thus deviates from the locally tree-like behavior of most standard random graph models. For the process of bond percolation, we derive analytical approximations for large percolation probabilities and the critical percolation value. Interestingly, these derivations show that when the average degree of a vertex is large, the influence of the deviations from the locally tree-like structure is small. In our simulations, this insensitivity to local clique structures often already kicks in for networks with average degrees as low as 6. Furthermore, we show that the different behavior of bond percolation on clustered networks compared to tree-like networks that was found in previous works can be almost completely attributed to differences in degree sequences rather than differences in clustering structures. We finally show that these results also extend to completely different types of dynamics, by deriving similar conclusions and simulations for the Kuramoto model on the same types of clustered and non-clustered networks.

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.

The Impact of Edge Correlations in Random Networks

2021 ◽  
Vol 30 (4) ◽  
pp. 525-537
Author(s):  
András Faragó ◽  
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Real Life ◽  
Graph Model ◽  
Random Networks ◽  
Random Graph Model ◽  
Special Cases ◽  
Graph Parameters ◽  
The Impact ◽  
Good Number

Random graphs are frequently used models of real-life random networks. The classical Erdös–Rényi random graph model is very well explored and has numerous nontrivial properties. In particular, a good number of important graph parameters that are hard to compute in the deterministic case often become much easier in random graphs. However, a fundamental restriction in the Erdös–Rényi random graph is that the edges are required to be probabilistically independent. This is a severe restriction, which does not hold in most real-life networks. We consider more general random graphs in which the edges may be dependent. Specifically, two models are analyzed. The first one is called a p-robust random graph. It is defined by the requirement that each edge exist with probability at least p, no matter how we condition on the presence/absence of other edges. It is significantly more general than assuming independent edges existing with probability p, as exemplified via several special cases. The second model considers the case when the edges are positively correlated, which means that the edge probability is at least p for each edge, no matter how we condition on the presence of other edges (but absence is not considered). We prove some interesting, nontrivial properties about both models.

scholarly journals Ramsey properties of randomly perturbed graphs: cliques and cycles

2020 ◽  
Vol 29 (6) ◽  
pp. 830-867 ◽  
Author(s):  
Shagnik Das ◽  
Andrew Treglown
Keyword(s):  
Random Graph ◽  
High Probability ◽  
Graph Model ◽  
Significant Progress ◽  
Random Choice ◽  
Ramsey Problem ◽  
Random Graph Model ◽  
Dense Graph ◽  
Precise Solution ◽  
Analogous Question

AbstractGiven graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem.We also give a precise solution to the analogous question in the case when both H1 = Cs and H2 = Ct are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs, Kt) -Ramsey (for odd s ≽ 5 and t ≽ 4).To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

scholarly journals A theoretical and empirical comparison of the temporal exponential random graph model and the stochastic actor-oriented model

2019 ◽  
Vol 7 (1) ◽  
pp. 20-51 ◽  
Author(s):  
Philip Leifeld ◽  
Skyler J. Cranmer
Keyword(s):  
Network Analysis ◽  
Random Graph ◽  
Graph Model ◽  
Real Data ◽  
The Other ◽  
Exponential Random Graph Model ◽  
Empirical Comparison ◽  
Random Graph Model ◽  
Exponential Random Graph ◽  
Longitudinal Network Analysis

AbstractThe temporal exponential random graph model (TERGM) and the stochastic actor-oriented model (SAOM, e.g., SIENA) are popular models for longitudinal network analysis. We compare these models theoretically, via simulation, and through a real-data example in order to assess their relative strengths and weaknesses. Though we do not aim to make a general claim about either being superior to the other across all specifications, we highlight several theoretical differences the analyst might consider and find that with some specifications, the two models behave very similarly, while each model out-predicts the other one the more the specific assumptions of the respective model are met.

Data-Driven Dynamic Network Modeling for Analyzing the Evolution of Product Competitions

2020 ◽  
Vol 142 (3) ◽  
Author(s):  
Jian Xie ◽  
Youyi Bi ◽  
Zhenghui Sha ◽  
Mingxian Wang ◽  
Yan Fu ◽  
...  
Keyword(s):  
Survey Data ◽  
Random Graph ◽  
Network Modeling ◽  
Dynamic Network ◽  
Graph Model ◽  
Exponential Random Graph Model ◽  
Random Graph Model ◽  
Exponential Random Graph ◽  
Dynamic Network Modeling ◽  
The Impact

Abstract Understanding the impact of engineering design on product competitions is imperative for product designers to better address customer needs and develop more competitive products. In this paper, we propose a dynamic network-based approach for modeling and analyzing the evolution of product competitions using multi-year buyer survey data. The product co-consideration network, formed based on the likelihood of two products being co-considered from survey data, is treated as a proxy of products’ competition relations in a market. The separate temporal exponential random graph model (STERGM) is employed as the dynamic network modeling technique to model the evolution of network as two separate processes: link formation and link dissolution. We use China’s automotive market as a case study to illustrate the implementation of the proposed approach and the benefits of dynamic network models compared to the static network modeling approach based on an exponential random graph model (ERGM). The results show that since STERGM takes preexisting competition relations into account, it provides a pathway to gain insights into why a product may maintain or lose its competitiveness over time. These driving factors include both product attributes (e.g., fuel consumption) as well as current market structures (e.g., the centralization effect). With the proposed dynamic network-based approach, the insights gained from this paper can help designers better interpret the temporal changes of product competition relations to support product design decisions.

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