Statistical Properties of a Generalized Threshold Network Model
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Abstract
The threshold network model is a type of finite random graph. In this paper, we introduce a generalized threshold network model. A pair of vertices with random weights is connected by an edge when real-valued functions of the pair of weights belong to given Borel sets. We extend several known limit theorems for the number of prescribed subgraphs and prove a uniform strong law of large numbers. We also prove two limit theorems for the local and global clustering coefficients.
Keywords
Complex networks Threshold network models Random graphsAMS 2000 Subject Classifications
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References
- Albert R, Barabási A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74(1):47–97CrossRefGoogle Scholar
- Arcones MA, Giné E (1993) Limit theorems for U-processes. Ann Probab 21(3):1494–1542MATHCrossRefMathSciNetGoogle Scholar
- Billingsley P (1995) Probability and measure, 3rd edn. In: Wiley series in probability and mathematical statistics. A Wiley-Interscience Publication. Wiley, New YorkGoogle Scholar
- Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang D-U (2006) Complex networks: structure and dynamics. Phys Rep 424(4–5):175–308CrossRefMathSciNetGoogle Scholar
- Boguñá M, Pastor-Satorras R (2003) Class of correlated random networks with hidden variables. Phys Rev E 68:036112CrossRefGoogle Scholar
- Caldarelli G, Capocci A, De Los Rios P, Muñoz MA (2002) Scale-free networks from varying vertex intrinsic fitness. Phys Rev Lett 89:258702CrossRefGoogle Scholar
- Dudley RM (1999) Uniform central limit theorems. In: Cambridge studies in advanced mathematics, vol 63. Cambridge University Press, CambridgeGoogle Scholar
- Hagberg A, Schult DA, Swart PJ (2006) Designing threshold networks with given structural and dynamical properties. Phys Rev E 74:056116CrossRefMathSciNetGoogle Scholar
- Konno N, Masuda N, Roy R, Sarkar A (2005) Rigorous results on the threshold network model. J Phys A Math Gen 38(28):6277–6291MATHCrossRefMathSciNetGoogle Scholar
- Masuda N, Konno N (2006) VIP-club phenomenon: emergence of elites and masterminds in social networks. Soc Netw 28(4):297–309CrossRefGoogle Scholar
- Masuda N, Miwa H, Konno N (2004) Analysis of scale-free networks based on a threshold graph with intrinsic vertex weights. Phys Rev E 70:036124CrossRefMathSciNetGoogle Scholar
- Masuda N, Miwa H, Konno N (2005) Geographical threshold graphs with small-world and scale-free properties. Phys Rev E 71:036108CrossRefGoogle Scholar
- Najim CA, Russo RP (2003) On the number of subgraphs of a specified form embedded in a random graph. Methodol Comput Appl Probab 5(1):23–33MATHCrossRefMathSciNetGoogle Scholar
- Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256MATHCrossRefMathSciNetGoogle Scholar
- Peskir G (2000) From uniform laws of large numbers to uniform ergodic theorems. In: Lecture notes series (Aarhus), vol 66. University of Aarhus, Department of Mathematics, AarhusGoogle Scholar
- Serfling RJ (1980) Approximation theorems of mathematical statistics. In: Wiley series in probability and mathematical statistics. Wiley, New YorkGoogle Scholar
- Servedio VDP, Caldarelli G, Buttá P (2004) Vertex intrinsic fitness: how to produce arbitrary scale-free networks. Phys Rev E 70:056126CrossRefGoogle Scholar
- Söderberg B (2002) General formalism for inhomogeneous random graphs. Phys Rev E 66:066121CrossRefMathSciNetGoogle Scholar
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