SCIENCE CHINA Physics, Mechanics & Astronomy Print-CrossMark February 2017 Vol. 60 No. 2: 020521 doi: 10.1007/s11433-016-0446-3 c ⃝ Science China Press and Springer-Verlag Berlin Heidelberg 2016 phys.scichina.com link.springer.com . Letter to the Editor . Criticality of networks with long-range connections ZiQing Yang 1 , MaoXin Liu 2 , and XiaoSong Chen 1,3* 1 Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China; 2 Beijing Computational Science Research Center, Beijing 100084, China; 3 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Received November 8, 2016; accepted November 18, 2016; published online December 23, 2016 Citation: Z. Q. Yang, M. X. Liu, and X. S. Chen, Criticality of networks with long-range connections, Sci. China-Phys. Mech. Astron. 60, 020521 (2017), doi: 10.1007/s11433-016-0446-3 The formation of giant clusters, namely the percolation phase transition, is one of the most widely studied critical phenom- ena on networks. The critical behaviors of percolation in one- and two-dimensional lattices have been given in the book [1]. For d-dimensional lattices, the critical exponents of percola- tion change with d until the upper critical dimension d u = 6, above which they are independent of d and become mean- field like. It is also well known that the critical behaviors of percolation on Erd ˝ os-R´ enyi (ER) networks are also mean- field like [2, 3]. For spin models on d-dimensional lattices with interac- tions between spins decaying with distance r as J (r) ∼ r −(d+σ) , the critical behaviors depend on both d and σ [4, 5]. For long-range (LR) interactions with 0 <σ< 2, the up- per critical dimension of spin models d u = 2σ. When 0 <σ< d/2, the critical exponents of spin models are mean- field like and independent of σ. In the intermediate regime d/2 <σ< 2 − η SR , where η SR is the critical exponent of the spin model with short-range (SR) interactions, the critical ex- ponents are continuous functions of σ. It was found by Sak [6] that the spin models with σ> 2 − η SR have the critical exponents of SR interactions . It is natural to introduce networks on d-dimensional lat- tices with connection probabilities decaying with distance r as P(r) ∼ r −(d+σ) . For two-dimensional lattice network with LR connections, Moukarzel [7] focused on the topological properties such as shortest-paths and chemical dimensions. *Corresponding author (email: chenxs@itp.ac.cn) Li et al. [8] measured the fractal dimensions of the percola- tion clusters and Fisher exponent τ in one-dimensional chains and two-dimensional lattices. In this letter, we investigate the universality class of per- colation in one- and two-dimensional lattices with the con- nection probabilities P(r) of general σ. At σ = −d, the net- work becomes ER network. If σ →∞, only the nodes in the nearest neighborhood can be linked and we have the nor- mal lattice network. The critical behaviors of percolation can be studied by the largest cluster [9, 10] and its largest size gap during network evolution [11, 12]. In an evolution, edges are added one by one to the network. At an evolution step i of network with N nodes, we first choose a distance r with prob- ability P(r), and then connect two randomly chosen nodes separated by r. At this step the largest cluster has a reduced size gap δ i = [S 1 (i) − S 1 (i − 1)] /N. In m-th evolution of net- work, we can get the largest gap Δ (m) = max{δ i } at a reduced evolution step p (m) = T (m) /N, where T (m) = arg max{δ i }. After finishing M evolutions of network, we can calculate ¯ Δ(N) = ⟨Δ⟩ = 1 M ∑ M m=1 Δ (m) and ¯ p(N) = ⟨ p⟩ = 1 M ∑ M m=1 p (m) . For N = L d ≫ 1, they follow the finite-size scaling forms [11, 12] ¯ Δ(N) = a 1 L −β/ν = a 1 N −1/(dν) and ¯ p(N) = p c + a 2 N −1/(dν) , where p c is the critical point. The root mean squares of fluctuations δΔ=Δ (m) − ¯ Δ and δ p = p (m) − ¯ p sat- isfy also finite-size scaling forms [11, 12] χ Δ = √ ⟨ [δΔ] 2 ⟩ = a 3 N − ˆ β/(d ˆ ν) and χ p = √ ⟨ [δ p] 2 ⟩ = a 4 N −1/(d ˆ ν) . The universality class of percolation is characterized by the critical exponents β, ν, ˆ β, ˆ ν. We anticipate that β = ˆ β and ν = ˆ ν. We have performed Monte Carlo simulations of network