Fundamental Structural Constraint of Random Scale-Free Networks

Abstract

We study the structural constraint of random scale-free networks that determines possible combinations of the degree exponent γ and the upper cutoff kc in the thermodynamic limit. We employ the framework of graphicality transitions proposed by [Del Genio and co-workers, Phys. Rev. Lett. 107, 178701 (2011)], while making it more rigorous and applicable to general values of kc. Using the graphicality criterion, we show that the upper cutoff must be lower than kc ∼ N 1/γ for γ < 2, whereas any upper cutoff is allowed for γ > 2. This result is also numerically verified by both the random and deterministic sampling of degree sequences.PACS numbers: 89.75. Hc, 02.10.Ox, 89.75.Da, 64.60.aq Complex networks [1] are found in diverse natural and artificial systems, which consist of heterogeneous elements (nodes) coupled by connections (links) markedly different from those of ordinary lattices. In particular, many systems [2-6] can be interpreted as scale-free networks, in which the fraction of nodes with degree k (i.e., k links) obeys the power-law distribution P (k) ∼ k −γ over a broad range of values bounded by k m ≤ k ≤ k c where γ is called the degree exponent, k m is the lower cutoff, and k c is the upper cutoff. There have been interests in topological and dynamical properties induced by the degree distribution, which have been examined through various studies on random scale-free networks [7][8][9][10][11][12][13].Random scale-free networks refer to an ensemble of networks constrained only by the parameters γ, k m , and k c . In general, k m is set as a constant, while k c is assumed to increase with the number of nodes N as k c ∼ N α with 0 ≤ α ≤ 1. Besides, self-loops or multiple links between a pair of nodes are often disallowed. Under the circumstances, the degree exponent γ and the cutoff exponent α determine various properties of networks in the thermodynamic limit, N → ∞. It is known that γ contributes to the resilience against node failures [7], the epidemic threshold [8], the consensus time of opinion dynamics [9], etc. Meanwhile, α affects the expected value of the generated maximum degree [10], degree correlations [11], finite-size scaling at criticality [12,13], etc.The studies of random scale-free networks characterized by γ and α must be based on the knowledge that such networks actually exist in the thermodynamic limit. Hence, it is necessary to understand the constraint on the possible values of γ and α. This problem is exactly equivalent to the issue of the graphicality of random scale-free networks. A degree sequence {k 1 , k 2 , . . . , k N } is said to be graphical if it can be realized as a network without self-loops or multiple links. As an indicator of the existence of graphical sequences, the graphicality fractionis defined as the fraction of graphical sequences among the sequences with an even degree sum generated by P (k). Note that the degree sequences with an odd degree sum are left out, since such sequences are trivially nongraphical. The constraint on the possible values of γ and ...

Lower bound of assortativity coefficient in scale-free networks

“…(B2), the condition is reduced to N (x M ) ≥ x M + 1. Actually for x M we have 25,31]. Thus in the thermodynamical limit (i.e., N → ∞), N (x M ) ≥ x M + 1 holds.…”

“…Thus far, the characterization of networks is still a challenging task, because some seemingly standalone structural properties are indeed statistically dependent to each other, resulting in many non-trivial structural constrains that are rarely understood [23][24][25][26]. Let's look at the five fundamental structural features of general networks: (i) the density ρ, quantified by the ratio of the number of edges M to the possibly maximum value N (N − 1)/2, where N is the number of nodes; (ii) the degree k i of a node i (defined as the number of i's associated edges) as well as the degree distribution p(k) [27]; (iii) the average distance d over all pairs of nodes in a connected network [28]; (iv) the assortativity coefficient r that quantifies the degree-degree correlation (with math- * Electronic address: zhutou@ustc.edu ematical definition shown later) [29,30]; (v) the clustering coefficient c i of a node (defined as the ratio of the number of edges between i's neighbors to the possibly maximum value) and the average clustering coefficient c over all nodes with degree larger than 1 [28].…”

Graphicality conditions for general scale-free complex networks and their application to visibility graphs

“…Due to the strong model dependence of κ [5], an analysis using both exponents γ and κ would be a good starting point in the characterization of a network structure. Studying how these exponents behave in particular models, or when some kind of structure exists, is a methodology followed in many papers [5][6][7][8][9][10][11][12][13][14][15][16]. In general, one obtains structural bounds in the form of inequalities involving the exponents γ and κ.…”

“…This bound is not due to structural constraints but it is an inherent property of any finite degree sequence with a power-law distribution. Bounds due to structural constraints have been calculated, mainly for uncorrelated networks, using different methods: through properties of the degree-degree correlation [9][10][11], using statistical methods on network ensembles [12,13], or imposing graphicality conditions [14,15]. The structural bound for uncorrelated networks in the range γ > 2 is, from Refs.…”

“…The structural bound for uncorrelated networks in the range γ > 2 is, from Refs. [9,12], κ 1/2, whereas it is κ < 1/γ in the range 1 < γ 2 [11,15]. Since natural and structural bounds arise from independent conditions, the exponent κ of the effective bound should be taken as min{1/2,1/(γ − 1)} in the range γ > 2.…”