We propose a fluctuation analysis to quantify spatial correlations in complex networks. The approach considers the sequences of degrees along shortest paths in the networks and quantifies the fluctuations in analogy to time series. In this work, the Barabasi-Albert (BA) model, the Cayley tree at the percolation transition, a fractal network model, and examples of real-world networks are studied. While the fluctuation functions for the BA model show exponential decay, in the case of the Cayley tree and the fractal network model the fluctuation functions display a power-law behavior. The fractal network model comprises long-range anti-correlations. The results suggest that the fluctuation exponent provides complementary information to the fractal dimension.
Deep Dive into Quantifying long-range correlations in complex networks beyond nearest neighbors.
We propose a fluctuation analysis to quantify spatial correlations in complex networks. The approach considers the sequences of degrees along shortest paths in the networks and quantifies the fluctuations in analogy to time series. In this work, the Barabasi-Albert (BA) model, the Cayley tree at the percolation transition, a fractal network model, and examples of real-world networks are studied. While the fluctuation functions for the BA model show exponential decay, in the case of the Cayley tree and the fractal network model the fluctuation functions display a power-law behavior. The fractal network model comprises long-range anti-correlations. The results suggest that the fluctuation exponent provides complementary information to the fractal dimension.
Introduction. -Networks, consisting of simple elements, its nodes and links, can display complex properties, such as a broad degree distribution, clustering, modularity, and many others. This work focuses on the correlation between node degrees -the number of links attached to a node. Degree correlations, measuring the likelihood that nodes of a given degree are connected, help to explain important features of complex networks that are beyond the degree distribution or clustering. For instance, it has been found that many networks, such as the coauthorship, film actor (IMDb -Internet Movie Database), and company directors networks display assortative mixing between degrees, indicating that nodes of like degree tend to be connected. On the other hand, the Internet (autonomous system), the WWW, and some biological networks, exhibit disassortative mixing, where there is a high tendency for high degree nodes (hubs) to be connected to low degree nodes [1,2].
Different ways to quantify degree correlations are defined and used in the literature. One measure is the Pearson correlation coefficient of all linked pairs of nodes [1] which derives from the probability distribution p(k 1 , k 2 ) that two nodes of degree k 1 and k 2 are connected through a link [3,4]. Another measure of degree correlations is the average degree of neighbors of a degree-k node [3]. This measure may also be obtained as a particular case of the matrix p(k 1 , k 2 ), and classifies networks into assortative (disassortative) when this quantity increases (decreases) with k. Furthermore, it has been shown that disassortativity reflected in p(k 1 , k 2 ) is a tightly related property to fractality in complex networks [5,6].
The different measures of degree correlations consider only correlations between nearest neighbor nodes, i.e. only correlations between the degrees of nodes at distance 1, but not further. Because of this limitation, these measures fail to capture much of the rich topological information of the network. In this work we introduce an approach that extends the idea of degree correlations to larger distances. An immediate extension of previous methods could consist on simply considering the correlation coefficient from nearest neighbors to the second, the third, . . . d-th neighbors. However, in complex networks the direct calculation of such a function is not feasible. Here we introduce a fluctuation analysis that overcomes this issue. We study the fluctuations of the degree along shortest paths between two nodes and consider the distance between nodes analogous to time in time-series analysis. Our approach can be adapted to study many topological and dynamical correlation properties in networks, such as node activity, node weights, time of node addition, etc. In this work we in- troduce the formalism and focus on degree correlations, leaving other properties for further investigation.
Fluctuation analysis. -In order to quantify longrange degree (anti-) correlations at distances larger than nearest neighbors, we propose a fluctuation analysis. The method consists of the following steps:
- Find the shortest path between all pairs of nodes in the network (if the shortest path between a pair of nodes is not unique, then we consider an arbitrary one), see Fig. The fluctuation function describes the correlations of node degrees along shortest paths [7][8][9][10]. When an uncorrelated time series is partitioned into segments of size s, the standard deviation of the segments averages decays as
Because the overall distance between nodes in a network is short (compared to the time span of time series), we do not partition the degree sequences into segments. Instead, we consider all shortest path of any length d. Thus, if the covariance, C(d), between the degree of nodes at distance d scales as
where k is the average degree of the network, then for the fluctuation function we expect
where α k = -γ/2. Notice that because the average (calculated in step 2) involves a division by d, the degree fluctuation exponent α k differs by 1 from the usual Hurst-like exponent α [11] of time series analysis: α k = α -1. For asymptotical α k = -1/2 (α = 1/2) the degrees are uncorrelated.
Starting from the average degree along a shortest path, K d = 1 d+1 d+1 j=1 k j , we can express the variance as follows [11]:
where K d is the average over all values of K d . Therefore, assuming K d ≃ k , we find
In the case of (positive) long-range correlations with correlation exponent γ, the second term dominates. In the limit of large d we can integrate the sum, leading to the approximation:
With Eq. ( 1) and
While positive long-range correlations are characterized by fluctuation exponents -1/2 < α k < 0 [11], negative long-range correlations (long-range anti-correlations) are characterized by fluctuation exponents -1 < α k < -1/2 [12][13][14]. For the former, the covariance scales as C(d) ∼ d -γ and for the latter we assume
Results. -Barabasi-Albert m
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