In a network, we define shell $\ell$ as the set of nodes at distance $\ell$ with respect to a given node and define $r_\ell$ as the fraction of nodes outside shell $\ell$. In a transport process, information or disease usually diffuses from a random node and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the study of the transport property of networks. For a randomly connected network with given degree distribution, we derive analytically the degree distribution and average degree of the nodes residing outside shell $\ell$ as a function of $r_\ell$. Further, we find that $r_\ell$ follows an iterative functional form $r_\ell=\phi(r_{\ell-1})$, where $\phi$ is expressed in terms of the generating function of the original degree distribution of the network. Our results can explain the power-law distribution of the number of nodes $B_\ell$ found in shells with $\ell$ larger than the network diameter $d$, which is the average distance between all pairs of nodes. For real world networks the theoretical prediction of $r_\ell$ deviates from the empirical $r_\ell$. We introduce a network correlation function $c(r_\ell)\equiv r_{\ell+1}/\phi(r_\ell)$ to characterize the correlations in the network, where $r_{\ell+1}$ is the empirical value and $\phi(r_\ell)$ is the theoretical prediction. $c(r_\ell)=1$ indicates perfect agreement between empirical results and theory. We apply $c(r_\ell)$ to several model and real world networks. We find that the networks fall into two distinct classes: (i) a class of {\it poorly-connected} networks with $c(r_\ell)>1$, which have larger average distances compared with randomly connected networks with the same degree distributions; and (ii) a class of {\it well-connected} networks with $c(r_\ell)<1$.
Deep Dive into Structure of shells in complex networks.
In a network, we define shell $\ell$ as the set of nodes at distance $\ell$ with respect to a given node and define $r_\ell$ as the fraction of nodes outside shell $\ell$. In a transport process, information or disease usually diffuses from a random node and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the study of the transport property of networks. For a randomly connected network with given degree distribution, we derive analytically the degree distribution and average degree of the nodes residing outside shell $\ell$ as a function of $r_\ell$. Further, we find that $r_\ell$ follows an iterative functional form $r_\ell=\phi(r_{\ell-1})$, where $\phi$ is expressed in terms of the generating function of the original degree distribution of the network. Our results can explain the power-law distribution of the number of nodes $B_\ell$ found in shells with $\ell$ larger than the network diameter $d$, which is the average distance between all pairs
Many complex systems can be described by networks in which the nodes are the elements of the system and the links characterize the interactions between the elements. One of the most common ways to characterize a network is to determine its degree distribution. A classical example of a network is the Erdős-Rényi (ER) [1,2] model, in which the links are randomly assigned to randomly selected pairs of nodes. The degree distribution of the ER model is characterized by a Poisson distribution
where k is the average degree of the network. Another simple model is a random regular (RR) graph in which each node has exactly k = ψ links, thus P (k) = δ(k -ψ). The Watts-Strogatz model (WS) [3] is also well-studied, where a random fraction β of links from a regular lattice with k = ψ are rewired and connect any pair of nodes. Changing β from 0 to 1, the WS network interpolates between a regular lattice and an ER graph. In the last decade, it has been realized that many social, computer, and biological networks can be approximated by scale-free (SF) models with a broad degree distribution characterized by a power law
with a lower and upper cutoff, k min and k max [4,5,6,7,8,9]. A paradigmatic model that explains the abundance of SF networks in nature is the preferential attachment model of Barabási and Albert (BA) [4].
The degree distribution is not sufficient to characterize the topology of a network. Given a degree distribution, a network can have very different properties such as clustering and degree-degree correlation. For example, the network of movie actors [4] in which two actors are linked if they play in the same movie, although characterized by a power-law degree distribution, has higher clustering coefficient compared to the SF network generated by Molloy-Reed algorithm [10] with the same degree distribution.
Besides the degree distribution and clustering coefficient, a network is also characterized by the average distance between all pairs of nodes, which we refer to as the network diameter d. Random networks with a given degree distribution can be “small worlds” [2] d ∼ ln N
or “ultra-small worlds” [8] d ∼ ln ln N.
The diameter d depends sensitively on the network topology.
Another important characteristic of a network is the structure of its shells, where shell ℓ is defined as the set of nodes that are at distance ℓ from a randomly chosen root node [11].
The shell structure of a network is important for understanding the transport properties of the network such as the epidemic spread [12], where the virus spread from a randomly chosen root and reach nodes shell after shell. The structure of the shells is related to both the degree distribution and the network diameter. The shell structure of SF networks has been recently studied Ref. [11], which have introduces a new term “network tomography” referring to various properties of shells such as the number of nodes and open links in shell ℓ, the degree distribution, and the average degree of the nodes in the exterior of shell ℓ.
Many real and model networks have fractal properties while others are not [13]. Recently
Ref. [14] reported a power law distribution of number of nodes B ℓ in shell ℓ > d from a randomly chosen root. They found that a large class of models and real networks although not fractals on all scales exhibit fractal properties in boundary shells with ℓ > d. Here we will develop a theory to explain these findings.
In this paper, we extend the study of network tomography describing the shell structure in a randomly connected network with an arbitrary degree distribution using generating functions. Following Ref. [11], we denote the fraction of nodes at distance equal to or larger than ℓ as
and the nodes at distances equal or larger than ℓ as the exterior E ℓ of shell ℓ. Similarly, we define the “r-exterior”, E r , as the rN nodes with the largest distances from a given root node. To this end, we list all the nodes in ascending order of their distances from the root node. In this list, the nodes with the same distance are positioned at random. The last rN nodes in this list which have the largest distance to the root are called the E r . Notice that E r = E ℓ if r = r ℓ . Introducing r as a continuous variable is a new step compared to
Ref. [11], which allows us to apply the apparatus of generating functions to study network tomography.
The behavior of B ℓ for ℓ < d can be approximated by a branching process [15]. In shells with ℓ > d, the network will show different topological characteristics compared to shells with ℓ < d. This is due to the high probability to find high degree nodes (“hubs”) in shells with ℓ < d, so there is a depletion of high degree nodes in the degree distribution in E ℓ with ℓ > d. Indeed, the average degree of the nodes in shells with ℓ < d is greater than the average degree in the shells with ℓ > d [11,14].
Here, we develop a theory to explain the behavior of the degree distribution P r (k) in E r and the b
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
