Degree difference: a simple measure to characterize structural heterogeneity in complex networks

Despite the growing interest in characterizing the local geometry leading to the global topology of networks, our understanding of the local structure of complex networks, especially real-world networks, is still incomplete. Here, we analyze a simple, elegant yet underexplored measure, ‘degree difference’ (DD) between vertices of an edge, to understand the local network geometry. We describe the connection between DD and global assortativity of the network from both formal and conceptual perspective, and show that DD can reveal structural properties that are not obtained from other such measures in network science. Typically, edges with different DD play different structural roles and the DD distribution is an important network signature. Notably, DD is the basic unit of assortativity. We provide an explanation as to why DD can characterize structural heterogeneity in mixing patterns unlike global assortativity and local node assortativity. By analyzing synthetic and real networks, we show that DD distribution can be used to distinguish between different types of networks including those networks that cannot be easily distinguished using degree sequence and global assortativity. Moreover, we show DD to be an indicator for topological robustness of scale-free networks. Overall, DD is a local measure that is simple to define, easy to evaluate, and that reveals structural properties of networks not readily seen from other measures.

In this paper, we studied the DD distribution in a number of synthetic and real-world networks. Moreover, we have used this dataset for an empirical analysis of the properties of DD measure, its significance in revealing topological features of networks, and its correlation with other network measures. The synthetic networks considered in our analysis are as follows: • Erdös-Rényi (ER) random graphs 1 : This model generates network with n vertices and between any given pair of vertices, there exists an edge with probability p.
• Watts-Strogatz (WS) small-world graphs 2 : This model generates network by starting with a k-regular (ring) lattice with n vertices, and then, each edge is randomly rewired with probability β . The model gives networks with small-world property, i.e., with small average path length and high clustering coefficient.
• Barabási-Albert (BA) scale-free graphs 3 : This model generates network via a preferential attachment scheme, wherein at each step, β new edges connect a new vertex to existing vertices v with probability proportional to deg(v). These networks display power-law degree distribution and scale-free property.
• Random geometric (RG) graphs 4 : This model generates network with n vertices, each taking a random position in a 2-dimensional Euclidean plane. Thereafter, by fixing a radius parameter ε for the network, each vertex v is connected to all other vertices that fall inside the ball B ε (v) centered at v.
We have also analyzed the following undirected real-world networks: • Actor 5 : This is a co-stardom network with 702388 actors as its vertices and 29397908 edges connecting those actors who appeared in at least one movie together.
• Collaboration 6 : Condensed Matter Physics collaboration network with 23133 vertices corresponding to authors who authored papers posted on arXiv during the period from January 1993 to April 2003. This network has 93439 edges with each edge between two vertices (authors) signifying co-authorship in at least one paper.
• Phone calls 8 : This network captures phone calls between a sample of active cell phone users. In this network, there are 36595 users represented as vertices with 56853 edges between them. Two vertices are connected with an undirected edge if the corresponding users have at least once made a phone call to each other over the observed time interval.
• Power grid 2 : This network represents the power grid in western states of USA. Vertices are power plants and edges represent direct connections between power plants via a cable. In this network, there are 4941 vertices and 6594 edges between them.
• Protein 9 : This network is a human protein-protein interaction network with 2018 proteins as vertices and 2930 edges which represent mutual engagement of a pair of proteins in an interaction.
In addition to the above-mentioned undirected real networks, we have also analyzed the following directed real networks: • Citation 10 : This is a network of citations between 449673 papers (vertices) published in APS journals. A directed edge points from a vertex v to a vertex u if v cites u. There are 4685576 directed edges in this network.
• Email 11 : This network is based on Email communications at the University of Kiel, Germany over 112 days. There are 57194 vertices, which are the email addresses, and there is a directed link from vertex i to vertex j if i has sent at least one email to j. Overall, there are 93090 directed edges in this network.
• Metabolic 12 : This is a network of metabolic reactions in bacterium E. coli where vertices are metabolites and directed edges are reactions linking reactants to products of reactions. This network contains 1039 metabolites as vertices and 4741 reactions as directed edges.
• WWW 13 : This is a network of hyperlinks within nd.edu domain. In this network, vertices are webpages and there is a directed edge from a webpage v to a webpage u if v includes at least one hyperlink to u. This network contains 325729 vertices and 1117563 edges.   Figure S2. DD distributions for a given synthetic network and two rewired networks with same degree sequence as the given network and with pairwise global assortativity difference of ≤ 0.025 with respect to the given (starting) network. (a) Erdös-Rényi (ER) networks with n = 500 and p = 0.012. (b) Barabási-Albert (BA) networks with n = 500 and β = 5. In each subfigure, we show the average and standard deviation of the DD values over an ensemble of 20 networks as dots and error bars, respectively. In each plot, the legend gives the average and standard deviation of the global assortativities for the ensemble of 20 networks. Interestingly, although the global assortativity of each rewired network is ≤ 0.025 different from the starting network, and the difference between global assortativities of the two rewirings is ≤ 0.05, difference in DD distribution is clearly visible.  Figure S3. The evolution of DD distribution of Barabási-Albert (BA) networks created with n = 100 and β = 5 wherein the degree sequence is kept fixed while global assortativity differs. We start with a disassortative network and gradually increase the assortativity through a targeted rewiring scheme. Briefly, this heuristic to increase the assortativity of a given network is as follows. Given the graph at time step t, G t (V, E), we randomly pick two edges, {v, u} and {w, z} from E. We then remove the edges out of the network to obtainĜ(V ,Ê), and relabel v, u, w and z to v 1 , v 2 , v 3 , and v 4 where the vertices are indexed in the decreasing order of their degree. We next add a pair of edges {v 1 , v 2 } and {v 3 , v 4 } toÊ. If the assortativity ofĜ(V ,Ê) is greater than that of G t , we accept the change, and initialize G t+1 toĜ(V ,Ê), otherwise, we discard the change and initialize G t+1 to G t . We continue this process for a fixed number of time steps to obtain a more assortative network in comparison to the starting network. In this figure, we show the evolution of DD distributions for an ensemble of 20 BA networks with n = 100 and β = 5 starting as disassortative networks in subfigure (a) and evolving to assortative networks in (h). In each subfigure, we show the average and standard deviation of the DD values over an ensemble of 20 networks as dots and error bars, respectively. The ensemble average of the assortativity values are specified in the legend of each plot from (a)-(h).