Computing Degree Distribution in Complex Networks

Computing degree distribution in complex networks is one of the fundamental tasks in network science, helping researchers understand network topology and characteristics. Degree distribution represents the statistical features of node degrees in a network, which is crucial for studying scale-free properties, randomness, and other network attributes. This article demonstrates how to implement this calculation using MATLAB with practical code examples.

To compute degree distribution in MATLAB, the first step is to obtain the network's adjacency matrix representation. The adjacency matrix describes connection relationships between nodes, where matrix elements indicate the existence of edges between node pairs. For undirected networks, the adjacency matrix is symmetric, while for directed networks, it may not be symmetric. In MATLAB, this can be represented as a sparse or full matrix using functions like sparse() or by loading network data directly.

The initial calculation step involves determining each node's degree. For undirected networks, node degree corresponds to the number of edges connected to a node, which can be computed by summing rows or columns of the adjacency matrix using sum(A,1) or sum(A,2). For directed networks, in-degree and out-degree need to be calculated separately using sum(A,1) for in-degree and sum(A,2) for out-degree.

After obtaining all node degrees, the next step is to count the frequency of each possible degree value. This can be achieved using MATLAB's statistical functions like histcounts() or accumarray() to get raw degree distribution data. To visualize the distribution effectively, results are typically plotted with degree values on the x-axis and corresponding probability on the y-axis using plot() or loglog() functions.

Plotting degree distribution in logarithmic coordinates helps identify scale-free characteristics. If the distribution approximates a straight line in log-log scale, it suggests potential scale-free properties. MATLAB's loglog() function is particularly useful for this analysis, while polyfit() can help determine the slope of the distribution.

In practical applications, degree distribution often requires normalization to form a probability distribution using degree_prob = degree_count / total_nodes. For large networks, techniques like logarithmic binning may be necessary to address data sparsity issues using histcounts() with logarithmic bins. These methods provide comprehensive understanding of network degree distribution characteristics, forming a foundation for subsequent network analysis and research.