Calculating Node Degrees from Connectivity Matrices

When analyzing network structures, connectivity matrices (adjacency matrices) serve as fundamental tools for representing relationships between nodes. For undirected networks, symmetric matrix elements indicate connection existence between nodes; directed networks utilize asymmetric matrices to represent directional relationships, while weighted networks store connection weights within matrix elements.

Node Degree Calculation Undirected graphs: A node's degree equals the sum of non-zero elements in its corresponding row (or column), equivalent to its number of connections. Code implementation typically involves summing non-zero entries using functions like sum(A(i,:)) in MATLAB or np.sum(adj_matrix[i]) in Python. Directed graphs: Requires distinguishing between in-degree (number of connections pointing to the node) and out-degree (number of connections originating from the node), calculated as column and row sums of non-zero elements respectively. Algorithm implementation should handle asymmetric matrices by separating row and column operations. Weighted networks: Degree calculation can incorporate weight information by summing row or column elements directly (termed "strength" rather than degree). Implementation uses element-wise summation rather than binary counting.

Degree Distribution Analysis Statistical analysis of all node degrees plotted as histograms or probability distributions reveals overall network connectivity patterns. For example: power-law distributions suggest scale-free properties, while Poisson distributions correspond to random networks. Implementation involves collecting degree values using vectorized operations and visualization libraries like matplotlib.

Betweenness Centrality (Weighted Directed Networks) Measures a node's importance as a "bridge" using the following computational steps: Employ algorithms like Dijkstra's to obtain shortest paths between all node pairs (treating weights as distances). Implementation requires path-finding algorithms with priority queues for efficient computation. Calculate the proportion of shortest paths passing through the target node. When multiple equal-length shortest paths exist, distribute contributions proportionally. Code implementation tracks path counts and node occurrences using accumulation matrices. For directed networks, consider path directionality by separately calculating influences from incoming and outgoing edges. Algorithm modification ensures directional constraints during path exploration.

Important Considerations Weight normalization: When weights represent connection strength (e.g., traffic flow), betweenness centrality calculations may require reciprocal transformation (weight→distance conversion). Implementation should include weight preprocessing steps. Sparse matrix optimization: For large networks, utilize sparse matrix storage formats and parallelized algorithms to accelerate computations. Libraries like SciPy.sparse offer efficient data structures for memory optimization.