Graph Theory

Overview

Graph theory studies structures made of nodes and edges, providing a compact language for connectivity, routing, influence, and resource allocation problems. In analytics and engineering, graph models represent systems such as transportation networks, communication topologies, supply chains, and dependency graphs. This category focuses on practical network analysis tasks that can be computed directly from edge lists inside spreadsheet workflows. For background, see Graph theory on Wikipedia.

The unifying ideas are paths, weights, connectivity, and capacity constraints. Path-based measures quantify reachability and efficiency, centrality measures identify structurally important nodes, and flow formulations optimize movement through constrained networks. Many of these methods are linked by shared optimization principles; for example, shortest paths and spanning trees minimize cumulative edge cost, while max-flow and min-cut expose dual views of bottlenecks. A useful identity in flow analysis is the max-flow min-cut equivalence, expressed as \max f(s,t)=\min_{(S,T)} c(S,T), which connects feasible throughput to cut capacity.

Implementation is powered by NetworkX, a widely used Python library for graph construction, algorithms, and network analysis. NetworkX provides robust reference implementations for centrality scoring, shortest-path routines, flow optimization, and spanning-tree computation. These tools expose those algorithms in a tabular, spreadsheet-friendly interface while preserving core graph-theoretic behavior.

The centrality functions rank node importance from complementary perspectives: DEGREE_CENTRALITY measures local connectivity, CLOSENESS_CENT measures average proximity to other nodes, and BETWEENNESS_CENT measures brokerage along shortest paths. Spectral influence is captured by EIGENVECTOR_CENT, which rewards connections to already important nodes, while PAGERANK adds a random-surfer weighting model often used for ranking in directed networks. Together, these metrics support use cases such as identifying critical routers, key counterparties, influential users, or single points of failure.

The network-flow functions analyze constrained transport and bottlenecks. MAX_FLOW_VALUE computes the greatest feasible throughput from source to sink under edge capacities, and MIN_CUT_VALUE reports the corresponding smallest separating capacity. MIN_COST_FLOW_COST extends this perspective by minimizing total transport cost while satisfying node demands, which is central in logistics and allocation planning. In practice, this group is used for capacity planning, route balancing, and cost-efficient fulfillment design.

The path-and-structure functions support efficient connectivity design and routing decisions. SHORTEST_PATH finds minimum-length routes between specific node pairs in weighted or unweighted graphs, making it useful for navigation, dependency resolution, and latency estimation. MIN_SPANNING_TREE finds a minimum-cost backbone that connects all reachable nodes without cycles, which is valuable for network build-out, cable layout, and infrastructure simplification. Used together with centrality and flow metrics, these functions provide a complete toolkit for both descriptive network analysis and prescriptive optimization.

Centrality

Tool	Description
BETWEENNESS_CENT	Calculate the shortest-path betweenness centrality for nodes.
CLOSENESS_CENT	Calculate the closeness centrality for nodes.
DEGREE_CENTRALITY	Calculate the degree centrality for nodes in a graph.
EIGENVECTOR_CENT	Calculate the eigenvector centrality for nodes.
PAGERANK	Calculate the PageRank of the nodes in the graph.

Network Flow

Tool	Description
MAX_FLOW_VALUE	Calculate the maximum flow value between a source and sink.
MIN_COST_FLOW_COST	Find the minimum cost to satisfy all demands in a network.
MIN_CUT_VALUE	Calculate the capacity of the minimum (source, sink) cut.

Shortest Path

Tool	Description
MIN_SPANNING_TREE	Find the minimum spanning tree (MST) of an undirected graph.
SHORTEST_PATH	Find the shortest path and its length between two nodes in a network.