Nodessmall

Nodessmall is a term used in graph theory and network science to describe nodes that have relatively small degree within a given graph. It is not a formal mathematical term with a single universal definition, but rather a contextual descriptor used when analyzing network structure and dynamics.

Definition and scope

In a graph G = (V, E) with degree function deg(v), a node v is considered nodessmall with respect to a chosen threshold t if deg(v) ≤ t. The threshold t can be an absolute integer or defined in relation to the graph’s average degree, median degree, or a percentile. The concept helps distinguish lower-degree nodes from higher-degree hubs that often play different roles in network processes.

Applications

Nodessmall is used in studies of network resilience, diffusion and contagion models, and sampling or reconstruction algorithms. Low-degree nodes frequently participate in local or peripheral interactions, and identifying them can illuminate how information or influence propagates through a network, or how robust a network is to targeted removals of nodes.

Relation to other concepts

Nodessmall contrasts with hub nodes, which have high degree and often central positions in networks. It relates to leaf nodes in trees (degree 1) and to broader ideas such as degree distributions, k-core decomposition, and community structure, which may treat nodes differently based on their connectivity.

Computation and considerations

Determining nodessmall nodes requires calculating vertex degrees, typically achievable in linear time relative to the graph size. Caution is advised, as degree alone may not capture a node’s functional importance in certain networks; complementary measures such as betweenness, closeness, or core number can provide a fuller picture.