Living Temporal Networks Beyond Static Topology: Operational Small-World and Scale-Free Structure in Living Temporal Graphs

Classical complex-network theory has largely been formulated through static topology:vertices and edges are treated as available while properties such as clustering, short paths,degree heterogeneity, centrality, and hub formation are measured on a fixed graph. Many realsystems, however, operate under a different kind of constraint. Their components may becomeinactive, fail partially, recover, transmit without receiving, receive without transmitting, orinteract only during admissible time windows. In such settings, the static projection of anetwork can give a misleading picture of how the system actually communicates. A graph mayappear connected, clustered, or hub-dominated in aggregate form, while its time-respectingand state-dependent structure is fragmented or inefficient. This paper develops a livingtemporal extension of small-world and scale-free analysis. Building on the formal language ofLiving Temporal Graphs, we introduce operational versions of distance, clustering, centrality,and hubness that account for vertex vitality, directional availability, and the distinctionbetween continuous-flow and discrete-transport interactions. We show that familiar staticsignatures need not survive under living-temporal constraints: static short paths may becomeoperationally long or unavailable, and static hubs may lose their effective role when theirtemporal activity is limited. We also formulate a living preferential-attachment mechanismin which attachment is shaped by temporal vitality rather than degree alone, indicatinghow heavy-tailed structure can persist only after availability is properly accounted for. Twoexamples illustrate the gap between static and operational structure: one where the apparenthub is not the effective temporal hub, and another where a small-world support graph loses itsfunctional connectivity under activity-window restrictions. The resulting framework clarifieshow small-world and scale-free ideas can be transferred from static graph theory to livingtemporal systems, with potential relevance to communication networks, transport systems,infrastructure resilience, epidemic pathways, and adaptive socio-technical networks.