A temporal network, also known as a time-varying network, is a network whose links are active only at certain points in time. Each link carries information on when it is active, along with other possible characteristics such as a weight. Time-varying networks are of particular relevance to spreading processes, like the spread of information and disease, since each link is a contact opportunity and the time ordering of contacts is included. Examples include communication networks with short-lived links, such as phone calls or emails. Information and some computer viruses spread over such networks. Networks of physical proximity, encoding who encounters whom and when, can be represented as time-varying networks. Some diseases, such as airborne pathogens, spread through physical proximity. Real-world data on time resolved physical proximity networks has been used to improve epidemic modeling. Neural networks and brain networks can be represented as time-varying networks since the activation of neurons are time-correlated. Time-varying networks are characterized by intermittent activation at the scale of individual links. This is in contrast to various models of network evolution, which may include an overall time dependence at the scale of the network as a whole.
Applicability Time-varying networks are inherently dynamic, and used for modeling spreading processes on networks. Whether using time-varying networks will be worth the added complexity depends on the relative time scales in question. Time-varying networks are most useful in describing systems where the spreading process on a network and the network itself evolve at similar timescales. Let the characteristic timescale for the evolution of the network be t N {\displaystyle t_{N}} , and the characteristic timescale for the evolution of the spreading process be t P {\displaystyle t_{P}} . A process on a network will fall into one of three categories:
Static approximation – where t N ≫ t P {\displaystyle t_{N}\gg t_{P}} . The network evolves relatively slowly, so the dynamics of the process can be approximated using a static version of the network. Time-varying network – where t N ∼ t P {\displaystyle t_{N}\sim t_{P}} . The network and the process evolve at comparable timescales so the interplay between them becomes important. Annealed approximation – where t N ≪ t P {\displaystyle t_{N}\ll t_{P}} . The network evolves relatively rapidly, so the dynamics of the process can be approximated using a time averaged version of the network. The flow of data over the internet is an example for the first case, where the network changes very little in the fraction of a second it takes for a network packet to traverse it. The spread of sexually transmitted diseases is an example of the second, where the prevalence of the disease spreads in direct correlation to the rate of evolution of the sexual contact network itself. Behavioral contagion is an example of the third case, where behaviors spread through a population over the combined network of many day-to-day social interactions.
Representations There are three common representations for time-varying network data.
Contact sequences – if the duration of interactions are negligible, the network can be represented as a set C {\displaystyle C} of contacts ( i , j , t ) {\displaystyle (i,j,t)} where i {\displaystyle i} and j {\displaystyle j} are the nodes and t {\displaystyle t} the time of the interaction. Alternatively, it can be represented as an edge list E {\displaystyle E} where each edge e {\displaystyle e} is a pair of nodes and has a set of active times T e = { t 1 , … , t n } {\displaystyle T_{e}=\{t_{1},\ldots ,t_{n}\}} . Interval graphs – if the duration of interactions are non-negligible, T e {\displaystyle T_{e}} becomes a set of intervals over which the edge e {\displaystyle e} is active. T e = { ( t 1 , t 1 ′ ) , … , ( t n , t n ′ ) } {\displaystyle T_{e}=\{(t_{1},t_{1}'),\ldots ,(t_{n},t_{n}')\}}
Snapshots – time-varying networks can also be represented as a series of static networks, one for each time step.
Properties The measures used to characterize static networks are not immediately transferable to time-varying networks. See Path, Connectedness, Distance, Centrality. However, these network concepts have been adapted to apply to time-varying networks.
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