Temporal node centrality in complex networks

Many networks are dynamic in that their topology changes rapidly—on the same time scale as the communications of interest between network nodes. Examples are the human contact networks involved in the transmission of disease, ad hoc radio networks between moving vehicles, and the transactions between principals in a market. While we have good models of static networks, so far these have been lacking for the dynamic case. In this paper we present a simple but powerful model, the time-ordered graph, which reduces a dynamic network to a static network with directed flows. This enables us to extend network properties such as vertex degree, closeness, and betweenness centrality metrics in a very natural way to the dynamic case. We then demonstrate how our model applies to a number of interesting edge cases, such as where the network connectivity depends on a small number of highly mobile vertices or edges, and show that our centrality definition allows us to track the evolution of connectivity. Finally we apply our model and techniques to two real-world dynamic graphs of human contact networks and then discuss the implication of temporal centrality metrics in the real world.

Figure 1

Comparison of aggregated graph representation (left) and time series representation (right) of the contacts in Table .

Figure 2

The corresponding time-ordered graph $\mathcal{G}$ of Table . The path consisting of dashed red edges represents a temporal shortest path from $A$ to $B$ on the time interval $\left[0,3\right]$.

Figure 3

An example of a time-ordered graph $\mathcal{G}$ to explain our design philosophy for temporal closeness and betweenness. If we consider only the time interval $\left[i,j\right]$, the temporal shortest paths between all nodes are determined during the time interval $\left[0,1\right]$ alone regardless of subsequent changes, which does not help us measure the dynamic characteristics of this graph.

Figure 4

The comparison between the centrality—degree (left), closeness (middle), and betweenness (right)—of the resident and merchant nodes in the aggregated graph, the average centrality of those in the time series of graphs, and the temporal centrality of those for TMGs with $\eta =1$, $\nu =5$, $\gamma =6$, $p=0.4$, $b=0.1$, and $d=0.1$.

Figure 5

Temporal centrality distribution of nodes: closeness distribution in CAMBRIDGE (top left), betweenness distribution in CAMBRIDGE (top right), closeness distribution in MIT (bottom left), and betweenness distribution in MIT (bottom right). The computed values are sorted in descending order. For improved visualization, we use the same range on the $y$ axis except for the closeness centrality distribution in MIT (bottom left), since the centrality values in MIT are totally different from those in the other cases.

Figure 6

Kendall's tau (rank) correlation coefficients between ${\lambda }_{\text{from}}$ (or ${\lambda }_{\text{sans}}$) and closeness (or betweenness) centrality: closeness in CAMBRIDGE (top left), betweenness in CAMBRIDGE (top right), closeness in MIT (bottom left), and betweenness in MIT (bottom right).

Figure 7

Kendall's tau (rank) correlation coefficients between ${\lambda }_{\text{from}}$ (or ${\lambda }_{\text{sans}}$) and the closeness (or betweenness) centrality by varying $w$ while fixing $\tau =w(\lfloor T/2\rfloor +1)$: closeness in CAMBRIDGE (top left), betweenness in CAMBRIDGE (top right), closeness in MIT (bottom left), and betweenness in MIT (bottom right).