2.1 Temporal graphs and paths

We use the following formal definition and notation for temporal graphs.

For a temporal graph $\mathcal{G}$ , we use $V(\mathcal{G})$ to denote the set of vertices, $E(\mathcal{G})$ for the set of time edges, and $E_t(\mathcal{G})$ to denote the set of edges of $\mathcal{G}$ which are present at time step $t$ , i.e., $E_t(\mathcal{G}) \,:\!=\, \{\{u,v\} \mid (\{u,v\},t)\in E(\mathcal{G})\}$ . For a time edge $e$ , we use $t(e)$ to denote the time label of $e$ . We call $V(\mathcal{G}) \times [T]$ the set of vertex appearances.

We only consider undirected temporal graphs. However, temporal paths and walks are implicitly directed because of the ascending time labels. Hence, we need a notion for directed transitions on a temporal path or walk which indicates not only which time an edge is used but also in which direction. For any time edge $e = (\{v,w\},t)$ we call $(v,w,t)$ the transition from $v$ to $w$ at time step $t$ . We call $v$ the starting point and $w$ the endpoint of the transition.

Using this, we can now define temporal walks and temporal paths.

A temporal walk may visit the same vertex more than once. In contrast to that, a temporal path visits each vertex at most once. This is analogous to the definitions for static graphs.

For readability, we use the notation $v\overset{t}{\rightarrow }{w}$ instead of the triple $(v,w,t)$ . Since the endpoint of a transition is equal to the starting point of the next one for any walk, we use a shortened notation omitting the doubled vertices. For instance, we denote the “middle” temporal path from $s$ to $z$ in Fig. 1 by:

\begin{equation*} P = \left(s\overset {1}{\rightarrow }{b_1}\overset {2}{\rightarrow }{b_2}\overset {3}{\rightarrow }{b_3}\overset {4}{\rightarrow }{z}\right). \end{equation*}

In this example, $P$ is a temporal path since all involved vertices are visited only once. Moreover, $P$ is a strict temporal path because the time labels are strictly ascending.

2.2 Optimality of temporal walks and paths

In static graphs, shortest paths are a central concept. In temporal graphs, there are different concepts of optimal paths. Fig. 1 illustrates three of the most common optimization criteria: shortest, foremost, and fastest (Bui-Xuan et al., Reference Bui-Xuan, Ferreira and Jarry2003).

• $P_1 = \left(s\overset{1}{\rightarrow }{a}\overset{5}{\rightarrow }{z}\right)$ is shortest,

• $P_2 = \left(s\overset{1}{\rightarrow }{b_1}\overset{2}{\rightarrow }{b_2}\overset{3}{\rightarrow }{b_3}\overset{4}{\rightarrow }{z}\right)$ is foremost, and

• $P_3 = \left(s\overset{3}{\rightarrow }{c_1}\overset{4}{\rightarrow }{c_2}\overset{5}{\rightarrow }{z}\right)$ is fastest.

More formally, we use the following definitions:

Figure 1.

This temporal graph features a shortest, a foremost, and a fastest temporal path from $s$ to $z$ .

A temporal path is shortest, foremost, or fastest if it is a shortest, foremost, or fastest temporal walk, respectively. Note that for the optimality criteria, “foremost” and “fastest” optimal temporal walks are not necessarily temporal paths. Imagine a temporal graph where all time edges incident to $s$ have the same time label and also all time edges incident to $z$ have the same time label. Then, every walk from $s$ to $z$ (if it exists) is foremost and fastest, since all walks leave $s$ at the same time and arrive at $z$ at the same time.

Together with the distinction between strict and non-strict, this gives us six different types of optimal temporal paths. In the following, we use the term “ $\star$ -optimal” temporal path, where $\star$ denotes the type.

Next, we introduce terminology and notation for counting optimal temporal paths.

We set $\sigma ^{(\star )}_{vv}\,:\!=\,1$ . We further introduce terminology and notation for counting optimal temporal paths that visit a certain vertex or a certain vertex appearance.

We set $\sigma ^{(\star )}_{sz}(s)\,:\!=\,\sigma ^{(\star )}_{sz}$ and $\sigma ^{(\star )}_{sz}(z)\,:\!=\,\sigma ^{(\star )}_{sz}$ . We define $\sigma ^{(\star )}_{sz}(s,0)\,:\!=\,\sigma ^{(\star )}_{sz}$ , $\sigma ^{(\star )}_{sz}(s,t)\,:\!=\,0$ for all $t\neq 0$ , and $\sigma ^{(\star )}_{sz}(v,0)\,:\!=\,0$ for all $v\neq s$ . Defining these corner cases as above allows us to keep certain proofs simpler by avoiding to discuss the corner cases explicitly. Intuitively, we assume that we have a dummy vertex appearance $(s,0)$ for every temporal path starting at $s$ that we consider to be the vertex appearance that the path arrives at time step zero.

2.3 Temporal betweenness centrality

In static graphs, the betweenness centrality of a vertex measures how often this vertex is passed on shortest paths between pairs of vertices in the graph. Freeman (Reference Freeman1977) defines the betweenness centrality $C_B(v)$ of a vertex $v$ (in a connected graph) as

\begin{equation*} C_B(v) \,:\!=\, \sum _{s \neq v \neq z} \frac {\sigma _{sz}(v)}{\sigma _{sz}}.\end{equation*}

As we have seen above, there are different notions of optimal paths (for example, fastest, shortest, foremost) in temporal graphs. Thus, there are several options how to define temporal betweenness centrality based on any of these notions. Moreover, we do not want to assume that there is a temporal path from any vertex to any other vertex in the graph. That is, we assume that there are vertex pairs $s,z$ with $\sigma _{sz}=0$ which we want to leave out when summing over all vertex pairs. To formalize this, we use a connectivity matrix $A$ of the temporal graph: let $A$ be a $\lvert V\rvert \times \lvert V\rvert$ matrix, where for every $v,w\in V$ we have that $A_{v,w}=1$ if there is a temporal path from $v$ to $w$ , and $A_{v,w}=0$ otherwise. Note that $A_{s,z}=1$ implies that $\sigma _{sz}\neq 0$ .

Formally, temporal betweenness based on these different concepts of path optimality is defined as follows.

Definition 7 (Temporal Betweenness). The temporal betweenness of any vertex $v\in V$ is given by:

\begin{equation*} C^{(\star )}_B(v) \,:\!=\, \sum _{s \neq v \neq z \text { and } A_{s,z}=1} \frac {\sigma ^{(\star )}_{sz}(v)}{\sigma ^{(\star )}_{sz}}. \end{equation*}

For our work, we use a slightly different version of temporal betweenness which is defined for vertex appearances and show that the temporal betweenness as defined above can be easily computed from the modified version. This allows us to simplify some of our proofs. We drop the condition that, when summing over all vertex pairs, these vertices have to be different from the vertex of which we want to know the betweenness. Formally, we define

\begin{equation*} \hat {C}^{(\star )}_B(v,t) \,:\!=\, \sum _{s,z \in V \text { and } A_{s,z}=1} \frac {\sigma ^{(\star )}_{sz}(v,t)}{\sigma ^{(\star )}_{sz}}. \end{equation*}

We can observe that using the connectivity matrix $A$ for the temporal graph, we can compute the temporal betweenness $C^{(\star )}_B(v)$ from the modified temporal betweenness values of the vertex appearances $(v,t)$ .

Lemma 1. For any vertex $v \in V$ it holds

\begin{equation*} C^{(\star )}_B(v) = \sum _{t\in [T]\cup \{0\}} \hat {C}^{(\star )}_B(v,t) - \sum _{w\in V} (A_{v,w} + A_{w,v}) +1. \end{equation*}

Proof. We show the claim as follows.

\begin{align*} C^{(\star )}_B(v) =& \sum _{s \neq v \neq z \text{ and } A_{s,z}=1} \frac{\sigma ^{(\star )}_{sz}(v)}{\sigma ^{(\star )}_{sz}}\\ =& \sum _{s,z \in V \text{ and } A_{s,z}=1} \frac{\sigma ^{(\star )}_{sz}(v)}{\sigma ^{(\star )}_{sz}} - \sum _{s \in V \text{ and } A_{s,v}=1} \frac{\sigma ^{(\star )}_{sv}(v)}{\sigma ^{(\star )}_{sv}}\\ & - \sum _{z \in V \text{ and } A_{v,z}=1} \frac{\sigma ^{(\star )}_{vz}(v)}{\sigma ^{(\star )}_{vz}} + \frac{\sigma ^{(\star )}_{vv}(v)}{\sigma ^{(\star )}_{vv}}\\ =& \sum _{s,z \in V \text{ and } A_{s,z}=1} \sum _{t\in [T]\cup \{0\}} \frac{\sigma ^{(\star )}_{sz}(v,t)}{\sigma ^{(\star )}_{sz}} - \sum _{s \in V} A_{s,v} - \sum _{z \in V} A_{v,z} + 1\\ =& \sum _{t\in [T]\cup \{0\}} \hat{C}^{(\star )}_B(v,t) - \sum _{w\in V} (A_{v,w} + A_{w,v}) +1 \end{align*}

Figure 2.

Assume $V_1$ and $V_2$ are sets of vertices that are adjacent to $v_1$ and $v_2$ at time steps $1$ and $T$ , respectively. Then, $v_1$ and $v_2$ are vertices with high temporal betweenness based on temporal paths, whereas $v_3$ has very low temporal betweenness based on temporal paths. However for, example, foremost temporal walks, $v_3$ would also have a high temporal betweenness.

We remark that in contrast to the static setting, temporal walks (visiting vertices multiple times) can also be optimal for certain canonical optimality criteria. In our definition of temporal betweenness centrality, we use the number of optimal temporal paths as opposed to the number of optimal temporal walks. This is natural for static graphs because static shortest walks are always paths. In the temporal case, this is not always true—it is possible that there is a non-path temporal walk (visiting vertices multiple times) that arrives at the same time as the foremost temporal path, so the number of foremost temporal paths and foremost temporal walks between two vertices can be different. Consider the graph shown in Fig. 2. Clearly, every walk from the left half to the right passes either $v_1$ or $v_2$ and since the edges on the right are only present at exactly one time step, every walk going from left to right is a foremost walk. Intuitively, $v_1$ and $v_2$ should have very similar, high betweenness scores, whereas $v_3$ should be close to zero. But if we use walks instead of paths for our definitions, we get a very high number of walks alternating between $v_2$ and $v_3$ before arriving on the right side, so $v_2$ and $v_3$ would get a high centrality score. We conclude that paths are more suitable than walks for defining temporal betweenness centrality.

4.1 Counting shortest temporal paths

We first show how we can count shortest temporal paths efficiently. Our results from Section 3 indicate that this is not possible for the case of foremost and fastest temporal paths.

As it turns out, what we need to count is a slightly different version of shortest temporal paths, which we call $t$ -shortest temporal paths. A temporal path $P$ from $s$ to $z$ is a $t$ -shortest temporal path if it arrives at time $t$ and if there is no temporal path $P^{\prime}$ from $s$ to $z$ which arrives at time $t$ and is shorter than $P$ . We show that similarly to the static case, $t$ -shortest temporal paths $P$ have the property that every prefix is a $t^{\prime}$ -shortest temporal path, where $t^{\prime}$ is the time when the prefix arrives at its last vertex. Formally, we show the following.

Lemma 6 allows us to formulate a recursive relation of the number of shortest temporal paths. To do this, we need to define the predecessor of a vertex appearance $(v,t)$ on a temporal path.

If we want to use the predecessor relation to formulate recursive function, then we need that the relation is acyclic; otherwise, we are not guaranteed to reach a base case. To show this formally, we define the directed predecessor graph for $\star$ -optimal temporal paths starting at some vertex $s\in V$ as follows. Given a temporal graph $\mathcal{G}$ , the vertex set of the predecessor graph is the set of vertex appearances in $\mathcal{G}$ . The arc set is given by the ordered pairs of vertex appearances such that there is a $\star$ -optimal temporal path starting at $s$ that arrives in these vertex appearances in that order, that is, we add an arc from a vertex appearance $(v,t)$ to another vertex appearance $(w,t^{\prime})$ if $(v,t)\in P^{(\star )}_{s}(w,t^{\prime})$ .

Definition 9 (Predecessor Graph). Let $\mathcal{G}=(V,{\mathcal{E}},T)$ be a temporal graph. Let $s \in V$ . The predecessor graph of $\mathcal{G}$ is the directed static graph given by

\begin{equation*} G^{(\star )}_{\textrm {pre}}(s) \,:\!=\, (V \times ([T]\cup \{0\}), E), \end{equation*}

where

\begin{align*} E \,:\!=\, \left\{((v,t),(w,t^{\prime})) \mid (v,t)\in P^{(\star )}_{s}(w,t^{\prime})\right\}. \end{align*}

It is easy to observe that the predecessor graph is acyclic for all strict optimal temporal path versions, since the time stamps of the vertex appearances are strictly increasing along the arcs of the graph. In the case of $t$ -shortest temporal paths, we show next that this also holds in the non-strict case.

With the help of this notation and Lemma 6 (and Lemma 7 for the non-strict case), we can formulate the following recursion for the number of $t$ -shortest temporal paths.

Corollary 8 (Counting of $t$ -Shortest Temporal Paths). Let $s$ be a source and let $z$ be a vertex with $s \neq z$ . The number of $t$ -shortest temporal paths ( $t$ -sh) from $s$ to $z$ is given by:

\begin{align*} \sigma ^{{(t-{\textrm{sh}})}}_{sz} = \sum _{({v},{t^{\prime}})\in P_{s}^{t-{\textrm{sh}}}(z,t)} \sigma ^{{(t^{\prime}-{\textrm{sh}})}}_{sv}. \end{align*}

By an analogous argument as in the proof of Lemma 7, we get the following, which we need in the next subsection.

4.2 Temporal dependencies

Similar to Brandes (Reference Brandes2001), we introduce so-called dependencies to obtain an alternative formulation for the betweenness which we then use for the actual computation. Recall the definition of (static) betweenness centrality

\begin{equation*} C_B(v) \,:\!=\, \sum _{s, z\in V} \frac {\sigma _{sz}(v)}{\sigma _{sz}},\end{equation*}

where $\sigma _{sz}$ is the number of all shortest paths from $s$ to $z$ and $\sigma _{sz}(v)$ is the fraction of these paths which pass through $v$ . Brandes (Reference Brandes2001) calls the latter the pair-dependency of $s$ and $z$ on $v$ . Brandes’ central observation is that the betweenness centrality can be computed faster by first computing partial sums of the form

\begin{equation*} \delta _{s\bullet }(v) \,:\!=\, \sum _{z \in V} \frac {\sigma _{sz}(v)}{\sigma _{sz}},\end{equation*}

the dependency of $s$ on $v$ . He showed that the dependencies obey a recursive relation which can be exploited algorithmically.

Both concepts, dependencies and pair-dependencies, naturally generalize to optimal paths in temporal graphs. In some cases, we may be interested in the dependency on a vertex at a specific time instead of the whole lifetime. Hence, we introduce temporal dependencies of vertex appearances:

Definition 10 (Temporal Dependency). Let $s$ be a source and let $z$ be a target vertex. Let $v \neq s$ and $v \neq z$ . For any $t\in [T]$ , the temporal pair-dependency of $s$ and $z$ on $(v,t)$ and the dependency of $s$ on $(v,t)$ are given by:

\begin{align*} &\delta ^{(\star )}_{sz}(v,t) \,:\!=\, \begin{cases} 0, & if\ \sigma ^{(\star )}_{sz} = 0 \\[5pt] \dfrac{\sigma ^{(\star )}_{sz}(v,t)}{\sigma ^{(\star )}_{sz}}, & otherwise, \end{cases} \\[5pt] &\delta ^{(\star )}_{s\bullet }(v,t) \,:\!=\, \sum _{z\in V} \delta ^{(\star )}_{sz}(v,t). \end{align*}

From our definition of the modified temporal betweenness of a vertex appearance, it follows that we can use the temporal dependencies to compute it.

Observation 10. For any vertex appearance $({v},{t}) \in V\times [T]$ it holds:

\begin{equation*} \hat {C}^{(\star )}_B(v,t) = \sum _{s\in V} \delta ^{(\star )}_{s\bullet }(v,t). \end{equation*}

Now we state our central result allowing us to design algorithms to compute the temporal betweenness centrality based on shortest temporal paths.

Lemma 11 (Temporal Dependency Accumulation). Fix a source $s\in V$ . For any vertex appearance $({v},{t}) \in V\times [T]$ it holds:

\begin{align*} \delta ^{{({\textrm{sh}})}}_{s\bullet }(v,t) &= \delta ^{{({\textrm{sh}})}}_{sv}(v,t) + \sum _{({w},{t^{\prime}}):({v},{t})\in P_{s}^{{\textrm{sh}}}(w,t^{\prime})} \frac{\sigma ^{{(t-{\textrm{sh}})}}_{sv}}{\sigma ^{{(t^{\prime}-{\textrm{sh}})}}_{sw}}\cdot \delta ^{{({\textrm{sh}})}}_{s\bullet }(w,t^{\prime}). \end{align*}

Proof. We start by rewriting the pair-dependency of a vertex appearance as the summed fractions of shortest temporal paths using a specific transition from $v$ to some vertex $w$ at some time step:

\begin{align*} \delta ^{{({\textrm{sh}})}}_{s\bullet }(v,t) & = \sum _{z\in V} \delta ^{{({\textrm{sh}})}}_{sz}(v,t) \\ & = \delta ^{{({\textrm{sh}})}}_{sv}(v,t) + \sum _{z\in V} \sum _{({w},{t^{\prime}}):({v},{t}) \in P_{s}^{{\textrm{sh}}}(w,t^{\prime})} \delta ^{{({\textrm{sh}})}}_{sz}((v,t),(\{v,w\},t^{\prime})), \end{align*}

where $\delta ^{{({\textrm{sh}})}}_{sz}((v,t),(\{v,w\},t^{\prime}))$ is the fraction of shortest temporal paths from vertex $s$ to $z$ that use the transitions $v^{\prime}\overset{t}{\rightarrow }{v}\overset{t^{\prime}}{\rightarrow }{w}$ for some $v^{\prime}$ . Note that for the case that $z=v$ , there is no vertex appearance $(w,t^{\prime})$ such that $({v},{t}) \in P_{s}^{{\textrm{sh}}}(w,t^{\prime})$ ; hence, we need $\delta ^{{({\textrm{sh}})}}_{sv}(v,t)$ as an additional summand. Furthermore, we have that

\begin{equation*} \delta ^{{({\textrm {sh}})}}_{sz}((v,t),(\{v,w\},t^{\prime})) = \frac {\sigma ^{{(t-{\textrm {sh}})}}_{sv}}{\sigma ^{{(t^{\prime}-{\textrm {sh}})}}_{sw}} \cdot \frac {\sigma ^{{({\textrm {sh}})}}_{sz}(w,t^{\prime})}{\sigma ^{{({\textrm {sh}})}}_{sz}}. \end{equation*}

Inserting these cases into the double sum above yields the following:

\begin{align*} & \sum _{z\in V} \sum _{({w},{t^{\prime}}):({v},{t}) \in P_{s}^{{\textrm{sh}}}(w,t^{\prime})} \delta ^{{({\textrm{sh}})}}_{sz}((v,t),(\{v,w\},t^{\prime}))\\ =& \sum _{({w},{t^{\prime}}):({v},{t}) \in P_{s}^{{\textrm{sh}}}(w,t^{\prime})} \sum _{z \in V} \delta ^{{({\textrm{sh}})}}_{sz}((v,t),(\{v,w\},t^{\prime}))\\ =& \sum _{({w},{t^{\prime}}):({v},{t}) \in P_{s}^{{\textrm{sh}}}(w,t^{\prime})} \left (\frac{\sigma ^{{(t-{\textrm{sh}})}}_{sv}}{\sigma ^{{(t^{\prime}-{\textrm{sh}})}}_{sw}} \cdot \sum _{z\in V} \frac{\sigma ^{{({\textrm{sh}})}}_{sz}(w,t^{\prime})}{\sigma ^{{({\textrm{sh}})}}_{sz}}\right )\\ =& \sum _{({w},{t^{\prime}}):({v},{t}) \in P_{s}^{{\textrm{sh}}}(w,t^{\prime})} \frac{\sigma ^{{(t-{\textrm{sh}})}}_{sv}}{\sigma ^{{(t^{\prime}-{\textrm{sh}})}}_{sw}}\cdot \delta ^{{({\textrm{sh}})}}_{s\bullet }(w,t^{\prime}). \end{align*}

The lemma statement now follows immediately.

Lemma 1 together with Observation 10 show that we can use temporal dependencies to compute temporal betweenness centrality. Lemma 11 shows that the temporal dependencies obey a recursive relation, and Lemma 9 shows that it is acyclic. This together with Corollary 8 allows us to design a polynomial-time algorithm to compute temporal betweenness centrality based on shortest temporal paths. Before we give the specific algorithm description and running time analysis, we describe some other temporal betweenness concepts that can be computed in a similar fashion.

4.3 Specialized optimality concepts

So far, our results allow us to efficiently compute the temporal betweenness for strict and non-strict shortest temporal paths. For all other of the six canonical variants, we have shown computational hardness results in Section 3. In the following, we show that for some specialized optimality concepts for temporal paths, we can also efficiently compute the corresponding temporal betweenness. Specifically, we consider shortest foremost temporal paths and prefix-foremost temporal paths.

Shortest foremost temporal paths. A shortest foremost path is a foremost temporal path that is not longer than any other foremost temporal path. It may not be a shortest temporal path overall; however, as even shorter temporal paths with higher arrival times may exist. Shortest foremost temporal paths can be regarded as temporal paths that prioritize a low arrival time and use a low edge count as a tie-breaker.

We observe that a shortest foremost temporal path is a $t$ -shortest temporal path for the earliest possible arrival time. Hence, our findings in the previous subsection translate very easily to shortest foremost temporal paths.

Prefix-foremost temporal paths. Now we consider the class of foremost temporal paths for which every prefix of the temporal path is foremost as well. That is, every vertex that is visited by such a temporal path is visited as soon as possible. These temporal paths are called prefix-foremost paths.

Wu et al. (Reference Wu, Cheng, Ke, Huang, Huang and Wu2016) showed that there is always at least one prefix-foremost path between any pair of vertices $s$ and $z$ unless $z$ is not reachable from $s$ at all. We can observe that in the straightforward reduction from Paths (Valiant, Reference Valiant1979) that yields Proposition 2 we also get that counting non-strict prefix-foremost temporal paths is computationally hard.

The main difference between strict and non-strict prefix-foremost paths is that in the former case, the predecessor relation is acyclic, and in the latter it is not. Intuitively, this is the reason why we get hardness in the non-strict case and tractability for the strict version.

For strict prefix-foremost paths, one can show that the counting problem and the computation of the corresponding betweenness are solvable in polynomial time by very similar arguments as for shortest temporal paths. The main difference is that the time stamp of a predecessor is unique; hence, we can define the set of predecessors as a set of vertices as opposed to a set of vertex appearances; otherwise, the arguments are analogous to the ones used in the proof of Lemma 11. This allows us to simplify the recursive function for the temporal dependency for prefix-foremost temporal paths which also results in a faster computation.

Lemma 13 (Strict Prefix-Foremost Dependency Accumulation). Fix a source $s\in V$ . For any vertex $v \in V, v \neq s$ it holds:

\begin{align*} & \delta ^{{(\textrm{pfm})}}_{s\bullet }(v) = 1 + \sum _{w:v\in P_{s}^{(\textrm{pfm})}(w)} \frac{\sigma ^{{(\textrm{pfm})}}_{sv}}{\sigma ^{{(\textrm{pfm})}}_{sw}}\cdot \delta ^{{(\textrm{pfm})}}_{s\bullet }(w). &\\ \end{align*}

Proof. We start by rewriting the pair-dependency of a vertex appearance as the summed fractions of prefix-foremost temporal paths using a specific transition from $v$ to some vertex $w$ at some time step:

\begin{align*} \delta ^{{(\textrm{pfm})}}_{s\bullet }(v) &= \sum _{z\in V} \delta ^{{(\textrm{pfm})}}_{sz}(v) \\ &= \delta ^{{(\textrm{pfm})}}_{sv}(v) + \sum _{z\in V} \sum _{w:v \in P_{s}^{(\textrm{pfm})}(w)} \delta ^{{(\textrm{pfm})}}_{sz}(v,\{v,w\}), \end{align*}

where $\delta ^{{(\textrm{pfm})}}_{sz}(v,\{v,w\})$ is the fraction of prefix-foremost temporal paths from vertex $s$ to $z$ that use the transitions $v\overset{t}{\rightarrow }{w}$ for some $t$ . Note that for the case that $z=v$ there is no vertex $w$ such that $v \in P_{s}^{(\textrm{pfm})}(w)$ , hence we need $\delta ^{{(\textrm{pfm})}}_{sv}(v)$ as an additional summand. Furthermore, we have that

\begin{equation*} \delta ^{{(\textrm {pfm})}}_{sz}(v,\{v,w\}) = \frac {\sigma ^{{(\textrm {pfm})}}_{sv}}{\sigma ^{{(\textrm {pfm})}}_{sw}} \cdot \frac {\sigma ^{{(\textrm {pfm})}}_{sz}(w)}{\sigma ^{{(\textrm {pfm})}}_{sz}}. \end{equation*}

Inserting these cases into the double sum above yields the following:

\begin{align*} & \sum _{z\in V} \sum _{w:v \in P_{s}^{(\textrm{pfm})}(w)} \delta ^{{(\textrm{pfm})}}_{sz}(v,\{v,w\})\\ =& \sum _{w:v \in P_{s}^{(\textrm{pfm})}(w)} \sum _{z \in V} \delta ^{{(\textrm{pfm})}}_{sz}(v,\{v,w\})\\ =& \sum _{w:v \in P_{s}^{(\textrm{pfm})}(w)} \left (\frac{\sigma ^{{(\textrm{pfm})}}_{sv}}{\sigma ^{{(\textrm{pfm})}}_{sw}} \cdot \sum _{z\in V} \frac{\sigma ^{{(\textrm{pfm})}}_{sz}(w)}{\sigma ^{{(\textrm{pfm})}}_{sz}}\right )\\ =& \sum _{w:v\in P_{s}^{(\textrm{pfm})}(w)} \frac{\sigma ^{{(\textrm{pfm})}}_{sv}}{\sigma ^{{(\textrm{pfm})}}_{sw}}\cdot \delta ^{{(\textrm{pfm})}}_{s\bullet }(w). \end{align*}

The lemma statement now follows immediately.

5.1 Algorithms based on adaptations of Brandes’ algorithm to the temporal setting

Brandes’ algorithm (Brandes, Reference Brandes2001) (for static graphs without edge weights) is based on breadth-first search. For each vertex, the Single-Source Shortest Paths problem is solved once. At the end of each iteration, the dependency of the respective source $s$ on all other vertices $v$ is added to the betweenness score of $s$ .

Algorithm 1

Shortest (foremost) betweenness in temporal graphs

Our algorithms for temporal betweenness will follow roughly the same structure. Instead of breadth-first search, the appropriate algorithm for the respective temporal path class is used as the base for the algorithm. To describe our algorithm formally, we need the notion of temporal neighborhoods.

Shortest (foremost) temporal betweenness. We have shown that the temporal dependencies for shortest (foremost) paths follow a similar recursion as for static graphs.

Algorithm 1 uses a $\lvert V\vert \times T$ -table to store the number of shortest temporal paths to all vertex appearances. The overall structure of the algorithm is similar to Brandes’ algorithm (Brandes, Reference Brandes2001)—a single-source-all-shortest-paths traversal from each vertex to all vertex appearances is performed and the count of shortest temporal paths is stored in the aforementioned table. At the end of each iteration, we add the dependencies found for each vertex appearance to the betweenness score of the respective vertex.

In order to count shortest foremost temporal paths instead of shortest temporal paths, only the dependency accumulation needs to be changed: instead of checking whether the paths have minimal length, the algorithm checks for minimal arrival time.

Strict prefix-foremost temporal betweenness. Algorithm 2 modifies Brandes’ algorithm (Brandes, Reference Brandes2001) to count strict prefix-foremost temporal paths. The overall structure of the algorithm remains the same. Instead of iterating over all neighbors, however, we add all temporal edges of a vertex into a priority queue (prioritizing early time labels). This allows us to traverse the graph in a time-respecting manner and find the prefix-foremost temporal paths.

Notably, since the time labels of predecessors are unique in prefix-foremost temporal paths, we do not have to consider vertex appearances in this case. Intuitively, this is the main reason why we can achieve both a faster running time a lower space consumption.

Algorithm 2

Strict prefix-foremost betweenness

5.2 Algorithms based on the static expansion

In this section, we discuss the presumably more straightforward and “out-of-the-box” way of computing temporal betweenness, namely through the use of so-called static expansions or time-expanded temporal graphs, which are a key tool in temporal graph algorithmics (Zschoche et al., Reference Zschoche, Fluschnik, Molter and Niedermeier2020; Akrida et al., Reference Akrida, Czyzowicz, Gasieniec, Kuszner and Spirakis2019; Kempe et al., Reference Kempe, Kleinberg and Kumar2002; Mertzios et al., Reference Mertzios, Michail and Spirakis2019; Wu et al., Reference Wu, Cheng, Ke, Huang, Huang and Wu2016). Static expansions are directed graphs that have a vertex for every vertex appearance of the corresponding temporal graph and they preserve the connectivity between vertex appearances of the temporal graph.

The main idea is to (naïvely) use Brandes’ algorithm (Brandes, Reference Brandes2001) for (directed) static graphs on the static expansion. Of course, we need to make sure that each shortest path in the static expansion corresponds to exactly one optimal temporal path in the temporal graph for the optimality criterion that we are interested in. To do this, it is necessary to exclude some paths from the betweenness computation. To this end, we first give a generalized version of static betweenness. Then we present the details on how to construct static expansions for the optimality concepts used in our work, which are slightly tailored versions of the static expansions used in the literature (Zschoche et al., Reference Zschoche, Fluschnik, Molter and Niedermeier2020; Akrida et al., Reference Akrida, Czyzowicz, Gasieniec, Kuszner and Spirakis2019; Kempe et al., Reference Kempe, Kleinberg and Kumar2002; Mertzios et al., Reference Mertzios, Michail and Spirakis2019; Wu et al., Reference Wu, Cheng, Ke, Huang, Huang and Wu2016). Notably, we do not present a static expansion for the strict prefix-foremost case, since for this case our algorithm is significantly faster than for the other optimality concepts.

Let $S\subseteq V$ be a set of “source” vertices and $Z\colon S\rightarrow 2^V$ be a function that, for any $s\in S$ , returns the set of “terminal” vertices for that source. Now we define the $S$ - $Z$ -Betweenness of a vertex $v\in V$ to be

\begin{equation*} C_B^{(SZ)}(v) = \sum _{s\in S\setminus \{v\}}\sum _{z\in Z(s)\setminus \{v\}}\delta _{sz}(v). \end{equation*}

The problem of computing the $S$ - $Z$ -betweenness for each vertex in an arbitrary (weighted) directed graph can be solved with a straightforwardly generalized version Brandes’ algorithm (Brandes, Reference Brandes2001). Informally speaking, only vertices in $S$ are considered as sources in the outermost loop of the algorithm. Furthermore, when the dependencies are accumulated, only shortest paths from the current source to vertices in $Z$ are considered. Formally, we have the following.

Static expansion for shortest temporal betweenness. We now describe how to construct a static expansion of a temporal graph in order to compute (strict) shortest temporal betweenness. Let $\mathcal{G}=(V,{\mathcal{E}},T)$ be a temporal graph. We then define the static expansion graph for the (strict) shortest betweenness as the directed unweighted graph $G = (\tilde{V}, \tilde{E})$ , with the following sets of vertices and edges:

For each vertex $v\in V$ of the original graph, we define $T + 2$ vertices in the static expansion—one for each appearance and two special vertices: a “source” and a “sink.” Formally,

\begin{equation*} \tilde {V} \,:\!=\, \bigcup _{v\in V}\left \{v_t \mid t\in [T+1]\cup \{0\}\right \}. \end{equation*}

For the edges, we have three cases. The vertices $v_{T+1}$ are used as sentinel “terminal” vertices that will signify the end of a path in our betweenness computation. Hence, they do not have any outgoing edges. Now, for any temporal edge $(\{v, w\}, t)\in{\mathcal{E}}$ and every $v_{t^{\prime}}$ with $t^{\prime}\leq t$ , we either add the directed edge $(v_{t^{\prime}}, w_t)$ , or the edge $(v_{t^{\prime}}, w_{t+1})$ —the former in the non-strict case, the latter in the strict case. Moreover, we also add an edge to the corresponding terminal vertex, that is, we add the edge $(v_{t^{\prime}}, w_{T+1})$ . Naturally, we also go through the same process with $w$ as we did with $v$ . Formally, we have in the non-strict case:

\begin{equation*} \tilde {E} \,:\!=\, \bigcup _{(\{v, w\}, t)\in {\mathcal {E}}}\bigcup _{0\leq t^{\prime} \leq t}\big\{(v_{t^{\prime}}, w_t), (v_{t^{\prime}}, w_{T+1}), (w_{t^{\prime}}, v_t), (w_{t^{\prime}}, v_{T+1})\big\}, \end{equation*}

and in the strict case:

\begin{equation*} \tilde {E} \,:\!=\, \bigcup _{(\{v, w\}, t)\in {\mathcal {E}}}\bigcup _{0\leq t^{\prime} \leq t}\big\{(v_{t^{\prime}}, w_{t+1}), (v_{t^{\prime}}, w_{T+1}), (w_{t^{\prime}}, v_{t+1}), (w_{t^{\prime}}, v_{T+1})\big\}. \end{equation*}

Now, to use the static expansion to compute the betweenness values in the temporal graph, we first compute the $S$ - $Z$ -betweenness on the static expansion with

\begin{align*} S & \,:\!=\, \{v_0\mid v\in V\}, \\ Z(s) & \,:\!=\, \big\{v_{T+1}\mid v\in V\setminus \{s\}\big\}. \end{align*}

We can now compute the temporal betweenness values using the following (where $\star$ is the corresponding optimality concept):

\begin{equation*} C^{(\star )}_B(v) = \sum _{t\in \left [T\right ]}C_B^{(SZ)}(v_t). \end{equation*}

Static expansion for shortest foremost temporal betweenness. This variant of static expansion will be similar to the static expansion from the previous paragraph. Indeed, for a temporal graph $\mathcal{G}=(V,{\mathcal{E}},T)$ we define the static expansion for (strict) shortest foremost betweenness to be the directed weighted graph $G = (\tilde{V}, \tilde{E}, \omega )$ , with $\tilde{V}$ and $\tilde{E}$ defined exactly as in the static expansion for (strict) shortest betweenness (see previous paragraph). Additionally, we use the edge weight function $\omega$ to “simulate” the foremost aspect of the paths. We let all “internal” edges be of equal weight, but let the “terminal” edges have a weight related to their timestamps. We set the weight in such a way as to make it so that any path to an “appearance” $v_t$ is better than any path to an “appearance” $v_{t^{\prime}}$ for $t \lt t^{\prime}$ . Formally, we have

\begin{equation*} \omega (v_t, w_{t^{\prime}}) = \begin {cases} 1, & \text {if } t^{\prime} \leq T, \\[5pt] (n + 1)\cdot (t^{\prime} + 1), & \text {otherwise.}\end {cases} \end{equation*}

Note that this weight function works in both the strict and the non-strict case.

The further procedure, that is, the computation of $S$ - $Z$ -betweenness and then the temporal betweenness values can be done in exactly the same way as in for shortest temporal betweenness (see the previous paragraph).

We can observe that $\lvert \tilde{V}\rvert \in O(\lvert V\rvert \cdot T)$ , $\lvert \tilde{E}\rvert \in O(\lvert{\mathcal{E}}\rvert \cdot T)$ , and $\lvert S\rvert \in O(\lvert V\rvert )$ . Using Observation 16, we obtain the following running time and space bound.

Comparing the theoretical running time bounds of Corollary 17 to the ones from Proposition 15, we can see that using static-expansion-based algorithms required essentially the same running time but more space. However, we remark that the main purpose is to compare the approach of working directly on the temporal graph data to an “naïve” or “out-of-the-box” use of Brandes’ algorithm (Brandes, Reference Brandes2001). The static-expansion-based algorithm might be improved in terms of space consumption, which we leave for future research. In the next section, we investigate empirically how the two approaches compare in practice.

6.1 Setup and statistics

We implementedFootnote ⁵ Algorithms 1 and 2 and algorithms using the static expansions (see Subsection 5.2) in C++ and we compiled our source code with GCC 7.5.0. We carried out experiments on an Intel Xeon W-2125 computer with four cores clocked at 4.0 GHz and with 256 GB RAM, running Linux 4.15. We did not utilize the parallel-processing capabilities.

We used the following freely available data sets for temporal graphs: Physical-proximity networksFootnote ⁶ between high school students (“highschool-2011,” “highschool-2012," “highschool-2013” Gemmetto et al., Reference Gemmetto, Barrat and Cattuto2014; Stehlé et al., Reference Stehlé, Voirin, Barrat, Cattuto, Isella, Pinton, Quaggiotto, Van den Broeck, Régis, Lina, Vanhems and Viboud2011; Fournet and Barrat, Reference Fournet and Barrat2014), children and teachers in a primary school (“primary school” Stehlé et al., Reference Stehlé, Voirin, Barrat, Cattuto, Isella, Pinton, Quaggiotto, Van den Broeck, Régis, Lina, Vanhems and Viboud2011), patients and health-care workers (“hospital-ward” Vanhems et al., Reference Vanhems, Barrat, Cattuto, Pinton, Khanafer, Régis and Voirin2013), attendees of the Infectious SocioPatterns event (“infectious” Isella et al., Reference Isella, Stehlé, Barrat, Cattuto, Pinton and Van den Broeck2011), and conference attendees of ACM Hypertext 2009 (“hypertext” Isella et al., Reference Isella, Stehlé, Barrat, Cattuto, Pinton and Van den Broeck2011), and an email communication networks of the Karlsruhe Institute of Technology (KIT) (“karlsruhe” Goerke, Reference Goerke2011),Footnote ⁷ and one social communication network (“Facebook-like” Opsahl and Panzarasa, Reference Opsahl and Panzarasa2009; Panzarasa et al., Reference Panzarasa, Opsahl and Carley2009).Footnote ⁸ We summarize some important statistics about the different data sets in Table 2.

Table 2.

Statistics for the data sets used in our experiments. The lifetime $T^\star$ of a graph is the difference between the largest and smallest time stamp on an edge in the graph. The resolution $r$ indicates how often edges were measured. Note that the number of times steps is $T=T^\star/r$

6.2 Experimental results

We now discuss the results of our experiments. We analyzed the influence of the temporal betweenness type on the running time, the distribution of the temporal betweenness values, and the ranking of the ten vertices with the highest temporal betweenness values.

Running time. As indicated by our theoretical running time bounds (see Table 1), Algorithm 1 (computing shortest and foremost shortest betweenness) is several orders of magnitudes slower than Algorithm 2 (computing prefix-foremost betweenness). Algorithm 1 solved all instances except for (which was not solved within the timeout of five hours) within 45 minutes (keep in mind that Algorithm 1 computes shortest and foremost shortest temporal betweenness simultaneously). Algorithm 2 solved all instances except “karlsruhe” within 30 s. We show the specific running times in Table 3. It is noticeable that in small instances, the algorithms based on static expansions are roughly a factor of two faster. However, on large instances, the approach directly working on the temporal graph seems to be better, see for example “Facebook-like,” where we are faster by a factor of roughly six, and “infectious," where the algorithm based on static expansions did not finish within five hours. We believe that the static-expansion-based algorithm has higher memory requirements and more inefficient cache usage and hence becomes significantly slower for larger instances.

Table 3.

Running times in seconds of our implementation, where the last two columns correspond to the algorithms based on static expansions. From left to right: non-strict shortest and shortest foremost betweenness, strict shortest and shortest foremost betweenness, strict prefix-foremost betweenness, non-strict shortest and shortest foremost betweenness computed with static-expansion-based algorithms, strict shortest and shortest foremost betweenness computed with static-expansion-based algorithms. A -1 indicates that the instance was not solved within five hours

Impact of temporal betweenness type on the temporal betweenness distribution. In Fig. 5 we exemplarily show histograms of the temporal betweenness values of the “highschool-2013” data set, the “Facebook-like” data set, and the “primary school” data set. We can observe that the temporal betweenness centrality has a power-law-like distribution and the temporal betweenness type does not have a strong influence on the distribution in the “highschool-2013” data set and the “Facebook-like” data set (which is the case in most data sets). In the “primary school” data set, however, the distributions for non-strict shortest and strict shortest temporal betweenness are very similar but the other ones are quite different from each other.

Impact of temporal betweenness type on the vertex ranking. In Table 4, we present Kendall’s tau correlation measureFootnote ⁹ (Kendall, Reference Kendall1938; Knight, Reference Knight1966) for each pair of vertex rankings induced by the different temporal betweenness versions. For the temporal betweenness variants “shortest” and “shortest foremost,” we can observe that their respective non-strict and strict variants produce very similar rankings. We can further see that the betweenness variants “non-strict shortest foremost," “strict shortest foremost," and “strict prefix foremost” are pairwise similar. Recall that strict prefix-foremost temporal betweenness was computed significantly faster by our algorithms, hence we can conclude that if foremost temporal paths are of interest, then the prefix-foremost temporal betweenness is much easier to compute and yields very similar results.

Figure 5.

Top: temporal betweenness histogram of “highschool-2013” data set. Vertices are collected in 10 evenly distributed buckets between 0 and the highest temporal betweenness value. The $y$ -axis corresponds to the number of vertices. Middle: temporal betweenness histogram of “Facebook-like” data set. The $y$ -axis is on a log-scale. Bottom: temporal betweenness histogram of “primary school” data set. The temporal betweenness types from left to right are as follows: non-strict shortest, non-strict shortest foremost, strict shortest, strict shortest foremost, strict prefix foremost. Variants of shortest betweenness are colored in shades of red; foremost betweenness is colored in shades of blue.

These findings are supported by the pairwise comparisons of the sets of the ten vertices with the largest temporal betweenness values, which are presented in Table 5. Notably, the “primary school” has quite different top ten vertex sets for all pairs of betweenness variants that do not fall into the above-mentioned pairings. We can also observe that “non-strict shortest” and “strict prefix foremost” produce the most dissimilar vertex rankings and top ten sets.