A Higher-Order Temporal H-Index for Evolving Networks

The H-index of a node in a static network is the maximum value $h$ such that at least $h$ of its neighbors have a degree of at least $h$. Recently, a generalized version, the $n$-th order H-index, was introduced, allowing to relate degree centrality, H-index, and the $k$-core of a node. We extend the $n$-th order H-index to temporal networks and define corresponding temporal centrality measures and temporal core decompositions. Our $n$-th order temporal H-index respects the reachability in temporal networks leading to node rankings, which reflect the importance of nodes in spreading processes. We derive natural decompositions of temporal networks into subgraphs with strong temporal coherence. We analyze a recursive computation scheme and develop a highly scalable streaming algorithm. Our experimental evaluation demonstrates the efficiency of our algorithms and the conceptional validity of our approach. Specifically, we show that the $n$-th order temporal H-index is a strong heuristic for identifying super-spreaders in evolving social networks and detects temporally well-connected components.


INTRODUCTION
Measuring the centrality of nodes is one of the fundamental operations in network analysis and has various applications [11,68].The H-index was originally proposed by J. E. Hirsch [20] for measuring the productivity and impact of scientists.It is defined as the maximum value ℎ such that at least ℎ of the scientist's publications have been cited at least ℎ times.The corresponding centrality measure for static networks defines the H-index of a node as the maximum value ℎ, such that at least ℎ of the node's neighbors have a degree of (a) A small temporal network.The edges only exist on the indicated time stamps.
(b) The underlying aggregated, static graph without temporal restrictions.
(c) The evolving network changes over time.The connectivity under the temporal constraints is restricted and limits the possibilities for the flow of information.Figure 1: An example for the -th order temporal H-index.In the temporal network in (a), the 1-st order temporal Hindex of node  is four because  has four outgoing edges at time one, and each of these edges can be extended by four different edges, each with a strictly later time stamp.Nodes , , and  have a value of two and  of one.In the aggregate static graph (b), all nodes have a H-index value of three.at least ℎ.Recent work has shown that the index yields a successful heuristic for finding super-spreaders in static networks [16,38,39].Recently, Lü et al. [39] formally linked the established concepts of degree centrality, H-index, and the coreness of a node by introducing the -th order H-index.The -th order H-index is defined recursively using the H operator, which, given a multiset  of integers, returns the maximum integer  such that at least  elements in  are not smaller than .So far, the (-th order) H-index has been applied primarily to static networks.
However, recently there has been an increasing focus on dynamically changing networks [22,33,45].This trend is motivated by the fact that real-world processes are often inherently dynamic, implying possible causalities that must be taken into account to obtain valid interpretations [22].Figure 1a shows an example of a small temporal network representing a communication network.The nodes represent persons, and each edge is a bidirectional communication at a specific time shown at the edge.Figure 1c shows the evolution of the network over time.There is a static graph containing the edges with the corresponding time stamp for each time point.Note how temporal evolution restricts the possible flow of information.For example, person  communicates with  at time two and  with  at time four, there is a possibility that information flows from  to  via  but not in the reverse direction due to the chronological order of communications-all communications between  and  are after the communications between  and .If we only consider the underlying aggregated graph, in which the time stamps are removed, and resulting multi-edges are merged (see Figure 1b), there are no restrictions on the possible flow of information.Therefore, in this work, we focus on analyzing temporal networks consisting of a set of nodes and a set of temporal edges endowed with time stamps indicating their time of existence.Such temporal networks are prime models of real-world interactions like human contact, communication, and social networks [7-9, 12, 24].Current work: We generalize the -th order H-index for temporal networks.First, we introduce the time-dependent in-and outdegrees of a node  at time , counting the number of incoming or outgoing edges before or after .On this basis, we define the -th order temporal H-index, respecting temporal reachability and possible implied causality.We obtain a family of (1) temporal centrality measures and (2) corresponding core-like temporal decompositions.
The centrality measures based on the -th order temporal Hindex capture the importance of a node by considering its spatial and temporal position in a neighborhood of depth  in the network.We propose two variants-the first measures a node's ability to influence or distribute information, and the second measures how well a node can be influenced or obtain information.The 1-st order temporal H-index corresponds to the classic static variant with the critical difference that the temporal restrictions in terms of possible incoming or outgoing information flows are respected.For example, in Figure 1a, node  has four outgoing edges at time  = 1.Each of the four neighbors also has four outgoing edges at times  > 1.Hence, any information leaving node  over one of the four edges at the time one can be further distributed via four edges.Therefore, the 1-st order outward temporal H-index is four.With similar arguments, we see that node , , and  have a 1-st order outward temporal H-index of two (e.g.,  has two outgoing edges to  at times four and five, and  has two outgoing edges later than five).Finally, node  has a value of one.However, suppose we ignore the temporal information and calculate the conventional static H-index in the underlying aggregated static graph shown in Figure 1b.In that case, all nodes obtain the (traditional) H-index value of three, which is uninformative.Furthermore, if we compute the 1-st order inward temporal H-index Figure 1a, we obtain a different ranking of the nodes in which  has the smallest indexvalue and  the highest.Note that for the static H-index, there is no such distinction in undirected networks.Going from the 1-st order to higher orders, we can include a deeper neighborhood into the index computation and can identify nodes that have a strong influence in large parts of the network (or can be influenced by large parts of the network in the inward case).
Moreover, a recursive application of the temporal H-index leads to a core-like decomposition of the network.Core decompositions are an essential concept in the study of graph properties with many important applications, e.g., (social) network analysis, community detection, or network visualization [31,43].We define the (, )pseudocore in a temporal network G as the maximal induced temporal subgraph such that each node has an -th order temporal H-index of at least  with respect to G. We again propose two variants based on the in-and outward temporal H-index.Our evaluation shows that our -th order temporal H-index leads to pseudocores characterized by high temporal reachability.Our contributions are: (1) We introduce the -th order temporal H-index measuring the importance of nodes based on the structure of their neighborhood respecting temporal reachability.We propose two variants-the first measures a node's ability to influence or distribute information, and the second measures how well a node can be influenced or obtain information.(2) Based on the -th order temporal H-index, we introduce a corresponding decomposition of the network into (, )-pseudocores, describing components with high communication capabilities.(3) The -th order temporal H-index can be straightforwardly computed by directly implementing its recursive formulation.However, this approach is not scalable, and we propose a highly scalable streaming algorithm operating on the chronologically ordered edge stream using only a single pass over the edges.(4) Our evaluation on real-world temporal networks shows that our algorithms are highly efficient even for networks with millions of nodes and tens of millions of temporal edges.The -th order temporal H-index effectively identifies central nodes and temporally cohesive subgraphs.We demonstrate this in the use case of identifying super-spreaders in epidemic processes in real-world networks.The omitted proofs are provided in Appendix A.
Centrality for Temporal Networks.Several works [29,50,70,71] examine properties of temporal networks, including various temporal centrality measures, and discuss the necessity of considering temporal dynamics.Variants of temporal closeness have been introduced and discussed in [10,41,51,56,67].Similar to static closeness, the temporal closeness is often defined as the inverse of the sum of optimal temporal distances.Another path-based centrality measure is betweenness.Buß et al. [6] evaluate the theoretical complexity and practical hardness of computing several variants of temporal betweenness centrality.Santoro and Sarpe [66] discuss the estimation of temporal betweenness.Tsalouchidou et al. [72] extend Brandes' algorithm [5] for distributed computation of betweenness centrality in temporal networks.Temporal closeness and betweenness only consider optimal temporal paths.Our new centralities, in contrast, consider the general temporal reachability of the node itself and its neighborhood, which is more meaningful in settings where information does not necessarily spread along the shortest paths.The Katz centrality [27] measures node importance in terms of the weighted random walks starting (or arriving) at a node.The authors of [4,19] adapt the walk-based Katz centrality to temporal networks.Rozenshtein and Gionis [65] adapt PageRank for temporal networks by using temporal walks.Oettershagen et al. [54] generalized the random walk betweenness for temporal networks.
These centrality measures are based on counting temporal walks, while our -th order temporal H-index is based on counting neighbors with high temporal reachability through  recursion steps.Recent works try to identify influential nodes in spreading processes under the influence maximization model for temporal networks [13,14].
Temporal Core Decomposition.Wu et al. [76] proposed (, ℎ)core decompositions for temporal networks, where each node in a (, ℎ)-core has at least  neighbors and at least ℎ temporal edges to each neighbor.The concept can be interpreted as a weighted static core decomposition and does not take temporal dynamics such as reachability into account.A recent work by Galimberti et al. [15] introduced a notion of temporal span-cores where each core is associated with a time span, i.e., a time interval, for which the coreness property holds.Let  be the length of the time interval spanned by a temporal graph G with edge set E, then there is a quadratic number of time intervals for which a temporal span-core can exist, and the asymptotic running times of the proposed algorithms are in O ( 2 • |E |) which is prohibitive for large networks.A notable difference to our new approach is that the span-cores decompose a temporal network in its temporal domain such that each vertex may have several distinct span-core values in different time intervals.Whereas our decomposition, similar to the conventional -core, leads to a single core number for each vertex.
Core-like concepts are also used in temporal community mining.Hung and Tseng [25] define a temporal community as a (, )lasting core, which is a -core existing for at least  consecutive time steps and discuss the problem of finding the maximum (, )lasting core.Qin et al. [61] consider the densest subgraph problem and define a temporal community as an (, )-maximal dense core which has an average degree of at least  in a time interval of length at least .Qin et al. [60] mine stable communities in temporal networks using the concept of (, , )-stable cores.Here, a node is in a (, , )-stable core if it has no less than  neighbors that have a similarity of at least  to the node in at least  snapshots of the temporal network.Li et al. [36] propose as community model the (, )-persistent -core variant.It is defined for a time interval  such that a -core of size  exists in any subinterval of  with length  .Computing maximum (, )-persistent -cores is NP-hard.
Our -th order temporal H-index leads to a decomposition that has only a single parameter and allows efficient computation (see Table 2).Core decompositions are valuable tools, e.g., in social network analysis [69], community detection [63], and network visualization [1].The previously mentioned approaches do not take temporal reachability explicitly into account, whereas ours supports this naturally via the -th order temporal H-index.

PRELIMINARIES
We introduce basic definitions and our notation.We refer to the natural numbers without zero by N, and define N 0 = N ∪ {0}.A (static) graph  = ( , ) consists of a finite set of nodes  and a finite set  ⊆ {{,  } ⊆  |  ≠  } of undirected edges.A node  ∈  is incident to  ∈  if  ∈ .The degree  () of a node  ∈  is the number of edges incident to .

Temporal Networks
An undirected temporal network G = ( , E) consists of a finite set of nodes  and a finite set E of undirected temporal edges  = ({,  }, , ) with  and  in  ,  ≠ , availability time (or time stamp)  ∈ N and transition time (or traversal time)  ∈ N. The availability time specifies when the transition from  to  or  to  via the edge is possible.In a directed temporal network, a temporal edge has the form (, , , ) and can be traversed from  to  only.For convenience, we primarily consider directed temporal networks in the following, which allow to model undirected edges by means of two symmetric directed edges.The transition time of an edge is the time required to traverse the edge, e.g., in a temporal transportation network the time a vehicle needs between two stops, or in a human contact network the time to give information from one person to another person.Notice that it is reasonable and common to assume a strictly positive transition time as information in standard models cannot be transferred instantaneously.The in-and out-degrees of a node  count the temporal edges arriving or leaving, resp., at .We use  − max (G) to denote the maximal in-degree and  + max (G) for the maximal out-degree of G.We write  − max or  + max for short if G is clear from the context.
Given a (directed or undirected) temporal network G, removing all time stamps and traversal times, and merging resulting parallel edges, we obtain the (directed or undirected) aggregated graph Given a temporal network G = ( , E), it is common to restrict research questions to a given time interval  = [, ], such that only the temporal subgraph G ′ = ( , E ′ ) with E ′ = {(, , , ) ∈ E |  ≥  and  +  ≤ } needs to be considered.We do not include restrictive intervals in the following definitions for better readability.However, our definitions and algorithms can be easily adapted to consider only temporal edges starting and arriving within the interval.A temporal walk of length ℓ in a temporal network G is an alternating sequence ( 1 ,  1 , . . .,  ℓ ,  ℓ+1 ) of nodes and temporal edges connecting consecutive nodes, such that for 1 ≤  < ℓ,   = (  ,  +1 ,   ,   ) ∈ E and  +1 = ( +1 ,  +2 ,  + ,  +1 ) ∈ E, and the time constraint   +   ≤  +1 holds.Let  (G) be the temporal diameter of G, i.e., the length of the longest (in terms of number of edges) temporal walk in G.

The H-Index and k-Core Decomposition
A -core of a graph  is a maximal subgraph   of , such that every node in   has at least  neighbors in   .A node  has core number  () =  if  belongs to a -core but not the  + 1-core of the graph [43].The core numbers of all nodes can be computed in Lü et al. [39] showed a relationship between degree centrality, H-index, and core numbers.To this end, they introduced the H operator, which essentially represents the calculation of the Hindex.Let S be the set of finite multisets of integers.The function H : S → N 0 returns for a finite multiset of integers  ⊆ {{ |  ∈ N 0 }} the maximum integer  such that there are at least  elements  in  with  ≥ .We define H (∅) = 0.
Using the H operator, the H-index of a node  is defined as of a node  in a static graph is defined recursively [39].
(a) Temporal network G with the availability times at the edges ( = 1 for all edges).The outpseudocore-decomposition for  = 2 is shown.1a shows the corresponding values for all vertices.
Let  (0)  =  () the degree of node , then For  = 0 and  = 1, the value of  ()  corresponds to the degree and H-index of , respectively.For increasing , it converges to the core number of  [39].

THE N-TH ORDER TEMPORAL H-INDEX
We now define the -th order H-index for temporal networks.To this end, we start by generalizing the degree of a node to incorporate the temporal aspects of the local network topology.The time-dependent out-degree of  reflects the number of ways to extend a temporal walk from  at or after time  by an edge.The time-dependent in-degree counts the number of edges via which a temporal walk can reach  no later than at time .Furthermore, we define sets of nodes with time stamps that allow extending walks.Definition 2. We define the multiset N + (, ) that contains all pairs of nodes and times (,   ) such that there is a temporal edge (, ,  ′ , ) ∈ E with arrival time   =  ′ +  and  ′ ≥ .Analogously, we define the multiset N − (, ) that contains all pairs of nodes and starting times (,   ) such that there exists a temporal edge (, ,   , ) ∈ E with   +  ≤  .
Using the Definitions 1 and 2 together with the H operator, we define the -th order temporal H-index.

We call ℎ (𝑛)
,+ the outward -th order temporal H-index and ℎ () ,− the inward -th order temporal H-index.Note that also for undirected temporal networks typically ℎ ,− due to the direction implied by the temporal information.
The -th order temporal H-index equals the (in-/out-) degree centrality in the case of  = 0.For  = 1, we obtain time-directed variants of the classical H-index.For larger , intuitively, a node  has a high index value if it is in a temporally well-connected position in the network.More precisely, for the outward variant,  can reach many other nodes with temporal walks of length  + 1, and this recursively holds for the reachable nodes (with reduced walk length).Therefore, nodes with high outward temporal H-index can strongly influence their neighborhood, e.g., spread information easily, and are connected to similarly influential nodes.Analogously, a node  has a high inward index value if the node can be reached by many other nodes with temporal walks of length  + 1, and this recursively holds for the nodes reaching  (with reduced walk length).Hence, nodes with high inward temporal H-index can easily obtain information, i.e., be influenced, and are also connected to nodes with a similar ability.For both variants, the influence of temporal walk counts is controlled by the iterated application of the H operator based on the temporal structure of the neighborhood.We determine the exact lower bounds of the number of reachable nodes in Theorem 1. Figure 2a gives an example of a temporal network, and Table 1a shows the outward temporal H-index values of the nodes for  ≤ 4.
Table 1b shows the values of the static -th order H-index for the underlying static graph (G), ignoring the temporal information.The values of the H-index ( = 1) and -core numbers ( = 2) differ for all nodes.Likewise, for  = 1, the rankings of the nodes disagree, as in the static case, the nodes are only ranked in three groups (i.e., nodes with value 2, 3, or 4) and in the temporal case in four (i.e., nodes with values 0, 1, 2, or 3, respectively).For  = 2, the decompositions differ in the nodes , ℎ, and .As Figure 2a shows, the most inner core G (,1) (for  = 2, highlighted in green) contains both  and , but not ℎ.The static -core decomposition decomposes the graph into two groups of nodes with core numbers two and three.The most inner core, i.e., induced by nodes with the core number  = 3, contains ℎ but not  and  (highlighted in red).
Table 1: The values of the static and temporal  -index for the example shown in Figure 2. -th Order Temporal  -index ℎ

Properties
In the following, we investigate the properties of the -th order temporal H-index.To this end, we first define a reachability tree that represents all outgoing (or incoming) temporal walks from a node  ∈  .The reachability tree is not a data structure used in our algorithms but a valuable tool for illustrating and discussing the -th order temporal H-index.All results in this section hold for the outward and inward variants of the -th order temporal H-index.Let  () be is the depth of  ∈  ( ★ ()).The tree  + () contains a node  = (, ) for each node that we can reach at time  in a temporal walk starting at node  using  () consecutive temporal edges.For example, Figure 2b shows  + ( ) for the node  of the temporal network in Figure 2a.Similarly,  − () contains a node  = (, ) for each node that can reach  with a temporal walk starting at a time  using  () consecutive temporal edges.Definition 5. Let ★ ∈ {−, +},  ★ () = ( , ,  ),  ∈  , and  () ⊆  be the children of .For  ∈ N 0 , we define   :  → N 0 as The usefulness of  ★ () together with   is given by the following result.In static networks, the static -th order H-index of a node  ∈  converges to the -core value of the node  for sufficiently large  [39].This does not hold for the -th order temporal H-index.Therefore, neither the outward nor the inward -th order temporal H-index converges to the -core composition of the network.

Temporal Pseudocore Decomposition
The two variants of the -th order temporal H-index provide two natural ways of decomposing the temporal network.We distinguish inand out-pseudocores for ★ = − and ★ = +, resp.The following observations and results hold for both variants.For a node  in a (, )-pseudocore G (, ) , the inequality ℎ () ,★ ≥  does not hold necessarily with respect to G (, ) .However, each (, )pseudocore is a subset of nodes that implies a temporal subgraph containing nodes with similar temporal activity and importance in the network G.In Section 5.4, we show that the pseudocore can be used to identify temporally well-connected subgraphs.
Definition 7.For a fixed  ∈ N, the temporal pseudo-degeneracy   of a temporal network G is defined as the maximum  for which G contains a non-empty (, )-pseudocore.Similarly, for a fixed  ∈ N, we define the order pseudo-degeneracy   as the maximum  for which G contains a non-empty (, )-pseudocore.
We show the following containment properties.1b shows the core numbers of the (conventional) -core decomposition of the underlying aggregated graph (G) that ignores the temporal information.Note the differences for nodes , , and ℎ between the rankings of the static -core and the temporal out-pseudocore (note that we compare the relative rankings and not the core values, which are not directly comparable) resulting from the conceptual differences of the out-pseudocore and the conventional -core.

Computation
We discuss two algorithms with different properties and running times for computing the -th order temporal H-index.Both algorithms can be used to compute the inward and outward variants.
The first algorithm Recurs is a straightforward implementation of the recursive formulation.We additionally use memoization to avoid redundant computations.However, the running time of Recurs still is unsatisfactory.Therefore, we introduce a highly efficient one-pass streaming algorithm based on the edge stream representation of temporal networks, which represents the temporal network as a chronologically ordered list of the temporal edges (ties are broken arbitrarily).The edge stream representation is commonly used for temporal graph streaming algorithms, e.g., [49,54,75].Our streaming algorithm leads to significantly improved running times compared to the recursive algorithm (see Section 5.1).It only supports temporal networks with equal transition times for all temporal edges.However, this is a common property for many real-world temporal networks, e.g., face-to-face contact or email networks.Table 2 shows an overview of the algorithms and their properties.We provide the pseudocode and a discussion of the running time of Recurs in Appendix B. Both algorithms can be easily adapted to only consider edges in a given interval  .Alternatively, we can remove all edges not starting or arriving in  in a preprocessing step in O (|E |) time.Moreover, we assume that no isolated nodes exist, such that | | ≤ 2|E |.Because the degree of an isolated node  is zero, it follows ℎ ( )  = 0 for all 0 ≤  ≤ .Hence, we can safely remove all isolated nodes in a preprocessing step in O (| |) time.
Table 7 in the appendix gives an overview of the complexities of related temporal centrality measures and core decompositions, showing the competitiveness of our streaming algorithm.
In the following, we assume that  = 1 for all  ∈ E. Algorithm 1 shows the pseudocode of our streaming algorithm.It needs only a single pass over the edges in reverse chronological order, where temporal edges with equal availability times are processed in an arbitrary order.The algorithm computes ℎ A key idea of the algorithm is to keep a single counter for the time-dependent degree for each node.To this end, we use two arrays  and ; the former stores the last time  was updated, and  contains the current time-dependent degree of node .

Furthermore, the algorithm manages a list 𝜋 [𝑣]
[] of pairs ( ′ , ) of -th order  -indices for 0 ≤  ≤  + 1 and for all  ∈  .Here, the time  ′ is stored to exclude values during the computation of the H  operator depending on some time  (see line 1).

EXPERIMENTAL EVALUATION
We experimentally evaluate the efficiency of our algorithms and the efficacy of the -th order temporal H-index.Data Sets: We use a wide range of real-world temporal network data sets from various online and offline domains for our evaluation.Table 3 shows the properties and statistics of the data sets.All data sets are publicly available. 1234The transition times of the temporal edges are one for all data sets.Algorithms: We implemented the following algorithms in C++ using the GNU CC Compiler 10.3.0.
• Recurs is the implementation of naive recursive algorithm Algorithm 2 (see Appendix B). • Stream is the implementation of Algorithm 1.
All experiments were performed on a computer cluster.Each experiment had an exclusive node with an Intel(R) Xeon(R) Gold 6130 CPU @ 2.10GHz and 192 GB of RAM.The source code is available at https://gitlab.com/tgpublic/tgh.Figure 4: Effect of increasing  on the number of pseudocores.

Running Time and Memory Usage
Table 4 shows the running times for Recurs and Stream for  ∈ {2  |  ∈ {0, . . .7}}.We used a time limit of twelve hours.Our streaming algorithm Stream performs very well even on data sets with tens of millions of edges, e.g., the Wikipedia data set.Stream is faster than Recurs for all datasets and all .In most cases, the speedup is significant and up to several orders of magnitude.In general, the running time increases roughly linearly with increasing , as expected for both algorithms.Recall that Stream computes the -th order H-index for all 0 ≤  ≤ , where Recurs only computes the value for  = .Hence, computing the remaining values for 0 ≤  ≤ −1 for Recurs would require additional running time.As expected, the speed-up of Stream compared to Recurs is higher for the data sets with a high average degree because the asymptotic running time of Recurs contains the node degree as a quadratic factor (see Theorem 5), whereas for Stream it has the node degree as a linear factor (see Theorem 4).Recurs cannot finish the computations for the Wikipedia data set and  ≥ 64 within the time limit.Stream has competitive running times compared to other state-of-the-art temporal centrality measures and core-decompositions-a further discussion can be found in Section 5.2.The memory usage of both algorithms asymptotically behaves similarly.Figure 3 compares the memory usage for Hospital and Email.Recurs needs slightly more memory because the multisets  are maintained on the stack for recursive calls.

Additional Efficiency Comparisons
We additionally compare the running times of our new approach to related centrality measures and core decompositions.We provide the running times of the following methods: • H-Index is the static H-index and -core is the static -core.
Both were implemented in C++.• Tc, Twc, Tkatz, and Tpr are the temporal closeness, walk, Katz, and PageRank centrality.We used the C++ implementations provided by [53].• Tb is the temporal betweenness centrality using strict temporal path (i.e., temporal paths that are strictly advancing in time for each edge) from [6].The C++ implementation was provided by the authors.• (, ℎ)-core with ℎ ∈ {2, 4, 8} is the (, ℎ)-core described in [76].We used the C++ implementations provided by [53].• Span-core is temporal core decomposition from [15].We used the Cython implementation computing the maximum span-cores provided by the authors.
See Table 7 in the appendix for an overview of the time and space complexities of the centrality measures and core decompositions showing that our -temporal H-index has competitive complexities.Which is also verified by the following empirical evaluation.Table 5 shows the running times in seconds.We see that our streaming algorithm for the temporal H-index is for low , often significantly faster than the algorithms for the computation of the more demanding temporal closeness and betweenness centrality measures.For  = 1, Stream has similar running times as computing the static H-index in the underlying aggregated graph, where the temporal variant is slightly faster in almost all cases.Our algorithm needs more time than the temporal Katz and PageRank computations, which is expected as the former are computable in linear time (see Table 7 in the appendix).In the case of the core decompositions, our streaming algorithm Stream has higher running times compared to the static  core and the (, ℎ)-cores.The reason is that the latter can be computed in linear time.The implementation of the span cores cannot finish the computations for most data sets due to outof-memory errors.The reason is that the algorithm needs memory in length of the time interval spanned by the temporal graph.

Effect of the Parameter 𝑛
Figure 4 shows the effect of increasing  ∈ {2  |  ∈ {0, . . .10}} on the number of out-pseudocores, i.e., distinct node values for the outward -th order temporal H-index. Figure 4a shows the results for the data sets with less than 150 pseudocores for  = 1, and Figure 4b shows the results for the remaining data sets.In both cases, we see how the numbers of pseudocores approach one, i.e., the -th order temporal H-index is zero for all nodes, as expected due to Theorem 2. For the data sets with the three highest numbers of average degree, i.e., Hospital, Malawi, and Highschool, the number of pseudocores decreases slower than for the other     data sets.The reason is that these data sets allow high numbers of temporal walks.In the case of the Malawi data set, we see a slight increase in the number of pseudocores from 82 for  = 1 to 83 for  = 2.This does not conflict with Theorem 2 because even though for individual nodes, the -th order temporal H-index approaches zero for increasing , the number of different values of the -th order temporal H-index for the nodes can increase.
Algorithm 1 efficiently computes the -th order temporal Hindex for all 0 ≤  ≤  in one pass.The result allows choosing a decomposition with a number of cores best suited for a downstream task or application.Note that we obtain all non-trivial index values

Temporal Characteristics and Use Case
We first discuss the temporal characteristics of the -th order temporal H-index.Due to the space restrictions, we focus on the outward variant.See Appendix C.2 for a comparison of node rankings computed with the inward and outward -th order temporal H-indices.Figure 5 shows an example of the outward -th order temporal H-index applied to the Malawi network, which is a human face-toface contact network [55].Figures 5a to 5d show the aggregated underlying static graph, and the nodes are colored according to their centrality/core value.Figures 5a and 5b show the results for the outward -th order temporal H-index, and  = 32 and  = 512, respectively.The values and range of values are higher for  = 32 compared to  = 512, as expected due to Theorem 2. Figures 5c  and 5d show the values of the static H-index and -core numbers of the aggregated graph (G) for comparison.First, we note that the -th order temporal H-index differentiates more nodes than the static H-index and -cores.Respecting the temporal reachability constraints leads to different node rankings compared to the static versions, and the nodes are assigned different (relative) importance.
More specifically, using the outward -th temporal H-index, nodes with a strong ability to reach other nodes are ranked high.To verify this property, we measure the local and global reachability of out-pseudocores.Let G = ( , E) be a temporal graph, and  :  ×  → {0, 1} the indicator function for temporal reachability, i.e.,  (, ) = 1 iff  can reach  via a temporal walk.We define the global and local reachability scores   and  ℓ of a pseudocore G (, ) = ( ′ , E ′ ) as , and  ℓ = , ∈ ′  (,) | ′ | 2 , respectively.
Figure 6 shows the global and local reachability scores for the Malawi contact network and the FacebookMsg communication network.We show the scores for the -th temporal H-index and the static -core.We choose two values for , where the first equals 2  where  is chosen such that there are at least 20 different ranks, i.e.,  = 256 for Malawi and  = 8 for FacebookMsg, and additionally, we choose 2 +1 .Furthermore, as an additional baseline, we use the temporal (, ℎ)-core from [76] where we choose ℎ ∈ {4, 8}.In a (, ℎ)-core, each node has at least  neighbors and at least ℎ temporal edges to each neighbor.The -axes in Figure 6 show cores by ordered rank from zero to the number of cores minus one, where  = 0 corresponds to the highest-ranked core.The results verify that our outward -th order temporal H-index leads to pseudocores characterized by high local and global temporal reachability.We discuss the sizes of the cores in Appendix C.3.The nodes in a highranked pseudocore, i.e., with a high -th order H-index, can, on average, reach many of the nodes of the graph (global), and the nodes in the pseudocore (local).We verify the usefulness of this important property with the following use case.

Use
Case: Influential Spreader Identification.We evaluate how well the outward -th order temporal H-index captures the node influences of spreading processes.Here, we use the stochastic susceptible-infected-recovered (SIR) spreading model in temporal networks, which is commonly applied to analyze information or disease dissemination [23,59,77].We give a detailed description in Appendix C.4.We follow the standard practice of recent works, e.g., [37][38][39]42], to compare the rankings obtained by centrality measures with the ranking obtained by applying the SIR model.We chose two face-to-face (Malawi and Infectious) and two communication networks (FacebookMsg and Email) for the experiment.We chose different infection probabilities  and computed for each  and node  ∈  the mean node influence   over 1000 independent SIR simulations leading to the SIR node rankings.We then compare the SIR rankings with those obtained by the centrality measures using the Kendall   rank correlation measure (Appendix C.1).As baselines, we use the static variants of the degree centrality, H-index, and -core, which are common and popular heuristics for identifying strong spreaders [39].Furthermore, we used the temporal closeness centrality, which was suggested as a strong heuristic for the case of temporal networks in [51,52], and the recently proposed temporal walk centrality [54].Finally, we use the state-of-the-art heuristics (local) gravity centrality [35,40] and extended neighborhood coreness (coreness+) [2], which belong to the best performing -core based heuristics as shown in a recent survey [42].For the -th order temporal H-index (Thi), we set  = 32 and  = 64, to include a sufficient neighborhood sizes and to consider sufficient depths of the computation.Figure 7 shows the results.Node rankings obtained from the -th order temporal H-index have, in almost all cases, a higher mean correlation to the SIR ranking than the node rankings from the baselines.Only for the Malawi data set, the temporal closeness achieves higher correlations.In conclusion, the temporal H-index often indicates the nodes' capabilities for spreading disease or information better than the baselines.

CONCLUSION
We generalized the static -th order H-index for temporal networks.We obtained meaningful variants by taking into account the restricted reachability caused by temporal dynamics.Our streaming algorithm for the common case of uniform transition times is highly scalable.In our experiments, we demonstrated the efficiency and effectiveness of our new approaches.Finally, we showed that the -th order temporal H-index could successfully identify super-spreaders in a use-case.In future work, we plan to utilize the pseudocore decomposition to detect cohesive temporal communities and visualize temporal networks.similar rankings, close to zero no correlation, and close to minus one a strong negative correlation.

C.2 Inward vs Outward 𝑛-th Temporal H-Index
Figure 8 shows the Kendall   correlations between the node rankings obtained by the inward and outward -th order temporal Hindex for  ∈ {64, 128, 256}.The correlation between the rankings using inward or outward -th order temporal H-index for different  is high, which we expected.The correlation between rankings obtained from the inward and the outward -th order temporal H-index is lower.However, we still observe correlations of at least 0.6 (Malawi) and 0.44 (FacebookMsg) as there are nodes with high (or low) inward and outward -th order temporal H-index in the data sets.

C.3 Sizes of the Pseudocores
Figure 9 shows boxplots of the vertex set sizes | | of the cores for the temporal H-index, the static  core, and the temporal (, ℎ)cores.We show the results for Malawi, FacebookMsg, Email, and Enron.For the temporal H-index, the pseudocore sizes increase with increasing  because the number of pseudocores decreases (Theorem 2).Hence, the sizes are controllable using the parameter .

C.4 SIR Model
The stochastic susceptible-infected-recovered (SIR) spreading model is a standard model for analyzing spreading processes [23,59].In the model, each node is either susceptible, infected, or recovered.We used the SIR model as described in [23] and the implementation for temporal networks provided by the author.In the beginning, a single node  ∈  is infected.The infection can spread along temporal edges with infection probability .An infected node will recover after time  sampled with probability  () =  exp(−) from an exponential distribution, where  is set to 20 time units of the network.At the start of the simulation, a single node is infected.The node influence   of the initially infected node  ∈  is the number of nodes that are either infected or recovered after the simulation finishes, i.e., when no further infections are possible.
(b) The reachability tree  + ( ) for vertex  in the temporal network shown in (a).

Figure 2 :
Figure 2: Toy example for the -th order temporal H-index.Table1ashows the corresponding values for all vertices.

Figure
Figure 2ashows an example of the out-pseudocore decomposition for  = 2, and Table1bshows the core numbers of the (conventional) -core decomposition of the underlying aggregated graph (G) that ignores the temporal information.Note the differences for nodes , , and ℎ between the rankings of the static -core and the temporal out-pseudocore (note that we compare for all 0 ≤  ≤  and  ∈  (note that Algorithm 2 only computes ℎ () ,★ for a fixed ).In Appendix B.2, we show how to transform the input to use Algorithm 1 to compute ℎ ( ) ,− for all 0 ≤  ≤  and  ∈  .

Figure 5 :
Figure 5: Comparison of the outward -th temporal H-index, the static H-index, and the static -core values where a darker color means a higher ranking.The shown network is based on human face-to-face contacts in a village in Malawi.
high rank ← core num.→ low rank high rank ← core num.→ low rank

Figure 7 :
Figure 7: Kendall   correlations between the rankings of different centrality measures and the ranking according to the SIR model for infection probabilities  ∈ {0.1, 0.2, . . ., 0.8}.Thi denotes the -th order temporal H-index.

Figure 9 :
Figure 9: Box-plots of the number of vertices in the computed cores.The -axes are logarithmic.
The values

Table 2 :
Overview of the algorithms for computing the -th order temporal H-index and their properties (★ = − for the inward variant and ★ = + for the outward variant.)

Table 4 :
Running times in seconds (s).OOT: out of time (time limit 12h).

Table 5 :
Running times of related centrality measures and core decompositions in seconds (s).OOT: Out of time (time limit 12h).OOM: Out of memory (available memory 196GB). ≤  (G) and that  (G), i.e., the maximal length of a temporal walk, is at most the number of availability times in G.

Table 7 :
Overview of related centrality measures and core decompositions, with T = { | (, , , ) ∈ E},   the largest cardinality of availability or arrival times at a node,  the total length of the time interval spanned by the temporal graph, and  the length of a time window.The time complexities are for computing the centrality or core number all nodes.