Attributed Stream Hypergraphs: temporal modeling of node-attributed high-order interactions

Recent advances in network science have resulted in two distinct research directions aimed at augmenting and enhancing representations for complex networks. The first direction, that of high-order modeling, aims to focus on connectivity between sets of nodes rather than pairs, whereas the second one, that of feature-rich augmentation, incorporates into a network all those elements that are driven by information which is external to the structure, like node properties or the flow of time. This paper proposes a novel toolbox, that of Attributed Stream Hypergraphs (ASHs), unifying both high-order and feature-rich elements for representing, mining, and analyzing complex networks. Applied to social network analysis, ASHs can characterize complex social phenomena along topological, dynamic and attributive elements. Experiments on real-world face-to-face and online social media interactions highlight that ASHs can easily allow for the analyses, among others, of high-order groups’ homophily, nodes’ homophily with respect to the hyperedges in which nodes participate, and time-respecting paths between hyperedges.


Introduction
Complex networks provide a lens through which to illustrate plenty of behaviors that characterize humans as social animals.The elements of graph theory constituted the most helpful toolbox to represent and analyze social networks, with the intention to study complex behavior by mapping any possible kind of human contact, interaction, or relation as pairs of edges between unit elements called nodes.Network science, founded on such a basis, has been able to unravel many social patterns hidden at several scales of human relationships.Global network structures such as rich-clubs [14] and core-periphery structures [20], together with meso-scale organizations in blocks or communities [19], give an idea to the extent to which graphs are useful to grasp the knowledge of complex social architectures.However, the intrinsic nature of graphs to map dyadic patterns does not allow encoding explicitly group connectivity or high-order relations, which are fundamental in the social sphere.An increasing number of works recently started to address the mathematical tools of hypergraph theory [1] and simplicial complexes [23,2] to implement multi-body representations of social systems [44].Such new lines of hyper-network science aim to point out the importance of high-order interactions when studying the social dynamics of groups [46,41] or nodes embedded in groups rather than within neighborhoods built upon pairwise connections [18].
Parallel to this new interest in augmented topologies, other lines of research in network science focus on representations combining the structure with the large amount of domain-specific elements often available from a social system, like people's qualities or preferences, or with any kind of information external to the system that can be related to the structure, e.g., the flow of time that could affect topological changes.Mining such semantically augmented networks helps to unhearth many interesting social properties, from assortative mixing patterns based on common preferences [33] to the rules hidden in the formation and evolution of groups [34,39].The term 'feature-rich' networks [24] unifies all these augmented implementations that aim to add external, semantic information to a complex structure.Originally designed for pairwise networks, we believe that any complex topology could benefit from a feature-rich implementation, thus also networked representations built upon hypergraphs and simplicial complexes.
Hence, the objective of this work is to address the analysis of high-order patterns together with feature-rich elements.Generalizing the feature-rich framework, we aim to represent and analyze complex social phenomena along the following three dimensions: topology, dynamic features, and node attributes.To this purpose we introduce ASH, an Attributed Stream-Hypernetwork implementation for representing high-order temporal networks with attributive information on nodes.
The rest of the work is organized as follows.Section 2 sums up the principal literature on the three main complex network contexts surrounding this work, namely the dynamic, the node-attributed, and the high-order representations for networks.Section 3 introduces a formalism for the Attributed Stream-Hypergraph, our framework for addressing node-attributed evolving high-order topologies.Section 4 discusses our main results on real-world scenarios, from face-to-face contacts to user interactions on online platforms.Section 5 concludes the work.Finally, in the Appendix we introduce a Python library to work with Attributed Stream Hypergraphs.

Related Work
In the following, we provide an overview of the main enriched/augmented network implementations that are addressed in the work.First, we discuss dynamic and node-attributed network representations; then, we sum up the emerging contributions about high-order representations for complex systems.

Dynamics of networks.
Many network data that represent human activity have an intrinsic dynamic nature, from e-mail exchanges [27] and financial transactions [48], which are instantaneous forms of connections, to face-to-face interactions, that involve a certain duration, and friendships, that are generally stable and persistent over time.Hence, choosing a proper representation for modeling the dynamics of all these different social behaviors is not a straightforward task.Different temporal semantics impose different representations [22,39], being possible to categorize them according to the following properties: i) stability, e.g., when dynamic data are represented as a snapshot sequence from a time-window aggregation [38,6]; ii) duration, e.g., when data are represented as interval graphs [22]; and iii) immediacy, e.g., when data are represented as a stream graph of temporal nodes and connections [28].In this work, we will mainly focus on such stream graphs, that have been proven to extend and generalize classic centrality measures [42], and multi-layer structure as well [35].More generally, among the most interesting and cutting-edge analyses on dynamic networks, we can mention community detection [39], link prediction [17], and mixing pattern estimation [13], as well as works extending properties like reciprocity [9] and structures like rich-clubs [36] to dynamic environments.
Networks with attributes.Attributes or metadata often describe the properties of the nodes involved in networked data.Node attributes can be fruitfully used for improving results on classic network tasks, e.g., in community detection, where both tight connectivity and label homogeneity within communities need to be guaranteed [11].Attribute-enriched implementations can support analyses on the combined structural and attributive dimensions, searching for possible relations between the properties of nodes and how they are likely to connect [30,33].Node attributes can be leveraged for estimating homophily and heterogeneous mixing patterns [37,40].Other tasks oriented to machine learning points of view can leverage on node metadata -e.g., the distribution of values within the adjacent neighborhood of a target node -for node classification and link prediction purposes [4].There is also an emerging effort toward the exploration of global patterns of connectivity in attributed data, which still is an unexplored topic in the literature on feature-rich networks.Attributed backboning, for instance, is the task of finding the subtree of a graph that spans over the nodes with a minimized connection cost, where such cost is determined by node affinitive attributes [21].Similarly, a (k,r)-core structure is a subgraph that is cohesive with respect to both node connectivity and similarity [47].
High-order networks.Although traditional network science mostly addressed pairwise network representations, many dynamics can be better thought of as high-order representations involving relations between groups of nodes.As an emerging line of research [2,25,44], the expressive power of such high-order relations is yet largely unexplored.The interest in the physics of high-order interactions is growing [3], being extensively explored in the area of diffusive processes on networks, e.g., for studying social contagion with simplicial complexes [23], in time-varying settings as well [8].High-order structures varying in time are an important and emerging trend of research [5,15].They have been applied to study the network structure of scientific revolutions [26], or the evolution of high-order linguistic networks in scientific texts [10].There is also an increasing interest in the analysis of high-order interactions with attributes, e.g., measures for estimating homophily in hypergraphs and simplicial complexes [46,41], or integrating node attributes through annotated high-order models [7].The high-order structure of static/dynamic networks is often addressed by investigating datasets originally designed for graph-based analysis, thus one of the most intriguing future challenges is the inference of statistically significant high-order interactions from complex systems [32].Finally, some lines of works tend to be more conservative, as in the case of the s-line graph analysis for hypergraphs [1], where the hyperedge-projection of the hypergraph is used to apply, for instance, graph-based centrality measures to characterize hyperedges rather than nodes.

Attributed Stream Hypergraphs
To study dynamic high-order social interactions, simply borrowing results from the existing literature is not enough.Hypergraphs and/or simplicial complex has been not adequately defined in the presence of evolving topologies.Moreover, individuals embedded in a social system can often be characterized by multiple featuresprofiles that contextualize some of the key properties playing a role in social interactions (e.g., nationality, gender, age. . .).In this Section we introduce ASH, our Attributed Stream Hypergraph model, adequately defined for evolving high-order interactions with semantically enriched nodes.We formally define ASHs as follows: Definition 1 (ASH) Let S = (T, V, W, E, L) be a stream hypergraph, where:

is the set of discrete time instants, with A and Ω the initial and final instants;
• V is the set of the nodes of the temporally flattened hypergraph, namely the set of all nodes appearing during the ASH's lifespan; is the set of m node attributes such that l (t,u) with (t, u) ∈ W and t ∈ T , identifies the categorical value of the attribute l associated to u at time t.
ASHs bring together high-order interactions, temporal dynamics, and node attributes.It should be noted that other modeling frameworks can be thought of as particular instances of an ASH, where one of the three dimensions is switched off.For instance, given an ASH S = (T, V, W, E, L), it is possible to switch off a dimension that results in one of the following representations: • an attributed stream graph [13] for |N | = 2, ∀ (t, N ) ∈ E, where |N | identifies the number of nodes included in hyperedge (t, N ) ∈ E; • a static node-attributed hypergraph [46] for |T | = 1 (i.e., there is no temporal dynamics), which implies W = V and E ⊆ V n ; • a stream hypergraph (without node attributes) for L = ∅.

Inheriting from stream graphs and hypergraphs
ASHs are a conservative extension of stream graphs [28] and hypergraphs [2,1], thus inheriting from such frameworks their peculiar concepts.For instance, ASHs inherit from stream graphs the peculiarities of temporal nodes and temporal edges, since the nature of nodes and edges is analyzed with respect to the times they appear in the temporal stream.Nodes/edges can be thought of as temporal entities that can be present or absent at a certain time in the stream, so that the contribution of a node/edge is said to be equal to 1 -i.e., represented as a whole quantity -only if it is present all the time in the stream.With a rapid example, the contribution of an edge uv is computed as follows: , where |T uv | represents the number of time instants where uv is present, and |T | is the overall number of time instants.Naturally, the main difference with stream graphs is that, in an ASH, the temporal presence of an interaction is accounted for hyperedges.This aspect captures the fact that nodes/edges might not be present all the time, thus |W |, the sum of active nodes across all temporal instants, and |T × V |, the sum of all possible active nodes across all temporal instants, might differ significantly.The contribution of temporal hyperedges is computed under the same rationale, i.e., the sum of active hyperedges across all temporal instants over the sum of all possible active hyperedges across all temporal instants.Finally, in the case when all nodes are present all the time in the stream, the representation is called link stream, and it is a possibility allowed for ASHs as well.
Another key concept that can be generalized to ASHs is that of path.Paths on graphs have already been extended to hypergraphs within the s-analysis framework [1].This frameowrk builds on the idea that hyperedge paths (or any walk, equivalently) not only have a length, i.e., the number of hyperedges crossed during the walk, but also a width, i.e., the cardinality of the minimum intersection between subsequent hyperedges.For instance, an s-walk of width 3 (3-walk, equivalently) is a sequence of hyperedges where each edge intersects on at least 3 nodes with its predecessor (except for the hyperedge at the beginning) as well as with its successor (except for the hyperedge at the end).However, the dynamic nature of ASHs comes with the added constraint of temporal contiguity.In other words, in a temporal setting, each subsequent hyperedge along an s-walk must come with non-decreasing, adjacent time instants.This also implies that, aside from length and width, a temporal s-walk also has a duration, namely the number of time instants occurring between the beginning and the end of the walk.Hence, we define a time-respecting s-walk as follows: Definition 2 (Time-respecting s-walk) A time-respecting s-walk of length k, width s, and duration d is a sequence P = {(t 0 , N 0 ), (t 1 , N 1 ), . . ., (t k−1 , N k−1 )} such that: for all is, where i ∈ Z + ∧ i < k identifies the position of a hyperedge along the walk; By leveraging the above formulation, the notions of shortest, fastest, fastest-shortest, shortest-fastest, and foremost time-respecting s-walks can be deduced as already done for stream graphs [28], e.g., shortest paths are the ones with minimal length k, fastest paths are the ones with minimal duration t k − t 0 , fastest-shortest are the fastest paths among the shortest ones, and shortest-fastest, viceversa; foremost paths, independently from length and duration, are the ones that reach first the destination.
Another concept that can be extended dynamically is that of node's star, namely the set of hyperedges where the node is present.This can be limited to include only hyperedges that are active at a specific point in time.
Definition 3 (Temporal Star) Let u ∈ V be a node in the ASH.The temporal star of u at time t is the set of temporal hyperedges that include u in t, and is denoted Temporal star analysis allows quantifying node-level properties of temporal high-order structures [15], as well as eventually extending to the temporal dimension concepts like hyperego-network density and overlap [29].

Towards temporal mixing patterns estimation
Apart from combining the stream graph's evolutionary nature with the hypergraph's high-order structure, ASHs can integrate time-evolving node attributes, i.e., labels that (may) change in time.This peculiarity allows studying not only how individuals' characteristics change (e.g., opinions, political leaning) but also how such changes relate/affect the topological structure surrounding them.As a node's attribute values might vary through time, one can quantify the extent of their consistency, namely to what extent a node's attribute value remains constant over time.
Definition 4 (Consistency) Let u ∈ V be a node, and l ∈ L be an attribute such that l (t,u) denotes the attribute value of u at time t; let T u identify the set of time instants where u is present.The Consistency of u with regards to l ranges in [0, 1] and is computed as: Consistency can be extended to the whole ASH by computing the average as follows: which quantifies the extent to which an ASH's nodes keep their attribute value constant in time.
Henceforth, we may be interested in quantifying (a) hyperedges' homogeneity, which is a rising hot topic in attributed high-order analyses [46,41], and also (b) the level of homogeneity of a target node with respect to the set of hyperedges it belongs.In case (a), several options are possible, e.g., as cleanly proposed in [46,41] with statistically validated measures.
More straightforward ways to measure hyperedges' homogeneity can consist in finding an aggregate value that sums up the characteristics of a hyperedge with respect to the labels carried by the nodes within it.For instance, a characteristic value can be the frequency of the most frequent class within a hyperedge.Hence, we can use hyperedges' purity [12] as follows: Definition 5 (Temporal Purity) Let (t, N ) ∈ E be a temporal hyperedge and l ∈ L be a node attribute.Let max l∈L ( n∈N l (t,n) ) be the most frequent categorical value within (t, N ).The temporal purity of (t, N ) is the relative frequency of the most frequent value and ranges in [0, 1]: Similarly, another characteristic value that can be used to describe a hyperedge is entropy, which quantifies the degree of disorder related to the nodes' attribute values within the hyperedge.
Definition 6 (Entropy) Let (t, N ) ∈ E be a temporal hyperedge and l ∈ L be a node attribute.Let A (t,N ),l be the set of the attribute values of l in (t, N ).The entropy of (t, N ) with respect to l ranges in [0, 1] and is computed as follows: In case (b), our focus is on a target node u ∈ V aiming to analyze u's homogeneity with respect to its attribute value l (t,u) .We can still associate each hyperedge in D(t, u) with a characteristic value.The ones described so far, i.e., purity and entropy, result in continuous values.However, we can also characterize a hyperedge by means of a categorical value.Here, for instance, we describe each hyperedge in the temporal star of a target node u by means of the most frequent attribute value within the hyperedge, namely max l∈L ( n∈N l(t, n)), with (t, N ) ∈ D(t, u).Having such categorical value can allow us to compute the relative frequency of such characteristic values with respect to the label of the target node.
Definition 7 (Homogeneity) Let u ∈ V be a node with attribute value l (t,u) , l ∈ L. Let be D(t, u) the temporal star of u, and max l∈L ( n∈N l (t,n) ), with (t, N ) ∈ D(t, u) the most frequent categorical value of a hyperedge belonging to the star of u.The star homogeneity of u with respect to l t,u is the relative frequency of the hyperedges in D(t, u) that share with u its same attribute value l t,u .It ranges in [0, 1] and is defined as follows: Star homogeneity quantifies a node's degree of embeddedness across all of the contexts/interactions it finds itself in.
Finally, purity, entropy, and star homogeneity, can be averaged to capture the global behavior of the ASH: Note that Eq. 8 averages over the number of nodes, as star homogeneity quantifies node behavior; conversely, Eqq.6 and 7 average over the number of hyperedges.patients tend to be disassortative all the time, coherently with the fact that they stay in different rooms [45] and they are visited only by nurses and medical staff.

Homophilic Behaviors in Pairwise and Group Political Discussions on Reddit
We focus on data collected from the debate between Trump supporters and anti-Trump citizens during the first two and half years of Donald Trump's presidency, covering a period between January 2017 and July 2019.The debates cover both controversial/polarizing sociopolitical issues and broader discussions within the US political ideologies, as follows: • Gun Control: this topic is identified by collecting lists of subreddits that either support gun legalization or are against it; • Minorities Discrimination: identified by considering groups that promote gender/racial/sexual equality and groups with more conservative attitudes; • Political Sphere: identified by covering different US political ideologies such as Republicans, Democrats, Liberals, and Populists.
Data collection, users' ideology inference, and network construction are properly described in the reference paper [31], being able to identify three users' families, protrump, antitrump, and neutral classes, that we use as our categorical attribute values.Leveraging the original temporal network 3 , here we infer the hypergraph structure by means of all the maximal cliques.As in the reference analysis, [31], we consider a time window of six months when analyzing system interactions' dynamics.Average statistics for the pairwise graphs are shown in Table 1.

Analytical setting
We set a four-fold framework to analyze ideological homogeneity from different network-based perspectives as in the following: i. we promote an analysis on dyadic interactions, measuring how much users are homogeneously embedded in their pairwise ego-networks; ii. we shift the focus from individual users to groups, and we measure the homophily of such groups represented as hyperedges; iii. we come back to individual users, adopting user's point of view by measuring how much a user is embedded in the hyperedges where he/she participates; iv.we introduce a time-aware analysis to track stability or variations in ideological homogeneity.
As a preliminary question, we aim to explore whether different behaviors emerge among individual users (i) and groups (ii): can high-order interactions capture patterns that are invisible to dyadic interactions?Then, we aim to understand the role of single users in the several hyperedges where they participate (iii), as a meeting point between the two previous issues: can high-order neighborhoods capture patterns that graph ego-networks cannot?Finally, the focus on interactions' dynamics (iv) would allow us to track stable or mutable patterns as time goes by.
It should be noted that computations in (i) and (iii) are different from (ii).In (i) and (iii) we aim to measure the homogeneity of users' contexts with respect to the political leaning of a specific target node.In (i), a context is represented by the set of adjacent nodes in the ego-network of a target node, while in (iii) the context is the set of hyperedges where the target node participates in.We use a measure of homogeneity to estimate target nodes' similarity within nodes' own contexts.We can use Eq. 7 for both pairwise and high-order nodes' ego-network, since in the former case we compute the relative frequency of the attribute values among the node's first-order neighborhood, and in the latter case we use the most frequent value as the characteristic values of a hyperedge.Conversely, in (ii) the focus is on hyperedges' homogeneities.Thus, we use Eq. 3, which computes the relative frequency of the most frequent attribute value within the hyperedge.
Figure 2 outlines graph ego-networks' homogeneities in the three topics considered.Results are aggregated over the semesters.The analysis of pairwise interactions captures both homophilic and heterophilic patterns, telling us that such political discussions manifest heterogeneity.For instance, in Politics, protrump and neutral users show heterophilic behavior, while antritrump are more homogeneous.Minority is overall more homophilic than GunControl, where interactions seem to be also more randomly mixed.These observations are coherent with the analyses performed on the original data paper [31], where in Minority and Politics it is more likely to observe echo-chambers -oriented  Results are aggregated over the semesters.The discussions in Minority are the purest ones, a result which is coherent to what already observed at the meso-scale graph-based community level in the original paper [31]: GunControl does not present strongly polarized communities (i.e., echo chambers) among different semesters [31] as well as it seems that only a bunch of contexts present quite perfect purity (Figure 3 (a), leftmost); in Minority on average, more than half of total users are trapped in echo chambers [31], and hyperedge purities show a quite similar pattern as well, with  Moreover, we analyze these patterns with respect to the hyperedge size.Figure 3 (b-c) highlight, respectively, the number (b) and the average purity (c) of pure groups in function of the group size.For instance, in Minority we observe that only protrump pure discussions involve groups with more than 7 participants, and that they are quite pure, 0.9.The same does not happen in GunControl, while in Politics the biggest contexts involve antitrump users only but with a lower purity than the one of protrump users in Minority.

Users are involved in heterogeneous debates.
As can be observed in Figure 4, the topics show diversified behaviors when the analysis shifts to star egos.Indeed, there is no more trace of the heterogeneous patterns observed in Figure 3(a).The key insight, however, relates to another type of heterogeneity in user debates.While engaging in relatively homogeneous contexts (Figure 3), it seems that users find themselves in rather mixed collections of debates.That is to say, although homophilic behavior is highlighted in most debates (i.e., hyperedges), the set of contexts a node is involved in (i.e., its star) is generally diversified with respect to ideology/political leaning.This is especially true in GunControl, where protrump users appear to engage in a more heterogeneous set of debates than their counterparts, as opposed to what was noted in Figure 3 (a).The same holds for Politics, which displays a peak in heterogeneous protrump stars while the antitrump ones show more homophilic behavior.Minority, instead, still shows strong homogeneity traits for both antitrump and protrump users, thus confirming previous observations.

Interactions' dynamics: users' preferences tend to be consistent in time
As far as the temporal dimension is concerned, a certain degree of consistency w.r.t.debates homogeneity/heterogeneity can be observed.As a matter of fact, the average star homogeneity outlines almost-flat trends (Figure 5), indicating minor variations.Here, GunControl and Minority reveal near-constant heterogeneity/homogeneity for both political alignments; lastly, Politics displays only a small bump during the third semester concerning protrump and neutral users.What makes this result so interesting is the fact that only ∼ 11% of the nodes stay in the network for more than a semester (see Table 2), but still coherence is observed regardless of the continuous turnover of nodes.

Discussion and Conclusion
In this work, we proposed an Attributed Stream Hypergraph (ASH) representation for taking into account both highorder relations [2] and feature-rich information [24] through which to describe complex social systems.With ASH, social phenomena represented by means of high-order interactions can also be studied together with additional information that goes beyond the network structure, namely nodes' semantics and time.We have shown how this paradigm can be used to analyze social interactions along the (i) structural, (ii) attributive, and (iii) temporal dimensions.The high-order architecture inherited by hypergraphs (i) allows to more realistically model social interactions which naturally occur in groups of varying sizes.Node metadata can be used to construct node profiles (ii), which can be used to assess differences and similarities in the behaviors of different classes.The temporal dimension (iii) can shed light on recurring patterns over time.
The novelty of ASH stands in the possibility to combine all these aspects together.The time-respecting s-walks introduced in Section 3, for instance, open the way to a broad characterization of high-order time-changing networks, e.g., by identifying stable, densely-connected sub-hypergraphs, or to generalize dynamic centrality scores to hypergraphs [42].While applying the ASH framework on US-politics-bound communities on Reddit we observed strong homophilic behaviors among groups/hyperedges with respect to users' political leaning.However, while focusing on the preferences of single nodes, namely on how much a target node is homogeneously embedded with respect to the representative political leaning of the groups/hyperedges it belongs to, we mostly observe a relevant decrease in nodes' homophilic behaviors.As a consequence, we observe that users prefer participating in contexts whose representative leaning is different than the target node's own label, although hyperedges are strongly homophilic per se.Interestingly, this pattern can not be observed when looking at the pairwise ego-networks only.
In future works, we plan to focus on the constraints that stream hypergraphs could eventually raise, such as the issues of under/overfitting social data or the robustness of the measures to missing data.Our findings can highlight how different temporal aggregations and graph vs. hypergraph representations deeply affect the output of analytical pipelines.Thus, some of the most interesting challenges in the future will be understanding the impact of different representations (e.g., graphs vs. hypergraphs), of high-order structure inference methods (e.g., via cliques [5], overlapping communities, or other statistical methods [16]), and of different measures to study mixing behaviors.We also plan to introduce synthetic ASH generators to be used in the validation of analysis results.
Lastly, we plan to update and maintain the ASH Python library, hoping it will simplify and make more accessible to researchers and practitioners feature-rich hypernetwork analysis.and its implementation.GR coordinated and supervised all of the research.All authors read and approved the final manuscript.

Figure 1 :
Figure 1: Duration of shortest s-paths (a) and temporal trends of average star homogeneity (b) for Primary School and Hospital Ward at different aggregation windows.

Figure 2 :
Figure 2: KDE distributions of pairwise ego-networks' homogeneity among the three different Reddit communities.

Figure 3 :
Figure 3: KDE distributions of hyperedges' purity (a), number of pure hyperedges (b), and average purity (c) in function of hyperedge size among the three different Reddit communities.

Figure 3 (
Figure 3 (a)  shows ideological homogeneity within the hyperedges in the three topics considered, captured by purity.Results are aggregated over the semesters.The discussions in Minority are the purest ones, a result which is coherent to what already observed at the meso-scale graph-based community level in the original paper[31]: GunControl does not present strongly polarized communities (i.e., echo chambers) among different semesters[31] as well as it seems that only a bunch of contexts present quite perfect purity (Figure3(a), leftmost); in Minority on average, more than half of total users are trapped in echo chambers[31], and hyperedge purities show a quite similar pattern as well, with

Figure 4 :
Figure 4: KDE distributions of hypergraph star ego-networks' homogeneity among the three different Reddit communities.

Figure 5 :
Figure 5: Average hypergraph star ego-networks' homogeneity over time among the three different Reddit communities.

Table 1 :
Reddit Data Network statistics (averaged across semesters).size of the network in terms of nodes and edges, number of users with a Pro-Trump, Anti-Trump or Neutral leaning score.

Table 2 :
Reddit Data Network statistics (averaged across semesters).Nodes that stay in the network for more than two semesters/timestamps; mean and std of the consistency values for such nodes.