Introduction
Related work
Attributed Stream Hypergraphs
Symbol | Description |
---|---|
\(\mathcal{S}\)
| Stream Hypergraph |
T
| Set of time instants |
t
| A time instant belonging to T |
V
| Set of nodes |
u
| A node u, belonging to V |
W
| Set of temporal nodes |
(t, u) | A temporal node observed at time t, belonging to W |
E
| Set of temporal hyperedges |
N
| A subset of nodes |
(t, N) | A temporal hyperedge observed at time t, belonging to E |
L
| Set of node attributes |
l
| Node attribute value |
\(l_{(t,u)}\)
| Attribute value of node u at time t |
P
| Sequence of hyperedges |
\(T_u\)
| Set of time instants where node u is present |
\(V_t\)
| Set of nodes active at time t |
D(t, u) | Temporal star of u at time t |
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\(T = [\textrm{A}, \Omega ]\) is the set of discrete time instants, with \(\textrm{A}\) and \(\Omega\) the initial and final instants, and \(t \in T\) identifies a time instant belonging to T;
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V is the set of the nodes of the temporally flattened hypergraph, namely the set of all nodes appearing during the ASH’s lifespan;
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\(W \subseteq T \times V\) is the set of temporal nodes such that \((t,u) \in W\) identifies a node u observed at time t;
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\(E \subseteq T \times V^n\) is the set of temporal hyperedges such that \((t,N) \in E\) implies that \(N \subseteq V\) and \(\forall \, u_i \in N, (t,u_i) \in W\);
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\(L=\{l_1,...l_m\}\) is the set of m node attributes such that \(l_{(t,u)}\) with \((t,u) \in W\) and \(t \in T\), identifies the categorical value of the attribute l associated to u at time t.
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an attributed stream graph (Citraro et al. 2022) for \(|N| = 2, \forall \, (t, N) \in E\), where |N| identifies the number of nodes included in hyperedge \((t, N) \in E\);
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a static node-attributed hypergraph (Veldt et al. 2023) for \(|T| = 1\) (i.e., there is no temporal dynamics), which implies \(W = V\) and \(E \subseteq V^n\);
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a stream hypergraph (without node attributes) for \(L = \emptyset\).
Inheriting from stream graphs and hypergraphs
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\(P \subseteq E\);
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\((t_i, N_i) \in E\);
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\(t_i \le t_{i+1}\) for all is, where \(i \in \mathbf {Z^+} \wedge i < k\) identifies the position of a hyperedge along the walk, with \(\mathbf {Z^+}\) identifying the set of positive integers;
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\(s \in \mathbf {Z^+} \wedge s \le |(N_i) \cap (N_{i+1})|\);
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\(d = t_{k-1} - t_0\);
Towards temporal mixing patterns estimation
Experiments
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We build a case study on online social network discussions about political topics (Morini et al. 2021) to describe aspects related to pairwise-based vs. high-order-based representations.
High-order temporal dynamics in face-to-face contacts
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Primary School (Stehlé et al. 2011): this dataset contains face-to-face interactions between children during the whole school day: node metadata include children’s gender and class;
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Hospital (Vanhems et al. 2013): this dataset contains the temporal contact data between medical doctors (MED), nurses and paramedics (NUR), administrative staff (ADM), and patients (PAT) in a short-stay geriatric unit of a University hospital. Data were collected for a week.
Time-respecting paths in face-to-face contacts
Temporal mixing patterns in face-to-face contacts
Homophilic behaviors in pairwise and group political discussions on Reddit
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Gun Control: this topic is identified by collecting lists of subreddits that either support gun legalization or are against it;
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Minorities Discrimination: identified by considering groups that promote gender/racial/sexual equality and groups with more conservative attitudes;
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Political Sphere: identified by covering different US political ideologies such as Republicans, Democrats, Liberals, and Populists.
Topic | # nodes | # edges | # Pro-Trump | # Anti-Trump | # Neutral |
---|---|---|---|---|---|
Gun Control | 4991 | 15298 | 3346 | 1645 | – |
Minorities Discrimination | 5540 | 12605 | 3318 | 2222 | – |
Political Sphere | 4509 | 7079 | 1280 | 2395 | 834 |
Analytical setting
Pairwise ego-networks reveal both homophilic and heterophilic users’ preferences
Hyperedges’ purity emphasizes heterogeneity
Users are involved in heterogeneous debates
Interactions’ dynamics: users’ preferences tend to be consistent in time
Topic | #stay | #stay_pct | mean_consistency (stay) | std_consistency (stay) |
---|---|---|---|---|
GunControl | 580 | 0.116 | 0.591 | 0.478 |
Minority | 735 | 0.132 | 0.715 | 0.441 |
Politics | 574 | 0.127 | 0.748 | 0.310 |
Discussion and conclusion
ASH
Python library, hoping it will simplify and make more accessible to researchers and practitioners feature-rich hypernetwork analysis.