COLLECTIVE BEHAVIOUR IN TEMPORAL NETWORKS DOOCN Satellite, CCS18 27th September 2018 Andrew Mellor Mathematical Institute University of Oxford
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 4. Email Correspondence 2. Telephone Calls 1. Twitter/Social Networks Examples: 3. Proximity Networks A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 4. Email Correspondence 2. Telephone Calls 1. Twitter/Social Networks Examples: 3. Proximity Networks A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 4. Email Correspondence 2. Telephone Calls 1. Twitter/Social Networks Examples: 3. Proximity Networks A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 4. Email Correspondence 2. Telephone Calls 1. Twitter/Social Networks Examples: 3. Proximity Networks A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 4. Email Correspondence 2. Telephone Calls 1. Twitter/Social Networks Examples: 3. Proximity Networks A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 4. Email Correspondence 2. Telephone Calls 1. Twitter/Social Networks Examples: 3. Proximity Networks A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 4. Email Correspondence 2. Telephone Calls 1. Twitter/Social Networks Examples: 3. Proximity Networks A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 3. Proximity Networks 2. Telephone Calls 1. Twitter/Social Networks Examples: 4. Email Correspondence A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5 ( A k ) T k =1
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 3. Proximity Networks 2. Telephone Calls 1. Twitter/Social Networks Examples: 4. Email Correspondence A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5 ( A k ) T k =1
DOOCN Satellite, CCS18 duration Andrew Mellor Collective Behaviour in Temporal Networks 3. Adjacency tensors 2. Time-node graphs 1. Adjacency matrices Other Representations: 3. Proximity Networks 2. Telephone Calls 1. Twitter/Social Networks Examples: 4. Email Correspondence A sequence of temporal events , time target source TEMPORAL NETWORKS 2 e i = ( u i , v i , t i , δ i ) ���� ���� ���� ���� 6 C E F 7 2 15 1 D A B 5 ( A k ) T k =1
DOOCN Satellite, CCS18 time Not all interactions are pairwise, or dyadic. In these cases we can consider temporal hyper-events , Andrew Mellor Collective Behaviour in Temporal Networks sources hyper-events can also be defined). Here a set of sources can interact with a set of targets (undirected targets duration HYPER-EVENTS 3 e i = ( U i ) , V i , t i , δ i ���� ���� ���� ���� 8 F C H 6 1 12 A E 6 1 3 10 G B 3 10 D
DOOCN Satellite, CCS18 has no explicit dependence on the set of events then it is denoted . Andrew Mellor Collective Behaviour in Temporal Networks is the inter-event time. where This amounts to using a weighted joining function only that edges are weighted by the time between the two events. The weighted event graph is topologically equivalent to the event graph Weighted Event Graph: If function which prescribes the edges of the graph. is a binary is a set of temporal events, and where is a directed static graph given by the tuple An event graph Event Graph: set of temporal events. EVENT GRAPH 4 Consider a temporal network G T = ( V, T, E ) where E ⊆ V 2 × T is the
DOOCN Satellite, CCS18 The weighted event graph is topologically equivalent to the event graph Andrew Mellor Collective Behaviour in Temporal Networks is the inter-event time. where This amounts to using a weighted joining function only that edges are weighted by the time between the two events. Weighted Event Graph: function which prescribes the edges of the graph. Event Graph: set of temporal events. EVENT GRAPH 4 Consider a temporal network G T = ( V, T, E ) where E ⊆ V 2 × T is the An event graph G is a directed static graph given by the tuple G = ( E, f E ) where E is a set of temporal events, and f E : E × E → [0 , 1] is a binary If f E has no explicit dependence on the set of events then it is denoted f .
DOOCN Satellite, CCS18 The weighted event graph is topologically equivalent to the event graph Andrew Mellor Collective Behaviour in Temporal Networks This amounts to using a weighted joining function only that edges are weighted by the time between the two events. Weighted Event Graph: function which prescribes the edges of the graph. Event Graph: set of temporal events. EVENT GRAPH 4 Consider a temporal network G T = ( V, T, E ) where E ⊆ V 2 × T is the An event graph G is a directed static graph given by the tuple G = ( E, f E ) where E is a set of temporal events, and f E : E × E → [0 , 1] is a binary If f E has no explicit dependence on the set of events then it is denoted f . f τ ( e i , e j ) = τ ij f ( e i , e j ) , where τ ij is the inter-event time.
DOOCN Satellite, CCS18 Collective Behaviour in Temporal Networks Andrew Mellor EVENT GRAPH 5 6 C E F 7 2 15 1 D A B 5 AB-1 AB-1 AB-1 14 14 14 1 1 4 3 3 AC-2 DA-5 BF-15 AC-2 DA-5 BF-15 AC-2 DA-5 BF-15 4 5 4 4 1 1 1 CE-6 EC-7 CE-6 EC-7 CE-6 EC-7 (a) ∆ t -adjacency (b) Node-subsequent (c) Walk-forming ∆ t -adjacency ∆ t -adjacency
DOOCN Satellite, CCS18 Collective Behaviour in Temporal Networks Andrew Mellor EVENT GRAPHS FOR HYPER-EVENTS 6 A:BC-1 8 5 7 2 F C H 6 1 12 GF:A-6 B:ED-3 C:H-8 A E 6 1 3 2 7 10 G B 3 D:BE-10 H:C-10 10 D (b) Node-subsequent (a) Temporal Hypergraph ∆ t -adjacent event graph
DOOCN Satellite, CCS18 Collective Behaviour in Temporal Networks Andrew Mellor DECOMPOSITION 7 6 Temporal Network γ δ ζ 11 (a) 13 3 2 , 8 5 15 4 1 α β ϵ ∆ t = 4 Components 6 γ γ ζ δ δ 11 (b) 13 3 2 , 8 15 5 4 1 α β α β ϵ 3 Temporal Barcode Component 2 (c) 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Time
DOOCN Satellite, CCS18 Reciprocal Andrew Mellor Collective Behaviour in Temporal Networks Red events occur before blue events. sequential Non ABCA Broadcasting ABAC passing Message ABBC Receiving ABCB ABBA Repeated ABAB possible (3, 4 events). Higher-order motifs are phonecalls/SMS). tweets/retweets or motifs (e.g. distinguish between We can incorporate coloured particular behaviour. Each motif is associated with a 6 node patterns, or motifs. Two adjacent events can have one of patterns observed in the network. Temporal motifs are repeatable TEMPORAL MOTIFS AND BEHAVIOUR 8
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