Weg2Vec: Event Embedding for Temporal Networks Márton Karsai
Temporal Networks (a) (b) (c) • Interactions between entities are not present always but varying in time (Holme, Saramaki 2012) • Calls, SMS, f2f, @mentions, collaborations, transportation networks…
Representation of temporal networks a 1. Temporal Graph : G t = ( V, E, T e ) b • V : set of vertices c a • E : set of edges d • T e ={t 1 , t 2,…, t n } : set of times when edge e is active
Representation of temporal networks a 1. Temporal Graph : G t = ( V, E, T e ) b • V : set of vertices c a • E : set of edges d • T e ={t 1 , t 2,…, t n } : set of times when edge e is active 2. Contact sequence (Similar representation is called link streams, (Latapy et al. 2018) t 1 a b Y A e t 2 a c E ⊂ T × V × V ( × i × L ) t 4 e f i where t 10 f a • T is the set of time stamps … • V is the set of interacting entities • A e are event attribute set e.g. duration, cost, etc, • L is a location set • sequence of events ev ∈ E ev ( t , u beg , v end , a 1 , a 2 , , . . . , loc beg , loc end )
Representation of temporal networks a 1. Temporal Graph : G t = ( V, E, T e ) b • V : set of vertices c a • E : set of edges d • T e ={t 1 , t 2,…, t n } : set of times when edge e is active 2. Contact sequence (Similar representation is called link streams, (Latapy et al. 2018) t 1 a b Y A e t 2 E ⊂ T × V × V ( × i × L ) a c t 4 e f i t 10 f a 3. Graphlet or snapshot representation … • Set of graphs representing aggregated interactions happening at the same time or interval • Can be represented as a dynamic adjacency matrix A ij (t) • Can be represented as a multiplex network e t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7
Representation of temporal networks a 1. Temporal Graph : G t = ( V, E, T e ) b • V : set of vertices c a • E : set of edges Computational difficulties: d • T e ={t 1 , t 2,…, t n } : set of times when edge e is active • Expensive to measure temporal centralities and similarities 2. Contact sequence • any node/link character vary in time (Similar representation is called link streams, (Latapy et al. 2018) t 1 a b • Expensive to compute time respecting paths Y A e t 2 E ⊂ T × V × V ( × i × L ) a c • depends on time and seed t 4 e f i • Expensive to detect causal correlations t 10 f a 3. Graphlet or snapshot representation • interactions are not independent but form local correlated patterns … • Set of graphs representing aggregated interactions happening at • Expensive to simulate dynamical/epidemic processes the same time or interval • They must be seeded from every time point and every node • Can be represented as a dynamic adjacency matrix A ij (t) • Can be represented as a multiplex network e t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7
Temporal Event Graphs
Time-respecting paths • Temporal equivalent of topological path in static graphs • Consider a temporal contact network (for simplicity without durations) • Any path between node has to respect the timing and ordering of events! Definition • Time-respecting path between node a and b is a set of events {( a , v , t 1 ), ( v , w , t 2 ), . . . , ( y , b , t n )} such that t 1 <t 2 <…<t n and consecutive events are adjacent (i.e. time ordered and share at least one node)
Time-respecting paths • Temporal equivalent of topological path in static graphs • Consider a temporal contact network (for simplicity without durations) • Any path between node has to respect the timing and ordering of events! Definition b b j j 2, 8 • Time-respecting path between node a and b is a set of 3, 7 7 events 1 i i k k {( a , v , t 1 ), ( v , w , t 2 ), . . . , ( y , b , t n )} 11 5, 9 1, 2 such that t 1 <t 2 <…<t n and consecutive events are adjacent 6 8, 10 l l (i.e. time ordered and share at least one node) 2, 4 a m a m t=2 t= ∞ static path temporal path Properties • Reachable set of nodes are limited • No reciprocity: the existence of the path a-b does not guarantee the existence of a b-a path • No transitivity: the existence of the path a-b and b-c does not guarantee that there is a a-b-c path • Time dependency: paths begin and end at certain times; if there is a path a-b that begins at t , this doesn’t guarantee a path at t’>t • They determine the spread of information thus the outcome of any collective phenomena
Weighted event graphs Temporal networks ork G = ( V, E, T ) ents E ⇢ V ⇥ V ⇥ [0 , T ] with events , we allow no self-edges Adjacent events e = ( a , b , t ) • Share at least one common node that e ! e 0 Events are adjacent if and t < t 0 . e ′ = ( b , c , t ′ ) • Temporal network Weighted event graphs Kivelä, Cambe, Saramäki, Karsai , Sci. Rep. (2018) representation of a Mellor J. Complex Netw. (2017). b D = ( E, E D , w ) t where = t=1 6 e’ e • nodes are events in G in and the edges a c w=5 in e D 2 E D ents e D = e ! e 0 • links are adjacent events 0 with weights as w ( e D ) = t 0 � t . Weighted even graph • weights are paths in the network. • δ t threshold for weights: keeping adjacent events which are closer in time than δ t • Static and lossless representation of all temporal and structural information • It is a weighted directed acyclic graph (DAG) • Superposition of every ( δ t-connected) time-respecting paths • Its connectedness determines the outcome of any dynamical process
Weg2Vec: Event Embedding for Temporal Networks
Temporal network embedding • Learn low-dimensional representations • Capture temporal and structural regularities in the network • Various applications: node classification, link prediction... �������������������������� �������������������������� … or the prediction of spreading outcome ������������������������������������������������������� ������������������������������������������������������� ���������������������������������������������� ���������������������������������������������� �������������� ��������������
Weg2Vec pipeline ���������������� ��������� ������� �������������������������� ��������� ������ ���������� ������� �������������� ���� ������������������ ���������� �������������������� ���� �������� ����������������������������������� ����������������������������������
Weg2Vec pipeline ���������������� ���������������� ���������������� ��������� ������ ������� �������������� ���������������� ��������� ������ ������� �������������� ������������������ �������������������� Event embedding ��������� ������ ��������� ������ ������� �������������� ������������������ �������������������� ������� �������������� ������������������ �������������������� �������� ������������������� Event graph representation ���������������������� Path (temporal) weight: • ����������������������������������� 1 ���������������������������������� w path ( e k , e l ) = 1 + | t k − t l | ����������������������������������� ����������������������������������� ���������������������������������� ����������������������������������
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