Link Streams Matthieu Latapy complexnetworks.fr Analysis of Temporal Interactions Context with Approach Basics Link Streams and Stream Graphs Degrees Density Paths Matthieu Latapy , Tiphaine Viard, Clémence Magnien Further http://complexnetworks.fr latapy@complexnetworks.fr LIP6 – CNRS and Sorbonne Université Paris, France 1/23
Link Streams interactions over time Matthieu Latapy complexnetworks.fr a Context b Approach c Basics Degrees d Density 0 2 4 6 8 time Paths Further • a , b , c , and d for 10 time units 2/23
Link Streams interactions over time Matthieu Latapy complexnetworks.fr a Context b Approach c Basics Degrees d Density 0 2 4 6 8 time Paths Further • a , b , c , and d for 10 time units • a always present, b leaves from 4 to 5, c present from 4 to 9, d from 1 to 3 2/23
Link Streams interactions over time Matthieu Latapy complexnetworks.fr a Context b Approach c Basics Degrees d Density 0 2 4 6 8 time Paths Further • a , b , c , and d for 10 time units • a always present, b leaves from 4 to 5, c present from 4 to 9, d from 1 to 3 • a and b interact from 1 to 3 and from 7 to 8; b and c from 6 to 9; b and d from 2 to 3. 2/23
Link Streams interactions over time Matthieu Latapy complexnetworks.fr a Context b Approach c Basics Degrees d Density 0 2 4 6 8 time Paths Further • a , b , c , and d for 10 time units • a always present, b leaves from 4 to 5, c present from 4 to 9, d from 1 to 3 • a and b interact from 1 to 3 and from 7 to 8; b and c from 6 to 9; b and d from 2 to 3. e.g., social interactions, network traffic, money transfers, chemical reactions, etc. 2/23
Link Streams interactions over time Matthieu Latapy complexnetworks.fr a Context b Approach c Basics Degrees d Density 0 2 4 6 8 time Paths Further • a , b , c , and d for 10 time units • a always present, b leaves from 4 to 5, c present from 4 to 9, d from 1 to 3 • a and b interact from 1 to 3 and from 7 to 8; b and c from 6 to 9; b and d from 2 to 3. e.g., social interactions, network traffic, money transfers, chemical reactions, etc. how to describe such data? 2/23
Link Streams structure or dynamics Matthieu Latapy complexnetworks.fr a Context b graph theory Approach c network science d Basics − → structure e Degrees f 0 2 4 6 8 10 12 14 16 18 20 22 time Density Paths Further signal analysis, time series − → dynamics 3/23
Link Streams structure and dynamics? Matthieu Latapy complexnetworks.fr a Context b Approach c d Basics e Degrees f 0 2 4 6 8 10 12 14 16 18 20 22 time Density Paths Further time slices → graph sequence 3/23
Link Streams structure and dynamics? Matthieu Latapy complexnetworks.fr a Context b graph theory Approach c network science d Basics − → structure e Degrees f 0 2 4 6 8 10 12 14 16 18 20 22 time Density Paths Further time slices signal analysis, time series − → dynamics → graph sequence information loss what slices? graph sequences? 3/23
Link Streams structure and dynamics Matthieu Latapy complexnetworks.fr a Context b c Approach d Basics e Degrees f 0 2 4 6 8 10 12 14 16 18 20 22 time Density ... Paths MAG / temporal graphs 4−8 12−16 2−6 18−20 Further 4−6 TVG 10−12 22−24 8−10 12−14 6−8 20−22 12−22 0−4 10−12 20−24 lossless but graph-oriented + ad-hoc properties (mostly path-related) + contact sequences + relational event models + ... 4/23
Link Streams structure and dynamics Matthieu Latapy complexnetworks.fr a Context b c Approach d Basics e Degrees f 0 2 4 6 8 10 12 14 16 18 20 22 time Density ... Paths MAG / temporal graphs 4−8 12−16 2−6 18−20 Further 4−6 TVG 10−12 22−24 8−10 12−14 6−8 20−22 12−22 0−4 10−12 20−24 lossless but graph-oriented + ad-hoc properties (mostly path-related) + contact sequences + relational event models + ... 4/23
Link Streams what we propose Matthieu Latapy complexnetworks.fr deal with the stream directly Context Approach Basics stream graphs and link streams Degrees Density a Paths b graph theory c Further network science d e f 0 2 4 6 8 10 12 14 16 18 20 22 time signal analysis, time series wanted features: simple and intuitive, comprehensive, time-node consistent, generalizes graphs/signal 5/23
Link Streams what we propose Matthieu Latapy complexnetworks.fr deal with the stream directly Context Approach Basics stream graphs and link streams Degrees Density a Paths b graph theory c Further network science d e f 0 2 4 6 8 10 12 14 16 18 20 22 time signal analysis, time series wanted features: simple and intuitive, comprehensive, time-node consistent, generalizes graphs/signal 5/23
Link Streams graph-equivalent streams Matthieu Latapy complexnetworks.fr stream with no dynamics: Context nodes always present, ⇐ ⇒ graph Approach Basics either always or never linked Degrees a Density b Paths c Further ⇐ ⇒ d e 0 2 4 6 8 time 6/23
Link Streams graph-equivalent streams Matthieu Latapy complexnetworks.fr stream with no dynamics: Context nodes always present, ⇐ ⇒ graph Approach Basics either always or never linked Degrees a Density b Paths c Further ⇐ ⇒ d e 0 2 4 6 8 time stream properties = graph properties ֒ → generalizes graph theory 6/23
Link Streams our approach Matthieu Latapy complexnetworks.fr very careful generalization of the most basic concepts Context stream graphs and link streams Approach numbers of nodes and links Basics clusters and induced sub-streams Degrees density and paths Density Paths Further → buliding blocks for higher-level concepts ֒ neighborhood and degrees clustering coefficient betweenness centrality many others + ensure consistency with graph theory + ensure classical relations are preserved 7/23
Link Streams definition of stream graphs Matthieu Latapy complexnetworks.fr Graph G = ( V , E ) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked Context Approach Basics Stream graph S = ( T , V , W , E ) Degrees T : time interval, V : node set Density W ⊆ T × V , E ⊆ T × V ⊗ V Paths Further ( t , v ) ∈ W ⇔ v is present at time t T v = { t , ( t , v ) ∈ W } ( t , uv ) ∈ E ⇔ u and v are linked at time t T uv = { t , ( t , uv ) ∈ E } ( t , uv ) ∈ E requires ( t , u ) ∈ W and ( t , v ) ∈ W i.e. T uv ⊆ T u ∩ T v 8/23
Link Streams definition of stream graphs Matthieu Latapy complexnetworks.fr Graph G = ( V , E ) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked Context Approach Basics Stream graph S = ( T , V , W , E ) Degrees T : time interval, V : node set Density W ⊆ T × V , E ⊆ T × V ⊗ V Paths Further ( t , v ) ∈ W ⇔ v is present at time t T v = { t , ( t , v ) ∈ W } ( t , uv ) ∈ E ⇔ u and v are linked at time t T uv = { t , ( t , uv ) ∈ E } ( t , uv ) ∈ E requires ( t , u ) ∈ W and ( t , v ) ∈ W i.e. T uv ⊆ T u ∩ T v 8/23
Link Streams definition of stream graphs Matthieu Latapy complexnetworks.fr Graph G = ( V , E ) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked Context Approach Basics Stream graph S = ( T , V , W , E ) Degrees T : time interval, V : node set Density W ⊆ T × V , E ⊆ T × V ⊗ V Paths Further ( t , v ) ∈ W ⇔ v is present at time t T v = { t , ( t , v ) ∈ W } ( t , uv ) ∈ E ⇔ u and v are linked at time t T uv = { t , ( t , uv ) ∈ E } ( t , uv ) ∈ E requires ( t , u ) ∈ W and ( t , v ) ∈ W i.e. T uv ⊆ T u ∩ T v 8/23
Link Streams definition of stream graphs Matthieu Latapy complexnetworks.fr Graph G = ( V , E ) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked Context Approach Basics Stream graph S = ( T , V , W , E ) Degrees T : time interval, V : node set Density W ⊆ T × V , E ⊆ T × V ⊗ V Paths Further ( t , v ) ∈ W ⇔ v is present at time t T v = { t , ( t , v ) ∈ W } ( t , uv ) ∈ E ⇔ u and v are linked at time t T uv = { t , ( t , uv ) ∈ E } ( t , uv ) ∈ E requires ( t , u ) ∈ W and ( t , v ) ∈ W i.e. T uv ⊆ T u ∩ T v 8/23
Link Streams definition of stream graphs Matthieu Latapy complexnetworks.fr Graph G = ( V , E ) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked Context Approach Basics Stream graph S = ( T , V , W , E ) Degrees T : time interval, V : node set Density W ⊆ T × V , E ⊆ T × V ⊗ V Paths Further ( t , v ) ∈ W ⇔ v is present at time t T v = { t , ( t , v ) ∈ W } ( t , uv ) ∈ E ⇔ u and v are linked at time t T uv = { t , ( t , uv ) ∈ E } ( t , uv ) ∈ E requires ( t , u ) ∈ W and ( t , v ) ∈ W i.e. T uv ⊆ T u ∩ T v 8/23
Link Streams an example Matthieu Latapy complexnetworks.fr a b Context Approach c Basics d Degrees 0 2 4 6 8 time Density Paths V = { a , b , c , d } T = [ 0 , 10 ] Further W = T × { a } ∪ ([ 0 , 4 ] ∪ [ 5 , 10 ]) × { b } ∪ [ 4 , 9 ] × { c } ∪ [ 1 , 3 ] × { d } T b = [ 0 , 4 ] ∪ [ 5 , 10 ] T a = T T c = [ 4 , 9 ] T d = [ 1 , 3 ] E = ([ 1 , 3 ] ∪ [ 7 , 8 ]) × { ab } ∪ [ 6 , 9 ] × { bc } ∪ [ 2 , 3 ] × { bd } T ab = [ 1 , 3 ] ∪ [ 7 , 8 ] T ad = ∅ T bc = [ 6 , 9 ] T bd = [ 2 , 3 ] 9/23
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