Analysis of Temporal Interactions Context with Approach Basics - - PowerPoint PPT Presentation

SLIDE 1

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

1/23

Analysis of Temporal Interactions with Link Streams and Stream Graphs

Matthieu Latapy, Tiphaine Viard, Clémence Magnien

http://complexnetworks.fr latapy@complexnetworks.fr LIP6 – CNRS and Sorbonne Université Paris, France

SLIDE 2

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

2/23

interactions over time

a b c d

time 2 4 6 8

• a, b, c, and d for 10 time units

SLIDE 3

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

2/23

interactions over time

a b c d

time 2 4 6 8

• 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

SLIDE 4

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

2/23

interactions over time

a b c d

time 2 4 6 8

• 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.

SLIDE 5

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

2/23

interactions over time

a b c d

time 2 4 6 8

• 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.

SLIDE 6

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

2/23

interactions over time

a b c d

time 2 4 6 8

• 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?

SLIDE 7

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

3/23

structure or dynamics

2 4 6 8 10 12 14 16 18 20 22 time

a b c d e f

graph theory network science − → structure signal analysis, time series − → dynamics

SLIDE 8

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

3/23

structure and dynamics?

2 4 6 8 10 12 14 16 18 20 22 time

a b c d e f

time slices → graph sequence

SLIDE 9

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

3/23

structure and dynamics?

2 4 6 8 10 12 14 16 18 20 22 time

a b c d e f

graph theory network science − → structure signal analysis, time series − → dynamics time slices → graph sequence

information loss what slices? graph sequences?

SLIDE 10

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

4/23

structure and dynamics

2 4 6 8 10 12 14 16 18 20 22 time

20−22 2−6 18−20 8−10 12−14 6−8 12−22 10−12 22−24 4−6 4−8 12−16 20−24 0−4 10−12

a b c d e f MAG / temporal graphs TVG

...

lossless but graph-oriented + ad-hoc properties (mostly path-related) + contact sequences + relational event models + ...

SLIDE 11

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

4/23

structure and dynamics

2 4 6 8 10 12 14 16 18 20 22 time

20−22 2−6 18−20 8−10 12−14 6−8 12−22 10−12 22−24 4−6 4−8 12−16 20−24 0−4 10−12

a b c d e f MAG / temporal graphs TVG

...

lossless but graph-oriented + ad-hoc properties (mostly path-related) + contact sequences + relational event models + ...

SLIDE 12

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

5/23

what we propose

deal with the stream directly

stream graphs and link streams

2 4 6 8 10 12 14 16 18 20 22 time

a b c d e f

graph theory network science signal analysis, time series

wanted features: simple and intuitive, comprehensive, time-node consistent, generalizes graphs/signal

SLIDE 13

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

5/23

what we propose

deal with the stream directly

stream graphs and link streams

2 4 6 8 10 12 14 16 18 20 22 time

a b c d e f

graph theory network science signal analysis, time series

wanted features: simple and intuitive, comprehensive, time-node consistent, generalizes graphs/signal

SLIDE 14

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

6/23

graph-equivalent streams

stream with no dynamics: nodes always present, either always or never linked ⇐ ⇒ graph a b c d e

2 4 6 8 time

⇐ ⇒

SLIDE 15

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

6/23

graph-equivalent streams

stream with no dynamics: nodes always present, either always or never linked ⇐ ⇒ graph a b c d e

2 4 6 8 time

⇐ ⇒ stream properties = graph properties ֒ → generalizes graph theory

SLIDE 16

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

7/23

• ur approach

very careful generalization of the most basic concepts stream graphs and link streams numbers of nodes and links clusters and induced sub-streams density and paths ֒ → 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

SLIDE 17

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

8/23

definition of stream graphs

Graph G = (V, E) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked

Stream graph S = (T, V, W, E) T: time interval, V: node set W ⊆ T × V, E ⊆ T × V ⊗ V (t, v) ∈ W ⇔ v is present at time t Tv = {t, (t, v) ∈ W} (t, uv) ∈ E ⇔ u and v are linked at time t Tuv = {t, (t, uv) ∈ E} (t, uv) ∈ E requires (t, u) ∈ W and (t, v) ∈ W i.e. Tuv ⊆ Tu ∩ Tv

SLIDE 18

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

8/23

definition of stream graphs

Graph G = (V, E) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked

Stream graph S = (T, V, W, E) T: time interval, V: node set W ⊆ T × V, E ⊆ T × V ⊗ V (t, v) ∈ W ⇔ v is present at time t Tv = {t, (t, v) ∈ W} (t, uv) ∈ E ⇔ u and v are linked at time t Tuv = {t, (t, uv) ∈ E} (t, uv) ∈ E requires (t, u) ∈ W and (t, v) ∈ W i.e. Tuv ⊆ Tu ∩ Tv

SLIDE 19

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

8/23

definition of stream graphs

Graph G = (V, E) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked

Stream graph S = (T, V, W, E) T: time interval, V: node set W ⊆ T × V, E ⊆ T × V ⊗ V (t, v) ∈ W ⇔ v is present at time t Tv = {t, (t, v) ∈ W} (t, uv) ∈ E ⇔ u and v are linked at time t Tuv = {t, (t, uv) ∈ E} (t, uv) ∈ E requires (t, u) ∈ W and (t, v) ∈ W i.e. Tuv ⊆ Tu ∩ Tv

SLIDE 20

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

8/23

definition of stream graphs

Graph G = (V, E) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked

Stream graph S = (T, V, W, E) T: time interval, V: node set W ⊆ T × V, E ⊆ T × V ⊗ V (t, v) ∈ W ⇔ v is present at time t Tv = {t, (t, v) ∈ W} (t, uv) ∈ E ⇔ u and v are linked at time t Tuv = {t, (t, uv) ∈ E} (t, uv) ∈ E requires (t, u) ∈ W and (t, v) ∈ W i.e. Tuv ⊆ Tu ∩ Tv

SLIDE 21

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

8/23

definition of stream graphs

Graph G = (V, E) with E ⊆ V ⊗ V uv ∈ E ⇔ u and v are linked

Stream graph S = (T, V, W, E) T: time interval, V: node set W ⊆ T × V, E ⊆ T × V ⊗ V (t, v) ∈ W ⇔ v is present at time t Tv = {t, (t, v) ∈ W} (t, uv) ∈ E ⇔ u and v are linked at time t Tuv = {t, (t, uv) ∈ E} (t, uv) ∈ E requires (t, u) ∈ W and (t, v) ∈ W i.e. Tuv ⊆ Tu ∩ Tv

SLIDE 22

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

9/23

an example

a b c d

time 2 4 6 8

T = [0, 10] V = {a, b, c, d} W = T × {a} ∪ ([0, 4] ∪ [5, 10]) × {b} ∪ [4, 9] × {c} ∪ [1, 3] × {d} Ta = T Tb = [0, 4] ∪ [5, 10] Tc = [4, 9] Td = [1, 3] E = ([1, 3] ∪ [7, 8]) × {ab} ∪ [6, 9] × {bc} ∪ [2, 3] × {bd} Tab = [1, 3] ∪ [7, 8] Tbc = [6, 9] Tbd = [2, 3] Tad = ∅

SLIDE 23

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

10/23

a few remarks

works with... discrete time, continuous time, instantaneous interactions or with durations, directed, weighted, bipartite... if ∀v, Tv = T then S ∼ L = (T, V, E) is a link stream if ∀u, v, Tuv ∈ {T, ∅} then S ∼ G = (V, E) is a graph-equivalent stream

SLIDE 24

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

11/23

size of a stream graph

How many nodes? How many links? a b c d

time 2 4 6 8

|Ta| = 10 = |Td| = 2

SLIDE 25

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

11/23

size of a stream graph

How many nodes? How many links? a b c d

time 2 4 6 8

|Ta| = 10 = |Td| = 2 n =

v∈V |Tv| |T|

n = |Ta|

10 + |Tb| 10 + |Tc| 10 + |Td| 10 = 1 + 0.9 + 0.5 + 0.2 = 2.6 nodes

SLIDE 26

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

11/23

size of a stream graph

How many nodes? How many links? a b c d

time 2 4 6 8

|Ta| = 10 = |Td| = 2 n =

v∈V |Tv| |T|

m =

uv∈V⊗V |Tuv| |T|

n = |Ta|

10 + |Tb| 10 + |Tc| 10 + |Td| 10 = 1 + 0.9 + 0.5 + 0.2 = 2.6 nodes

m = |Tab|

10 + |Tbc| 10 + |Tbd| 10 = 0.3 + 0.3 + 0.1 = 0.7 links

SLIDE 27

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

12/23

clusters, sub-streams

Cluster in G = (V, E): a subset of V. Cluster in S = (T, V, W, E): a subset of W ⊆ T × V. a b c d

time 2 4 6 8

C = [0, 2] × {a} ∪ ([0, 2] ∪ [6, 10]) × {b} ∪ [4, 8] × {c} S(C) sub-stream induced by C S(C) = (T, V, C, EC) ֒ → properties of (sub-streams induced by) clusters

SLIDE 28

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

12/23

clusters, sub-streams

Cluster in G = (V, E): a subset of V. Cluster in S = (T, V, W, E): a subset of W ⊆ T × V. a b c d

time 2 4 6 8

C = [0, 2] × {a} ∪ ([0, 2] ∪ [6, 10]) × {b} ∪ [4, 8] × {c} S(C) sub-stream induced by C S(C) = (T, V, C, EC) ֒ → properties of (sub-streams induced by) clusters

SLIDE 29

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

13/23

neighborhood

in G = (V, E): N(v) = {u, uv ∈ E} in S = (T, V, W, E): N(v) = {(t, u), (t, uv) ∈ E}

a b c d

2 4 6 8 time

N(d) = ([2, 3] ∪ [5, 10]) × {b} ∪ [5.5, 9] × {c} N(v) is a cluster

SLIDE 30

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

14/23

degree

in G and in S: d(v) is the size of N(v)

a b c d

2 4 6 8 time

N(d) = ([2, 3] ∪ [5, 10]) × {b} ∪ [5.5, 9] × {c} d(d) = |[2,3]∪[5,10]|

10

+ |[5.5,9]|

10

= 0.6 + 0.35 = 0.95

• degree distribution, average degree, etc
• if graph-equivalent stream then graph degree
• relation with n and m

SLIDE 31

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

15/23

density

in G: proba two random nodes are linked δ(G) =

nb links nb possible links

=

2·m n·(n−1) random ?

in S: proba two random nodes are linked at a random time instant δ(S) =

nb links nb possible links

=

• uv∈V⊗V |Tuv|
• uv∈V⊗V |Tu∩Tv|

SLIDE 32

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

15/23

density

in G: proba two random nodes are linked δ(G) =

nb links nb possible links

=

2·m n·(n−1) random ?

in S: proba two random nodes are linked at a random time instant

? random

δ(S) =

nb links nb possible links

=

• uv∈V⊗V |Tuv|
• uv∈V⊗V |Tu∩Tv|
• if graph-equivalent stream then graph density
• relation with n, m, and average degree

SLIDE 33

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

16/23

cliques

in G: sub-graph of density 1 all nodes are linked together

in S: sub-stream of density 1 all nodes interact all the time time a b c d

2 6 4 8

SLIDE 34

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

17/23

clustering coefficient

in G and in S: density of the neighborhood cc(v) = δ(N(v))

a b c d

2 4 6 8 time

N(d) = ([2, 3] ∪ [5, 10]) × {b} ∪ [5.5, 9] × {c}

SLIDE 35

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

17/23

clustering coefficient

in G and in S: density of the neighborhood cc(v) = δ(N(v))

a b c d

2 4 6 8 time

N(d) = ([2, 3] ∪ [5, 10]) × {b} ∪ [5.5, 9] × {c} cc(d) = δ(N(d)) =

|[6,9]| |[5.5,9]| = 6 7

SLIDE 36

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

18/23

paths

in G:

d c b a

from a to d: (a, b), (b, c), (c, d) length: 3 → shortest paths

in S: from (1, d) to (9, c): (2, d, b), (3, b, a), (7.5, a, b), (8, b, c) length: 4 duration: 6 → shortest paths → fastest paths

SLIDE 37

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

18/23

paths

in G:

a b d c

from a to d: (a, b), (b, c), (c, d) length: 3 → shortest paths

in S: a b c d

2 4 6 8 time

from (1, d) to (9, c): (2, d, b), (3, b, a), (7.5, a, b), (8, b, c) length: 4 duration: 6 → shortest paths → fastest paths

SLIDE 38

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

19/23

betweenness centrality

in G: b(v) = fraction of shortest paths from any u to any w in V that involve v in S: b(t, v) = fraction of shortest fastest paths from any (i, u) to any (j, w) in W that involve (t, v)

shortest paths

SLIDE 39

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

19/23

betweenness centrality

in G: b(v) = fraction of shortest paths from any u to any w in V that involve v in S: b(t, v) = fraction of shortest fastest paths from any (i, u) to any (j, w) in W that involve (t, v)

shortest paths shortest fastest paths

SLIDE 40

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

20/23

many other concepts

a b c d

2 4 6 8 time

a b c d

2 4 6 8 10 time

a b c d e

2 4 6 8 time

a b c d

2 4 6 8 time

a b c d e

2 4 6 8 time

a b c d

2 4 6 8 time

A B C

2 4 6 8 time

a b c d

2 4 6 8 time

a b c d

2 4 6 8 time

a b c

1 2 time

a b c d

2 4 6 8 time

a b c

2 4 6 8 time

a u b v c

2 4 6 8 time

a b c d e f g

2 4 6 8 time

a b c

2 4 6 8 time

u v w

1 2 3 time

u x v y w

2 4 6 time

u v w

1 2 time

a b c

1 time

a b c d

2 4 6 8 time

a b c d

2 4 6 8 time

a b c d

2 4 6 8 time

a b c d

2 4 6 8 time

a b c d

2 4 6 8 time

a u b v c

2 4 6 8 time

u v w

2 4 6 8 time

a b c d

2 4 6 8 time

a b c

1 2 time

a b c d

2 4 6 8 time

a b c

2 4 6 8 time

∆ = 2

a b c d

2 4 6 8 time

a b c

1 time

ab bc ac

2 4 6 8 time

u x v y w

2 7 10 15 a b c d e f g h time

a b c

2 4 6 8 time

a b c d

2 4 6 8 time

a b c d

2 4 6 8 time

a b c d

2 4 6 8 time

a b c d

2 4 6 8 time

a b c

1 3 5 7 time

a b c d

2 4 6 8 time

arxiv preprint

SLIDE 41

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

21/23

relations vs interactions

graph/networks = relations (like friendship) dynamic graphs/networks = evolution of relations (like new friends) stream graphs / link streams = interactions (like face-to-face contacts) interactions = traces/realization of relations? link streams = traces of graphs/networks? relations = consequences of interactions? graphs/networks = traces of link streams?

SLIDE 42

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

22/23

conclusion

we provide a language (set of concepts) that:

• makes it easy to deal with interaction traces,
• combines structure and dynamics in a consistent way,
• generalizes graphs / networks ; signals / time series ?
• meets classical and new algorithmic challenges,
• opens new perspectives for data analysis,
• clarifies the interplay interactions ←

→ relations. studies in progress: internet traffic, financial transactions, mobility/contacts, mailing-lists, sales, etc.

SLIDE 43

Link Streams Matthieu Latapy

complexnetworks.fr

Context Approach Basics Degrees Density Paths Further

23/23

calls for papers

special issues of international journals Theoretical Computer Science (TCS) Link Streams: models and algorithms Computer Networks Link Streams: methods and case studies deadline: July 15th http://link-streams.com