jános kertész central european university pisa summer
TRANSCRIPT
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Dynamics of Networks
Pisa Summer School September 2019
János KertészCentral European University
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Mark Granovetter
The most pressing need for further
development of network ideas is a
move away from static analyses that observe a
system at one point in time and to pursue
instead systematic accounts of how such
systems develop and change. Only by careful
attention to this dynamic problem can social
network analysis fulfill its promise as a powerful
instrument in the analysis of social life.1983
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Dynamics on and of Networks
• Dynamic processes on networks - Diffusion, random walk- Transport- Packet transfer according to protocol- Synchronization- Spreading
• Dynamics of networks- Network growth and development- Network shrinkage and collapse- Network restructuring, network adaptation- Temporal networks
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Dynamics on Networks: Diffusion, random walk Example: PageRank
PR is an iterative procedure to determine the importance of web pages based on random walk
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Transport
htt
p:/
/ww
w.o
ps.
fhw
a.d
ot.
gov/
frei
ght/
Mem
ph
is/
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Packet switching
Information ischopped intopieces (packets),which travel on different routes and get reassembled finally
Circuit switchingFor communication a route has to be established and kept open throughout the exchange of information
www.tcpipguide.com
Packet transfer according to protocol
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Complex electrical circuit
http://www.networx.com/c.rescigno-electric
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Spreading
Medieval spreading of „Black Death”(short range interaction)
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p:/
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isto
ryo
fin
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/
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Spreading
Swine flu June 2009(long range interaction)
Wik
ipe
dia
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Dynamics of networks
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Network growth
See also network models, e.g., Barabási-Albert
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Network restructuring
Group (community) evolution
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Network adaptation
Network restructuring is coupled to an opinion dynamics mechanism
Nodes (people) look for more satisfactory connections.
The resulting community structure reflects the opinions
Inig
uez
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Time scales
In reality processes on the network and restructuring happen simultaneously.
Important: Time scales
If time scales separate, one can treat the dynamic degrees of freedom for the processes on the network separately from those of the network.Similar to the adiabatic approximation for solids.
E.g. road construction vs daily traffic
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No separation of time scales
Reason: • The characteristic times are similar (e.g., if the road is as frequently reconstructed as cars cross the static model of a network is meaningless.)
• There are no characteristic times (e.g., inter-event times are power-law distributed)
Even more so, if the network is defined by the events!E.g.: communication
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Temporal networks
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Aggregate networks
Consider all links over a period of time
Assuming that mobile phone calls represent social contacts, the aggregate network of call events is a proxy for the weighted human interaction network at sociatal level. On
nel
a et
al.
PN
AS
20
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Spreading (of rumor, disease etc.)
1
2
3
Incoming information (1) reaches everyone
Aggregation: information loss
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Spreading (of rumor, disease etc.)
1
2
3Incoming information (1) does not reach
The sequence of calls is crucial for the process
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Network definition
Networks (graphs) are defined aswhere V is the set of nodes (vertices) and E is the set of – possibly directed – links (edges). Given the number N of nodes, the network is uniquely defined by the 𝑁 × 𝑁 adjacency matrix Aij
indicating that there is a link from i to j: Aij = 1 or Aij = 0 otherwise for non-weighted networks.
G = V ,E
Wikipedia
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A temporal network (contact sequence) is defined as 𝒯 = 𝑉, 𝑆 where V is the set of nodes and S is the set of – possibly directed – event sequences assigned to pairs of nodes. For
sij S
sij = tij
(1),ij
(1);tij
(2),ij
(2);...;tij
(n ),ij
(n );...
Temporal network definition
continuous or discrete
A(i, j,t) =1 if i → j connected at t
0 otherwise
adjacency index
where tij-s are the beginnings and τij-s the durations of events i→ j within a time window
τij=0 can often be assumed
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Temporal network visualization
Figures are taken from that review if not indicated otherwise
Holme, Saramaki : Phys. Rep. 519, 97-125 (2012)
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When are temporal networks important?
From each temporal network a (weighted) static network can be constructed by aggregation.
wDC
This can be used to model dynamic phenomena if processes are simple (Poissonian).
Always, if sequence of events is important (spreading) or temporal inhomogeneities matter (jamming).
𝑤𝑖𝑗 = න𝑡min
𝑡max
𝐴 𝑖, 𝑗, 𝑡 𝑑𝑡: 𝑤𝑖𝑗 = # or total duration of events
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Relation to multiplex networks: discrete time
Blue lines are strictly directed
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Consequences of strong temporal inhomogeneities
Temporal behavior is often non-Poissonian, bursty. This may have different reasons from seasonalities to external stimuli and to intrinsic burstiness.
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Examples of temporal networks
• Communication networks• Physical proximity• Gene regulatory networks• Parallel and distributed computing• Neural networks• etc.
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Examples of temporal networks
• Communication networks• Physical proximity• Gene regulatory networks• Parallel and distributed computing• Neural networks• etc.
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Temporal communication networks
• One to one- face to face- phone - SMS- email- chat
• One to many- lecture - multi address SMS- multi address email- twit, blog
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• Many to many- meeting- conference call
IT related communication data are precious: Large in number and accurate
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Examples of temporal networks
• Communication networks• Physical proximity• Gene regulatory networks• Parallel and distributed computing• Neural networks• etc.
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Physical proximity
Human or animal proximity
Important, e.g., for spread of airborne pathogens or mobile phone viruses transmitted via bluetooth
Data: MIT Reality mining (Bluetooth), Barrat group (RFID), OtaSizzle (tower, WiFi), Copenhagen Network Study (CDR, Wi-Fi), traffic (GPS)
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Examples of temporal networks
• Communication networks• Physical proximity• Gene regulatory networks• Parallel and distributed computing• Neural networks• etc.
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Gene regulatory networks
Aggregate NW, in reality: Sequence of chemical reactions. Order pivotal! B
aláz
siet
al.
Sci.
Rep
. 20
11
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Examples of temporal networks
• Communication networks• Physical proximity• Gene regulatory networks• Parallel and distributed computing• Neural networks• etc.
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Parallel and distributed computing
DC: Put all resources together to solve a single task efficiently.
Problems similar toparallel computing,where many processors work simultaneously.
Data transfer: Processes use results of other units – timing is crucial.
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Examples of temporal networks
• Communication networks• Physical proximity• Gene regulatory networks• Parallel and distributed computing• Neural networks• etc.
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Neural networks
Neurons get stimulating or inhibitory impulses from other ones
Output heavily depends on the sequence of the inputs:s1, i1, s2, i2, s3, i3, s4,…. is totally different froms1, s2, s3, s4,…, i1, i2, i3,…
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Characterizing networks
Aggregated networks can be considered as static ones:An arsenal of concepts and measures exist:
- path, distance, diameter- degree- centrality measures- correlations (e.g., assortativity)- components- minimum spanning tree- motifs- communities
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Characterizing temporal networks
Similarities with directed networks – due to the arrow of time.Difference: sequential order matters
Need for generalization of concepts
- path, distance, diameter- centrality measures- components- motifs
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Paths vs reachability
A path in a graph consists of a series of subsequent edges without visiting a node more than once.
P (1,n) = e12,e23,e34 ,...,en−1,n eij E A path from i to j on the aggregate graph does not mean that j is reachable from i.
A
B
C
D
There is a path DA, which is symmetric for undirectedgraphs. A can be reached from D but not D from A.
Like for directed graphs
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Time respecting path (journey)
Time respecting paths define the set of influence of node i within this window:such that all times >t in -s are within the window.
F i(t) = j j V, J i→ j
J i→ j
Similarly, the source set is defined as the set of nodes from which i can be reached by t within the window
P i(t) = j j V, J j → i
Temporal networks should be studied with respect to a time window 𝑡𝑖𝑗 ∈ (𝑡min, 𝑡max).
where 𝑡𝑖𝑗-s are event times and the nodes 1,2, … , 𝑛
form a path in the aggregate network.
𝒥1→𝑛 = 𝑡12, 𝑡23, 𝑡34, … , 𝑡𝑛−1,𝑛|𝑡12 < ⋯ < 𝑡𝑛−1,𝑛 ,
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Journeys are non-transitive: A→B and B→C does not imply A→C.
ℱ𝐵 10 = {𝐴, 𝐶}
𝒫𝐶 5 = {𝐵, 𝐷}
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Journeys with max. waiting times
Similarly, sets of influence and source set can be defined with respect to ∆𝑐. Reachability ratio:
a) Mobile call datachar. time: 1-2db) Air trafficchar. time: 30 min (~transfer time)
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𝒥1→𝑛∆𝑐 = 𝑡12, 𝑡23, 𝑡34, … , 𝑡𝑛−1,𝑛|𝑡12 < ⋯ < 𝑡𝑛−1,𝑛; 𝑡𝑖,𝑖+1 − 𝑡𝑖−1,𝑖 < ∆𝑐
𝑓Finite(∆𝑐) =1
N
𝑘=1
𝑁
ℱ𝑘∆𝑐(𝑡min)
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Connectivity and components
For directed networks: Strongly connectedweakly conn.components
Analogously for temporal graphsWINDOW-DEPENDENCE!
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Shortest paths, fastest journeys
Length l of a path is the number of edges in it.
Distance d(i,j) is the length of the shortest path.
Duration δ(1,n) of a journey is the time Latency λ(i,j) is the duration of the fastest journey.
tn,n−1 − t1,2
l(C,D,B,A)=3
d(C,B,A)=2
δ(C,B,A)=15-8=7λ(C,D,B,A)=3-2+6-3=4
λ strongly depends on the time window
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Mean shortest path, average latencyDefined for a connected component (𝐸 is its edge set)
Generalization to average latency is non-trivial.1. Mean shortest path tells about spatial
reachability, latency is about time2. There are strong variations even for the
average over a single link.
Pan
an
d S
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Temporal boundary cond.
ҧ𝑑 = 1/|𝐸|
(𝑖,𝑗)
𝑑(𝑖, 𝑗)
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Centrality measures I.
Detect importance of elements
Closeness centrality in graphs: inverse average distance from i
CC (i) =N −1
d(i, j)i j
Temporal analogue
CC (i,t) =N −1
t (i, j)i j
t (i, j)where is the latency from i→ j at time t
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Centrality measures II.
Betweenness centrality in graphs: proportional to the number of shortest paths through element where is the number
of shortest paths through i and
i( j,k)
( j,k) = i( j,k)i
Temporal analogue
Possibilities:a) Shortest paths ratio conditioned by reachability
b) Fastest path ratioTemporal BC-s!
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Motifs
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Static motifs
Main task of studying (static) complex network is to understand the relation between topology and function.
Centrality measures try to identify most important elements.
What are the most important groups of elements?
Motif: set topologically equivalent (isomorphic) subgraphs
Cardinality of a motif shows its relevance with respect to a (random) null model.
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If the cardinality of a motif is significantly high, it is expected that the represented subgraphsare relevant for some kind of function.
Relevance of static motifs
If it is small, the related function is irrelevantNull model: Configuration model, no degree-degree correlations. The studied NW is a single sample, the null model is an ensemble leading to distributions in properties.
Measure: z-score
cardinality of motif m in the empirical NWaverage cardinality of motif m
is its standard deviation in the null model
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Example for static motifs
Milo
et
al. S
cien
ce (
20
02
)
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Induced subgraphs
Let’s take a situation, where a star subgraph exists in the static graph under consideration:
This would contribute to the following motifs: 3 × + 3 ×
+
Only the last one is ”real”, the others cause caunting and interpretation problems
Only induced subgraphs should be considered!
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Motifs: Temporal aspects
• Time dependence of static motifs
• Daily mobility patterns
• Trigger statistics (causality)
• Temporal motifs
• Analysis of role of tagged nodes in temporal networks
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Activity counts on static motifs
Data: Mobile phone time records
L. Kovanen 2014
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Activity counts on triangles
Kova
ne
net
al.
(20
10
)
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Activity counts on directed triangles
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Evolution of motifs
Data from Chinese and European mobile phone services
Time stamped, who calls whom (hashed)
Problem: Which link is representing real social tie?
(And not commercial or technical calls)
Statistically validated network
How are static motifs present in the aggregate form in time? What is the characteristic time scale?
Ming-Xia Li et al. NJP 2014
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Change of the participation of nodes in the largest connected component
Ch
EU
Strong effect, underlining the importance of filteringGiant component exists in the original but not in the filtered nw. As time windows grows giant comp. emerges.
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Relevance of morifs
Examples of evolution of overrepresented motifs (Ch)
𝜇 𝜎 𝜇ref 𝜎ref
daily
weekly
monthly
ref: shuffled network, keeping in/out degrees and bidirectional links
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Examples of evolution of underrepresented motifs (Ch)
𝜇 𝜎 𝜇ref 𝜎ref
daily
weekly
monthly
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Correlations and evolution of motifs:
arrows indicate conditional probabilities from day Monday → Monday + Tuesday
Ch EU
Closed triangles form on an intraday scale!
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Mobility patterns
Data: Mobile call records with tower position, surveys (Paris, Chicago)
C.M. Schneider et al. 2013Spatio-temporal resolution
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Distribution of the number of distinct visited sites
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Mobility motifs – these 16 motifs explain 90%
Groups: # nodes. Numbers: 𝑁0/𝑁𝑓 with 𝑁0 the # of
observed and 𝑁𝑓 the total number of Eulerian paths.
Chicago Paris
Mostly Eulerian cycles – home, sweet home
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Action triggers (order important)
Kovanen: Thesis (2013)
Data: Mobile call records
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Action triggers: characteristic reaction time
Ref: average first response
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Temporal motifs (formalism)
A temporal subgraph with respect to Δt is set of events, which are mutually Δt-connected.
sij and sjk are Δt-adjacent events if their time difference is not longer than Δt.
sij and snm are Δt-connected events if there is a sequence of Δt-adjacent events connecting i and m.(There is no ordering requested, m is not necessarily reachable from i.)
A temporal subgraph is valid if no event is skipped at any node to construct it. It is ”consecutive”.
A temporal motif is a set of isomorphic valid temporal subgraphs, where isomorphism is defined with respect to the order of events.
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a) Temporal graph (no durations) b), c) Maximal subgraphs
d) Valid subgraphsNon-valid subgraphs in (b)
∆𝑡 = 10
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Maximal temporal motifs
Δt = 15 Two maximal temporal subgraphs(max. set of pairwise Δt-connevents)
Temporal motifs based on maximal subgraphs are maximal temporal motifs.Importance of a temporal motif is measured by its cardinality.
Kova
ne
net
al.
J. S
tat.
Mec
h. (
20
11
)
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Results on maximal temporal motifs (different ∆𝑡-s)
Reference systems?
Mobilephone data(∆𝑡 = 10 min)
Kova
ne
net
al.
J. S
tat.
Mec
h. (
20
11
)
”Causal”
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Null modelsThe comparison of the empirical data with a statistics on a null model tells whether the properties of the null model give a good null hypothesis. (E.g., strong deviations from the configuration model suggest that topological correlations are important for static models.)
Simple time shuffling leads to relevance of too many motifs.
Better: Check relevance of temporal aspects for node properties (gender, age, type of user): Colored temporal networks
Kova
ne
net
al.
(20
12
)
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Simple randomizing the types of nodes does not give a good null model for their role, since weight may play a role.
A proper null model can be constructed if the weight distribution of the aggregate network is taken into account when randomizing the colors.The null model is created by counting the motifs assuming dependence only on edge weight but not on node type.
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Results on motifs as compared to null model
Most frequent motifsmpre fprempost fpost
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Example of temporal effects
Female, 42 ± 2 years old, prepaid user
Male, 50 ± 2 years old, postpaid user
Dyn. 1st pre to post
Outstar: Targetage same
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Observations:- Clear indication of temporal homophily. Very strong for prepaid – socioeconomic background
- Outstars with same category of target are overrepresented
- Chains and stars overrepresented for femails
- Local edge density correlates with temporal overrepresented motifs (temporal Granovetter effect)
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Temporal networks are important for dynamic processes on complex networks if links are defined by the events and events happen inhomogeneously in time and/or the sequence of events is crucial
Temporal networks are defined with respect to a time window of observation.
Many concepts can be generalized: path, distance, connectivity, motifs etc.Motifs: static in evolution, mobility, temporal
Broad field of applications
Summary
Burstiness has a decelerating effect on spreading (!)