Abstract — Inspecting the dynamics of networks opens a new dimension in understanding the interactions among the components of complex systems. Our goal is to understand the baseline properties to be expected from elementary random changes over time, in order to be able to assess the effects found in longitudinal data. In our earlier work, we created elementary dynamic models from classic random and preferential networks. Focusing on edge dynamics, we defined several processes changing networks of fixed size. We applied simple rules, including random, preferential or assortative modification of existing edges - or a combination of these. Starting from initial Erdos-Renyi or Barabasi-Albert networks, we examined various basic network properties (e.g., density, clustering, average path length, number of components, degree distribution, etc.) of both snapshot and cumulative networks (of various lengths of aggregation time windows). In the current paper, we extend this line of research by applying time-dependent edge creation and deletion algorithms. I.e., we model processes where edge dynamics is defined as a function of time. Our results provide a baseline for changes to be expected in dynamic networks. Also, they suggest that certain network properties have a strong, non-trivial dependence on the length of the sampling window.

1. Supportedbythe HungarianGovernment (KMOP-1.1.2-08/1-2008-0002 ) viatheEuropean Regional Development Fund (ERDF) and bytheEuropean Union's Seventh Framework Programme: DynaNets,FET-Open project no. FET-233847 (http://www.dynanets.org). Richárd O. Legéndi, László GulyásAITIA International, Inc, Loránd Eötvös University and Collegium Budapest rlegendi@aitia.ai, lgulyas@aitia.ai EFFECTS OF TIME-DEPENDENT EDGE DYNAMICS ON PROPERTIES OF CUMULATIVE NETWORKS ECCS 2011, EPNACS Satellite Vienna, September 12-16, 2011

2. Overview Complex Systems, ComplexNetworks DynamicNetworks Aggregationtimewindow ElementaryModels of DynamicNetworks Previousresults Furthermotivations ElementaryModels of Time-DependentEdge Dynamics Preliminaryresults

3. Complex SystemsComplexNetworks

4. 4 Complex Systems, Definitions Systems composed of interacting components Simpleentitiesyieldcomplicateddynamics Nonlinearity, self-organization (patterndevelopment) „The whole is more than the sum of its parts” Recursiveeffectsfrominteractions; pathdependence; dynamicallyemergentproperties Typically not amenable to analytic solutions Size and computationalcomplexity, explosion Nonexistence of „solution”: infititelylonglivedtransients, nonequilibriumcascades, sensitivedependencies, etc.

5. 2011.09.15. Complex Networks, BIOINF 5 InteractionStructurematters Network Science Focusontheinteractionstructure Similarities and common properties Network as a general abstraction. Common properties and consequences.

6. Static Network versusDynamic Network Dynamics ofthenetwork(versus dynamicsonthenetwork) ThereareNOstaticnetworks Real life processeshappenintime(i.e., aredynamic) Wetakestaticsamples of them…

7. A PracticalProbleminModelingDynamicNetworks ∆t The importance of thesamplingwindow...

8. ElementaryModels ofDynamicNetworks

9. Elementary (Models of) DynamicNetworks GrowingNetworks (posteronMonday) Shrinking Networks(robustness studies, earlier publications) Networks of ConstantSize(posteronTuesday, earlierpublications)

10. Definitions Snapshotnetwork (@t) The networkatanysingletmomentintime.(Usingthefinestpossiblegranularityavailableinthemodel) Cumulativenetwork (@[t, t+T]) The union of snapshotnetworks(collected over thespecifiedinterval of time) Typically over the [0,T] intervalinourstudies Summationnetwork (@[t, t+T]) The sum of snapshotnetworks(collected over thespecifiedinterval of time) Typicallyyieldsmulti-nets

11. Definitions t=0 t=1 t=2 t=2 Snapsot ∆t Cumulative Summation

12. ElementaryDynamicNetworks @ ConstantDensity (earlierresults) Wecreatesimpledynamicmodels Similarinveintomodelslike Erdős-Rényi Watts-Strogatzor Barabási-Albert (planned) Explorevarioussamplingwindows Wecomparesnapshot and cumulativenetworks

13. Sensitivity to aggregation

14. dEGREE DISTRIBUTION RADICALLY CHANGES

15. sENSITIVITY OF DEGREE DISTRIBUTION Normal, lognormal,even power law distribution For the same model Using different time frames

16. Time-DependentEdge Dynamics

17. FurtherMotivations Incertaindomains (e.g., inchemicalreactions) interactionsareforshorttimeonly Human interactionsarealsotemporal „(…), the very behavior that makes these people important tovaccinate can help us finding them. People you have met recently are more likely to be socially active andthus central in the contact pattern, and important to vaccinate. We propose two immunization schemesexploiting temporal contact patterns.” (S .Lee, L.E.C. Rocha, F.Liljeros, P.Holme. Exploiting temporal networkstructures of human interaction to effectively immunize population. arXiv:q-bio/1011.3928, 2010.)

19. PreliminaryResults

20. ER4 – Density Directly connected to other properties(e.g., centralities) Increases linearly with edge lifetime (snapshot) Cumulative networks are identical Most measures include these observations

21. ER4 – Reaching the connected network

22. er4 – Clustering Clustering shows similiar trends for the cumulative network Snapshot may drastically change when groups found

23. ER5 – density Density changes linearly Average degree, components show the same transition rate

24. ER5 – Reaching the connected network

25. er5 - clustering Snapshot networks arestationary Cumulative networks drastically change High jumps Slow decreasing

26. Summaryand Future Works

27. Summary Studiedelementarydynamicnetworks Withtime-dependentedgedynamics Most statistics show expected values linearity Reaching the connected network is a tipping point betweenness, average path length Some properties may show wild oscillations clustering

28. Future Works More extensivestudies (e.g., parameterdependence) More extensivestudies of theeffectofsamplingfrequency Non-uniform samplingwindows Dedicatingparts of thenetworkasconstant (The last 3 stemfrompracticalissuesinreal-worldcases. E.g, inpharmaneutics.)

29. rlegendi@aitia.ai http://people.inf.elte.hu/legendi/ September 15th, 2011 Thankyou!