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Effects of Time-Dependent Edge Dynamics on Properties of Cumulative Networks

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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.

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Effects of Time-Dependent Edge Dynamics on Properties of Cumulative Networks

  1. 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 (<br />Richárd O. Legéndi, László GulyásAITIA International, Inc, Loránd Eötvös University and Collegium Budapest<br />,<br />EFFECTS OF TIME-DEPENDENT EDGE DYNAMICS ON PROPERTIES OF CUMULATIVE NETWORKS<br />ECCS 2011, EPNACS Satellite<br />Vienna, September 12-16, 2011<br />
  2. 2. Overview<br />Complex Systems, ComplexNetworks<br />DynamicNetworks<br />Aggregationtimewindow<br />ElementaryModels of DynamicNetworks<br />Previousresults<br />Furthermotivations<br />ElementaryModels of Time-DependentEdge Dynamics<br />Preliminaryresults<br />
  3. 3. Complex SystemsComplexNetworks<br />
  4. 4. 4<br />Complex Systems, Definitions<br />Systems composed of interacting components<br />Simpleentitiesyieldcomplicateddynamics<br />Nonlinearity, self-organization (patterndevelopment)<br />„The whole is more than the sum of its parts”<br />Recursiveeffectsfrominteractions; pathdependence; dynamicallyemergentproperties<br />Typically not amenable to analytic solutions<br />Size and computationalcomplexity, explosion<br />Nonexistence of „solution”: infititelylonglivedtransients, nonequilibriumcascades, sensitivedependencies, etc.<br />
  5. 5. 2011.09.15.<br />Complex Networks, BIOINF<br />5<br />InteractionStructurematters<br />Network Science<br />Focusontheinteractionstructure<br />Similarities and common properties<br />Network as a general abstraction.<br />Common properties and consequences.<br />
  6. 6. Static Network versusDynamic Network<br />Dynamics ofthenetwork(versus dynamicsonthenetwork)<br />ThereareNOstaticnetworks<br />Real life processeshappenintime(i.e., aredynamic)<br />Wetakestaticsamples of them… <br />
  7. 7. A PracticalProbleminModelingDynamicNetworks<br />∆t<br />The importance of thesamplingwindow...<br />
  8. 8. ElementaryModels ofDynamicNetworks<br />
  9. 9. Elementary (Models of) DynamicNetworks<br />GrowingNetworks (posteronMonday)<br />Shrinking Networks(robustness studies, earlier publications)<br />Networks of ConstantSize(posteronTuesday, earlierpublications)<br />
  10. 10. Definitions<br />Snapshotnetwork (@t)<br />The networkatanysingletmomentintime.(Usingthefinestpossiblegranularityavailableinthemodel)<br />Cumulativenetwork (@[t, t+T])<br />The union of snapshotnetworks(collected over thespecifiedinterval of time)<br />Typically over the [0,T] intervalinourstudies<br />Summationnetwork (@[t, t+T])<br />The sum of snapshotnetworks(collected over thespecifiedinterval of time)<br />Typicallyyieldsmulti-nets<br />
  11. 11. Definitions<br />t=0<br />t=1<br />t=2<br />t=2<br />Snapsot<br />∆t<br />Cumulative<br />Summation<br />
  12. 12. ElementaryDynamicNetworks @ ConstantDensity (earlierresults)<br />Wecreatesimpledynamicmodels<br />Similarinveintomodelslike<br />Erdős-Rényi <br />Watts-Strogatzor<br />Barabási-Albert (planned)<br />Explorevarioussamplingwindows<br />Wecomparesnapshot and cumulativenetworks<br />
  13. 13. Sensitivity to aggregation<br />
  15. 15. sENSITIVITY OF DEGREE DISTRIBUTION<br />Normal, lognormal,even power law distribution<br />For the same model<br />Using different time frames<br />
  16. 16. Time-DependentEdge Dynamics<br />
  17. 17. FurtherMotivations<br />Incertaindomains (e.g., inchemicalreactions) interactionsareforshorttimeonly<br />Human interactionsarealsotemporal<br /> „(…), 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.”<br /> (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.)<br />
  18. 18. Evaluated models<br />Two dynamic versions of the Erdős-Rényi model<br /><ul><li>T= 100, N=100, p0=0.02, each seed with 3 values, meaned results</li></ul>ER4 Edges have a time presence<br />Uniformly appear<br />For a given lifetime<br />ER5 Edges appear periodically <br />in each k * s time step (k = 1, 2, ...)<br />
  19. 19. PreliminaryResults<br />
  20. 20. ER4 – Density<br />Directly connected to other properties(e.g., centralities)<br />Increases linearly with edge lifetime (snapshot)<br />Cumulative networks are identical<br />Most measures include these observations<br />
  21. 21. ER4 – Reaching the connected network<br />
  22. 22. er4 – Clustering<br />Clustering shows similiar trends for the cumulative network<br />Snapshot may drastically change when groups found<br />
  23. 23. ER5 – density<br />Density changes linearly<br />Average degree, components show the same transition rate<br />
  24. 24. ER5 – Reaching the connected network<br />
  25. 25. er5 - clustering<br />Snapshot networks arestationary<br />Cumulative networks drastically change<br />High jumps<br />Slow decreasing<br />
  26. 26. Summaryand Future Works<br />
  27. 27. Summary<br />Studiedelementarydynamicnetworks<br />Withtime-dependentedgedynamics<br />Most statistics show expected values<br />linearity<br />Reaching the connected network is a tipping point<br />betweenness, average path length<br />Some properties may show wild oscillations<br />clustering<br />
  28. 28. Future Works<br />More extensivestudies (e.g., parameterdependence)<br />More extensivestudies of theeffectofsamplingfrequency<br />Non-uniform samplingwindows<br />Dedicatingparts of thenetworkasconstant<br /> (The last 3 stemfrompracticalissuesinreal-worldcases. E.g, inpharmaneutics.) <br />
  29. 29.<br /><br />September 15th, 2011<br />Thankyou!<br />