Embed Size (px)
DESCRIPTION
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.
Citation preview
EFFECTS OF TIME-DEPENDENT EDGE DYNAMICS ON PROPERTIES OF CUMULATIVE NETWORKS
Richárd O. Legéndi, László GulyásAITIA International, Inc, Loránd Eötvös University and Collegium [email protected], [email protected]
Supported by the Hungarian Government (KMOP-1.1.2-08/1-2008-0002 ) via the European Regional Development Fund (ERDF) and by the European Union's Seventh Framework Programme: DynaNets, FET-Open project no. FET-233847 (http://www.dynanets.org).
ECCS 2011, EPNACS SatelliteVienna, September 12-16, 2011
OVERVIEW
Complex Systems, Complex Networks Dynamic Networks Aggregation time window
Elementary Models of Dynamic Networks Previous results Further motivations
Elementary Models of Time-Dependent Edge Dynamics Preliminary results
COMPLEX SYSTEMSCOMPLEX NETWORKS
4
COMPLEX SYSTEMS, DEFINITIONS
Systems composed of interacting components Simple entities yield complicated dynamics Nonlinearity, self-organization (pattern development)
„The whole is more than the sum of its parts” Recursive effects from interactions; path
dependence; dynamically emergent properties Typically not amenable to analytic solutions
Size and computational complexity, explosion Nonexistence of „solution”: infititely long lived
transients, nonequilibrium cascades, sensitive dependencies, etc.
2023.04.13.Complex Networks, BIOINF
5
INTERACTION STRUCTURE MATTERS
Network Science Focus on the interaction structure
Similarities and common properties Network as a general abstraction. Common properties and consequences.
STATIC NETWORK VERSUS DYNAMIC NETWORK
Dynamics of the network (versus dynamics on the network)
There are NO static networks
Real life processes happen in time (i.e., are dynamic)
We take static samples of them…
A PRACTICAL PROBLEM IN MODELING DYNAMIC NETWORKS
The importance of the sampling window...
∆t
ELEMENTARY MODELS OFDYNAMIC NETWORKS
ELEMENTARY (MODELS OF) DYNAMIC NETWORKS
Growing Networks (poster on Monday)
Shrinking Networks (robustness studies, earlier publications)
Networks of Constant Size (poster on Tuesday, earlier publications)
DEFINITIONS Snapshot network (@t)
The network at any single t moment in time.(Using the finest possible granularity available in the model)
Cumulative network (@[t, t+T]) The union of snapshot networks
(collected over the specified interval of time) Typically over the [0,T] interval in our studies
Summation network (@[t, t+T]) The sum of snapshot networks
(collected over the specified interval of time) Typically yields multi-nets
DEFINITIONS
Snapsot
t=0
∆t
t=1
t=2
t=2
Cumulative
Summation
ELEMENTARY DYNAMIC NETWORKS @ CONSTANT DENSITY (EARLIER RESULTS)
We create simple dynamic models Similar in vein to models like
Erdős-Rényi Watts-Strogatz or Barabási-Albert (planned)
Explore various sampling windows We compare snapshot and cumulative
networks
SENSITIVITY TO AGGREGATION
DEGREE DISTRIBUTION RADICALLY CHANGES
SENSITIVITY OF DEGREE DISTRIBUTION
Normal, lognormal, even power law distribution
For the same model Using different time
frames
TIME-DEPENDENT EDGE DYNAMICS
FURTHER MOTIVATIONS
In certain domains (e.g., in chemical reactions) interactions are for short time only
Human interactions are also temporal„(…), the very behavior that makes these people important to vaccinate can help us finding them. People you have met recently are more likely to be socially active and thus central in the contact pattern, and important to vaccinate. We propose two immunization schemes exploiting temporal contact patterns.”
(S .Lee, L.E.C. Rocha, F. Liljeros, P. Holme. Exploiting temporal network structures of human interaction to effectively immunize population. arXiv:q-bio/1011.3928, 2010.)
EVALUATED MODELS
Two dynamic versions of the Erdős-Rényi model T= 100, N=100, p0=0.02, each seed with 3 values, meaned
results
ER4 Edges have a time presence Uniformly appear For a given lifetime
ER5 Edges appear periodically in each k * s time step (k = 1, 2, ...)
PRELIMINARY RESULTS
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
ER4 – REACHING THE CONNECTED NETWORK
ER4 – CLUSTERING
Clustering shows similiar trends for the cumulative network
Snapshot may drastically change when groups found
ER5 – DENSITY
Density changes linearly
Average degree, components show the same transition rate
ER5 – REACHING THE CONNECTED NETWORK
ER5 - CLUSTERING
Snapshot networks are stationary
Cumulative networks drastically change High jumps Slow decreasing
SUMMARYAND FUTURE WORKS
SUMMARY
Studied elementary dynamic networks With time-dependent edge dynamics
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
FUTURE WORKS
More extensive studies (e.g., parameter dependence)
More extensive studies of the effect of sampling frequency
Non-uniform sampling windows
Dedicating parts of the network as constant
(The last 3 stem from practical issues in real-world cases. E.g, in pharmaneutics.)
