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.

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

EFFECTS OF TIME-DEPENDENT EDGE DYNAMICS ON PROPERTIES OF CUMULATIVE NETWORKS

Richárd O. Legéndi, László GulyásAITIA International, Inc, Loránd Eötvös University and Collegium Budapestrlegendi@aitia.ai, lgulyas@aitia.ai

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

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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

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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.)

rlegendi@aitia.aihttp://people.inf.elte.hu/legendi/

September 15th, 2011

THANK YOU!