Comparing Snapshots of Networks Shah Jamal Alam and Ruth Meyer Centre for Policy Modelling 28 th March, 2007 – CAVES Bi-annual Meeting, IIASA, Laxenburg, Austria

A General Problem … Consider a simulated network S (N, E) at the start of a simulation run, i.e. at time t=0. N is the set of nodes and E is the set of edges through which the nodes of the graph are connected. The network could be of any structure, e.g. unconnected (E = {Φ}), a small-world network, or a scale-free network. At every time step, x: x  (0,  ] new agents become a member of N, while y already present agents leave: y  (0, |N|].

Agent-based Social Networks (ABSN) – I The majority of social simulation models assume a fixed number of agents, usually for ‘simplicity sake’. Many agent-based social simulation models have shown that agents’ local interactions can lead to structures such as power-law and small-world networks, where the network’s configuration stabilizes after a sufficient number of simulation runs.

Agent-based Social Networks (ABSN) – II It is usually the case, there are inherent local mechanisms that bias towards such structures. For models where the agent population is constrained by social processes and not on a fixed size, this might not be the case... Even different networks with similar global properties such as clustering coefficient, can have varying local structures

Example from two Agent-based Models The first is by Jin et al. (2001): Structure of growing social networks. Phys. Rev. E. The American Physical Society 64(4) Second is based on from the South African case study concerning the impact of HIV/AIDS. The two different models are different in terms of scale, context, purpose, granularity (level of abstraction) We are looking on the network signatures from the simulation

The model for growing friendship networks by Jin et al. (2001) Assumes a fix number of agents (250) Based on three rules: –“meetings take place between pairs of individuals at a rate which is high if a pair has one or more mutual friends and low otherwise; –acquaintances between pairs of individuals who rarely meet decay over time –there is an upper limit on the number of friendships an individual can maintain.” Demonstrates the evolution of ‘small-world’ like structure from agents’ acting on the above rules.

The model for growing friendship networks by Jin et al. (2001)

Comparing the network’s snapshots … We compare network snapshots, P= {P 0, P 0+ , P 0+2 ,…}, where each element P i represents a distinct population of agents, P 0 being the population at t=0.  defines the time lag. A 2-population (independent) Kolmogorov- Smirnov (K-S) test is performed for the degree distribution each consecutive population pair (P i, P i+  ). The idea is to determine if two datasets from two population samples differ significantly.

Comparing the network’s snapshots … Y-axis: P-score of the K-S test for the degree distribution of the friendship network compared for consecutive snapshots taken at every (left) 100 th tick and (right) 25 th time step.

The agents’ network grew in the initial steps and the similarity in the network configuration continued to increase From the figures in the previous slide, we can see that the network structure stabilizes around 100th step

A Model for Impact of HIV/AIDS – I The model focuses on the behaviour of individual agents as well as that of households and thus attempts to take into account both the individual interactions and the decisions taken by the households. Agents are born and die during the simulation … the population size depends upon social processes and not based on any fixed mechanism… the mode details have been presented already

A Model for Impact of HIV/AIDS – II The simulations were initialized with 30 households at the start, with the number of adult and senior agents around 150 at the start. Every agent has an expected life … but may die due to malnutrition, AIDS… Birth of agents depends upon the marriages and the fertility of a mother infected from HIV/AIDS … In sum: The size and the network composition cannot be assumed in advance. The three rules from Jin et al. was plugged in the model for the agents’ friendship network.

First suit of simulation: Three different settings: Case – I: we removed the random removal rule from Jin et al. Case – II: incorporated all the three rules. Agents also developed friendship ties when they join the same club for the above two cases. Case – III: just the three rules from Jin et al. to make friendship ties …

First suit of simulation: Three different settings

Second suit of simulation: Three different settings Case – I: no birth of an agent was possible, while all other processes continued in the model. Case – II: Case-II was set with only the new agents joining the simulation through marriages and birth processes but no agent was set to die during the simulation run. Case – III: was when the initial population of the agents continued throughout the run and no new agents joined or existing agents died at all.

Second suit of simulation: Three different settings

Outlook – I … Without the birth and death of agents during the simulation, the agents’ friendship network shows the similar signature, in our context, the clustering coefficient as in the first model. This supports the conjecture that network signatures are affected when the agents’ population size is governed by the underlying social processes.

Outlook – II The main motivation has been to look for techniques that can help keep track of the network structure over the course of a simulation run. Global measures aggregate the network characteristics … the snapshots are compared on the output number Other techniques, such as, use of the Kolmogorov-Smirnov statistic can give us more insight about the changes in the network structure…

Outlook – III Networks sharing similar global properties can have significantly varying local structures.

‘motifs’ are subgraph structures that occur in them network far more often than in randomized network ‘complex networks’ community ‘fragments’ are the building blocks of social ties … starting from dyads to simplest forms of social networks  triads social networks community Networks sharing similar global properties can have significantly varying local structures.