Degree Dependence

Adams, Hreinsson, Weber

March 26, 2015

Abstract

In this project the dependence of the survival rate of Masaai herders in a Watts Strogatzring network using di↵erent Osotua asking policies was investigated.Generally a high mean degree turns out to be favorable combined a rewiring coe�cient ofapproximately 0.2 and ordered asking of the richest Osotua partner first. The obtained resultfor the best asking policy contradicts a previous result for closed networks favoring to ask theleast rich partner first. Furthermore the e↵ect of the di↵erent network parameters on herd sizedistributions was investigated as well as the e↵ect of rewiring the network after certain timeperiods.

1 Preliminaries

We used python-igraph to generate the Wattz Strogatz graph (built in function). For the osotuaexchange we started by randomizing the list of herders that were below the minimum herd size.We then went through the list, one by one and tried di↵erent approaches of asking the herderneighbors:

• No exchange is the standard reference method were no exchange takes place.

• Ask once randomly picks a single herder randomly from the list of neighbors, if that herderhas enough calf (by standard Osotua rules) he will give the required amount, otherwise theasker “gives up” and remains below minimum herd size.

• In ask often randomly the herder in need goes in random order between his friends askingfor calf. If a neighbor can provide calf (not falling below the minimum himself while neversupplying less than requested) exchange takes place. If no neighbor can provide calf theasking herder remains below minimum herd size.

• In ask least rich we assume the asker has perfect knowledge of all his neighbors herd sizes.If any neighbors have enough calf to give, he will ask the neighbor that “just barely” hasenough calf to give.

• Ask richest will try to ask the richest neighbor for calf, if the neighbor has enough to giveexchange takes place, otherwise the asker remains below the limit.

The program ran 200 trials for each combination of (approach, mean-degree, rewiring-coe�cient)with every trial and combination of parameters being a completely independent experiment.

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Figure 1: Comparing survival rate of di↵erent exchange approaches and mean degrees.

2 Comparing exchange approaches

Figure 1 compares the di↵erent asking strategies for networks with di↵erent mean degrees. Notsurprisingly, no exchange is the worst and independent of the network mean degree. Asking ran-domly once comes second and benefits from higher degree, since low mean degree networks havemore chance of having completely disconnected herders (reducing survivability). The approachesthat ask randomly multiple times or selectively go for the neighbor that has enough calf to giveprove to be the best. This is not surprising, you utilize your neighbors to the full extent, wherenobody dies if anyone has enough calf to o↵er. High mean degrees are clearly beneficial, o↵eringlarger “support networks”. For the rest of this discussion we will focus on asking the richest, seeingas it was the most successful approach.

3 Survivability with mean degree and rewiring coe�cient

It can be observed that a higher mean degree generally improves survival after 50 years mostlyindependent of the rewiring coe�cient. The dependence between mean degree and survival rate isnonlinear and increases less for higher mean degrees.

To visualize the already presented results heat maps are a useful tool. The optimum parametersin terms of survivability after 50 years are: mean degree = 10, rewiring coe�cient = 0.2 for askingthe richest person first. All other asking policies perform worse as shown before.

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4 Network Fragmentation

Herders die during the course of a trail, erasing nodes from the Osotua network. This may lead tofragmentation of the network and forming independent, isolated Osotua networks. The probabilityof fragmentation depends on mean degree and rewiring coe�cient since these factors determine thenumber of neighbors nodes need to be isolated from in order to isolate a fragment of the network.Intuitively for a high rewiring coe�cient almost no fragmentation takes place at a mean degreeof 4 but initial fragmentation occurs after the process of rewiring. There seems to be a thresholdin network degree below which fragmentation takes place at all. Furthermore fragmentation satu-rates. This could be an interesting result choosing starting conditions for a realistic Osotua model.Considering these results it is probably realistic to assume a fragmented Osotua network at meandegrees below 5.

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5 Herd size (wealth) distribution

In general the herd size distributions resemble the previously obtained result for Osotua pairsof the form of a normal distribution and an exponential decay. All displayed histograms arenormalized. Higher mean degrees result in faster decaying curves with a higher proportion ofsmall herds while changes in the rewiring coe�cients barely influence the herd size distribution.Asking the richest Osotua partners first not only increases survival rates but also yields more ”flat”herd size distributions and therefor a better general wealth of the Masaai herders.

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6 Rewiring the network during the trial

Considering social dynamics in the Osotua system, rewiring of the system very likely occursas time passes. For these investigations, we used the poorest asking the least rich askingrules. We assume the asker has complete information about the wealth of everyone within theirown neighborhood. We investigated the influence of rewiring the network once after 25 years(which is a possible time interval for a herder generation change and introduction of new so-cial connections) as well as rewiring the network once every 10 years. Rewiring the network ineach instance is done with the same rewiring coe�cient used to create the initial small-worldnetwork. Figure 11 is a visual example of how a single rewiring is completed. For this por-tion of experiments, the initial small-world networks were generated in Mathematica 10 with theRandomGraph[WattsStrogatzGraphDistribution[networksize, n, m]] function, where n is therewiring coe�cient and m is the mean degree.

Figure 11

From Figures 12-16, it is apparent that changing the network once at t=25 has little to no e↵ecton the herd survival rate. In a usual scenario after 25 years the network should have developeda certain stability since herd growth should have brought many farmers into a zone of big herdsizes and therefore increase the strength of the Osotua society. The network may also developstronger neighborhoods no longer interconnected with the rest of the network as well as weakerneighborhoods slowly dying o↵. Therefore this result is a rather interesting one implying thatsurvival rates are una↵ected by reconnecting di↵erent isolated subnetworks. Death rates are alwayshighest in the early years of an Osotua simulation. Because of this, rewiring the network every 10years significantly alters the survival rate which is likely due to the first network rewiring. Thisreinforces the notion that the network dynamics are most unstable during the earlier years of theOsotua simulation and relax after most of the weak neighborhoods die.

1 2 3 4 5Mean Degree

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Survival prop.Herd Survival at t=50, �=0.2

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Change at t=25Change every 10 years

Figure 12

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1 2 3 4 5Mean Degree

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Survival prop.Herd Survival at t=50, �=0.4

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1 2 3 4 5Mean Degree

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Survival prop.Herd Survival at t=50, �=0.6

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1 2 3 4 5Mean Degree

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Survival prop.Herd Survival at t=50, �=0.8

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1 2 3 4 5Mean Degree

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Survival prop.Herd Survival at t=50, �=1

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Change at t=25Change every 10 years

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