In this work, study the mean first saturation time (MFST), a generalization to the mean first passage time, on networks and show an application to the 2015 Burundi refugee crisis. The MFST between a sink node j, with capacity s, and source node i, with n random walkers, is the average number of time steps that it takes for at least s of the random walkers to reach a sink node j. The same concept, under the name of extreme events, has been studied in previous work for degree biased-random walks [2]. We expand the literature by exploring the behaviour of the MFST for node-biased random walks [1] in Erdős–Rényi random graph and geographical networks. Furthermore, we apply MFST framework to study the distribution of refugees in camps for the 2015 Burundi refugee crisis. For this last application, we use the geographical network of the Burundi conflict zone in 2015 [3]. In this network, nodes are cities or refugee camps, and edges denote the distance between them. We model refugees as random walkers who are biased towards the refugee camps which can hold s_j people. To determine the source nodes (i) and the initial number of random walkers (n), we use data on where the conflicts happened and the number of refugees that arrive at any camp under a two-month period after the start of the conflict [3]. With such information, we divide the early stage of the Burundi 2015 conflict into two waves of refugees. Using the first wave of refugees we calibrate the biased parameter β of the random walk to best match the distribution of refugees on the camps. Then, we test the prediction of the distribution of refugees in camps for the second wave using the same biased parameters. Our results show that the biased random walk can capture, to some extent, the distribution of refugees in different camps. Finally, we test the probability of saturation for various camps. Our model suggests the saturation of one or two camps (Nakivale and Nyarugusu) when in reality only Nyarugusu camp saturated.

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networks.” Physical review letters 99.9 (2007): 098701.

[2] Kishore, Vimal, M. S. Santhanam, and R. E. Amritkar. ”Extreme event-size fluctuations in biased

random walks on networks.” arXiv preprint arXiv:1112.2112 (2011).

[3] Suleimenova, Diana, David Bell, and Derek Groen. ”A generalized simulation development approach

for predicting refugee destinations.” Scientific reports 7.1 (2017): 13377.