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Convergence Properties of Optimal Transport-Based Temporal Networks

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Complex Networks & Their Applications X (COMPLEX NETWORKS 2021)

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We study network properties of networks evolving in time based on optimal transport principles. These evolve from a structure covering uniformly a continuous space towards an optimal design in terms of optimal transport theory. At convergence, the networks should optimize the way resources are transported through it. As the network structure shapes in time towards optimality, its topological properties also change with it. The question is how do these change as we reach optimality. We study the behavior of various network properties on a number of network sequences evolving towards optimal design and find that the transport cost function converges earlier than network properties and that these monotonically decrease. This suggests a mechanism for designing optimal networks by compressing dense structures. We find a similar behavior in networks extracted from real images of the networks designed by the body shape of a slime mold evolving in time.

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The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Diego Baptista.

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Correspondence to Diego Baptista .

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Supplementary Information (SI)

1 Synthetic Data

Details of the Studied Transport Problems. As mentioned in the main manuscript, we consider a set of points \(S = \{s_0,s_1,...,s_M\}\) in the space \(\varOmega = [0,1]^2,\) and \(0<r\) a positive number, and we use this to define the distributions \(f^+\) and \(f^-\) as

$$ f^+(x) \propto \mathbbm {1}_{R_0}(x), \ \ f^-(x) \propto \sum _{i>0} \mathbbm {1}_{R_i}(x) $$

where \(R_i\) is the circle of center \(s_i\) and radius r. The points \(s_1,...,s_M\), the support of the sink, are sampled uniformly at random from a regular grid. The used grid and different realizations of the sampling are shown in Fig. 7.

Fig. 7.
figure 7

Support of \(f^-\). The nodes of the grid constitutes the set of candidates from which the support of \(f^-\)

Fig. 8.
figure 8

Total length and Lyapunov cost. Top row: from left to right we see \(\beta =1.1, 1.3\) and 1.4. Bottom row: from left to right we see \(\beta =1.6, 1.7\) and 1.9. Mean and standard deviation of the total length l(G) as function of time t; Bottom plot: Mean and standard deviation of the Lyapunov cost \(\mathcal {L}\), energy dissipation \(\mathcal {E}\) and structural cost \(\mathcal {M}\) of transport densities. Red and blue lines denote \(t_P\) and \(t_\mathcal {L}\) for \(p = 1.05\).

Total Length and Lyapunov Cost. We show in this section a figure like the one presented in the Fig. 2 of the main manuscript, for other values of \(\beta .\) As mentioned in there, the properties show decreasing behaviors for which is always true that \(t_P>t_\mathcal {L}\) (see Fig. 8).

Network Properties and Lyapunov Cost. We show in this section a figure like the one presented in the Fig. 2 of the main manuscript, for the other network properties (see Fig. 9).

Fig. 9.
figure 9

Other network properties and Lyapunov cost. From left to right: \(\beta =1.2, 1.5\) and 1.8. From top to bottom: Mean and standard deviation of the average degree, number of nodes |N|,  number of bifurcations bif(G), and the Lyapunov cost \(\mathcal {L}\), energy dissipation \(\mathcal {E}\) and structural cost \(\mathcal {M}\). Red and blue lines denote \(t_P\) and \(t_\mathcal {L}\) for \(p = 1.05\).

2 P. polycephalum Networks

Data Information. In this section, we give further details about the used real data. As mentioned in the main manuscript, the images are taken from the Slime Mold Graph Repository [16]. The number of studied sequences \(\{G_i\}_i^T\) equals 4. Every sequence’s length T changes depending on the amount of images provided in the repository, since different experiments need more o less shots. An experiment, as explained in the repository’s documentation, consists of placing a slime mold inside a Petri dish with a thin sheet of agar and no sources of food. The idea, as explained by the creators, is to let the slime mold fully explore the Petri dish. Since the slime mold is initially lined up along one of the short side of the dish, the authors stop capturing images once the plasmodium is about to reach the other short side.

Fig. 10.
figure 10

Network properties for P. polycephalum sequences. From top to bottom: motion12, motion24, motion40 and motion79. Subfigures show the evolution of the properties |E|,  average degree and |N| for every sequence as a function of time.

Fig. 11.
figure 11

P. polycephalum total length evolution. From top to bottom: motion24, motion40 and motion79. Plots are separated in couples. For every couple, the plots on top show both P. polycephalum images and networks extracted from them. The network at the lower leftmost plot is a subsection of the graph shown inside the red rectangle on top. The plot at the bottom shows the total length as a function of time. The red shade in this plot highlights a tentative consolidation phase towards optimality.

Network Extraction. The studied network sequences are extracted from the image sets motion12, motion24, motion40 and motion79, which are stored in the repository. Each image set contains a number of images ranging from 60 to 150, thus, obtained sequences exhibit diverse lengths. Every network is extracted using the Img2net algorithm described in [20]. The main parameters of this algorithms are N_runs, t2, t3 and new_size. N_runs controls how many times the algorithm needs to be run; t2 (and t3) are the minimum value (and maximum) a pixel’s grayscale value must be so it is considered as a node; new_size is the size to which the input image must be downsampled before extracting the network from it. For all the experiments reported in this manuscript, the previously mentioned parameters are set to be N_runs = 1, t2 = 0.25, t3 = 1 and new_size = 180.

More Network Properties. Other network properties are computed for the real systems referenced in this manuscript. Similar decreasing behaviors, like the one shown for the total length property in the main manuscript, are found for these properties; see Figs. 10 and 11.

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Baptista, D., De Bacco, C. (2022). Convergence Properties of Optimal Transport-Based Temporal Networks. In: Benito, R.M., Cherifi, C., Cherifi, H., Moro, E., Rocha, L.M., Sales-Pardo, M. (eds) Complex Networks & Their Applications X. COMPLEX NETWORKS 2021. Studies in Computational Intelligence, vol 1072. Springer, Cham.

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