La evaluación de la satisfacción materna en el nacimiento de sus bebés

Recall from section 2.2 that a temporal network represents interactions that occurred at several different instances of time. In such a network, each period of time is represented by a layer and the nodes in each layer are connected to the instances of the same node in the previous and subsequent layers (hence encoding the chronological sequence of the layers’ interactions).

In this section, we focus on a synthetic test used in Pamfil et al. [17] to generate temporal networks. The generative model, first used by Ghasemian et al. [29], is as follows.

First, we generate a ground-truth community membership in the first layer by splitting evenly betweenKavailable community labels. Then, for each subsequent layer, the community label is 2_{Recall from subsection 4.2.4 that a stable partitionσwith respect to a set of partitionsΣwill also be stable in any} subset ofΣthat includesσ. Hence, since this 4-community partition was stable in Figure 5.1, it is also stable when we restrict consideration to partitions with exactly 4 communities.

copied from the previous layer with probabilityηand randomly assigned from allKpossible labels with probability1₋η.

Using this ground-truth community assignment, edges are independently placed between pairs of nodes in each layer with probabilitypinif the nodes are in the same ground-truth community and

with probabilitypoutotherwise. A model parameterε=pout/pinis used to control the strength of the

community structure in these layers (smaller values ofεplace more edges within communities than between communities).

In our tests here, we generated multi-layer networks with “copying probability”η= 0.7, edge probability ratioε= 0.4, number of layersT = 15, number of communitiesK = 2and 150 nodes per layer.3 The behavior of Pamfil et al.’s iterative procedure on this network is shown in Figure 5.3.

0.4 0.6 0.8 1.0 1.2 1.4 1.6 ω 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 γ

Synthetic Network (Gamma, Omega) Estimates from Louvain

Figure 5.3: The behavior of the iterative procedure introduced in Pamfil et al. [17] on our synthetic network. The parameter values for the ground truth community are shown as a blue point near

(ω, γ)_≈(0.98,0.94). Over a grid of the(ω, γ)plane, arrows indicate the direction of the updated resolution parameter estimates after maximizing modularity with the Louvain algorithm for this choice of(ω, γ), averaged over five trials. As in [17], arrow sizes are scaled down for clarity (here, shown as 10% their actual update movement).

3

Note that this choice ofK= 2means the copying of labels from one layer to the next actually occurs with probability

η+1

2(1−η) = 0.85. This choice is used in both [17] and [29], so we have also chosen to use only two ground truth

communities.

This iterative scheme converges close to the ground truth resolution parameter estimates for much of the(ω, γ)plane, but notably there are regions for which the scheme diverges away from the ground truth values (e.g. most initializations withγ >1.1fail to converge to the ground truth).

We now compare these results to our method, which in part will explain why the above scheme can diverge in this way.

After generating a network using the above model, we obtained 50,625 partitions by running the Louvain algorithm in a225_×225uniform grid ofγ _∈[0.0,2.0],ω_∈[0.0,2.0]. Of these partitions were 33,410 unique partitions with more than one community. We start by pruning with CHAMP with the number of communities left unconstrained.

Across the entire resolution parameter areaγ _∈[0.0,2.0],ω _∈[0.0,2.0], our pruned subset from CHAMP has 77 partitions with more than one community. The domains of optimality and associated resolution parameter estimates are shown in Figure 5.4.

0.4 0.6 0.8 1.0 1.2 1.4 1.6 ω 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 γ

Synthetic Network Domains and (Gamma, Omega) Estimates

Figure 5.4: Domains of optimality for the partitions in CHAMP’s pruned subset (approximately 25 partitions are somewhere dominant in the region of the(ω, γ)plane shown). For each partition, an arrow is drawn from the centroid of the partition’s domain of optimality to its resolution parameter estimate(ω, γ).

Qualitatively, these domains exhibit the same behavior as the iterative scheme shown in Figure 5.3 with many partitions’ estimates lying close to the ground truth resolution parameter values of

(ω, γ)_≈(0.98,0.94). Once again, the partitions dominant in the approximate regionγ >1.1do not converge near this ground truth. To see why this is the case, we plot the number of communities in these partitions and their AMIs4with the ground truth 2-community partition in Figure 5.5.

0.4 0.6 0.8 1.0 1.2 1.4 1.6 ω 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 γ

AMI of Domains with Ground Truth

0.0 0.2 0.4 0.6 0.8 1.0 AMI 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ω 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 γ

Domains with Number of Communities

2 4 6 8 10 12 14 16 18 20 22 24 Num b er of Comm unities

Figure 5.5: Top: Domains of optimality from CHAMP’s pruned subset, colored by AMI with the ground truth partition. Bottom: Domains of optimality from CHAMP’s pruned subset, colored by number of communities.

4_{In short, the adjusted mutual information (AMI) is a measure of how closely two partitions agree where values closer} to 1 indicate stronger alignment and 0 indicates “no alignment”. In contrast to normalized mutual information (NMI), AMI is adjusted for chance so that random clustering have an expected AMI of 0.

Here, we see the issue is precisely the one discussed in section 2.5 where the resolution parameter estimates depend heavily on the number of communities in the partition. In fact, the transition around γ _≈1.1is marked by an incredibly rapid increase in the number of communities fromK = 2to K ≈15! Unsurprisingly, this transition also coincides with a sharp decrease in alignment with the ground truth partition.

Now, we analyze the stable partitions from CHAMP’s pruned subset (once again, with the number of communities unconstrained). 5 out of the 77 partitions with more than one community are stable and the details of these partitions are given in Table 5.2.

Number of communitiesK AMI with ground truth (ω, γ)estimate

2 0.95 (0.94, 0.95) 12 0.12 (1.21, 1.23) 13 0.13 (1.30, 0.93) 16 0.11 (1.32, 0.89) 21 0.09 (1.50, 0.85) 28 0.07 (1.62, 0.84)

Table 5.2: Details of the stable partitions from CHAMP’s pruned subset on this synthetic network. Recall that the ground truth resolution parameter estimates here are(ω, γ)≈(0.98,0.94).

Notably, the 2-community stable partition has very strong alignment with the ground truth (agreeing for∼99.5%of the network’s nodes). In fact, it has the highest AMI value with the ground truth group membership among any of the partitions in CHAMP’s pruned subset.

The domains of all other stable partitions lie beyond the transition where the number of com- munities increases atγ _≈1.1and do not closely match the ground truth. Indeed, in a network with two ground truth communities and only 15 layers, the relatively large number of communities in the other stable partitions (K = 12,13,16,21,28) suggest that they are only dominant for unreasonable choices of the intralayer resolution parameterγ.

Finally, we reproduce Figure 5.4 when CHAMP’s pruned subset only considers partitions with K = 2, constituting 6,794 partitions from the original runs of the Louvain algorithm. We show the domains of optimality and(ω, γ)estimates in Figure 5.6.

0.4 0.6 0.8 1.0 1.2 1.4 1.6 ω 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 γ

Synthetic Network Domains and (Gamma, Omega) Estimates,K= 2

Figure 5.6: Domains of optimality for the partitions in CHAMP’s pruned subset when we restrict K = 2. For each partition, an arrow is drawn from the centroid of the partition’s domain of optimality to its resolution parameter estimate(ω, γ).

In this case, CHAMP’s pruned subset has 23 partitions that are somewhere dominant and exactly one of these is stable. This is the same high-AMI, stable 2-community partition that was discovered whenK was left unconstrained (since as we’ve seen before, stability of a partition persists when restricting focus to a smaller subset of partitions).