Schematic illustration of our method for a simple network. (*A*) Example network. (*B*) Modularity landscape. For the example network, there are 15 distinct groupings of nodes into modules. Each large colored circle represents a partition, which we draw inside the circle, with different colors indicating different modules. For clarity, we label each partition with a number from 1 to 15. The color of the partition circle indicates the modularity of that partition following the color code on the bottom right-hand side of the diagram. For simplicity, we consider only single node changes; thus, we connect two partitions, for instance 1 and 2, because the change of a node to a new module in partition 1 generates partition 2. The arrows show the direction of increasing modularity. Local maxima correspond to those partitions that do not point to any other partition; that is, the change of a single node does not increase the modularity. In the example, there are two local maxima: partition 1 and partition 15. To illustrate the concept of basin of attraction, we show next to each partition a colored bar (black and white) that represents the probability that a walker that starts from, for instance, partition 2 and only moves to partitions with larger modularity ends up in either of the local maxima. We use white to indicate partition 15 and black to indicate partition 1. (*C*) Coclassification matrix. We show the number of times two nodes are classified in the same module, starting from a random partition. Note that nodes a, c and b, d are always classified together because they are in the same module in both local maxima (partitions 1 and 15). In contrast, nodes a and b are only in the same module for one of the maxima (partition 1); therefore, the coclassification is lower than one, but larger than zero. (*D*) Comparison with randomized networks. In this case, this is the only network that one can build keeping the same degree distribution and not allowing for self-loops. Therefore, the average modularity for the local maxima of the randomized networks and that of the network under analysis are the same. Thus, our conclusion is that this network has no internal organization. (*E*) Representation of the hierarchical organization for the example network. We show the ordered coclassification matrix on the *Left*, and on the *Right* is the tree showing the organization of the nodes into modules. In this case, the network has no significant structure; thus, we show a bar of a single color indicating that there is a single module. Note that a modularity maximization algorithm would have a certain chance (the probability depending on the specific algorithm) of finding partition 15 as the optimal partition and would thus conclude that the network does have a modular structure.

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