Table 1.
The different types of messages mi,h used in the AnB algorithm.
Fig 1.
Schematic flowchart depicting the finite state machine of each node of the network executing the AnB algorithm.
Note that the colors of the circles correspond to the colors of the section in Algorithm An overview of the AnB algorithm. Also, the steps outlined in the yellow box are carried out by all nodes irrespective of their state.
Fig 2.
Demonstration of the AnB algorithm on a typical network.
Panel (A) shows the initial state of a network of size 13. Panels (B) through (E) show the successive steps in the execution of the aggregate phase of the algorithm. The nodes in Active, Leaf, Residue and Inactive states are shown in red, green, cyan and grey colors respectively. The bracket beside each node shows the id i of the node, the number of its active neighbors ei, and its current local count ci, as an ordered tuple. For inactive nodes, only the id of the node is shown because the quantities ei and ci are not required in the inactive state. Once the network reaches a state where only residue and inactive nodes are present in the network (as seen in Panel E), the AnB algorithm enters its broadcast phase and the individual local counts of the residue nodes are broadcast throughout the network.
Fig 3.
Numerically estimated time costs of the AnB algorithm.
The left panel shows, on a log-linear scale, the total number of iterative steps taken by the AnB algorithm for different random networks (solid lines). The dashed lines show the scaling of the time for the All-2-All method, which corresponds to the network diameter D from Table 3. The diameter is known up to a scaling factor, here we report curves scaled to values comparable to AnB’s execution time to ease the comparison. In fact, the intersection of same-colour curves indicates that for large networks, the AnB algorithm is asymptotically slower than the All-2-All method. This is the case for all the analyzed network topologies but the Random Geometric networks. In RG networks, All-2-All shows a steeper curve that would slow down the process for very large networks (see inset on a log-log scale). The right panel shows the fraction of residue nodes in the network. Low x implies low r and hence better performance of AnB algorithm in terms of memory and communications cost (see Table 2). For each network size, we report the average results for the simulation of 1,000 independent random networks. (95% confidence intervals are reported in the left panel as shades but often are smaller than the line width).
Table 2.
Exact costs for the two algorithms for a general network with diameter D, average degree d, and r residue nodes.
For memory cost, we indicate the individual degree di for the generic node i. The AnB algorithm is more efficient than the All-2-All and the ST methods in terms of memory and communication. Analytical solution for time is out of reach and we provide numerical results in Fig 3.
Table 3.
The analyzed networks.
Fig 4.
The AnB is the most efficient algorithm in terms of communication and memory costs, compared with the All-2-All and ST algorithms.
The left panel shows the total number of messages sent by the nodes. The right panel shows the corresponding memory requirements per node with average connectivity degree d. In both panels, the dashed lines show the scaling for the All-2-All and ST algorithms, whereas the solid lines of various colors show the scaling for the AnB algorithm. Note that the number of messages sent and the memory requirements depends only on the network size for All-2All and ST algorithms and hence, are independent of the network topology. However, the number of messages sent and the memory requirements for AnB algorithm depends on the number of residue nodes which in turn depends on the topology of the network. Therefore, their dependence on the network topology is also explicitly shown.