Figure 1.
An example of the network representing method.
A: A random 3-regular graph of six vertices, where each vertex has three neighbors randomly selected. B: A snapshot of the process of BFS: after vertex 3 has been explored, the pointer of QueueT moves to vertex 2. We explore the neighbors of 2 in a random order 3, 5, 6. Only untouched vertex 6 is pushed into QueueT and assigned coordinates (5/6, 2/6). To preserve all linking information of 2, we push the copies of 3, 5 and 6 into QueueG and assign them coordinates (5/6, 2/6), (6/6, 2/6) and (7/6, 2/6), respectively. Then the pointer moves on to 4. C: BFS-tree. D: BFS-graph, we highlight the copies in black for their first appearances in QueueG. The line with one right angle represents an edge connecting two vertices or copies. For example, in panel D, polylines (2/6, 1/6)-(2/6, 2/6)-(6/6, 2/6) and (4/6, 1/6)-(4/6, 4/6)-(12/6, 4/6) represent an undirected (bidirectional) edge connecting two vertices 2 and 5. So a vertex can see all its neighbors through a mirror placed on the line Y = X. The dotted polylines (red) represent a pathway 3 - 4 - 1.
Figure 2.
Diagrams of a random -regular graph of size
and
.
A: BFS-tree, where vertices are closely located around the curve . Each small square (green) represents the last vertex of its tree level of the BFS tree. B: BFS-graph, where copies of vertices are closely located around the curve
. In the two diagrams, the shaded areas (yellow) represent the edges, and the polylines with right angles (red) represent a same shortest path between the root and a destination node.
Figure 3.
BFS-trees, BFS-graphs and auxiliary views of four example networks.
Random regular graph (RRG, ), Poisson-distributed random graph (PoissonRG, average degree
), LERRG (
,
,
) and power-law distributed random graph (PLRG,
,
,
,
) with edges not shown. In BFS-trees (panel A) and BFS-graphs (panel B), each solid line represents the vertices or copies resulted from one run of BFS on the associated network of size
, and the dots are the theoretical values. C:
, the expected tree degree of a vertex on BFS tree varies with its position
in QueueT. D:
, the expected graph degree. E: Here the expected search efficiency
is defined as
measuring the efficiency of a vertex exploring new ones through its edges (the
of vertices with one degree are set to zero). In panels C–E, the tiny dots are sampled uniformly from simulated results averaged over
runs of BFS on the associated networks, and the lines are the analytic results.
Table 1.
Classification accuracy (%) for four network models.
Figure 4.
Measure the degree of local clustering.
A: The first few steps of a BFS on a two-dimensional lattice. The blue, pink and white vertices stand for the explored, pending and untouched vertices, respectively. B: A BFS on a 4-RRG. C: Search efficiency. D: Search efficiencies of a network (lattice) with its random counterpart (RRG) vary with the vertices' position in QueueT. E: Average shortest path length
, average clustering coefficient
and
vary with the random rewiring probability
for a family of small-world networks, where the
transforms a regular ring lattice (
and each vertex has
nearest neighbors) to random graphs from
to
. Each data point is averaged over
random realizations of the rewiring process, and have been normalized by the values
,
and
of the regular lattice. F: Clustering coefficient
and
vary with
, the dimension of regular lattice. Each data point is averaged over
realizations of lattices with network size
.
Figure 5.
The degree distribution of a BFS tree in a LERRG (,
).
The power-law behavior extends up to a cutoff at degree
. The hollow dots are results from one numerical simulation on a network of size
, and the solid dots are our analytic results.
Figure 6.
Comparisons between the model networks and the Drosophila PPI network for .
A–C: BFS-graphs. In each diagram, the thick red curve represents Drosophila's network, and the thin blue curves represent the generated model networks. D: The size distribution of the giant components of the
DMC networks. In Drosophila's network,
of the vertices are in the giant component (red vertical bar). E: Graph distance distributions. Each vertical bar represents a median graph distance
which is a value separating the closer half from the farther half to the center graph, i.e., the Drosophila network. F: Degree distributions. Each distribution of the three models is averaged over the
generated networks. Although their degree distributions are similar to that of the Drosophila, their curves vary widely.
Figure 7.
Robustness test against noises for Drosophila PPI network ().
A fraction of edges in Drosophila network are replaced by random ones (noise1, panel A) or randomly rewired while maintaining the degree distribution of the original network (noise2, panel B). Classify the noised network as one of the four classes which has the closest median graph distance. Each data point is averaged over different realizations of the randomization procedure. As validation, the networks are confidently classified as a PoissonRG with the increasing of noise1.
Table 2.
Median graph distance between the model networks and the Drosophila PPI network for different confidence thresholds
, and the model with the minimal distance wins.
Table 3.
The Drosophila network is classified as a DMC network over DMR and LPA by a network classification method based on subgraph census [29].