Abstract
Background
Understanding the structure of complex networks is a continuing challenge, which calls for novel approaches and models to capture their structure and reveal the mechanisms that shape the networks. Although various topological measures, such as degree distributions or clustering coefficients, have been proposed to characterize network structure from many different angles, a comprehensive and intuitive representation of large networks that allows quantitative analysis is still difficult to achieve.
Methodology/Principal Findings
Here we propose a mesoscopic description of large networks which associates networks of different structures with a set of particular curves, using breadth-first search. After deriving the expressions of the curves of the random graphs and a small-world-like network, we found that the curves possess a number of network properties together, including the size of the giant component and the local clustering. Besides, the curve can also be used to evaluate the fit of network models to real-world networks. We describe a simple evaluation method based on the curve and apply it to the Drosophila melanogaster protein interaction network. The evaluation method effectively identifies which model better reproduces the topology of the real network among the given models and help infer the underlying growth mechanisms of the Drosophila network.
Conclusions/Significance
This curve-shaped description of large networks offers a wealth of possibilities to develop new approaches and applications including network characterization, comparison, classification, modeling and model evaluation, differing from using a large bag of topological measures.
Introduction
Networks have been widely used as a concise mathematical representation of the structure of systems with interacting objects [1]–[4]. Protein-protein interaction networks, brain networks, scientific collaboration networks, the Internet and the World Wide Web are a few examples.
Decades ago, the study of graph theory focused on the analysis of small networks, or regular graphs such as a lattice. One could easily lay out the network on a piece of paper and visually investigate its features. However, real-world networks studied in recent years often involve thousands or millions of vertices and edges. Networks on this scale cannot be easily represented in a way that allows quantitative analysis to be conducted by eye [5]. Instead of network drawing, the current understanding of network structure relies mainly on specific properties, measures or statistics, such as degree distributions [6], [7], community structure measurements [8]–[10], or motif counts [11]. But one may note that specific properties characterize the structure of networks point-by-point. We are used to carrying a large bag of measures to describe a network. A good description or representation of network which holds more complete topological information in one bag may provide a clear intuitive understanding of network and reflect some special structural features, such as the curved landscape of the World Wide Web [12], cartographic representation of complex networks [13] and circular perspective drawings of protein interaction networks [14].
With this view in mind, we propose a mesoscopic description of large networks by using breadth-first search. It serves as a bridge linking networks of different structures with a set of particular curves. We use curves of this kind to represent the corresponding networks and refer to them as the characteristic curves. Then we apply this curve shaped description to both random graphs and lattice embedded random regular graphs, and derive the expressions of their curves. The curve expression possesses a number of network properties in one bag, such as the size of the giant component and the local clustering. Interestingly, it shows that not only homogeneous random graphs appear to have a power-law degree distribution 
under traceroute sampling [15], [16], but a small-world-like network also does.
Moreover, characteristic curves or functions shaped by network structures can be used to compare networks comprehensively, e.g., the mesoscopic response function [17] resembling fingerprints. The network structural comparison has many applications. A useful one is to evaluate how well a network model fits a real-world network by comparing the network generated by the model with that of the real world. In recent years, network modeling has been attracting tremendous attention. Various models have been proposed to reproduce the topology of the real-world networks to infer their underlying growth mechanisms. Among the notable ones are the preferential attachment model [18], [19] and the small-world model [20]. Even a specific real-world network often has a variety of well-fitting models. Take protein-protein interaction (PPI) networks as an example, there are multiple models of widely varying mechanisms (e.g. [21]–[25],) that perfectly fit the real PPI data in terms of selected network properties, such as the degree distributions or the clustering coefficients. However, questions arise: among so many good models, which one best reproduces the structure of the real data? Which one best reveals the underlying growth mechanisms? It's clear that comparing the well fitted network properties mentioned above is not sufficient to identify the best-fitting model. It needs a discriminative method for network comparison to evaluate the fit of the models to the data.
Recent studies of structural comparison for PPI networks show that the comparison methods based on local structural properties, such as graphlet counts [26]–[28] or subgraph census [29], have a strong power in discriminating the differences between networks. However, the methods paying too much attention on local network properties may fail to distinguish some obvious global differences between two networks (see section “Evaluation Results” for detailed discussions), and they usually require a large amount of computation time and will be computationally infeasible for large networks with high average degree.
To deal with these issues, we use a fast method to compare large networks that works by comparing their characteristic curves, which are shaped by both the local and global structures of the network. First, we introduce a simple graph distance to evaluate the structural difference between two networks by comparing their curves. The graph distance can then be used to evaluate the fit of a network model to the real data. We apply this evaluation method to the Drosophila melanogaster PPI network [30] along with three network models, including linear preferential attachment model [19] and two biologically motivated network models [21], [22]. The evaluation results then determine which model better reproduces the topology of Drosophila's network. We also compare our results with that achieved by a method using subgraph census and machine learning techniques [29]. And at the same time, we examine the strengths and weaknesses of the two methods.
Methods
In this section, we first describe a network representing method. Then we apply the method to random graphs and lattice embedded random regular graphs, and derive the expressions of their characteristic curves. For the structural comparison between large networks, we introduce a graph distance based on the curve, and apply it to the Drosophila PPI network to evaluate the fit of the selected models to it.
Network Representing Method
Consider a network of 
vertices and 
edges (the terms network/graph, vertex/node and edge/link are interchangeable in this paper). For the convenience of description, we assume that the network is undirected and connected in this section, i.e., every edge in the network is undirected and every pair of distinct vertices can be connected through some path. The proposed representing method is based on the algorithm of breadth-first search (BFS) [31], where the root vertex is selected by taking one end of a randomly chosen edge (different root selection schemes yield different outputs, the affects of root selection are discussed in details in section 3 in Supporting Information S1). One can consider the process of BFS as exploring the graph one vertex at a time in the order of first touch, first explore. At the beginning, the root vertex is labeled pending, and all other vertices are untouched. As an ongoing process (see Figure 1B), a pending vertex will be explored and all its untouched neighbors will be labeled pending and pushed into a queue named QueueT in a random order. Each of them is assigned a position
which is the ratio of its sequence in the queue to 
, and stores 
, the position of its parent who brings it to the queue, i.e., who touches it at first during the process of search. Taking these two sets of positions as the coordinates 
of the vertices, the search tree is mapped into a two-dimensional plane (see Figure 1C) and we refer to it as BFS-tree, where each edge is represented by a straight line with one right angle and parallel to each other.
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.
Note that the BFS-tree is not a full representation of the original graph since it has lost too many edges. To get the full linking information, we now record all links of the graph during BFS. Create 
copies for each vertex of degree 
, and replace each undirected edge with two opposite directed edges connecting two copies owned by the corresponding vertices. Unlike QueueT which only accepts untouched neighbors of the vertex on exploring, another queue named QueueG accepts the copies of all its neighbors to preserve full linking information (see Figure 1B). Meanwhile, it is similar to the vertices of QueueT that each copy of QueueG is assigned a position 
(the ratio of its order in QueueG to 
) and stores 
(the position of its parent copy). Thus the coordinates 
help to map a network into a two-dimensional plane (see Figure 1D) which is referred to as BFS-graph.
Both the BFS-tree and BFS-graph are in the two-dimensional plane, and every vertex or copy can see its neighbors through a mirror placed on the line 
or 
. By associating vertex and edge with optical element and light beam, respectively, such a simple layout has potential applications in manufacturing large-scale optical networks. For a large network, as illustrated in Figure 2, the global picture becomes very clear where the vertices or copies line up, and automatically forms a particular curve. Since the BFS-graph holds more linking information than the BFS-tree, we here use the curve of the BFS-graph to represent the corresponding network and refer to it as the characteristic curve.
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.
Characteristic Curves
It is desirable to find the exact expressions of the characteristic curves for various networks, and see whether the curves indeed identify networks of different structures. To proceed, let us first track the states of QueueT and QueueG. During the process of BFS, network is explored one vertex at a time (can also be explored one edge at a time, the conclusions are consistent, see section 1 B in Supporting Information S1 for details). Consider a vertex 
to be explored at time 
has graph degree 
, and also 
is 
's position in QueueT. After 
is explored at time 
, it has one parent and 
newly touched children, where 
is 
's degree on the search tree. The states of QueueT and QueueG change as follows, probing the linking information of network:
 |
(1) |
where 
is the number of vertices that QueueT holds and 
is the number of copies that QueueG holds right before exploring 
at time 
. In the proposed representing method, each vertex or copy is assigned a coordinates 
or 
which records the positions of it and its parent. Thus, when the network is explored one vertex at a time, Eq.1 can be written as:
 |
(2) |
where the initial values of 
and 
are all zeroes, and 
increases at a rate of 
per time step. Hence, knowing the values of every vertex's graph degree 
, tree degree 
and its position 
in QueueT are crucial for the derivation of the curve expressions.
We then apply this approach to two undirected networks. One is random graphs with arbitrary degree distributions, including random regular graph (RRG), Poisson-distributed random graph (PoissonRG) and power-law distributed random graph (PLRG). The other is lattice embedded random regular graph (LERRG) which is not only similar to many real-world networks, but also has practical applications. We use 
and 
to represent the function of the tree curve and graph curve, respectively, where root vertex is in the giant component of the graph (a giant component is a connected subgraph that contains a majority of the entire graph's vertices). In general, 
and 
are nondecreasing and satisfy: 
, 
, 
and 
, where 
is the average degree of the graph. The smallest positive root of 
is just the size of the giant component.
Random Graphs with Arbitrary Degree Distributions
Suppose the degree distribution of a random network is 
, defined as the probability that a randomly chosen vertex has 
edges. Meanwhile, consider the network is obtained from the configuration model [3]: create 
copies for each vertex of degree 
, and then choose pairs of these copies uniformly at random and connect them to form the edges. Such network is a multi-graph with self-loops and multiple edges permitted. To derive the curve expressions of BFS-tree and BFS-graph for this network, as Eq. 1 shows, we should at first know the values of 
and 
varying with 
.
During the process of BFS, QueueT accepts newly touched vertices one by one and assigns them positions. The term 
stands for the number of edges possessed by a vertex with position 
. To trace the value of 
varying with 
, consider a situation when QueueT has accepted 
vertices and is going to accept a new one 
. The new vertex 
will be pushed into QueueT and assigned position 
, our goal is to find 
's degree 
.
Vertex 
is selected from the 
untouched vertices. Because in a random network, the copies of vertices are coupled uniformly at random, the probability of vertex 
having degree 
is proportional to 
, where 
is the degree distribution of the 
untouched vertices. The distribution 
varies with 
when QueueT obtains untouched vertex one by one. For the technical convenience to describe the relationship between 
and 
, we use 
to represent 
, where 
is a variable changes as a function of 
: 
. Let
 |
(3) |
where 
(note that 
and 
, which is the average degree of the graph). Then we arrive at the distribution 
, where 
changes as a function of 
in the limit of large 
(the term 
is omitted):
 |
(4) |
Let 
be the expected graph degree of the newly touched vertex 
. Since the probability of vertex 
having degree 
is proportional to 
, we can write:
 |
(5) |
Next, we trace the value of the tree degree 
. Suppose 
vertices have been touched before exploring a vertex 
with position 
. In the limit of large 
, the expected number of untouched vertices that 
will meet through its 
edges (except one edge connecting its parent) is:
 |
(6) |
where 
is the total number of edges, see section 1 A in Supporting Information S1 for the detailed explanation of this equation. This equation is also valid for random graphs with extremely dense edges (
), which have numerous self-loops and multi-edges (see section 1 B in Supporting Information S1 for details).
In the limit of large 
, we use a mean-field approximation where 
and 
are represented by their expectations 
and 
, respectively. Substituting Eqs. 2 and 5 into Eq. 6 and associating it with Eqs.3 and 4, the curve function 
of BFS-tree satisfies (see section 1 C in Supporting Information S1 for the detailed derivation):
 |
(7) |
where 
, 
. 
is the smallest positive root of 
. Note that 
is simply the size of the giant component of the graph, which is consistent with the size derived in different forms by Molloy and Reed [32] and Newman et al.
[33] for random graphs with arbitrary degree distributions.
From Eqs.3 and 4, we get 
, substituting this into Eqs. 2 and 5, the curve function 
of the BFS-graph satisfies:
 |
(8) |
where 
, 
. As mentioned above, 
is the size of the giant component and 
is the smallest positive root of 
. When 
reaches 
, the BFS explored all vertices in the giant component and the mapping comes to the end (we here only consider the curves of the giant component since it retains the significant structural features of the graph).
As examples, we now introduce three commonly studied graphs.
- Poisson-distributed random graphs. This is one of the best studied graph models [34], and is also known as Erdös-Rényi random graph that has a Poisson degree distribution in the limit of large graph size, as given by
. The curve functions are (see section 1 D in Supporting Information S1 for the detailed derivation):
where
(10) 
, and 
is the smallest positive root of 
. 
is Lambert's function, defined as 
where 
. - Power-law distributed random graphs. It was found that a wide range of real networks, such as the Internet and science collaboration graph, display power-law degree distributions, also known as scale-free networks [1]. In Figure 3, we only consider a random graph possessing a power-law degree distribution given by
where
is a constant and 
. 
and 
are the minimal and maximal degree of the graph, respectively. The curve expressions are the same as Eqs.7 and 8.
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.
Lattice Embedded Random Regular Graphs
A graph of this type is formed from a superposition of an 
-RRG and a 
-dimensional finite lattice with periodic boundary conditions, i.e., each vertex has 
nearest lattice neighbors and 
long-range neighbors chosen uniformly at random from the lattice. This is similar to the small-world model proposed by Watts and Strogatz [20], in which there are many local links and a few long-range links connecting local clusters together. These links lead to both small path lengths and high clustering called small-world property and have been observed in a wide range of real-world networks, such as the collaboration graph of film actors and the power grid. Moreover, the LERRG is not only similar to a number of real-world networks, it also has practical applications. For example, Korniss et al.
[36] and Guclu et al.
[37] found that two typical graphs of LERRGs have remarkable advantages in constructing a parallel discrete-event simulation scheme since the processing elements can carry out the tasks distributed on them at a nonzero, near-uniform rate without requiring global synchronization.
In the LERRG, 
, and in the limit of large network size 
, 
, where 
is the largest real root of 
(see section 2 in Supporting Information S1 for the detailed derivation). In association with Eq. 2, the curve functions are:
 |
(11) |
Interestingly, they have a similar form as that of RRGs (Eq.9), and are consistent with Eq.9 when 
.
Graph Distance
Each of the example networks studied above corresponds to a particular curve. We here use the curve as a discriminating feature for network comparison. To evaluate the structural difference between two networks, we describe a simple graph distance 
by comparing their curves
 |
(12) |
where 
represents the characteristic curve. 
and 
stand for the curves of a pair of graphs to be compared. The distance 
is simply the area between the two curves. Note that Onnela et al. define a graph distance based on mesoscopic response function in a similar fashion and performs well for network taxonomy [17]. Because the BFS-graph holds more linking information than the BFS-tree, we chose the curve of the BFS-graph to calculate the difference. The two-tuple 
is the coordinates of vertex's copy in the BFS-graph, and 
increases at a rate of 
, where 
is the total number of edges in the graph. To align two graphs with different average degrees 
, we assign 
to 
until 
reaches 
, that is, until the BFS has explored all vertices in the giant component. To ease the calculation of the distance 
between two graphs with different sizes of the giant components, we assign 
to 
when the value of 
exceeds 
. We only consider the giant component since it retains the significant structural features of the graph. For graphs which consist of small isolated groups of connected vertices, that is, whose giant components are too small (e.g., 
) to represent the significant structural features of the entire graphs, the distance 
is not suitable to measure the structural difference for them.
As an example, we use 
to evaluate the differences between the four example graphs in Figure 3. If we take RRG as the center graph, PoissonRG is the most similar graph with 
from the RRG. The LERRG is the second similar with 
and PLRG is the most different with 
. The results agree with the common understanding of the four types of graphs.
Data Set
We use a protein-protein interaction data derived from Drosophila melanogaster based on yeast two-hybrid screening [30]. A PPI network can be constructed from the data by taking proteins as vertices, and observed interactions between proteins as undirected edges. The degree or connectivity of a protein is defined as the number of its interaction partners. Because the data has numerous false positives, Giot et al.
[30] assign each interaction a confidence score 
, measuring how likely the interaction occurs in vivo. To exclude unlikely interactions, they suggest a confidence threshold 
. An edge appears only if its confidence score 
. We also present results for a higher threshold 
which is suggested by Middendorf et al. in ref. [29], and 
which includes all interactions observed. After removing the multiple edges and self-loops from the network [38] and eliminating isolated vertices, the resulting networks consist of 3,279/4,508/6,823 vertices and 2,728/4,569/19,630 edges for 
, respectively.
Network Models
We select three network models and compare their generated networks with that of the Drosophila to determine which model better describes the evolutionary processes of the Drosophila. The first two models are biologically motivated, and have been argued as the best two models to reproduce the Drosophila network among seven candidate models [29] including the linear preferential attachment model and the small-world model. The last one is the linear preferential attachment model. All the three models start with a small seed graph and grow the network one vertex at a time following these steps:
Duplication-mutation-complementation model (DMC)
The model proposes a gene duplication followed by mutations (divergence) which preserve functional complementarity [22]. At each time step, a new vertex 
is added. It then chooses an existing vertex 
at random, and copies all links of 
, i.e., places edges between 
and all neighbors of 
. For each pair of their links connected to a same neighbor 
, one randomly selects one of the two links (
, 
) or (
, 
) and deletes it with a probability 
. It ensures that if one of the duplicate genes loses one of its functions (links), the other preserves the same function (the link to the same neighbor). The duplicate pair 
and 
are themselves connected with a probability 
, representing an interaction of a protein with its own copy. The parameters 
and 
are sampled uniformly in 
.
Duplication-mutation using random mutations model (DMR)
The model has a different duplication algorithm from that of the DMC. It emphasizes the creation of new advantageous functions by random mutations in gene and neglects possible interactions between duplicate pairs [21]. At each time step, a newly added vertex 
chooses an existing vertex 
at random, and copies all links of 
. For each link of 
inherited from 
, one deletes it with a probability 
. New links can be created between 
and any other existing vertices with a probability 
, where 
is the total number of existing vertices, introducing new viable interactions between proteins. The parameters 
and 
are sampled uniformly in 
.
Linear preferential attachment model (LPA)
At each time step, a newly added vertex preferentially attaches to existing vertices with probabilities proportional to their degrees [19]. This simple probabilistic model can give rise to scale-free degree distribution which is one of the most important features that many real-world networks exhibit, including the PPI networks.
Network Classification Method
Given a network 
and a set of network classes, a network classifier should find which class the network 
belongs to. The graph distance (Eq.12) can be used to design a simple and efficient network classifier. Consider a given set of network classes which is composed by network instances, that is, each class possesses a number of networks. If the given set of network classes is composed by network models, we generate a certain amount of network instances for each class. For each of the network classes, the classifier calculates the graph distance 
between 
and every network in this class. We simply use the median graph distance 
to represent the distance between 
and the class, where the median graph distance is a value separating the closer half from the farther half. Finally, the classifier obtains all the median graph distances and classifies 
as the class which has the minimal 
from 
.
To validate the proposed classification method, we use four network models, including the DMC, DMR, LPA and PoissonRG, by following steps. First, generate 
instances for each of the four models and obtain their graph curves. Second, build a network 
by using one of the four models, and calculate the graph distance between 
and the 
graphs generated in the first step. Classify 
as the class which has the minimal 
from 
. Repeat the second step 
times for each of the four models, and we obtain a classification accuracy table at last (see Table 1).
Table 1. Classification accuracy (%) for four network models.
| Classification | ||||
| Original | DMC | DMR | LPA | PoissonRG |
| DMC | 99.0 | 1.0 | 0.0 | 0.0 |
| DMR | 3.4 | 95.7 | 0.0 | 0.9 |
| LPA | 0.0 | 0.0 | 100.0 | 0.0 |
| PoissonRG | 0.0 | 0.0 | 0.0 | 100.0 |
The (i, j) entry is the probability of classifying class j given that the original class is i. The networks built by models are based on the size of the Drosophila protein network with a confidence threshold of 
.
The overall classification accuracy is high, around 
, and most of these networks can find back their generative models. Classification errors among DMC, DMR, and PoissonRG networks are due to equivalence of the models in specific parameter regimes and correspondingly show overlaps. For example, when the growth parameter 
of a DMR network approximates to zero, the growth of such a network is dominated by the duplication mechanisms, which is similar to that of the DMC model. Therefore, a small fraction of DMR networks are classified as DMC.
To test the robustness of our classification method against noise, we carried out a sensitivity analysis by perturbing the structure of the original networks by using two kinds of edge random mechanisms [17], [29]. The first is to replace some percentage of original edges in the network by random ones (noise1), and the second is to randomly rewire some percentage of edges while maintaining the degree distribution of the original network (noise2). The numerical results show that the classification performs well for small and intermediate amounts of the noises on the DMC, DMR and LPA networks (see section 4 in Supporting Information S1 for details). Meanwhile, the robustness again the second noise is better than the first one since the second noise maintains the degree distribution of the original network.
Results and Discussion
Properties of Graph Curves
As an example shown in Figure 3, the characteristic curves coupled with auxiliary views identify networks of different topologies and reflect several local and global structural features. Among the four example networks with close average degree 
, PLRG is the most special because it has an inhomogeneous degree distribution, where a small fraction of vertices (hubs) are richly connected while many other vertices are not. At an early stage of BFS on PLRG, a small fraction of vertices with high degree are firstly touched. They explore the majority of vertices and leave few opportunities for latter vertices to touch new ones. The 
, 
and 
of PLRG decline with 
much faster than those of the other three homogeneous networks, in which the vertices have approximately the same number of edges. Such decline of LERRG is the slowest due to its high local clustering, where 
and 
. The two homogeneous random graphs RRG and PoissonRG are the most similar.
Now we turn to characterize the structure of local clustering by the use of search efficiency 
. It is known that a highly clustered group of vertices has more links between them than expected by chance. A simple effect of such a structure related to BFS is that, the search explores many links but harvests less new vertices (see an example in Figure 4 A). In contrast, the search on a random graph gets more new vertices with the number close to the links explored (see Figure 4 B). Therefore, the search efficiency 
of a vertex in a lattice is smaller than that in a random graph at the first stage of search process.
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 
.
Furthermore, observe that the search efficiency 
of a lattice or an LERRG is lower than that of its random counterpart (a random network with the same degree distribution allowing self-loops and multiple edges, here it is an RRG) at the early stage of the search process, but becomes larger than its counterpart later (see Figures 3 E and 4 D). That is, although the search process catches less new vertices in a clustered network than its random counterpart at first, it still has chance to meet new ones much later for its local clustering structure.
Guided by these observations, we conjecture that the larger the difference of 
between a network 
with its random counterpart 
, the higher the degree of local clustering of 
is. We then use a relative difference of 
between 
and 
to measure the degree of local clustering of 
:
 |
(13) |
which is simply the area between two 
variation curves of 
and 
(see Figure 4 D for an example, corresponds to the shaded area between two curves), normalized by the sum of their areas.
To validate this measure, we apply it to small-world networks [20] and regular lattices which are known to have high local clustering, finding that the measure performs similar to the average clustering coefficient for small-world networks (see Figure 4 E), and can catch the local clustering of lattice network where its average clustering coefficient equals to zero (see Figure 4 F).
Next, we consider a series of characteristic curves:
 |
(14) |
where 
and 
. Both RRGs and LERRGs fall into this category (Eqs.9 and 11) though they are very different since the latter have high local clustering (
) but the former have none (
).
Eq. 14 can illustrate what the associated networks look like under a single-source, all-destination traceroute sampling. In the study of Internet mapping, traceroute sampling is widely used to infer the topology of the Internet, typically by collecting paths from a small number of sources to a large number of destinations through the network. However, Lakhina et al. [39], Petermann and De Los Rios [40] independently showed that traceroute sampling can significantly bias the observed degree distribution since it only samples a fraction of links. In particular, they found that the sampled subgraphs have power-law degree distributions while the substrate networks are Poisson distributed. Later, Clauset and Moore [15] presented an analytical approach to derive the power law observed in ref. [39]. Achlioptas et al. [16] and Dall'Asta et al. [41] studied the bias of traceroute sampling analytically and systematically for random graphs. Interestingly, Achlioptas et al. found that RRGs also have apparent power laws under traceroute sampling.
Here we find that even a LERRG, which is small-world-like and homogeneous, appears to have a power-law degree distribution 
under traceroute sampling (see Figure 5). Since this sampling essentially generates a BFS tree under the common assumption that Internet routing protocols approximate shortest paths [15], [16], we turn to calculate the degree distribution of the BFS-tree. Use 
to represent the expected tree degree of a vertex with position 
in QueueT. Eqs. 2 and 14 give
 |
(15) |
Since 
is a monotonic decreasing function, a rough estimate of the tree degree's density 
can be given by only considering the expected tree degrees during the search process
 |
(16) |
substituting from Eq. 15 and letting 
be the tree degree give
 |
(17) |
where 
. In the limit of large 
, 
. For RRGs with 
, this approximate result agrees with a more rigorous one derived by different means by Achlioptas et al.
[16]. Furthermore, our result is valid not only for RRGs but also for other networks described in Eq.14, including LERRGs. Therefore, even a LERRG, which is small-world-like and homogeneous, displays a power-law degree distribution 
(in the limit of large 
) under traceroute sampling.
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.
Evaluation Results
Comparing large networks by their graph curves gives an intuitive understanding of the topological differences between the networks of Drosophila and each of the three models (see Figure 6, Table 2, and section 5 in Supporting Information S1 for more details). The results suggest that the DMR model better reproduces the topology of Drosophila's network than the DMC and LPA for high confidence thresholds 
. To test the robustness of this result, we artificially introduce two kinds of noises into the original Drosophila network (
), finding that the result still holds for small and intermediate amounts of the noises (see Figure 7).
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.
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.
 = 0.65 |
 = 0.5 |
 = 0.0 |
||||
| Rank | Model |
|
Model |
|
Model |
|
| 1 | DMR | 0.0162 | DMR | 0.0183 | LPA | 0.0351 |
| 2 | LPA | 0.0230 | DMC | 0.0851 | DMR | 0.0651 |
| 3 | DMC | 0.0310a | LPA | 0.0963 | DMC | 0.2181 |
For 
, the DMC network consists of small isolated groups of connected vertices. Its giant component is too small (only around 
of the vertices are in the giant component, nine times smaller than that of Drosophila, 
) to represent the significant structural features of the entire graph. Though we give the distance value, the graph distance is not suitable for this case.
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.
For the DMC networks, their characteristic curves are far from that of the Drosophila, a result which indicates that the structures of the DMC networks are very different from that of the Drosophila. This result is completely opposite to the result achieved by a method based on subgraph census [29], which suggests that the DMC best reproduces Drosophila's network among seven candidate models, including the DMR and LPA (see Table 3).
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].
 = 0.65 |
 = 0.5 |
|
| Rank | Model | Model |
| 1 | DMC | DMC |
| 2 | DMR | DMR |
| 3 | LPA | LPA |
These contradictory results are due to the different angles from which subgraph census and BFS-graph characterize the structure of a network, where the former focuses on the substructures of the network, while the latter cares about a global view of the network. Because subgraph census counts every occurrence of a set of small subgraphs in the network, it's clear that the census can reveal more local network properties than the BFS-graph. However, subgraph census so deeply concerns the local network properties that it may fail to distinguish some obvious structural differences between two networks. The most obvious difference between the DMC and Drosophila is that the size of the giant component of the DMC network is much smaller than that of the Drosophila for high confidence thresholds 
, where the former is around 
(i.e., 
of the nodes are in the giant component), while the latter is more than nine/three times larger, 
(see Figure 6 D, and section 5 in Supporting Information S1). For the higher confidence threshold 
, the DMC network consists of small isolated groups of connected vertices, a structure which is very different from that of the Drosophila. This failure of subgraph census implies that although the census knows every occurrence of the particular subgraphs in the network, it lacks a general assembly drawing of how these amounts of subgraphs are assembled into the original large network. The same amount of subgraphs may form a network different from the original network, resembling using the same building blocks to construct different buildings.
On the other hand, the BFS-graph presents a global view of the network by assembling the vertices one by one, which reflects a complementary aspect of the network to that reflected by the degree distribution and subgraph census. The degree distribution counts the degrees of all vertices and shows their distribution. Similarly, subgraph census counts the occurrences of a set of small subgraphs and shows their distribution. The two clearly know the amounts of the building blocks, but lack a general assembly drawing of how to assemble them into the original network. In contrast, the BFS-graph possesses the assembling information of the network through BFS, which strings up the vertices one by one from the bottom up, and at last, gives a global view of the network. Thus, the structural information reflected by BFS-graph and subgraph census complement each other. Applying both of them can provide a more comprehensive understanding of the network structure, which will improve the accuracy of the structural comparison.
Except for the DMC, the two methods based on BFS-graph and subgraph census agree well on the DMR and LPA, that is, the DMR better reproduces the topology of the Drosophila network than the LPA for 
. It is worth noting that the method based on BFS-graph (with time complexity 
) is fast for large networks with high average degree 
, for which the subgraph census (with time complexity at least 
) may be computationally infeasible. For example, subgraph census will cost a great deal of time for the Drosophila network when it includes all interactions observed (
), which has many more vertices and edges than that for 
. But the BFS-graph can quickly figure out the differences between the Drosophila network and the networks generated by the models (see section 5 in Supporting Information S1 for details). It shows that the fits of the three models to the data are relatively poor for 
(see Table 2), a result probably due to the presence of strong additional noise in the data when including low confidence value interactions.
In summary, none of the three models is simultaneously ranked as the best by both the methods based on BFS-graph and subgraph census, implying that there is still room for improvement for these models. The DMC gets a higher rank than the DMR and LPA when using subgraph census, a result indicating that the gene duplications that preserve functional complementarity and facilitate the connections between duplicate pairs are good at reproducing the substructures of Drosophila's network. When using the BFS-graph, the DMR has a closer graph distance to Drosophila's network than those of the other two for high confidence thresholds 
, a result showing that the gene mutations that create new interactions between proteins are important for keeping the global connectivity of the PPI networks. These results suggest that a model integrating several mechanisms might be able to fit the Drosophila PPI network more accurately.
Conclusions
We have presented a mesoscopic description of large networks which associates networks with a set of curves. Specific examples show that the curves can reflect a number of structural features commonly shared by a series of networks. Moreover, the curve can be used to classify networks and evaluate the fit of network models to real-world networks. After evaluating the fit of three network models to the Drosophila protein interaction network, we found that the model DMR better reproduces the topology of the Drosophila network than the DMC and LPA, although there is still room to improve the three models. We also compared our evaluation method and results with that of Middendorf et al.'s in ref. [29], where they identify the best-fitting model based on subgraph census, and found that the structural information reflected by characteristic curve and subgraph census complement each other. Applying the two together can provide a more comprehensive understanding of the network structure, which will improve the accuracy of the structural comparisons and model evaluations.
Using the characteristic curve, we preliminarily investigated the network properties and the fit of network models. Our further work will include the relationship study between the network structure and the curves, conditions that make this relationship one-to-one and the general algorithm (if there is one) that could recover the networks from the characteristic curves. With this algorithm and the well designed curves or functions one could generate networks with required topological features. The network describing method, in essence, utilized the process of BFS and depicted its trace on the network to capture network structure. Other processes such as random walks may also be useful to develop new approaches and applications including network characterization, comparison, classification, modeling and model evaluation.
Supporting Information
Supporting Information S1 includes the detailed derivations of characteristic curves of random graphs and LERRGs, discussion on the effects of root selection, robustness test of the network classification method and more network comparison results.
(PDF)
Acknowledgments
We thank Jianjun Ma, Aaron Clauset, Haijun Zhou, Yingxian Zhao, Xiaolin Shi, Chengbin Peng, Yujie Wan and Hongying Gu for their helpful corrections, comments and suggestions on the manuscript, and Shifa Su, Yixiao Li, Jie Chang, Bo Peng, Lifeng An and Agata Fronczak for useful discussions. Special acknowledgment is due to the anonymous reviewer's valuable comments and suggestions, and Yue Chen for her excellent lectures on Numerical Analysis. We benefited from the KITPC 2008 program “Collective Dynamics in Information Systems”.
Footnotes
Competing Interests: The authors have declared that no competing interests exist.
Funding: This work was supported by the National Science Foundation of China grants 61070069 and 60803110 (http://www.nsfc.gov.cn/). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
References
- 1.Albert R, Barabási AL. Statistical mechanics of complex networks. Rev Mod Phys. 2002;74:47–97. [Google Scholar]
- 2.Dorogovtsev SN, Mendes JFF. Evolution of networks. Adv Phys. 2002;51:1079–1187. [Google Scholar]
- 3.Newman MEJ. The structure and function of complex networks. SIAM Rev. 2003;45:167–256. [Google Scholar]
- 4.Boccaletti S, Latora V, Moreno Y, Chavez M, Hwang DU. Complex networks: Structure and dynamics. Phys Rep. 2006;424:175–308. [Google Scholar]
- 5.Newman MEJ, Leicht EA. Mixture models and exploratory analysis in networks. Proc Natl Acad Sci U S A. 2007;104:9564–9569. doi: 10.1073/pnas.0610537104. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 6.Albert R, Jeong H, Barabási AL. Internet: Diameter of the world-wide web. Nature (London) 1999;401:130–131. [Google Scholar]
- 7.Faloutsos M, Faloutsos P, Faloutsos C. On power-law relationships of the internet topology. ACM SIGCOMM Comput Commun Rev. 1999;29:251–262. [Google Scholar]
- 8.Girvan M, Newman MEJ. Community structure in social and biological networks. Proc Natl Acad Sci U S A. 2002;99:7821–7826. doi: 10.1073/pnas.122653799. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 9.Zhou H. Distance, dissimilarity index, and network community structure. Phys Rev E. 2003;67:061901. doi: 10.1103/PhysRevE.67.061901. [DOI] [PubMed] [Google Scholar]
- 10.Fortunato S. Community detection in graphs. Phys Rep. 2010;486:75–174. [Google Scholar]
- 11.Alon U. Network motifs: Theory and experimental approaches. Nat Rev Genet. 2007;8:450–461. doi: 10.1038/nrg2102. [DOI] [PubMed] [Google Scholar]
- 12.Eckmann JP, Moses E. Curvature of co-links uncovers hidden thematic layers in the World Wide Web. Proc Natl Acad Sci U S A. 2002;99:5825–5829. doi: 10.1073/pnas.032093399. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 13.Guimerμa R, Amaral LAN. Cartography of complex networks: modules and universal roles. J Stat Mech: Theor Exp. 2005;2005:P02001. doi: 10.1088/1742-5468/2005/02/P02001. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 14.Li W, Kurata H. Visualizing global properties of large complex networks. PLoS ONE. 2008;3:e2541. doi: 10.1371/journal.pone.0002541. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 15.Clauset A, Moore C. Accuracy and scaling phenomena in internet mapping. Phys Rev Lett. 2005;94:018701. doi: 10.1103/PhysRevLett.94.018701. [DOI] [PubMed] [Google Scholar]
- 16.Achlioptas D, Clauset A, Kempe D, Moore C. STOC '05: Proceedings of the 37th ACM Symposium on Theory of Computing. New York, NY, USA: ACM Press; 2005. On the bias of traceroute sampling: or, power-law degree distributions in regular graphs. pp. 694–703. [Google Scholar]
- 17.Onnela JP, Fenn DJ, Reid S, Porter MA, Mucha PJ, et al. A taxonomy of networks. 2010. Available: http://arxiv.org/abs/1006.5731v1.
- 18.de Solla Price DJ. Networks of Scientific Papers. Science. 1965;149:510–515. doi: 10.1126/science.149.3683.510. [DOI] [PubMed] [Google Scholar]
- 19.Barabási AL, Albert R. Emergence of scaling in random networks. Science. 1999;286:509–512. doi: 10.1126/science.286.5439.509. [DOI] [PubMed] [Google Scholar]
- 20.Watts DJ, Strogatz SH. Collective dynamics of “small-world” networks. Nature (London) 1998;393:440–442. doi: 10.1038/30918. [DOI] [PubMed] [Google Scholar]
- 21.Solé RV, Pastor-Satorras R, Smith E, Kepler TB. A model of large-scale proteome evolution. Adv Complex Syst. 2002;5:43–54. [Google Scholar]
- 22.Vázquez A, Flammini A, Maritan A, Vespignani A. Modeling of protein interaction networks. Com Plex Us. 2003;1:38–44. [Google Scholar]
- 23.Wagner A. How the global structure of protein interaction networks evolves. Proc R Soc Lond B Biol Sci. 2003;270:457–466. doi: 10.1098/rspb.2002.2269. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 24.Berg J, Lässig M, Wagner A. Structure and evolution of protein interaction networks: a statistical model for link dynamics and gene duplications. BMC Evol Biol. 2004;4:51. doi: 10.1186/1471-2148-4-51. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 25.Ispolatov I, Krapivsky PL, Yuryev A. Duplication-divergence model of protein interaction network. Phys Rev E. 2005;71:061911. doi: 10.1103/PhysRevE.71.061911. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 26.Pržulj N, Corneil DG, Jurisica I. Modeling interactome: scale-free or geometric? Bioinformatics. 2004;20:3508–3515. doi: 10.1093/bioinformatics/bth436. [DOI] [PubMed] [Google Scholar]
- 27.Pržulj N. Biological network comparison using graphlet degree distribution. Bioinformatics. 2007;23:e177–e183. doi: 10.1093/bioinformatics/btl301. [DOI] [PubMed] [Google Scholar]
- 28.Memisevic V, Milenkovic T, Pržulj N. An integrative approach to modelling biological networks. J Integr Bioinform. 2010;7:120. doi: 10.2390/biecoll-jib-2010-120. [DOI] [PubMed] [Google Scholar]
- 29.Middendorf M, Ziv E, Wiggins CH. Inferring network mechanisms: The Drosophila melanogaster protein interaction network. Proc Natl Acad Sci U S A. 2005;102:3192–3197. doi: 10.1073/pnas.0409515102. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 30.Giot L, Bader JS, Brouwer C, Chaudhuri A, Kuang B, et al. A Protein Interaction Map of Drosophila melanogaster. Science. 2003;302:1727–1736. doi: 10.1126/science.1090289. [DOI] [PubMed] [Google Scholar]
- 31.Knuth DE. The Art Of Computer Programming, volume 1. Boston: Addison-Wesley, third edition; 1997. [Google Scholar]
- 32.Molloy M, Reed B. The size of the giant component of a random graph with a given degree sequence. Combin Probab Comput. 1998;7:295–305. [Google Scholar]
- 33.Newman MEJ, Strogatz SH, Watts DJ. Random graphs with arbitrary degree distributions and their applications. Phys Rev E. 2001;64:026118. doi: 10.1103/PhysRevE.64.026118. [DOI] [PubMed] [Google Scholar]
- 34.Bollobás B. Random Graphs. Cambridge, UK: Cambridge University Press, second edition; 2001. [Google Scholar]
- 35.Wormald NC. Models of random regular graphs. In: Lamb JD, Preece DA, editors. Surveys in Combinatorics, 1999. Cambridge, UK: Cambridge University Press, volume 267 of London Mathematical Society Lecture Note Series; 1999. pp. 239–298. [Google Scholar]
- 36.Korniss G, Novotny MA, Guclu H, Toroczkai Z, Rikvold PA. Suppressing Roughness of Virtual Times in Parallel Discrete-Event Simulations. Science. 2003;299:677–679. doi: 10.1126/science.1079382. [DOI] [PubMed] [Google Scholar]
- 37.Guclu H, Korniss G, Novotny MA, Toroczkai Z, Rácz Z. Synchronization landscapes in small-world-connected computer networks. Phys Rev E. 2006;73:066115. doi: 10.1103/PhysRevE.73.066115. [DOI] [PubMed] [Google Scholar]
- 38.Yu J, Pacifico S, Liu G, Finley R. Droid: the drosophila interactions database, a comprehensive resource for annotated gene and protein interactions. BMC Genomics. 2008;9:461. doi: 10.1186/1471-2164-9-461. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 39.Lakhina A, Byers JW, Crovella M, Xie P. INFOCOM 2003: Twenty-Second Annual Joint Conference of the IEEE Computer and Communications Societies. Piscataway, NJ, USA: IEEE Press, volume 1; 2003. Sampling biases in ip topology measurements. pp. 332–341. [Google Scholar]
- 40.Petermann T, De Los Rios P. Exploration of scale-free networks: Do we measure the real exponents? Eur Phys J B. 2004;38:201–204. [Google Scholar]
- 41.Dall'Asta L, Alvarez-Hamelin I, Barrat A, Vázquez A, Vespignani A. Exploring networks with traceroute-like probes: Theory and simulations. Theor Comput Sci. 2006;355:6–24. [Google Scholar]
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Supplementary Materials
Supporting Information S1 includes the detailed derivations of characteristic curves of random graphs and LERRGs, discussion on the effects of root selection, robustness test of the network classification method and more network comparison results.
(PDF)