网络模体

大家好，我们的公众号计划要推送一篇关于网络模体的综述文章，我们希望可以配套建议该重要概念：网络模体。现在希望可以大家一起协作完成这个词条。 翻译任务主要分为以下5个内容：

• 网络定义和历史 ---许菁
• 网络模体的发现算法 mfinder和FPF算法---李鹏
• 网络模体的发现算法 ESU和对应的软件FANMOD---Imp
• 网络模体的发现算法 G-Trie、算法对比和算法分类——Ricky（中英对照初稿在这里）
• 已有网络模体及其函数表示 --周佳欣
• 活动模体+批判 --- 孙宇
• 代码实现

大家可以在对应感兴趣的部分下面，写上姓名。我们的协作方式是石墨文档上翻译，最后再编辑成文。 对应的词条链接：https://en.wikipedia.org/wiki/Network_motif#Well-established_motifs_and_their_functions

截止时间：今晚12：00

All networks, including biological networks, social networks, technological networks (e.g., computer networks and electrical circuits) and more, can be represented as graphs, which include a wide variety of subgraphs. One important local property of networks are so-called network motifs, which are defined as recurrent and statistically significant sub-graphs or patterns.

所有网络，包括生物网络(biological networks)、社会网络(social networks)、技术网络（例如计算机网络和电路）等，都可以用图的形式来表示，这些图中会包括各种各样的子图(subgraphs)。网络的一个重要的局部性质是所谓的网络基序，即重复且具有统计意义的子图或模式(patterns)。

Network motifs are sub-graphs that repeat themselves in a specific network or even among various networks. Each of these sub-graphs, defined by a particular pattern of interactions between vertices, may reflect a framework in which particular functions are achieved efficiently. Indeed, motifs are of notable importance largely because they may reflect functional properties. They have recently gathered much attention as a useful concept to uncover structural design principles of complex networks. Although network motifs may provide a deep insight into the network's functional abilities, their detection is computationally challenging. 网络模体(Network motifs)是指在特定网络或各种网络中重复出现的相同的子图。这些子图由顶点之间特定的交互模式定义，一个子图便可以反映一个框架，这个框架可以有效地实现某个特定的功能。事实上，之所以说模体是一个重要的特性，正是因为它们可能反映出对应网络功能的这一性质。近年来这一概念作为揭示复杂网络结构设计原理的一个有用概念而受到了广泛的关注。 但是，虽然通过研究网络模体可以深入了解网络的功能，但是对于模体的检测在计算上是具有挑战性的。

Definition

Let G = (V, E) and G′ = (V′, E′) be two graphs. Graph G′ is a sub-graph of graph G (written as G′ ⊆ G) if V′ ⊆ V and E′ ⊆ E ∩ (V′ × V′). If G′ ⊆ G and G′ contains all of the edges 〈u, v〉 ∈ E with u, v ∈ V′, then G′ is an induced sub-graph of G. We call G′ and G isomorphic (written as G′ ↔ G), if there exists a bijection (one-to-one) f:V′ → V with 〈u, v〉 ∈ E′ ⇔ 〈f(u), f(v)〉 ∈ E for all u, v ∈ V′. The mapping f is called an isomorphism between G and G′.

设G = (V, E) 和 G′ = (V′, E′) 是两个图。若V′ ⊆ V且满足E′ ⊆ E ∩ (V′ × V′)）（即图{{math|G′ ⊆ G}的所有边和点都属于图G）则称图{{math|G′ ⊆ G}是图G的一个子图

若G′ ⊆ G，且对于顶点u、v及其连边，只要u和v存在于G′（即若U, V′ ⊆ V），那么G′ ⊆ G中就应该包含原图G中的所有u和V的对应连边（即〈u, v〉 ∈ E），则称此时图G′就是图G的导出子图。

如果存在一个双射（一对一）f:V′ → V，且对所有u, v ∈ V′都有〈u, v〉 ∈ E′ ⇔ 〈f(u), f(v)〉 ∈ E ，则称G&prime 是G的同构图（记作：G′ → G），映射f则称为G与G′之间的同构（isomorphism）。[2]

When G″ ⊂ G and there exists an isomorphism between the sub-graph G″ and a graph G′, this mapping represents an appearance of G′ in G. The number of appearances of graph G′ in G is called the frequency F_G of G′ in G. A graph is called recurrent (or frequent) in G, when its frequency F_G(G′) is above a predefined threshold or cut-off value. We use terms pattern and frequent sub-graph in this review interchangeably. There is an ensemble Ω(G) of random graphs corresponding to the null-model associated to G. We should choose N random graphs uniformly from Ω(G) and calculate the frequency for a particular frequent sub-graph G′ in G. If the frequency of G′ in G is higher than its arithmetic mean frequency in N random graphs R_i, where 1 ≤ i ≤ N, we call this recurrent pattern significant and hence treat G′ as a network motif for G. For a small graph G′, the network G and a set of randomized networks R(G) ⊆ Ω(R), where R(G) = N, the Z-Score that has been defined by the following formula:

[math]\displaystyle{ Z(G^\prime) = \frac{F_G(G^\prime) - \mu_R(G^\prime)}{\sigma_R(G^\prime)} }[/math]

当G″ ⊂ G，且G″与图G′之间存在同构时，该映射表示G′对于G存在。图G′在G的出现次数称为G′出现在G的频率F_G。当一个子图的频率F_G高于预定的阈值或截止值时，则称G′是G中的递归（或频繁）子图。

在接下来的内容中，我们交替使用术语“模式（motifs）”和“频繁子图（frequent sub-graph）”。

设从与G相关联的零模型（the null-model）获得的随机图集合为Ω(G)，我们从Ω(G)中均匀地选择N个随机图，并计算其特定频繁子图的频率。如果G′出现在G的频率高于N个随机图Ri的算术平均频率，其中1 ≤ i ≤ N，我们称此递归模式是有意义的，因此可以将G′视为G的网络模体。

对于一个小图G′，网络G和一组随机网络R(G) ⊆ Ω(R)，当R(G) = N时，由其Z分数（Z-score）的定义如下式：

[math]\displaystyle{ Z(G^\prime) = \frac{F_G(G^\prime) - \mu_R(G^\prime)}{\sigma_R(G^\prime)} }[/math]

where μ_R(G′) and σ_R(G′) stand for mean and standard deviation frequency in set R(G), respectively. The larger the Z(G′), the more significant is the sub-graph G′ as a motif. Alternatively, another measurement in statistical hypothesis testing that can be considered in motif detection is the P-Value, given as the probability of F_R(G′) ≥ F_G(G′) (as its null-hypothesis), where F_R(G′) indicates the frequency of G' in a randomized network. A sub-graph with P-value less than a threshold (commonly 0.01 or 0.05) will be treated as a significant pattern. The P-value is defined as

[math]\displaystyle{ P(G^\prime) = \frac{1}{N}\sum_{i=1}^N \delta(c(i)) ; c(i): F_R^i(G^\prime) \ge F_G(G^\prime) }[/math]

式中，μ_R(G′) 和 σ_R(G′)分别代表集合R(G)中的平均和标准偏差频率。. Z(G′)越大，子图G′作为模体的意义也就越大。

此外还可以使用统计假设检验中的另一个测量方法，可以作为模体检测中的一种方法，即P值（P-value），以 F_R(G′) ≥ F_G(G′)的概率给出（作为其零假设null-hypothesis），其中F_R(G′)表示随机网络中G′的频率。 当P值小于阈值（通常为0.01或0.05）时，该子图可以被称为显著模式。该P值定义为：

[math]\displaystyle{ P(G^\prime) = \frac{1}{N}\sum_{i=1}^N \delta(c(i)) ; c(i): F_R^i(G^\prime) \ge F_G(G^\prime) }[/math]

Different occurrences of a sub-graph in a graph. (M1 – M4) are different occurrences of sub-graph (b) in graph (a). For frequency concept F₁, the set M1, M2, M3, M4 represent all matches, so F₁ = 4. For F₂, one of the two set M1, M4 or M2, M3 are possible matches, F₂ = 2. Finally, for frequency concept F₃, merely one of the matches (M1 to M4) is allowed, therefore F₃ = 1. The frequency of these three frequency concepts decrease as the usage of network elements are restricted.

Where N indicates number of randomized networks, i is defined over an ensemble of randomized networks and the Kronecker delta function δ(c(i)) is one if the condition c(i) holds. The concentration of a particular n-size sub-graph G′ in network G refers to the ratio of the sub-graph appearance in the network to the total n-size non-isomorphic sub-graphs’ frequencies, which is formulated by

[math]\displaystyle{ C_G(G^\prime) = \frac{F_G(G^\prime)}{\sum_i F_G(G_i)} }[/math]

where index i is defined over the set of all non-isomorphic n-size graphs. Another statistical measurement is defined for evaluating network motifs, but it is rarely used in known algorithms. This measurement is introduced by Picard et al. in 2008 and used the Poisson distribution, rather than the Gaussian normal distribution that is implicitly being used above.

其中索引 i 定义在所有非同构 n 大小图的集合上。 另一种统计测量是用来评估网络主题的，但在已知的算法中很少使用。 这种测量方法是由 Picard 等人在2008年提出的，使用的是泊松分佈分布，而不是上面隐含使用的高斯正态分布。其中N表示随机网络的数目，i定义在随机网络的集合上，若条件c(i)成立，则Kroneckerδ函数δ(c(i))是1。在网络G中，一个特定的n维子图N′的集中度是指子图在网络中出现频率与n维非同构子图的总频率之比，其计算公式如下：

[math]\displaystyle{ C_G(G^\prime) = \frac{F_G(G^\prime)}{\sum_i F_G(G_i)} }[/math]

In addition, three specific concepts of sub-graph frequency have been proposed. As the figure illustrates, the first frequency concept F₁ considers all matches of a graph in original network. This definition is similar to what we have introduced above. The second concept F₂ is defined as the maximum number of edge-disjoint instances of a given graph in original network. And finally, the frequency concept F₃ entails matches with disjoint edges and nodes. Therefore, the two concepts F₂ and F₃ restrict the usage of elements of the graph, and as can be inferred, the frequency of a sub-graph declines by imposing restrictions on network element usage. As a result, a network motif detection algorithm would pass over more candidate sub-graphs if we insist on frequency concepts F₂ and F₃.

此外，他们还提出了子图频率的三个具体概念。 如图所示，第一频率概念 F₁考虑原始网络中图的所有匹配，这与我们前面介绍过的类似。第二个概念F₂定义为原始网络中给定图的最大不相交边的数量。最后，频率概念F₃包含与不相交边（disjoint edges）和节点的匹配。因此，两个概念F2和F3限制了图元素的使用，并且可以看出，通过对网络元素的使用施加限制，子图的频率下降。因此，如果我们坚持使用频率概念F₂和F₃，网络模体检测算法将可以筛选出更多的候选子图。

History

The study of network motifs was pioneered by Holland and Leinhardt who introduced the concept of a triad census of networks. They introduced methods to enumerate various types of subgraph configurations, and test whether the subgraph counts are statistically different from those expected in random networks. 霍兰（Holland）和莱因哈特（Leinhardt）率先提出了网络三合会普查（a triad census of networks）的概念，开创了网络模体研究的先河。 他们介绍了枚举各种子图配置的方法，并测试子图计数是否与随机网络中的期望值存在统计学上的差异。

这里对于网络三合会普查（a triad census of networks）这一概念的翻译存疑

This idea was further generalized in 2002 by Uri Alon and his group when network motifs were discovered in the gene regulation (transcription) network of the bacteria E. coli and then in a large set of natural networks. Since then, a considerable number of studies have been conducted on the subject. Some of these studies focus on the biological applications, while others focus on the computational theory of network motifs.

2002年，Uri Alon和他的团队[17]在大肠杆菌的基因调控（gene regulation network）（转录 transcription）网络中发现了网络模体，随后在大量的自然网络中也发现了网络模体，从而进一步推广了这一观点。自那时起，许多科学家都对这一问题进行了大量的研究。其中一些研究集中在生物学应用上，而另一些则集中在网络模体的计算理论上。

The biological studies endeavor to interpret the motifs detected for biological networks. For example, in work following, the network motifs found in E. coli were discovered in the transcription networks of other bacteria as well as yeast and higher organisms. A distinct set of network motifs were identified in other types of biological networks such as neuronal networks and protein interaction networks.

生物学研究试图解释为生物网络检测到的模体。例如，在接下来的工作中，文献[17]在大肠杆菌中发现的网络模体存在于其他细菌以及酵母和高等生物的转录网络中。文献在其他类型的生物网络中发现了一组不同的网络模体，如神经元网络和蛋白质相互作用网络。

The computational research has focused on improving existing motif detection tools to assist the biological investigations and allow larger networks to be analyzed. Several different algorithms have been provided so far, which are elaborated in the next section in chronological order.

另一方面，对于计算研究的重点则是改进现有的模体检测工具，以协助生物学研究，并允许对更大的网络进行分析。到目前为止，已经提供了几种不同的算法，这些算法将在下一节按时间顺序进行阐述。

Most recently, the acc-MOTIF tool to detect network motifs was released.

最近，还发布了用于检测网络基序的acc基序工具。

模体发现算法 Motif discovery algorithms

Various solutions have been proposed for the challenging problem of motif discovery. These algorithms can be classified under various paradigms such as exact counting methods, sampling methods, pattern growth methods and so on. However, motif discovery problem comprises two main steps: first, calculating the number of occurrences of a sub-graph and then, evaluating the sub-graph significance. The recurrence is significant if it is detectably far more than expected. Roughly speaking, the expected number of appearances of a sub-graph can be determined by a Null-model, which is defined by an ensemble of random networks with some of the same properties as the original network.

针对模体发现这一问题存在多种解决方案。这些算法可以归纳为不同的范式：例如精确计数方法，采样方法，模式增长方法等。但模体发现问题包括两个主要步骤：首先，计算子图的出现次数，然后评估子图的重要性。如果检测到的重现性远超过预期，那么这种重现性是很显著的。粗略地讲，子图的预期出现次数可以由零模型 Null-model 确定，该模型定义为具有与原始网络某些属性相同的随机网络的集合。

Here, a review on computational aspects of major algorithms is given and their related benefits and drawbacks from an algorithmic perspective are discussed.

接下来，对下列算法的计算原理进行简要回顾，并从算法的角度讨论了它们的优缺点。

mfinder 算法

mfinder, the first motif-mining tool, implements two kinds of motif finding algorithms: a full enumeration and a sampling method. Until 2004, the only exact counting method for NM (network motif) detection was the brute-force one proposed by Milo et al.. This algorithm was successful for discovering small motifs, but using this method for finding even size 5 or 6 motifs was not computationally feasible. Hence, a new approach to this problem was needed.

mfinder是第一个模体挖掘工具，它主要有两种模体查找算法：完全枚举 full enumeration 和采样方法 sampling method。直到2004年，用于NM（网络模体 networkmotif）检测的唯一精确计数方法是Milo等人提出的暴力穷举方法。该算法成功地发现了小规模的模体，但是这种方法甚至对于发现规模为5个或6个的模体在计算上都不可行的。因此，需要一种解决该问题的新方法。

Kashtan et al. presented the first sampling NM discovery algorithm, which was based on edge sampling throughout the network. This algorithm estimates concentrations of induced sub-graphs and can be utilized for motif discovery in directed or undirected networks. The sampling procedure of the algorithm starts from an arbitrary edge of the network that leads to a sub-graph of size two, and then expands the sub-graph by choosing a random edge that is incident to the current sub-graph. After that, it continues choosing random neighboring edges until a sub-graph of size n is obtained. Finally, the sampled sub-graph is expanded to include all of the edges that exist in the network between these n nodes. When an algorithm uses a sampling approach, taking unbiased samples is the most important issue that the algorithm might address. The sampling procedure, however, does not take samples uniformly and therefore Kashtan et al. proposed a weighting scheme that assigns different weights to the different sub-graphs within network. The underlying principle of weight allocation is exploiting the information of the sampling probability for each sub-graph, i.e. the probable sub-graphs will obtain comparatively less weights in comparison to the improbable sub-graphs; hence, the algorithm must calculate the sampling probability of each sub-graph that has been sampled. This weighting technique assists mfinder to determine sub-graph concentrations impartially.

Kashtan 等人首次提出了一种基于边缘采样的网络模体（NM）采样发现算法。该算法估计了所含子图 induced sub-graphs 的集中度 concentrations ，可用于有向或无向网络中的模体发现。该算法的采样过程从网络的任意一条边开始，该边连向大小为2的子图，然后选择一条与当前子图相关的随机边对子图进行扩展。之后，它将继续选择随机的相邻边，直到获得大小为n的子图为止。最后，采样得到的子图扩展为包括这n个节点在内的网络中存在的所有边。当使用采样方法时，获取无偏样本是这类算法可能面临的最重要问题。而且，采样过程并不能保证采到所有的样本（也就是不能保证得到所有的子图，译者注），因此，Kashtan 等人又提出了一种加权方案，为网络中的不同子图分配不同的权重。 权重分配的基本原理是利用每个子图的抽样概率信息，即，与不可能的子图相比，可能的子图将获得相对较少的权重；因此，该算法必须计算已采样的每个子图的采样概率。这种加权技术有助于mfinder公正地确定子图的集中度 concentrations 。

In expanded to include sharp contrast to exhaustive search, the computational time of the algorithm surprisingly is asymptotically independent of the network size. An analysis of the computational time of the algorithm has shown that it takes O(nⁿ) for each sample of a sub-graph of size n from the network. On the other hand, there is no analysis in on the classification time of sampled sub-graphs that requires solving the graph isomorphism problem for each sub-graph sample. Additionally, an extra computational effort is imposed on the algorithm by the sub-graph weight calculation. But it is unavoidable to say that the algorithm may sample the same sub-graph multiple times – spending time without gathering any information. In conclusion, by taking the advantages of sampling, the algorithm performs more efficiently than an exhaustive search algorithm; however, it only determines sub-graphs concentrations approximately. This algorithm can find motifs up to size 6 because of its main implementation, and as result it gives the most significant motif, not all the others too. Also, it is necessary to mention that this tool has no option of visual presentation. The sampling algorithm is shown briefly:

与穷举搜索形成鲜明对比的是，该算法的计算时间竟然与网络大小渐近无关。对算法时间复杂度的分析表明，对于网络中大小为n的子图的每个样本，它的时间复杂度为[math]\displaystyle{ O(n^n) }[/math]。另一方面，并没有对已采样子图的每一个子图样本判断图同构问题的分类时间进行分析。另外，子图权重计算将额外增加该算法的计算负担。但是不得不指出的是，该算法可能会多次采样相同的子图——花费时间而不收集任何有用信息。总之，通过利用采样的优势，该算法的性能比穷举搜索算法更有效；但是，它只能大致确定子图的集中度 concentrations 。由于该算法的实现方式，使得它可以找到最大为6的模体，并且它会给出的最重要的模体，而不是其他所有模体。另外，有必要提到此工具没有可视化的呈现。采样算法简要显示如下：

mfinder
Definitions: Esis the set of picked edges. Vs is the set of all nodes that are touched by the edges in E.
Init Vs and Es to be empty sets. 1. Pick a random edge e1 = (vi, vj). Update Es = {e1}, Vs = {vi, vj} 2. Make a list L of all neighbor edges of Es. Omit from L all edges between members of Vs. 3. Pick a random edge e = {vk,vl} from L. Update Es = Es ⋃ {e}, Vs = Vs ⋃ {vk, vl}. 4. Repeat steps 2-3 until completing an n-node subgraph (until |Vs| = n). 5. Calculate the probability to sample the picked n-node subgraph.

mfinder

定义：[math]\displaystyle{ E_{s} }[/math]是采集的边集合。[math]\displaystyle{ V_{s} }[/math]是[math]\displaystyle{ E }[/math]中所有边的顶点的集合。

初始化[math]\displaystyle{ V_{s} }[/math]和[math]\displaystyle{ E_{s} }[/math]为空集。 1. 随机选择一条边[math]\displaystyle{ e_{1} = (v_{i}, v_{j}) }[/math]，更新 [math]\displaystyle{ E_{s} = \{e_{1}\}, V{s} = \{v_{i}, v_{j}\} }[/math] 2. 列出[math]\displaystyle{ E{s} }[/math]的所有邻边列表[math]\displaystyle{ L }[/math]，从[math]\displaystyle{ L }[/math]中删除[math]\displaystyle{ V{s} }[/math]中所有元素组成的边。 3. 从[math]\displaystyle{ L }[/math]中随机选择一条边[math]\displaystyle{ e = \{v_{k},v_{l}\} }[/math]， 更新[math]\displaystyle{ E_{s} = E_{s} \cup \{e\} , V_{s} = V_{s} \cup \{v_{k}, v_{l}\} }[/math]。 4. 重复步骤2-3，直到完成包含n个节点的子图 ([math]\displaystyle{ \left | V_{s} \right | = n }[/math])。 5. 计算对选取的n节点子图进行采样的概率。

FPF (Mavisto)算法

Schreiber and Schwöbbermeyer proposed an algorithm named flexible pattern finder (FPF) for extracting frequent sub-graphs of an input network and implemented it in a system named Mavisto. Their algorithm exploits the downward closure property which is applicable for frequency concepts F₂ and F₃. The downward closure property asserts that the frequency for sub-graphs decrease monotonically by increasing the size of sub-graphs; however, this property does not hold necessarily for frequency concept F₁. FPF is based on a pattern tree (see figure) consisting of nodes that represents different graphs (or patterns), where the parent of each node is a sub-graph of its children nodes; in other words, the corresponding graph of each pattern tree's node is expanded by adding a new edge to the graph of its parent node.

Schreiber和Schwöbbermeyer 提出了一种称为灵活模式查找器（FPF）的算法，用于提取输入网络的频繁子图，并将其在名为Mavisto的系统中加以实现。 他们的算法利用了向下闭包特性，该特性适用于频率概念[math]\displaystyle{ F_{2} }[/math]和[math]\displaystyle{ F_{3} }[/math]。向下闭包性质表明，子图的频率随着子图的大小而单调下降；但这一性质并不一定适用于频率概念[math]\displaystyle{ F_{1} }[/math]。FPF算法基于模式树（见右图），由代表不同图形（或模式）的节点组成，其中每个节点的父节点是其子节点的子图；换句话说，每个模式树节点的对应图通过向其父节点图添加新边来扩展。

FPF算法中的模式树展示.

At first, the FPF algorithm enumerates and maintains the information of all matches of a sub-graph located at the root of the pattern tree. Then, one-by-one it builds child nodes of the previous node in the pattern tree by adding one edge supported by a matching edge in the target graph, and tries to expand all of the previous information about matches to the new sub-graph (child node). In next step, it decides whether the frequency of the current pattern is lower than a predefined threshold or not. If it is lower and if downward closure holds, FPF can abandon that path and not traverse further in this part of the tree; as a result, unnecessary computation is avoided. This procedure is continued until there is no remaining path to traverse.

首先，FPF算法枚举并维护位于模式树根部的子图的所有匹配信息。然后，它通过在目标图中添加匹配边缘支持的一条边缘，在模式树中一一建立前一节点的子节点，然后通过在目标图中添加匹配边支持的一条边，逐个构建模式树中前一个节点的子节点，并尝试将以前关于匹配的所有信息拓展到新的子图(子节点)中。下一步，它判断当前模式的频率是否低于预定义的阈值。如果它低于阈值且保持向下闭包，则FPF算法会放弃该路径，而不会在树的此部分进一步遍历；这样就避免了不必要的计算。重复此过程，直到没有剩余可遍历的路径为止。

The advantage of the algorithm is that it does not consider infrequent sub-graphs and tries to finish the enumeration process as soon as possible; therefore, it only spends time for promising nodes in the pattern tree and discards all other nodes. As an added bonus, the pattern tree notion permits FPF to be implemented and executed in a parallel manner since it is possible to traverse each path of the pattern tree independently. However, FPF is most useful for frequency concepts F₂ and F₃, because downward closure is not applicable to F₁. Nevertheless, the pattern tree is still practical for F₁ if the algorithm runs in parallel. Another advantage of the algorithm is that the implementation of this algorithm has no limitation on motif size, which makes it more amenable to improvements. The pseudocode of FPF (Mavisto) is shown below:

该算法的优点是它不会考虑不频繁的子图，并尝试尽快完成枚举过程；因此，它只花时间在模式树中用于有希望的节点上，而放弃所有其他节点。还有一点额外的好处，模式树概念允许 FPF 以并行方式实现和执行，因为它可以独立地遍历模式树的每个路径。但是，FPF对于频率概念[math]\displaystyle{ F_{2} }[/math]和[math]\displaystyle{ F_{3} }[/math]最为有用，因为向下闭包不适用于[math]\displaystyle{ F_{1} }[/math]。尽管如此，如果算法并行运行，那么模式树对于[math]\displaystyle{ F_{1} }[/math]仍然是可行的。该算法的另一个优点是它的实现对模体大小没有限制，这使其更易于改进。FPF（Mavisto）的伪代码如下所示：

Mavisto
Data: Graph G, target pattern size t, frequency concept F Result: Set R of patterns of size t with maximum frequency.
R ← φ, fmax ← 0 P ←start pattern p1 of size 1 Mp1 ←all matches of p1 in G While P ≠ φ do  Pmax ←select all patterns from P with maximum size.  P ← select pattern with maximum frequency from Pmax  Ε = ExtensionLoop(G, p, Mp)  Foreach pattern p ∈ E  If F = F1 then f ← size(Mp)  Else f ← Maximum Independent set(F, Mp)  End  If size(p) = t then  If f = fmax then R ← R ⋃ {p}  Else if f > fmax then R ← {p}; fmax ← f  End  Else  If F = F1 or f ≥ fmax then P ← P ⋃ {p}  End  End  End End

Mavisto

数据: 图 [math]\displaystyle{ G }[/math], 目标模式规模 [math]\displaystyle{ t }[/math], 频率概念 [math]\displaystyle{ F }[/math]。 结果: 以最大频率设置大小为 [math]\displaystyle{ t }[/math]的模式 [math]\displaystyle{ R }[/math].

[math]\displaystyle{ R \leftarrow \Phi , f_{max}\leftarrow 0 }[/math] [math]\displaystyle{ P \leftarrow }[/math] 开始于大小为1的模式 [math]\displaystyle{ p_{1} }[/math] [math]\displaystyle{ M_{p_{1}} \leftarrow }[/math] 图 [math]\displaystyle{ G }[/math] 中模式 [math]\displaystyle{ p_{1} }[/math] 的所有匹配 当 [math]\displaystyle{ P \neq \Phi }[/math] 时，执行：     [math]\displaystyle{ P_{max} \leftarrow }[/math] 从 [math]\displaystyle{ P }[/math] 中选择最大规模的所有模式     [math]\displaystyle{ P\leftarrow }[/math] 从 [math]\displaystyle{ P_{max} }[/math] 中选择最大频率的模式     [math]\displaystyle{ E = ExtensionLoop(G, p, M_{p}) }[/math]     对于 [math]\displaystyle{ p \in E }[/math] 的所有模式：         如果 [math]\displaystyle{ F = F_{1} }[/math] ，那么 [math]\displaystyle{ f \leftarrow size(M_{p}) }[/math]         其他[math]\displaystyle{ f \leftarrow }[/math] 最大独立集 [math]\displaystyle{ (F, M_{p}) }[/math]         结束         如果 [math]\displaystyle{ size(p) = t }[/math] ，那么             如果 [math]\displaystyle{ f = f_{max} }[/math] ，那么 [math]\displaystyle{ R \leftarrow R \cup \{p\} }[/math]             其他 如果 [math]\displaystyle{ f \gt f_{max} }[/math] ，那么 [math]\displaystyle{ R \leftarrow \{p\} }[/math]； [math]\displaystyle{ f_{max} \leftarrow f }[/math]             结束         其他             如果 [math]\displaystyle{ F = F_{1} or f \geq f_{max} }[/math] ，那么 [math]\displaystyle{ P \leftarrow P \cup \{p\} }[/math]             结束         结束     结束 结束

ESU (FANMOD)算法及对应的软件

The sampling bias of Kashtan et al. provided great impetus for designing better algorithms for the NM discovery problem. Although Kashtan et al. tried to settle this drawback by means of a weighting scheme, this method imposed an undesired overhead on the running time as well a more complicated implementation. This tool is one of the most useful ones, as it supports visual options and also is an efficient algorithm with respect to time. But, it has a limitation on motif size as it does not allow searching for motifs of size 9 or higher because of the way the tool is implemented.

由于Kashtan等学者发现的『采样偏差』问题为『the NM discovery problem』设计更好的算法提出了更高要求。虽然Kashtan等人尝试用加权法来解决这个弊端，但这个方法在运行上，消耗了过多的运行时间，且实现起来也变得更加复杂。但这个工具还是最好用的工具之一，因为它支持可视化选项，同时也『是个节约时间的算法』。但是，它在所支持的模体的规模大小还是有局限性。由于该工具在具体实施中，不允许搜索规模大小为9或者更大的模体。

Wernicke introduced an algorithm named RAND-ESU that provides a significant improvement over mfinder. This algorithm, which is based on the exact enumeration algorithm ESU, has been implemented as an application called FANMOD. RAND-ESU is a NM discovery algorithm applicable for both directed and undirected networks, effectively exploits an unbiased node sampling throughout the network, and prevents overcounting sub-graphs more than once. Furthermore, RAND-ESU uses a novel analytical approach called DIRECT for determining sub-graph significance instead of using an ensemble of random networks as a Null-model. The DIRECT method estimates the sub-graph concentration without explicitly generating random networks. Empirically, the DIRECT method is more efficient in comparison with the random network ensemble in case of sub-graphs with a very low concentration; however, the classical Null-model is faster than the DIRECT method for highly concentrated sub-graphs. In the following, we detail the ESU algorithm and then we show how this exact algorithm can be modified efficiently to RAND-ESU that estimates sub-graphs concentrations.

Weinicke引入了一种叫RAND-ESU的算法，这个新引入的算法比Mfinder软件有着更显著的提升。RAND-ESU基于精准的ESU算法，已有对应的软件FANMOD。RAND-ESU是一种[NM算法]，可应用于定向的或者不定向的网络中，能够有效的在网络中利用无偏差节点进行采样，以及保证了一个子图仅仅被搜索一次,且不会产生无意义的子图。并且，RAND-ESU采用了一个叫做DIRECT的全新的分析方式，从而来确定子图的重要性，而不是用随机网络的组合来建立『Null模型』。DIRECT方法可以不用大量生成随机网络就能估计子图的浓度。实际上，相较于用随机网络组合分析比较低集中度的子图来说，DIRECT这个方法更加的高效。但是，传统的『Null模型』又比DIRECT这个算法能更加快速地解决高度集中的子图。接下来，我们将详细讲述ESU算法和展示如何把这种精确的算法调整为RAND-ESU算法去估计子图的浓度。

The algorithms ESU and RAND-ESU are fairly simple, and hence easy to implement. ESU first finds the set of all induced sub-graphs of size k, let S_k be this set. ESU can be implemented as a recursive function; the running of this function can be displayed as a tree-like structure of depth k, called the ESU-Tree (see figure). Each of the ESU-Tree nodes indicate the status of the recursive function that entails two consecutive sets SUB and EXT. SUB refers to nodes in the target network that are adjacent and establish a partial sub-graph of size |SUB| ≤ k. If |SUB| = k, the algorithm has found an induced complete sub-graph, so S_k = SUB ∪ S_k. However, if |SUB| < k, the algorithm must expand SUB to achieve cardinality k. This is done by the EXT set that contains all the nodes that satisfy two conditions: First, each of the nodes in EXT must be adjacent to at least one of the nodes in SUB; second, their numerical labels must be larger than the label of first element in SUB. The first condition makes sure that the expansion of SUB nodes yields a connected graph and the second condition causes ESU-Tree leaves (see figure) to be distinct; as a result, it prevents overcounting. Note that, the EXT set is not a static set, so in each step it may expand by some new nodes that do not breach the two conditions. The next step of ESU involves classification of sub-graphs placed in the ESU-Tree leaves into non-isomorphic size-k graph classes; consequently, ESU determines sub-graphs frequencies and concentrations. This stage has been implemented simply by employing McKay's nauty algorithm, which classifies each sub-graph by performing a graph isomorphism test. Therefore, ESU finds the set of all induced k-size sub-graphs in a target graph by a recursive algorithm and then determines their frequency using an efficient tool.

ESU和RAND-ESU两种算法都比较简捷，所以实现起来都很容易。『ESU首先找到大小为k的所有诱导子图的集合』，并命名这个集合为Sk。因为EUS以递归函数的形式实现，该函数的运行可以演示为『k级』的树状结构，称为ESU-Tree（见图）。每一个在ESU-Tree上的节点都表示递归函数的状态，这个递归函数需要两个连续集合的SUB和EXT。『SUB指的是在目标网络的相邻节点上，并且是一部分的层级绝对值大小小于等于k的子图集合。』如果SUB集合层级的绝对值等于k，那么这个算法可以找到一个『完整的诱导子图』，所以在此情况下Sk等于SUB与Sk的并集。相反，如果它的绝对值小于k，那么这个算法必须把SUB扩大，才能实现基数为k。『EXT这个集合包含了所有的满足以下两个情况的节点。第一，每个在EXT的节点必须至少与在SUB的一个节点相邻。第二，他们的下标必须比在SUB的第一个元素大。』？？？第一个条件保证了『SUB节点的展开产生相关的图』，第二个条件能使ESU-Tree树状图上的分支变得离散。所以，这个方法可以避免过度计算。注意，EXT集合不是一个固定的集合。所以每一步都有可能扩展满足于以上两个条件的新节点。下一步包含了在ESU-Tree分支上的子图的分类，『将它们分为非同构的大小为k的图类』。因此，ESU决定了子图的『频率以及浓度』。这一阶段的实施仅通过运用McKay的nauty算法，这一算法可以通过图的同构测试来把每个子图进行分类。所以，ESU能够在目标图中通过递归算法，找到所有规模大小为k的诱导子图集合，且使用高效的工具来确定他们的『频率』。

The procedure of implementing RAND-ESU is quite straightforward and is one of the main advantages of FANMOD. One can change the ESU algorithm to explore just a portion of the ESU-Tree leaves by applying a probability value 0 ≤ p_d ≤ 1 for each level of the ESU-Tree and oblige ESU to traverse each child node of a node in level d-1 with probability p_d. This new algorithm is called RAND-ESU. Evidently, when p_d = 1 for all levels, RAND-ESU acts like ESU. For p_d = 0 the algorithm finds nothing. Note that, this procedure ensures that the chances of visiting each leaf of the ESU-Tree are the same, resulting in unbiased sampling of sub-graphs through the network. The probability of visiting each leaf is ∏_dp_d and this is identical for all of the ESU-Tree leaves; therefore, this method guarantees unbiased sampling of sub-graphs from the network. Nonetheless, determining the value of p_d for 1 ≤ d ≤ k is another issue that must be determined manually by an expert to get precise results of sub-graph concentrations. While there is no lucid prescript for this matter, the Wernicke provides some general observations that may help in determining p_d values. In summary, RAND-ESU is a very fast algorithm for NM discovery in the case of induced sub-graphs supporting unbiased sampling method. Although, the main ESU algorithm and so the FANMOD tool is known for discovering induced sub-graphs, there is trivial modification to ESU which makes it possible for finding non-induced sub-graphs, too. The pseudo code of ESU (FANMOD) is shown below: 运用RAND-ESU的过程十分的简单，这也是FANMOD的一个主要的优点。可以通过对ESU-Tree『树状图』的每个级别应用概率0 ≤ p_d ≤ 1并强制ESU以概率p_d遍历d-1级别中节点的每个子节点，来更改ESU算法使其仅搜索ESU-Tree分支的一部分。 这种新的演算方式叫RAND-ESU。显然，当所有阶段p_d = 1时，RAND-ESU等同于ESU。当p_d = 0时，在这个算法下没有任何意义。注意，这个过程只是确保了可以找到ESU-Tree上的每一分支的机会都是相同的，从而使网络中的子图采样无偏差。访问每个分支的概率为∏_dp_d，这对于所有ESU-Tree中的分支都是相同的； 因此，该方法可确保从网络中对子图进行无偏采样。但是，设置1 ≤ d ≤ k的p_d参数是另一个问题，必须由专家人工确定才能获得子图『浓度』的精确结果。尽管对此没有明确的规定，但是Wrenucke提出了一些一般性的观察结论，这些结论有可能可以帮助我们确定p_d值。总的来说，在诱导子图支持无偏采样方法的情况下，RAND-ESU是一个能快速解决『NM discovery problem』的算法。 尽管，ESU算法的主要部分和FANMOD工具是以用来寻找诱导子图而著称的，但只需对ESU进行细小的改动，就可以用来寻找诱导子图。ESU（FANMOD）的伪代码如下：

(a) A target graph of size 5, (b) the ESU-tree of depth k that is associated to the extraction of sub-graphs of size 3 in the target graph. Leaves correspond to set S3 or all of the size-3 induced sub-graphs of the target graph (a). Nodes in the ESU-tree include two adjoining sets, the first set contains adjacent nodes called SUB and the second set named EXT holds all nodes that are adjacent to at least one of the SUB nodes and where their numerical labels are larger than the SUB nodes labels. The EXT set is utilized by the algorithm to expand a SUB set until it reaches a desired sub-graph size that are placed at the lowest level of ESU-Tree (or its leaves).

Enumeration of ESU (FANMOD)
EnumerateSubgraphs(G,k) Input: A graph G = (V, E) and an integer 1 ≤ k ≤ |V|. Output: All size-k subgraphs in G. for each vertex v ∈ V do  VExtension ← {u ∈ N({v}) | u > v}  call ExtendSubgraph({v}, VExtension, v) endfor
ExtendSubgraph(VSubgraph, VExtension, v) if |VSubgraph| = k then output G[VSubgraph] and return while VExtension ≠ ∅ do  Remove an arbitrarily chosen vertex w from VExtension  VExtension′ ← VExtension ∪ {u ∈ Nexcl(w, VSubgraph) | u > v}  call ExtendSubgraph(VSubgraph ∪ {w}, VExtension′, v) return

NeMoFinder

Chen et al. introduced a new NM discovery algorithm called NeMoFinder, which adapts the idea in SPIN to extract frequent trees and after that expands them into non-isomorphic graphs. NeMoFinder utilizes frequent size-n trees to partition the input network into a collection of size-n graphs, afterward finding frequent size-n sub-graphs by expansion of frequent trees edge-by-edge until getting a complete size-n graph K_n. The algorithm finds NMs in undirected networks and is not limited to extracting only induced sub-graphs. Furthermore, NeMoFinder is an exact enumeration algorithm and is not based on a sampling method. As Chen et al. claim, NeMoFinder is applicable for detecting relatively large NMs, for instance, finding NMs up to size-12 from the whole S. cerevisiae (yeast) PPI network as the authors claimed.

NeMoFinder consists of three main steps. First, finding frequent size-n trees, then utilizing repeated size-n trees to divide the entire network into a collection of size-n graphs, finally, performing sub-graph join operations to find frequent size-n sub-graphs. In the first step, the algorithm detects all non-isomorphic size-n trees and mappings from a tree to the network. In the second step, the ranges of these mappings are employed to partition the network into size-n graphs. Up to this step, there is no distinction between NeMoFinder and an exact enumeration method. However, a large portion of non-isomorphic size-n graphs still remain. NeMoFinder exploits a heuristic to enumerate non-tree size-n graphs by the obtained information from the preceding steps. The main advantage of the algorithm is in the third step, which generates candidate sub-graphs from previously enumerated sub-graphs. This generation of new size-n sub-graphs is done by joining each previous sub-graph with derivative sub-graphs from itself called cousin sub-graphs. These new sub-graphs contain one additional edge in comparison to the previous sub-graphs. However, there exist some problems in generating new sub-graphs: There is no clear method to derive cousins from a graph, joining a sub-graph with its cousins leads to redundancy in generating particular sub-graph more than once, and cousin determination is done by a canonical representation of the adjacency matrix which is not closed under join operation. NeMoFinder is an efficient network motif finding algorithm for motifs up to size 12 only for protein-protein interaction networks, which are presented as undirected graphs. And it is not able to work on directed networks which are so important in the field of complex and biological networks. The pseudocode of NeMoFinder is shown below:

NeMoFinder

Input: G - PPI network; N - Number of randomized networks; K - Maximal network motif size; F - Frequency threshold; S - Uniqueness threshold; Output: U - Repeated and unique network motif set; D ← ∅; for motif-size k from 3 to K do  T ← FindRepeatedTrees(k);  GDk ← GraphPartition(G, T)  D ← D ∪ T;  D′ ← T;  i ← k;  while D′ ≠ ∅ and i ≤ k × (k - 1) / 2 do  D′ ← FindRepeatedGraphs(k, i, D′);  D ← D ∪ D′;  i ← i + 1;  end while end for for counter i from 1 to N do  Grand ← RandomizedNetworkGeneration();  for each g ∈ D do  GetRandFrequency(g, Grand);  end for end for U ← ∅; for each g ∈ D do  s ← GetUniqunessValue(g);  if s ≥ S then  U ← U ∪ {g};  end if end for return U;

Grochow–Kellis

Grochow and Kellis proposed an exact algorithm for enumerating sub-graph appearances. The algorithm is based on a motif-centric approach, which means that the frequency of a given sub-graph,called the query graph, is exhaustively determined by searching for all possible mappings from the query graph into the larger network. It is claimed that a motif-centric method in comparison to network-centric methods has some beneficial features. First of all it avoids the increased complexity of sub-graph enumeration. Also, by using mapping instead of enumerating, it enables an improvement in the isomorphism test. To improve the performance of the algorithm, since it is an inefficient exact enumeration algorithm, the authors introduced a fast method which is called symmetry-breaking conditions. During straightforward sub-graph isomorphism tests, a sub-graph may be mapped to the same sub-graph of the query graph multiple times. In the Grochow–Kellis (GK) algorithm symmetry-breaking is used to avoid such multiple mappings. Here we introduce the GK algorithm and the symmetry-breaking condition which eliminates redundant isomorphism tests.

(a) graph G, (b) illustration of all automorphisms of G that is showed in (a). From set AutG we can obtain a set of symmetry-breaking conditions of G given by SymG in (c). Only the first mapping in AutG satisfies the SynG conditions; as a result, by applying SymG in the Isomorphism Extension module the algorithm only enumerate each match-able sub-graph in the network to G once. Note that SynG is not necessarily a unique set for an arbitrary graph G.

The GK algorithm discovers the whole set of mappings of a given query graph to the network in two major steps. It starts with the computation of symmetry-breaking conditions of the query graph. Next, by means of a branch-and-bound method, the algorithm tries to find every possible mapping from the query graph to the network that meets the associated symmetry-breaking conditions. An example of the usage of symmetry-breaking conditions in GK algorithm is demonstrated in figure.

As it is mentioned above, the symmetry-breaking technique is a simple mechanism that precludes spending time finding a sub-graph more than once due to its symmetries. Note that, computing symmetry-breaking conditions requires finding all automorphisms of a given query graph. Even though, there is no efficient (or polynomial time) algorithm for the graph automorphism problem, this problem can be tackled efficiently in practice by McKay's tools. As it is claimed, using symmetry-breaking conditions in NM detection lead to save a great deal of running time. Moreover, it can be inferred from the results in that using the symmetry-breaking conditions results in high efficiency particularly for directed networks in comparison to undirected networks. The symmetry-breaking conditions used in the GK algorithm are similar to the restriction which ESU algorithm applies to the labels in EXT and SUB sets. In conclusion, the GK algorithm computes the exact number of appearance of a given query graph in a large complex network and exploiting symmetry-breaking conditions improves the algorithm performance. Also, GK algorithm is one of the known algorithms having no limitation for motif size in implementation and potentially it can find motifs of any size.

Color-coding approach

Most algorithms in the field of NM discovery are used to find induced sub-graphs of a network. In 2008, Noga Alon et al. introduced an approach for finding non-induced sub-graphs too. Their technique works on undirected networks such as PPI ones. Also, it counts non-induced trees and bounded treewidth sub-graphs. This method is applied for sub-graphs of size up to 10.

This algorithm counts the number of non-induced occurrences of a tree T with k = O(logn) vertices in a network G with n vertices as follows:

1. Color coding. Color each vertex of input network G independently and uniformly at random with one of the k colors.
2. Counting. Apply a dynamic programming routine to count the number of non-induced occurrences of T in which each vertex has a unique color. For more details on this step, see.
3. Repeat the above two steps O(e^k) times and add up the number of occurrences of T to get an estimate on the number of its occurrences in G.

As available PPI networks are far from complete and error free, this approach is suitable for NM discovery for such networks. As Grochow–Kellis Algorithm and this one are the ones popular for non-induced sub-graphs, it is worth to mention that the algorithm introduced by Alon et al. is less time-consuming than the Grochow–Kellis Algorithm.

MODA

Omidi et al. introduced a new algorithm for motif detection named MODA which is applicable for induced and non-induced NM discovery in undirected networks. It is based on the motif-centric approach discussed in the Grochow–Kellis algorithm section. It is very important to distinguish motif-centric algorithms such as MODA and GK algorithm because of their ability to work as query-finding algorithms. This feature allows such algorithms to be able to find a single motif query or a small number of motif queries (not all possible sub-graphs of a given size) with larger sizes. As the number of possible non-isomorphic sub-graphs increases exponentially with sub-graph size, for large size motifs (even larger than 10), the network-centric algorithms, those looking for all possible sub-graphs, face a problem. Although motif-centric algorithms also have problems in discovering all possible large size sub-graphs, but their ability to find small numbers of them is sometimes a significant property.

Using a hierarchical structure called an expansion tree, the MODA algorithm is able to extract NMs of a given size systematically and similar to FPF that avoids enumerating unpromising sub-graphs; MODA takes into consideration potential queries (or candidate sub-graphs) that would result in frequent sub-graphs. Despite the fact that MODA resembles FPF in using a tree like structure, the expansion tree is applicable merely for computing frequency concept F₁. As we will discuss next, the advantage of this algorithm is that it does not carry out the sub-graph isomorphism test for non-tree query graphs. Additionally, it utilizes a sampling method in order to speed up the running time of the algorithm.

Here is the main idea: by a simple criterion one can generalize a mapping of a k-size graph into the network to its same size supergraphs. For example, suppose there is mapping f(G) of graph G with k nodes into the network and we have a same size graph G′ with one more edge &langu, v〉; f_G will map G′ into the network, if there is an edge 〈f_G(u), f_G(v)〉 in the network. As a result, we can exploit the mapping set of a graph to determine the frequencies of its same order supergraphs simply in O(1) time without carrying out sub-graph isomorphism testing. The algorithm starts ingeniously with minimally connected query graphs of size k and finds their mappings in the network via sub-graph isomorphism. After that, with conservation of the graph size, it expands previously considered query graphs edge-by-edge and computes the frequency of these expanded graphs as mentioned above. The expansion process continues until reaching a complete graph K_k (fully connected with 模板:Frac edge).

As discussed above, the algorithm starts by computing sub-tree frequencies in the network and then expands sub-trees edge by edge. One way to implement this idea is called the expansion tree T_k for each k. Figure shows the expansion tree for size-4 sub-graphs. T_k organizes the running process and provides query graphs in a hierarchical manner. Strictly speaking, the expansion tree T_k is simply a directed acyclic graph or DAG, with its root number k indicating the graph size existing in the expansion tree and each of its other nodes containing the adjacency matrix of a distinct k-size query graph. Nodes in the first level of T_k are all distinct k-size trees and by traversing T_k in depth query graphs expand with one edge at each level. A query graph in a node is a sub-graph of the query graph in a node's child with one edge difference. The longest path in T_k consists of (k²-3k+4)/2 edges and is the path from the root to the leaf node holding the complete graph. Generating expansion trees can be done by a simple routine which is explained in.

MODA traverses T_k and when it extracts query trees from the first level of T_k it computes their mapping sets and saves these mappings for the next step. For non-tree queries from T_k, the algorithm extracts the mappings associated with the parent node in T_k and determines which of these mappings can support the current query graphs. The process will continue until the algorithm gets the complete query graph. The query tree mappings are extracted using the Grochow–Kellis algorithm. For computing the frequency of non-tree query graphs, the algorithm employs a simple routine that takes O(1) steps. In addition, MODA exploits a sampling method where the sampling of each node in the network is linearly proportional to the node degree, the probability distribution is exactly similar to the well-known Barabási-Albert preferential attachment model in the field of complex networks. This approach generates approximations; however, the results are almost stable in different executions since sub-graphs aggregate around highly connected nodes. The pseudocode of MODA is shown below:

Illustration of the expansion tree T4 for 4-node query graphs. At the first level, there are non-isomorphic k-size trees and at each level, an edge is added to the parent graph to form a child graph. In the second level, there is a graph with two alternative edges that is shown by a dashed red edge. In fact, this node represents two expanded graphs that are isomorphic.

MODA

Input: G: Input graph, k: sub-graph size, Δ: threshold value Output: Frequent Subgraph List: List of all frequent k-size sub-graphs Note: FG: set of mappings from G in the input graph G fetch Tk do  G′ = Get-Next-BFS(Tk) // G′ is a query graph  if |E(G′)| = (k – 1)  call Mapping-Module(G′, G)  else  call Enumerating-Module(G′, G, Tk)  end if  save F2  if |FG| > Δ then  add G′ into Frequent Subgraph List  end if Until |E(G')| = (k – 1)/2 return Frequent Subgraph List

Kavosh

A recently introduced algorithm named Kavosh aims at improved main memory usage. Kavosh is usable to detect NM in both directed and undirected networks. The main idea of the enumeration is similar to the GK and MODA algorithms, which first find all k-size sub-graphs that a particular node participated in, then remove the node, and subsequently repeat this process for the remaining nodes.

For counting the sub-graphs of size k that include a particular node, trees with maximum depth of k, rooted at this node and based on neighborhood relationship are implicitly built. Children of each node include both incoming and outgoing adjacent nodes. To descend the tree, a child is chosen at each level with the restriction that a particular child can be included only if it has not been included at any upper level. After having descended to the lowest level possible, the tree is again ascended and the process is repeated with the stipulation that nodes visited in earlier paths of a descendant are now considered unvisited nodes. A final restriction in building trees is that all children in a particular tree must have numerical labels larger than the label of the root of the tree. The restrictions on the labels of the children are similar to the conditions which GK and ESU algorithm use to avoid overcounting sub-graphs.

The protocol for extracting sub-graphs makes use of the compositions of an integer. For the extraction of sub-graphs of size k, all possible compositions of the integer k-1 must be considered. The compositions of k-1 consist of all possible manners of expressing k-1 as a sum of positive integers. Summations in which the order of the summands differs are considered distinct. A composition can be expressed as k₂,k₃,…,k_m where k₂ + k₃ + … + k_m = k-1. To count sub-graphs based on the composition, k_i nodes are selected from the i-th level of the tree to be nodes of the sub-graphs (i = 2,3,…,m). The k-1 selected nodes along with the node at the root define a sub-graph within the network. After discovering a sub-graph involved as a match in the target network, in order to be able to evaluate the size of each class according to the target network, Kavosh employs the nauty algorithm in the same way as FANMOD. The enumeration part of Kavosh algorithm is shown below:

Enumeration of Kavosh
Enumerate_Vertex(G, u, S, Remainder, i) Input: G: Input graph  u: Root vertex  S: selection (S = { S0,S1,...,Sk-1} is an array of the set of all Si)  Remainder: number of remaining vertices to be selected  i: Current depth of the tree. Output: all k-size sub-graphs of graph G rooted in u. if Remainder = 0 then  return else  ValList ← Validate(G, Si-1, u)  ni ← Min(|ValList|, Remainder)  for ki = 1 to ni do  C ← Initial_Comb(ValList, ki)  (Make the first vertex combination selection according)  repeat  Si ← C  Enumerate_Vertex(G, u, S, Remainder-ki, i+1)  Next_Comb(ValList, ki)  (Make the next vertex combination selection according)  until C = NILL  end for  for each v ∈ ValList do  Visited[v] ← false  end for end if
Validate(G, Parents, u) Input: G: input graph, Parents: selected vertices of last layer, u: Root vertex. Output: Valid vertices of the current level. ValList ← NILL for each v ∈ Parents do  for each w ∈ Neighbor[u] do  if label[u] < label[w] AND NOT Visited[w] then  Visited[w] ← true  ValList = ValList + w  end if  end for end for return ValList

Recently a Cytoscape plugin called CytoKavosh is developed for this software. It is available via Cytoscape web page [1].

G-Tries

2010年， Pedro Ribeiro 和 Fernando Silva 提出了一个叫做g-trie的新数据结构，用来存储一组子图。这个在概念上类似前缀树的数据结构，根据子图结构来进行存储，并找出了每个子图在一个更大的图中出现的次数。这个数据结构有一个突出的方面：在应用于模体发现算法时，主网络中的子图需要进行评估。因此，在随机网络中寻找那些在不在主网络中的子图，这个消耗时间的步骤就不再需要执行了。

g-trie 是一个存储一组图的多叉树。每一个树节点都存储着一个图节点及其对应的到前一个节点的边的信息。从根节点到叶节点的一条路径对应一个图。一个 g-trie 节点的子孙节点共享一个子图（即每一次路径的分叉意味着从一个子图结构中扩展出不同的图结构，而这些扩展出来的图结构自然有着相同的子图结构）。如何构造一个 g-trie 在中有详细描述。构造好一个 g-trie 以后，需要进行计数步骤。计数流程的主要思想是回溯所有可能的子图，同时进行同构性测试。这种回溯技术本质上和其他以模体为中心的方法，比如MODA 和 GK 算法中使用的技术是一样的。这种技术利用了共同的子结构，亦即在一定时间内，几个不同的候选子图中存在部分是同构的。

在上述算法中，G-Tries 是最快的。然而，它的一个缺点是内存的超量使用，这局限了它在个人电脑运行时所能发现的模体的大小

对比

下面的表格和数据显示了在各种标准网络中运行上述算法所获得的结果。这些结果皆获取于各自相应的来源 ，因此需要独立地对待它们。

Runtimes of Grochow–Kellis, mfinder, FANMOD, FPF and MODA for subgraphs from three nodes up to nine nodes.

Grochow–Kellis, FANMOD, 和 G-Trie 在5个不同网络上生成含3到9个节点子图所用的运行时间

网络	子图大小	原始网络数据			随机网络平均数据
		FANMOD	GK	G-Trie	FANMOD	GK	G-Trie
Dolphins	5	0.07	0.03	0.01	0.13	0.04	0.01
	6	0.48	0.28	0.04	1.14	0.35	0.07
	7	3.02	3.44	0.23	8.34	3.55	0.46
	8	19.44	73.16	1.69	67.94	37.31	4.03
	9	100.86	2984.22	6.98	493.98	366.79	24.84
Circuit	6	0.49	0.41	0.03	0.55	0.24	0.03
	7	3.28	3.73	0.22	3.53	1.34	0.17
	8	17.78	48.00	1.52	21.42	7.91	1.06
Social	3	0.31	0.11	0.02	0.35	0.11	0.02
	4	7.78	1.37	0.56	13.27	1.86	0.57
	5	208.30	31.85	14.88	531.65	62.66	22.11
Yeast	3	0.47	0.33	0.02	0.57	0.35	0.02
	4	10.07	2.04	0.36	12.90	2.25	0.41
	5	268.51	34.10	12.73	400.13	47.16	14.98
Power	3	0.51	1.46	0.00	0.91	1.37	0.01
	4	1.38	4.34	0.02	3.01	4.40	0.03
	5	4.68	16.95	0.10	12.38	17.54	0.14
	6	20.36	95.58	0.55	67.65	92.74	0.88
	7	101.04	765.91	3.36	408.15	630.65	5.17

mfinder, FANMOD, Mavisto 和 Kavosh 在3个不同网络上生成含3到10个节点子图所用的运行时间

	子图大小→	3	4	5	6	7	8	9	10
网络↓	算法↓
E. coli	Kavosh	0.30	1.84	14.91	141.98	1374.0	13173.7	121110.3	1120560.1
	FANMOD	0.81	2.53	15.71	132.24	1205.9	9256.6	-	-
	Mavisto	13532	-	-	-	-	-	-	-
	Mfinder	31.0	297	23671	-	-	-	-	-
Electronic	Kavosh	0.08	0.36	8.02	11.39	77.22	422.6	2823.7	18037.5
	FANMOD	0.53	1.06	4.34	24.24	160	967.99	-	-
	Mavisto	210.0	1727	-	-	-	-	-	-
	Mfinder	7	14	109.8	2020.2	-	-	-	-
Social	Kavosh	0.04	0.23	1.63	10.48	69.43	415.66	2594.19	14611.23
	FANMOD	0.46	0.84	3.07	17.63	117.43	845.93	-	-
	Mavisto	393	1492	-	-	-	-	-	-
	Mfinder	12	49	798	181077	-	-	-	-

算法的分类

正如表格所示，模体发现算法可以分为两大类：基于精确计数的算法，以及使用统计采样以及估计的算法。因为后者并不计数所有子图在主网络中出现的次数，所以第二类算法会更快，却也可能产生带有偏向性的，甚至不现实的结果。

更深一层地，基于精确计数的算法可以分为以网络为中心的方法以及以子图为中心的方法。前者在给定网络中搜索给定大小的子图，而后者首先根据给定大小生成各种可能的非同构图，然后在网络中分别搜索这些生成的图。这两种方法都有各自的优缺点，这些在上文有讨论。

另一方面，基于估计的方法可能会利用如前面描述过的颜色编码手段，其它的手段则通常会在枚举过程中忽略一些子图(比如，像在 FANMOD 中做的那样)，然后只在枚举出来的子图上做估计。

此外，表格还指出了一个算法能否应用于有向网络或无向网络，以及导出子图或非导出子图。更多信息请参考下方提供的网页和实验室地址及联系方式。

模体发现算法的分类

计数方式	基础	算法名称	有向 / 无向	导出/ 非导出
精确基数	以网络为中心	mfinder	皆可	导出
		ESU (FANMOD)	皆可	导出
		Kavosh (used in CytoKavosh)	皆可	导出
		G-Tries	皆可	导出
		PGD	无向	导出
	以子图为中心	FPF (Mavisto)	皆可	导出
		NeMoFinder	无向	导出
		Grochow–Kellis	皆可	Both
		MODA	皆可	皆可
采样估计	颜色编码	N. Alon et al.’s Algorithm	无向	非导出
	其他手段	mfinder	皆可	导出
		ESU (FANMOD)	皆可	导出

算法提出者的地址和联系方式

算法	实验室/研究组	学院	大学/研究所	地址	电子邮件
mfinder	Uri Alon's Group	Department of Molecular Cell Biology	Weizmann Institute of Science	Rehovot, Israel, Wolfson, Rm. 607	urialon at weizmann.ac.il
FPF (Mavisto)	----	----	Leibniz-Institut für Pflanzengenetik und Kulturpflanzenforschung (IPK)	Corrensstraße 3, D-06466 Stadt Seeland, OT Gatersleben, Deutschland	schreibe at ipk-gatersleben.de
ESU (FANMOD)	Lehrstuhl Theoretische Informatik I	Institut für Informatik	Friedrich-Schiller-Universität Jena	Ernst-Abbe-Platz 2,D-07743 Jena, Deutschland	sebastian.wernicke at gmail.com
NeMoFinder	----	School of Computing	National University of Singapore	Singapore 119077	chenjin at comp.nus.edu.sg
Grochow–Kellis	CS Theory Group & Complex Systems Group	Computer Science	University of Colorado, Boulder	1111 Engineering Dr. ECOT 717, 430 UCB Boulder, CO 80309-0430 USA	jgrochow at colorado.edu
N. Alon et al.’s Algorithm	Department of Pure Mathematics	School of Mathematical Sciences	Tel Aviv University	Tel Aviv 69978, Israel	nogaa at post.tau.ac.il
MODA	Laboratory of Systems Biology and Bioinformatics (LBB)	Institute of Biochemistry and Biophysics (IBB)	University of Tehran	Enghelab Square, Enghelab Ave, Tehran, Iran	amasoudin at ibb.ut.ac.ir
Kavosh (used in CytoKavosh)	Laboratory of Systems Biology and Bioinformatics (LBB)	Institute of Biochemistry and Biophysics (IBB)	University of Tehran	Enghelab Square, Enghelab Ave, Tehran, Iran	amasoudin at ibb.ut.ac.ir
G-Tries	Center for Research in Advanced Computing Systems	Computer Science	University of Porto	Rua Campo Alegre 1021/1055, Porto, Portugal	pribeiro at dcc.fc.up.pt
PGD	Network Learning and Discovery Lab	Department of Computer Science	Purdue University	Purdue University, 305 N University St, West Lafayette, IN 47907	nkahmed@purdue.edu

Well-established motifs and their functions

Much experimental work has been devoted to understanding network motifs in gene regulatory networks. These networks control which genes are expressed in the cell in response to biological signals. The network is defined such that genes are nodes, and directed edges represent the control of one gene by a transcription factor (regulatory protein that binds DNA) encoded by another gene. Thus, network motifs are patterns of genes regulating each other's transcription rate. When analyzing transcription networks, it is seen that the same network motifs appear again and again in diverse organisms from bacteria to human. The transcription network of E. coli and yeast, for example, is made of three main motif families, that make up almost the entire network. The leading hypothesis is that the network motif were independently selected by evolutionary processes in a converging manner, since the creation or elimination of regulatory interactions is fast on evolutionary time scale, relative to the rate at which genes change, Furthermore, experiments on the dynamics generated by network motifs in living cells indicate that they have characteristic dynamical functions. This suggests that the network motif serve as building blocks in gene regulatory networks that are beneficial to the organism.

The functions associated with common network motifs in transcription networks were explored and demonstrated by several research projects both theoretically and experimentally. Below are some of the most common network motifs and their associated function.

Negative auto-regulation (NAR)

Schematic representation of an auto-regulation motif

One of simplest and most abundant network motifs in E. coli is negative auto-regulation in which a transcription factor (TF) represses its own transcription. This motif was shown to perform two important functions. The first function is response acceleration. NAR was shown to speed-up the response to signals both theoretically and experimentally. This was first shown in a synthetic transcription network and later on in the natural context in the SOS DNA repair system of E .coli. The second function is increased stability of the auto-regulated gene product concentration against stochastic noise, thus reducing variations in protein levels between different cells.

Positive auto-regulation (PAR)

Positive auto-regulation (PAR) occurs when a transcription factor enhances its own rate of production. Opposite to the NAR motif this motif slows the response time compared to simple regulation. In the case of a strong PAR the motif may lead to a bimodal distribution of protein levels in cell populations.

Feed-forward loops (FFL)

文件:Feed-forward motif.GIF

Schematic representation of a Feed-forward motif

This motif is commonly found in many gene systems and organisms. The motif consists of three genes and three regulatory interactions. The target gene C is regulated by 2 TFs A and B and in addition TF B is also regulated by TF A . Since each of the regulatory interactions may either be positive or negative there are possibly eight types of FFL motifs. Two of those eight types: the coherent type 1 FFL (C1-FFL) (where all interactions are positive) and the incoherent type 1 FFL (I1-FFL) (A activates C and also activates B which represses C) are found much more frequently in the transcription network of E. coli and yeast than the other six types. In addition to the structure of the circuitry the way in which the signals from A and B are integrated by the C promoter should also be considered. In most of the cases the FFL is either an AND gate (A and B are required for C activation) or OR gate (either A or B are sufficient for C activation) but other input function are also possible.

Coherent type 1 FFL (C1-FFL)

The C1-FFL with an AND gate was shown to have a function of a ‘sign-sensitive delay’ element and a persistence detector both theoretically and experimentally with the arabinose system of E. coli. This means that this motif can provide pulse filtration in which short pulses of signal will not generate a response but persistent signals will generate a response after short delay. The shut off of the output when a persistent pulse is ended will be fast. The opposite behavior emerges in the case of a sum gate with fast response and delayed shut off as was demonstrated in the flagella system of E. coli. De novo evolution of C1-FFLs in gene regulatory networks has been demonstrated computationally in response to selection to filter out an idealized short signal pulse, but for non-idealized noise, a dynamics-based system of feed-forward regulation with different topology was instead favored.

Incoherent type 1 FFL (I1-FFL)

The I1-FFL is a pulse generator and response accelerator. The two signal pathways of the I1-FFL act in opposite directions where one pathway activates Z and the other represses it. When the repression is complete this leads to a pulse-like dynamics. It was also demonstrated experimentally that the I1-FFL can serve as response accelerator in a way which is similar to the NAR motif. The difference is that the I1-FFL can speed-up the response of any gene and not necessarily a transcription factor gene. An additional function was assigned to the I1-FFL network motif: it was shown both theoretically and experimentally that the I1-FFL can generate non-monotonic input function in both a synthetic and native systems. Finally, expression units that incorporate incoherent feedforward control of the gene product provide adaptation to the amount of DNA template and can be superior to simple combinations of constitutive promoters. Feedforward regulation displayed better adaptation than negative feedback, and circuits based on RNA interference were the most robust to variation in DNA template amounts.

Multi-output FFLs

In some cases the same regulators X and Y regulate several Z genes of the same system. By adjusting the strength of the interactions this motif was shown to determine the temporal order of gene activation. This was demonstrated experimentally in the flagella system of E. coli.

Single-input modules (SIM)

This motif occurs when a single regulator regulates a set of genes with no additional regulation. This is useful when the genes are cooperatively carrying out a specific function and therefore always need to be activated in a synchronized manner. By adjusting the strength of the interactions it can create temporal expression program of the genes it regulates.

In the literature, Multiple-input modules (MIM) arose as a generalization of SIM. However, the precise definitions of SIM and MIM have been a source of inconsistency. There are attempts to provide orthogonal definitions for canonical motifs in biological networks and algorithms to enumerate them, especially SIM, MIM and Bi-Fan (2x2 MIM).

Dense overlapping regulons (DOR)

This motif occurs in the case that several regulators combinatorially control a set of genes with diverse regulatory combinations. This motif was found in E. coli in various systems such as carbon utilization, anaerobic growth, stress response and others. In order to better understand the function of this motif one has to obtain more information about the way the multiple inputs are integrated by the genes. Kaplan et al. has mapped the input functions of the sugar utilization genes in E. coli, showing diverse shapes.

已知的模体及其功能

许多实验工作致力于理解基因调控网络中的网络模体。在响应生物信号的过程中，这些网络控制细胞中需要表达的基因。这样的网络以基因作为节点，有向边代表对某个基因的调控，基因调控通过其他基因编码的转录因子结合在DNA上的调控蛋白来实现。因此，网络模体是基因之间相互调控转录速率的模式。在分析转录调控网络的时候，人们发现某些相同的网络模体在不同的物种中不断地出现，从细菌到人类。例如，大肠杆菌和酵母的转录网络由三种主要的网络模体家族组成，它们可以构建几乎整个网络。主要的假设是在进化的过程中，网络模体是被以收敛的方式独立选择出来的。 因为相对于基因改变的速率，转录相互作用产生和消失的时间尺度在进化上是很快的。 此外，对活细胞中网络模体所产生的动力学行为的实验表明，它们具有典型的动力学功能。这表明，网络模体是基因调控网络中对生物体有益的基本单元。

一些研究从理论和实验两方面探讨和论证了转录网络中与共同网络模体相关的功能。下面是一些最常见的网络模体及其相关功能。

负自反馈调控(NAR)

Schematic representation of an auto-regulation motif

负自反馈调控是大肠杆菌中最简单和最冗余的网络模体之一，其中一个转录因子抑制它自身的转录。这种网络模体有两个重要的功能，其中第一个是加速响应。人们发现在实验和理论上， NAR都可以加快对信号的响应。这个功能首先在一个人工合成的转录网络中被发现， 然后在大肠杆菌SOS DAN修复系统这个自然体系中也被发现。 负自反馈网络的第二个功能是增强自调控基因的产物浓度的稳定性，从而抵抗随机的噪声，减少该蛋白含量在不同细胞中的差异。

正自反馈调控(PAR)

正自反馈调控是指转录因子增强它自身转录速率的调控。和负自反馈调节相反，NAR模体相比于简单的调控能够延长反应时间。 在强PAR的情况下，模体可能导致蛋白质水平在细胞群中呈现双峰分布。

前馈回路 (FFL)

文件:Feed-forward motif.GIF

Schematic representation of a Feed-forward motif

前馈回路普遍存在于许多基因系统和生物体中。这种模体包括三个基因以及三个相互作用。目标基因C被两个转录因子（TFs）A和B调控，并且TF B同时被TF A调控。由于每个调控相互作用可以是正的或者负的，所以总共可能有八种类型的FFL模体。 其中的两种：一致前馈回路的类型一（C1-FFL）（所有相互作用都是正的）和非一致前馈回路的类型一（I1-FFL）（A激活C和B，B抑制C）在大肠杆菌和酵母中相比于其他六种更频繁的出现。 除了网络的结构外，还应该考虑来自A和B的信号被C的启动子集成的方式。在大多数情况下，FFL要么是一个与门(激活C需要A和B)，要么是或门(激活C需要A或B)，但也可以是其他输入函数。

一致前馈回路类型一(C1-FFL)

具有与门的C1-FFL有“符号-敏感延迟”单元和持久性探测器的功能，这一点在大肠杆菌阿拉伯糖系系统的理论和实验上 都有发现。这意味着该模体可以提供脉冲过滤，短脉冲信号不会产生响应，而持久信号在短延迟后会产生响应。当持久脉冲结束时，输出的关闭将很快。与此相反的行为出现在具有快速响应和延迟关闭特性的加和门的情况下，这在大肠杆菌的鞭毛系统中得到了证明。在基因调控网络的重头进化中，对于滤除理想化的短信号脉冲作为进化压，C1-FFLs已经在计算上被证明可以进化出来。但是对于非理想化的噪声，不同拓扑结构前馈调节的动态系统将被优先考虑。

非一致前馈回路类型一(I1-FFL)

I1-FFL是一个脉冲生成器和响应加速器。I1-FFL的两种信号通路作用方向相反，一种通路激活Z，而另一种抑制Z。完全的抑制会导致类似脉冲的动力学行为。另外有实验证明，它可以类似于NAR模体起到响应加速器的作用。与NAR模体的不同之处在于，它可以加速任何基因的响应，而不必是转录因子。I1-FFL网络还有另外一个功能：在理论和实验上都有证明I1-FFL可以生成非单调的输入函数，无论在人工合成的还是自然的系统中。 最后，包含非一致前馈调控的基因生成物的表达单元对DNA模板的数量具有适应性，可以优于简单的组合本构启动子。 前馈调控比负反馈具有更好的适应性，并且基于RNA干扰的网络对DNA模板数具有最高的鲁棒性。

多输出前馈回路

在某些情况，相同的调控子X和Y可以调控同一系统中的多个Z基因。通过调节相互作用的强度，这些网络可以决定基因激活的时间顺序。这一点在大肠杆菌的鞭毛系统中有实验证据。

单一输入模块(SIM)

当单个调控子调控一组基因，并且没有其他的调控因素时，这样的模体叫做单一输入模块（SIM）。当很多基因合作执行某个功能时这是有用的，因为这些基因需要同步地被激活。通过调节相互作用的强度，可以编排它所调控的基因表达的时间顺序。

在文献中，多输入模块（MIM）来自于SIM的扩展。但是二者的精确定义并不太一致。有一些尝试给出生物网络中规范模体的正交定义，也有一些算法去枚举它们，特别是SIM，MIM和2x2 MIM等。

密集交盖调节网(DOR)

这种类型的网络存在于多个调节子结合起来控制一组基因的情形，并且有多种调控的组合。这种模体出现在大肠杆菌的多种系统中，例如碳利用、厌氧生长、应激反应等。 为了更好地理解这种网络，我们必须得到关于基因集成多种输入的方式的信息。Kaplan et al.绘制了大肠杆菌糖利用基因的输入函数，表现出各种各样的形状。

活动模体

有一个对网络模体的有趣概括：活动模体是在对节点和边都被量化标注的网络中可发现的【反复】斑图。例如，当新城代谢的边以相应基因的表达量或【时间】来标注时，一些斑图在给定的底层网络结构里【是反复的】。

批判

对拓扑子结构有一个（某种程度上隐含的）前提性假设是其具有特定的功能重要性。但该假设最近遭到质疑，有人提出在不同的网络环境下模体可能表现出多样性，例如双扇模体，故模体的结构不必然决定功能，网络结构也不当然能揭示其功能；这种见解由来已久，可参见【Sin 操纵子】。

大多数模体功能分析是基于模体孤立运行的情形。最近的研究表明网络环境至关重要，不能忽视网络环境而仅从本地结构来对其功能进行推论——引用的论文还回顾了对观测数据的批判及其他可能的解释。人们研究了单个模体模组对网络全局的动力学影响及其分析。而另一项近期的研究工作提出生物网络的某些拓扑特征能自然地引起经典模体的常见形态，让人不禁疑问：这样的发生频率是否能证明模体的结构是出于其对所在网络运行的功能性贡献而被选择保留下的结果？

模体的研究主要应用于静态复杂网络，而时变复杂网络的研究就网络模体提出了重大的新解释，并介绍了时变网络模体的概念。Braha和Bar-Yam研究了本地模体结构在时间依赖网络/时变网络的动力学，发现的一些反复模式有望成为社会互动周期的经验论据。他们证明了对于时变网络，其本地结构是时间依赖的且可能随时间演变，可作为对复杂网络中稳定模体观及模体表达观的反论，Braha和Bar-Yam还进一步提出，对时变本地结构的分析有可能揭示系统级任务和功能方面的动力学的重要信息。

See also

• Clique (graph theory)
• Graphical model

External links

• A software tool that can detect network motifs
• bio-physics-wiki NETWORK MOTIFS
• FANMOD: a tool for fast network motif detection
• MAVisto: network motif analysis and visualisation tool
• NeMoFinder
• Grochow–Kellis
• MODA
• Kavosh
• CytoKavosh
• G-Tries
• acc-MOTIF detection tool